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Active and passive microwave remote sensing synergy for geophysical parameter estimation

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Active and passive microwave remote sensing synergy for geophysical parameter estimation

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Content Active and Passive Microwave Remote Sensing Synergy for Geophysical Parameter Estimation by Ruzbeh Akbar A dissertation submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy (Ming Hsieh Department of Electrical Engineering) in The University of Southern California May 2016 Doctoral Committee: Professor Mahta Moghaddam, Chair Professor Shrikanth S. Narayanan Associate Professor Hossein Hashemi Associate Professor Michelle L. Povinelli Associate Professor Behnam Jafarpour To my parents, with endless love. ii TABLE OF CONTENTS DEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x LIST OF APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiii CHAPTER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Context and Motivation . . . . . . . . . . . . . . . . . . . . . 3 1.3 Soil Moisture Remote Sensing Heritage . . . . . . . . . . . . . 5 1.4 Soil Moisture Retrieval Process . . . . . . . . . . . . . . . . . 8 1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 iii II. Fundamental Concepts and Theoretical Background . . . . . 11 2.1 Radar Remote Sensing . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Generalized Radar Scattering Theory . . . . . . . . 11 2.1.2 Electromagnetic Radar Scattering Model . . . . . . 17 2.2 Radiometer Remote Sensing . . . . . . . . . . . . . . . . . . . 27 2.2.1 Black Body Radiation and Brightness Temperature 27 2.2.2 Radiative Transfer . . . . . . . . . . . . . . . . . . . 30 2.2.3 Geophysical Emission Model . . . . . . . . . . . . . 32 2.3 Radar vs. Radiometer . . . . . . . . . . . . . . . . . . . . . . 34 2.3.1 Measurement Spatial Resolution . . . . . . . . . . . 35 2.3.2 Soil Moisture Sensitivity & Estimation Accuracy . . 37 2.3.3 Information Theoretic Perspective . . . . . . . . . . 40 2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . 44 III. Single-Resolution Combined Radar-Radiometer Soil Mois- ture Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Soil Moisture Estimation: Numerical Simulations . . . . . . . 49 3.3 Soil Moisture Estimation: Application to Field Measurements 55 3.3.1 PALS SMEX02 Dataset . . . . . . . . . . . . . . . . 56 3.3.2 ComRad Dataset . . . . . . . . . . . . . . . . . . . 58 3.4 Multi-parameter Estimation and Self-regularization . . . . . . 69 3.4.1 Forward Model Ambiguity . . . . . . . . . . . . . . 70 3.4.2 Simultaneous Soil Moisture and Surface Roughness Estimation . . . . . . . . . . . . . . . . . . . . . . . 77 3.5 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . 92 iv IV. Multi-Resolution Combined Radar-Radiometer Soil Mois- ture Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 NASA SMAP Mission . . . . . . . . . . . . . . . . . . . . . . 98 4.3 Multi-Resolution Soil Moisture Estimation . . . . . . . . . . . 103 4.3.1 Perspective and Approach . . . . . . . . . . . . . . 104 4.3.2 Multi-Objective Active-Passive Formulation . . . . . 108 4.4 Soil Moisture Estimation Using SMAP Data . . . . . . . . . . 112 4.4.1 Global Comparison . . . . . . . . . . . . . . . . . . 112 4.4.2 Case Study: Tonzi Ranch Study Site . . . . . . . . 125 V. Forward Emission and Scattering Modeling Considerations . 134 5.1 Observation Geometry Eects . . . . . . . . . . . . . . . . . . 134 5.1.1 Spacecraft Observation View Angles . . . . . . . . . 135 5.1.2 Forward Model Comparison with SMAP Radar . . . 143 5.1.3 Radar-only Soil Moisture Estimation . . . . . . . . 147 5.2 Joint Modeling of Emission and Scattering . . . . . . . . . . . 151 5.2.1 Eects of Surface Roughness . . . . . . . . . . . . . 152 5.3 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . 158 APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 159 LIST OF ABBREVIATIONS : : : : : : : : : : : : : : : : : : : : : : : : : 165 BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 167 v LIST OF FIGURES Figure 1.1 Soil Moisture Measurement Platforms. . . . . . . . . . . . . . . . . 2 1.2 Radar and Radiometer Soil Moisture Estimation Heritage. . . . . . 7 1.3 Concept of Combined Active-Passive Soil Moisture Estimation . . . 7 1.4 Soil Moisture Retrieval Process. . . . . . . . . . . . . . . . . . . . . 9 2.1 Generalized Scattering Coordinate System. . . . . . . . . . . . . . . 12 2.2 Schematic Representation of Remote Sensing Targets . . . . . . . . 18 2.3 Discretized Target Structure . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Various Vegetation shapes and structures . . . . . . . . . . . . . . . 19 2.5 Discretized Radar Scattering Mechanisms . . . . . . . . . . . . . . . 21 2.6 DSA Model Simulation Example . . . . . . . . . . . . . . . . . . . . 24 2.7 DSA Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Planck's Law of Radiation . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Total Emission Contributions . . . . . . . . . . . . . . . . . . . . . 31 2.10 Spatial Resolution Comparison Between SAR and Radiometer. . . . 37 2.11 Radar and Radiometer Response to changes in Soil Moisture. . . . . 38 2.12 Complementarity of Radar and Radiometer measurements. . . . . . 39 2.13 Backscatter and Emission Probability Distributions . . . . . . . . . 42 2.14 Backscatter and Emission Mutual Information. . . . . . . . . . . . . 43 2.15 Remote Sensing Process . . . . . . . . . . . . . . . . . . . . . . . . 45 vi 3.1 Objective Function Cross-section plots. . . . . . . . . . . . . . . . 50 3.2 Baseline Radar-only, Radiometer-only and Combined Soil Moisture Estimation Comparison. . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 C-AP retrieval error vs. parameter. . . . . . . . . . . . . . . . . 54 3.4 C-AP Soil Moisture Estimation for PALS SEMX02 Data. . . . . . . 57 3.5 Radar Backscatter Model-Data Comparison for PALS SMEX02. . . 59 3.6 ComRAD Soil Moisture Temporal Dynamics. . . . . . . . . . . . . 60 3.7 Radar Backscatter Model-Data Comparison for ComRAD. . . . . . 63 3.8 Radiometer Brightness Temperature Model-Data Comparison for ComRad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.9 C-AP Soil Moisture Estimation for ComRad. . . . . . . . . . . . . . 65 3.10 Normalized ComRad Cost function cross-section plot. . . . . . . . 66 3.11 C-AP Biased Removed Soil Moisture Retrieval For ComRad. . . . . 68 3.12 Model Ambiguity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.13 2D Radar-only Cost Function Hyper-plane. . . . . . . . . . . . . . . 73 3.14 2D Radiometer-only Cost Function Hyper-plane. . . . . . . . . . . . 74 3.15 2D Active-Passive Cost Function Hyper-plane. . . . . . . . . . . . . 75 3.16 2D Active-Passive Cost Function Hyper-plane vs. VWC. . . . . . . 76 3.17 Active-Passive Permittivity and Roughness Estimation. . . . . . . . 80 3.18 Estimated Permittivity-Roughness pair scatter Plot. . . . . . . . . . 83 3.19 Estimated Permittivity Error Breakdown. . . . . . . . . . . . . . . . 85 3.20 Estimated Roughness Error Breakdown. . . . . . . . . . . . . . . . 86 3.21 Soil Moisture and Roughness Retrieval Errors vs. VWC.. . . . . . . 87 3.22 ComRAD C-AP Soil Moisture Estimation with unknown Roughness 88 3.23 ComRAD C-AP Taylor Plots. . . . . . . . . . . . . . . . . . . . . . 89 3.24 ComRAD C-AP Soil Moisture Scatter Plot for Corn. . . . . . . . . 91 3.25 ComRAD C-AP Soil Moisture Scatter Plot for Soybean. . . . . . . . 92 4.1 Single Resolution vs. Multi-Resolution . . . . . . . . . . . . . . . . 98 vii 4.2 Multi-Resolution Combined Active-Passive Concept. . . . . . . . . . 99 4.3 EASE 2.0 Grid Nested Grids. . . . . . . . . . . . . . . . . . . . . . 100 4.4 The SMAP L2SM AP Algorithm . . . . . . . . . . . . . . . . . . . 102 4.5 SMAP L2SM AP Data Product Example . . . . . . . . . . . . . . . 103 4.6 Model-Objective Function Space . . . . . . . . . . . . . . . . . . . . 106 4.7 MOEA Optimization Flowchart . . . . . . . . . . . . . . . . . . . . 111 4.8 Global IGBP Land Cover Classication. . . . . . . . . . . . . . . . 113 4.9 SMAP TBH Global 3-day Composite . . . . . . . . . . . . . . . . . 115 4.10 SMAP TBV Global 3-day Composite . . . . . . . . . . . . . . . . . 115 4.11 SMAP Sigma-HH Global 3-day Composite . . . . . . . . . . . . . . 116 4.12 SMAP Sigma-VV Global 3-day Composite . . . . . . . . . . . . . . 116 4.13 MOEA 3-day global soil moisture composite . . . . . . . . . . . . . 117 4.14 SMAP Active-Passive 3-day global soil moisture composite . . . . . 117 4.15 SMAP and MOEA soil moisture scatter plots for Bare surfaces . . . 118 4.16 SMAP and MOEA soil moisture scatter plots for Grasslands . . . . 118 4.17 SMAP and MOEA soil moisture scatter plots for Cropland surfaces 119 4.18 SMAP and MOEA soil moisture scatter plots for Grasslands . . . . 119 4.19 SMAP and MOEA soil moisture scatter plots for Savannas . . . . . 120 4.20 SMAP and MOEA soil moisture scatter plots for Woody Savannas . 120 4.21 SMAP and MOEA soil moisture scatter plots for Permanent Wetlands121 4.22 SMAP and MOEA soil moisture scatter plots for Evergreen Broad- leaf Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.23 SMAP and MOEA soil moisture scatter plots for Closed Shrublands 122 4.24 SMAP and MOEA soil moisture scatter plots for Mixed Forests . . 122 4.25 SMAP and MOEA soil moisture scatter plots for Deciduous Needle Leaf forests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.26 SMAP and MOEA soil moisture scatter plots for Deciduous Bread- leaf Leaf forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 viii 4.27 RMSD Plot for dierent land cover classications. . . . . . . . . . . 124 4.28 EASE-grid Nesting over the greater Tonzi Ranch Area. . . . . . . . 126 4.29 Temporal Plot of Tonzi Ranch Soil Moisture. . . . . . . . . . . . . . 127 4.30 MOEA Pareto Front For Tonzi Ranch . . . . . . . . . . . . . . . . 128 4.31 Tonzi Ranch Sub-pixel selection. . . . . . . . . . . . . . . . . . . . . 129 4.32 Active-Passive Soil Moisture Retrieval Over Tonzi Ranch . . . . . . 130 4.33 Tonzi Ranch 9 km 0 . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.34 Vegetation Eects on Tonzi Ranch Soil Moisture Estimation . . . . 133 5.1 SMAP Observation Geometry . . . . . . . . . . . . . . . . . . . . . 137 5.2 Tonzi Ranch Study Site . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3 SMAP 1 km Fore- vs. Aft-look 0 . . . . . . . . . . . . . . . . . . 139 5.4 SMAP 3 km Fore- vs. Aft-look 0 . . . . . . . . . . . . . . . . . . 141 5.5 Tonzi Ranch Soil Moisture and SMAP 3 km 0 . . . . . . . . . . . 142 5.6 Forward Model Comparison with SMAP 3 km 0 . . . . . . . . . . 144 5.7 Temporal Plot of Tonzi Ranch Soil Moisture and Radar-only Estimates148 5.8 Radar-only Soil Moisture Estimation over Tonzi Ranch. . . . . . . . 149 5.9 Joint-physics Emission-Scattering Model Concept . . . . . . . . . . 153 5.10 Emission Model Comparison . . . . . . . . . . . . . . . . . . . . . . 155 5.11 Bare surface Joint-physics Error Maps . . . . . . . . . . . . . . . . . 156 5.12 Joint-physics Retrieval Improvement . . . . . . . . . . . . . . . . . 157 A.1 SMAP Active-Passive Data Processing Flow Chart . . . . . . . . . 161 ix LIST OF TABLES Table 1.1 Soil Moisture Measurement Schemes . . . . . . . . . . . . . . . . . 4 1.2 Radar and Radiometer Heritage . . . . . . . . . . . . . . . . . . . . 6 2.1 Radar Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 SAR and Radiometer Satellite Missions. . . . . . . . . . . . . . . . 36 3.1 Active-Passive Soil Moisture Retrieval Simulation Model Parameters 48 3.2 Noisy Retrieval RMS Errors. . . . . . . . . . . . . . . . . . . . . . . 52 3.3 CAP Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Error Reduction due to Regularization . . . . . . . . . . . . . . . . 55 3.5 Datacube Roughness Parameter Fitting. . . . . . . . . . . . . . . . 61 3.6 Radiometer -! Optimum Model Parameters. . . . . . . . . . . . . 62 3.7 ComRad Model-Data Comparison. . . . . . . . . . . . . . . . . . . 65 3.8 Multi-parameter Estimation Model Parameters. . . . . . . . . . . . 77 3.9 Active-Passive Noise Scenarios. . . . . . . . . . . . . . . . . . . . . 78 3.10 Soil Permittivity Retrieval RMS Errors. . . . . . . . . . . . . . . . . 81 3.11 Surface Roughness Retrieval RMS Errors. . . . . . . . . . . . . . . . 82 3.12 Worst Case RMS Errors. . . . . . . . . . . . . . . . . . . . . . . . . 82 3.13 Worst Case RMS Errors. . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1 SAR and Radiometer Satellite Missions . . . . . . . . . . . . . . . . 97 4.2 C-AP Soil Moisture Retrieval Errors over Tonzi Ranch. . . . . . . . 131 x 5.1 SMAP Fore vs. Aft-look Radar Comparison. . . . . . . . . . . . . . 140 5.2 Model vs. SMAP 0 Comparison. . . . . . . . . . . . . . . . . . . . 145 5.3 Savanna Radar Scattering Model Parameters. . . . . . . . . . . . . 146 5.4 Radar-only Soil Moisture Retrieval Error Statistics. . . . . . . . . . 150 xi LIST OF APPENDICES Appendix A. Active-Passive Data Processing Flow Chart . . . . . . . . . . . . . . . 160 B. Equations for Scattering from Finite Dielectric Cylinder . . . . . . . . 162 xii ABSTRACT This dissertation presents a detailed investigation of combined and synergistic utilization of active radar and passive radiometer microwave remote sensing tech- niques to estimate surface soil moisture across multiple measurement platforms, i.e., tower-mounted, airborne and Spacecraft data. Chapter I provides a comprehensive introduction to Combined Active-Passive soil moisture estimation as well as listing the objectives and contributions of this disserta- tion. The theoretical basis of emission and scattering, along with radar and radiome- ter comparisons are discussed in Chapter III. Thereafter, Chapters III and IV present in detail Combined Radar-Radiometer soil moisture retrieval for multiple dierent measurement scenarios and instruments. Lessons learned and future consideration are then elaborated in the nal Chapter V. xiii CHAPTER I Introduction 1.1 Introduction Surface soil moisture (SM) is essential to the science community primarily as a driving force behind many of Earth's hydrological and hydrometeorological dynam- ics. Soil moisture changes have profound implications on terrestrial water, energy and carbon cycles, as well as evaporation and transpiration at the land-atmosphere boundary. Over continental and regional scales, soil moisture variations also aect weather and climate evolutions. The performance and prediction accuracy of cur- rent Numerical Weather Prediction (NWP) and Global Climate Models (GCMs) will signicantly improve with accurate knowledge and understanding of surface soil moisture, since it is a key initializing parameter. In addition, improved ood pre- diction, drought monitoring, and enhanced agricultural productivity are all possible with better understanding of soil moisture availability. Over the past few decades, science application requirements, especially in Hydro- climatology, Hydrometeorology, and carbon sciences, have converged towards three criteria for acceptable soil moisture estimates (a) higher spatial (<10 km) and tem- poral resolution (3 days), (b) global coverage, and (c) volumetric soil moisture prediction accuracy of 0.04 m 3 /m 3 or less [1]. Soil moisture predictions that meet these requirements will assist in resolving water, energy, and carbon uxes as well as 1 Figure 1.1: Soil Moisture measurements platforms: Probe, Tower-based, Airborne and Spacecraft. improving short and long term weather and climate forecast skills from a hydrome- teorological viewpoint. Over regional and global scales, measurements of soil moisture are only possible through the use of microwave remote sensing techniques, which include active radars, especially Synthetic Aperture Radar (SAR), as well as passive Radiometer (RAD). Changes in surface soil moisture and vegetation conditions result in detectable change in both the measured radar backscatter cross-section 0 and radiometer Brightness Temperature, TB. During the past decades, many radar-specic and radiometer- specic instruments and missions have demonstrated soil moisture estimation at various geographical scales and to varying degrees of accuracy, each meeting their own mission specic scientic goals and requirements. At the current state of global climate studies, historical methodologies to estimate soil moisture are no longer adequate since they can no longer meet evolving science application requirements especially in terms of resolution, coverage, sensing depth and accuracy. To brie y highlight this fact, in Table 1.1 a qualitative comparison between dier- ent existing soil moisture observation schemes across multiple platforms is presented 2 and schematically shown in Fig. 1.1. Tower or truck-mounted and airborne observa- tories yield higher resolution predictions, but their geographical coverages are very limited. For example, airborne systems such as the Passive and Active L- and S-band Sensor (PALS), Uninhabited Aerial Vehicle Synthetic Aperture Radar (UAVSAR), or Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) typically cover an approximate area of 20 x 100 km or less. Although on a regional scale the coverage is signicant, frequent global mapping becomes impractical. On the other hand, the near-daily global soil moisture coverage provided by spacecraft instruments are highly desirable. The NASA Soil Moisture Active Passive (SMAP) and Euro- pean Space Agency (ESA) Soil Moisture Ocean Salinity (SMOS) missions [2, 3, 4], for example, are capable of mapping the entire globe approximately every 2-3 days. From an instruments perspective, however, as well as soil moisture estimation abili- ties, neither SAR nor RAD alone are sucient to meet all of the science application requirements and standards at once. Nonetheless, the drawbacks in estimating soil moisture via active-only or passive- only remote sensing approaches can be signicantly overcome when the complemen- tary set of strengths between the two techniques are merged and viewed collectivity. Advancing current remote sensing methodologies to address the shortcomings discussed above is therefore of primary interest. Building upon a solid understanding of the nature of emission and scattering, new and dierent soil moisture estimation techniques can be developed to eectively use data from both active radar and passive radiometer instruments to estimate global soil moisture distributions that meets current scientic goals and requirements. 1.2 Context and Motivation The NASA Soil Moisture Active-Passive (SMAP) mission, in response to the National Research Council's 2007 Earth Science Decadal Survey [5] was initiated in 2008. The mission aims to provide the science community with global surface soil moisture estimates to address many of the pressing and current climate dynamics 3 Table 1.1: Soil Moisture Measurement Schemes Platform Instrument Method Spatial Res. Temporal Res. Coverage Accuracy Probe SM Sensor Direct Point Frequent Localized High Tower Radar Radiometer Indirect m Frequent Localized High Airborne Radar Radiometer Indirect m-km Days- Weeks Regional Medium Spaceborne Radar Radiometer Indirect km Daily Global Low- Medium questions and its eorts will yield unprecedented high resolution global soil moisture estimates at 9 km spatial resolution. Essentially, the goal of SMAP is to address the high spatial and temporal resolution global soil moisture science requirements. To meet mission objectives and scientic goals, SMAP incorporates a high reso- lution L-band (1.26 GHz) SAR along with an L-band (1.41 GHz) radiometer within a single observational platform. The SMAP conically scanning antenna, shared be- tween the SAR and RAD, provides a near-constant incidence angle of about 40 in a near-polar sun-synchronous orbit with an eight-day exact revisit cycle. Given the satellite's orbital conguration and processing schemes, Level-1 (L1) geolocated and Earth gridded radar backscatter (L1_S0_HiRes) and radiometer Brightness Temper- ature (L1C_TB) will be available, within a three-day global coverage temporal win- dow, as inputs to any retrieval and estimation algorithm. High-resolution backscatter information is provided at a 3-km resolution covering the outer 70% of the swath and Earth gridded Brightness Temperature at 36-km resolution. The primary data product of the mission, i.e., L2_SM_AP, is the high accuracy and high resolution sur- face soil moisture estimates obtained through the Combined Active-Passive (C-AP) algorithm which is produced by combining SMAPs complementary active radar and passive radiometer observations. 4 Therefore, within the context of the SMAP mission, developing new and novel C-AP methods is of keen interest and the focus of the work presented here. 1.3 Soil Moisture Remote Sensing Heritage A long and diverse history of radar and radiometer soil moisture remote sensing exists, demonstrating soil moisture retrieval using tower/truck mounted, airborne and spaceborne instruments. Early works in the mid-1960's and early 1970's [6, 7, 8] have demonstrated noticeable sensitivity of radars (scatterometer) and radiometers to changes in both surface soil moisture and dierent land cover types. In the years that followed, many eorts were made to develop various retrieval techniques based on these studies; various empirical, semi-analytical, and analytical methods have been formulated. Similarly, L-band radiometry has shown measurable sensitivity to changes in surface soil moisture for Vegetation Water Content (VWC) less than 5 kg/m 2 such that over a large range of soil moisture measured Brightness Temperature can vary up to tens of Kelvin [9]. Under similar conditions, radar backscatter cross-section can vary, at times, up to 10 dB or more; however such radar measurements exhibit high sensitivity to the geometry of vegetation and the amount of surface roughness. To capture, on a high level, the existing work in this eld a time-line of major contributers to soil moisture remote sensing as well as relevant missions is given in Fig. 1.2. Table 1.2 lists relevant work along with their estimation approaches (analytical, empirical, etc.). The important point to note here is that despise the long history of radars and ra- diometers, very few techniques have attempted to estimate soil moisture by combining both active and passive microwave remote sensing data. Only recently, and in support of SMAP, eorts have been made to investigate combined active-passive estimation techniques. Change-detection and time series methods [19, 22], as well as spatial variability techniques [20], have demonstrated 5 Table 1.2: Radar and Radiometer Heritage Platform Author Method/Approach Ref. # Radiometer Ulaby et. al. Analytical & Empirical [7] Njoku et. al. Analytical [10] Jackson et. al. Semi-Empirical [11] Owe et. al. Semi-Analytical [12] Kerr et. al. Analytical [3] Radar Ulaby et. al. Analytical & Empirical [6][8] Dubois et. al. Empirical [13] Oh et. al. Empirical [14] Durden et. al. Analytical [15] Moghaddam et. al. Analytical [16] Kim et. al. Analytical [17] Combined Radar-Radiometer Crow et. al. Bayesian Merging [18] Piles et. al. Change Detection [19] Zhan et. al. Semi-Empirical [14] Das et. al. Disaggregation [20] Akbar et. al. Joint Physics-based [21] the ability to retrieve soil moisture using radar and radiometer measurements in a unied fashion. Statistical methods such as [23], based on Bayesian merging tech- niques and Kalman ltering have also demonstrated soil moisture retrieval in the presence of measurement noise. Common themes amongst these recent methods are (1) establishing some relationship, mostly linear, between radar and radiometer mea- surements with soil moisture or changes in soil moisture and (2) developing an inter- mediate higher resolution radiometer product, the so-called disaggregated brightness temperature. However, almost all these methods perform single data-type retrievals using established radiometer-only algorithms. Therefore, an opportunity exists to systematically and fundamentally research combined radar-radiometer soil moisture estimation techniques were both measure- ment types are used simultaneously within a single retrieval framework. An extension of the method would be to take advantage of the larger measurement space provided by the two instruments and examine estimation of multiple unknowns rather than just soil moisture. 6 2005: Crow et al. 2009 Piles et al Zhan et al. 2008: SMAP* 2011: SMAP Baseline v.1 2014: SMAP Baseline v.2 Radiometer Radar 2009: SMOS 2011: Aquarius 2012: AirMOSS 1973-79: Ulaby et al. 1999: Njoku et al 1999-00: AMSR-E 1973-75: Ulaby et al. 2014: Kim et al. 1992: Oh et al. 1989: Durden et al. 2009: Kim et al. 1996: Moghaddam et al. 1995: Dubois et al. 1993: Jackson et al 2015: SMAP** 2001: Kerr et al. 2005: Owe et al. Mission Radar-only Radiometer-only Radar-Radiometer Key * Start of SMAP science ** Launched Jan 2015 Note: this is a non-exhausve list 1996: Wanger et al. 2006: ALOS PALSAR 2015: Akbar et al. Figure 1.2: Radar and Radiometer Soil Moisture Retrieval Heritage Figure 1.3 schematically presents the concept of combined radar-radiometer esti- mation. Unlike conventional and disjoint processes of radar-only and radiometer-only soil moisture retrieval, C-AP proposes to use both data types within a single opti- mization framework. : soil moisture, roughness and vegetation Radar Model Radiometer Model TB Inverse Problem Inverse Problem Conventional Methods (a) : soil moisture, roughness and vegetation Radar Model Radiometer Model TB Inverse Problem Proposed Method (b) Figure 1.3: (a) Conventional Radar-only or Radiometer-only soil moisture retrieval processes (b) New Combined Active-Passive Process ow. 7 1.4 Soil Moisture Retrieval Process As pointed out in the previous section, many dierent approaches to soil moisture exits. In this work, physics-based methods are used such that soil moisture retrieval is performed by minimizing specically dened Objective Functions, or Loss Func- tions, here generalized asL( x; 0 ;TB). x is the set of geophysical model parameters, including soil moisture; 0 and TB are the corresponding SAR and RAD remote sens- ing observations. Typically the objective function is anL 2 norm of model predictions minus observations, i.e.,jModel Dataj 2 . Given the complex analytical form of current physics-based scattering and emis- sion models they do not have closed-form inverse solutions. Thus, iterative optimiza- tion schemes are used such that optimum model parameters x opt which minimizeL( x) are reported as the desired geophysical parameter of interest. 1 A high level owchart demonstrating such a process is given in Fig. 1.4. Based on certain optimization rules, input model parameters x are iteratively adjusted certain termination criteria are met: 1. L( x) is minimized; that is for an error threshold , L( x) for a certain number of iterations. 2. Surpassing a certain number of function evaluations. 3. Surpassing a certain time limit. Throughout this work example outcomes will be compared and contrasted via numerical simulations, which mimic actual measurement scenarios, and tests on ac- tual tower, aircraft and satellite data. 1 0 and TB are implied within the Objective Function and dropped for brevity. 8 Remote Sensing Data (Radar & Radiometer) Figure 1.4: Soil Moisture Retrieval Process: The process is initiated by initial model parameters x 0 and then iteratively adjusted till L( x) is minimum. Other necessary model parameters are provided from ancillary data sources. 1.5 Objectives The objectives of this thesis, in lights of the motivation presented above stated as follows: 1. Development of physics-based active-passive soil moisture estimation techniques. 2. Detailed investigation of on the nature on microwave emission and radar scat- tering and their joint and complementary behavior. 3. Global combined radar-radiometer soil moisture estimation using SMAP satel- lite data. 9 4. Retrieval of multiple geophysical parameters, in addition to soil moisture, within a radar-radiometer framework. 5. Understanding of forward model performance limitations and outlining features to improve upon. The work presented in this thesis can also be viewed as an opening discussion and opportunity to further enhance Radar and Radiometer Synergy for Soil moisture remote sensing. 1.6 Contributions 1. Novel and rst time demonstration of simultaneous and combined Radar- Radiometer soil moisture estimation across three dierent measurement plat- forms (1) Tower-mounted instruments (2) Airborne data (3) Satellite data. 2. Enhanced understanding of the nature of emission and scattering models, their limitations and strengths in the context of optimization problems. 3. Application of Multi-objective Evolutionary Algorithms to multi-resolution Radar-Radiometer soil moisture retrieval using SMAP satellite data. 4. Quantication of spacecraft observation and geometry eects on radar mea- surements and their propagation through the soil moisture retrieval process. 5. Demonstration of estimation of multiple geophysical parameters using comple- mentary active and passive remote sensing observation. 6. Improved soil moisture prediction accuracy by developing and implementing joint-physics emission-scattering models. 10 CHAPTER II Fundamental Concepts and Theoretical Background At the heart of physics-based soil moisture retrieval methodologies lies forward scattering and emission models, which to varying degrees of accuracy predict the expected radar backscatter 0 or radiometer TB measurements. This chapter will present the basis of Radar (Active) and Radiometer (Passive) Remote Sensing and their associated Electromagnetic models. A concise discussion on the complementar- ity and interrelatedness of emission and scattering is also presented at the conclusion of this chapter. 2.1 Radar Remote Sensing 2.1.1 Generalized Radar Scattering Theory When an incident Electromagnetic plane wave illuminates an object, part of the incident energy is absorbed by the target and part of the energy is scattered. Denot- ing the incidence eld as E i and the scattered eld as E s they can be decomposed 11 into vertical and horizontal eld components generally written as E i = ^ v i E i v + ^ h i E i h (2.1a) E s = ^ v s E s v + ^ h s E s h (2.1b) A generalized schematic of an incident and scattered wave geometry, applicable to most remote sensing scenarios, is shown in Fig. 2.1. Incidence angles are dened as ( i ; i ) and scattering angle as ( s ; s ). Henceforth in this chapter superscripts or subscripts i and s refer to the incident and scattered waves respectively. The unit vectors ^ v and ^ h decompose the traveling wave into vertical and horizontal components with respect to the direction of propagation ^ k. Typically two coordinate system conventions are used to describe the relationship between the three orthogonal vectors ^ k, ^ v; and ^ h: (1) Backscatter Alignment (BSA) Convention (2) Forward Scatter Alignment (FSA) Convention. x z y E i q (θ i ,ϕ i ) θ i θ s Φ i Φ s E s p (θ s ,ϕ s ) Figure 2.1: Generalized Radar Scattering Coordinate System. In the wave-oriented FSA convention, the orthonormal wave coordinates are aligned with the Spherical Coordinate System and each unit vector, both in the 12 incidence and scattered directions, are dened as: ^ k i = ^ xcos i sin i + ^ ysin i sin i ^ zcos i (2.2a) ^ h i = ^ z ^ k i j^ z ^ k i j =^ xsin i + ^ ycos i (2.2b) ^ v i = ^ h i ^ k i =^ xcos i cos i ^ ysin i cos i ^ zsin i (2.2c) ^ k s = ^ xcos s sin s + ^ ysin s sin s + ^ zcos s (2.2d) ^ h s = ^ z ^ k s j^ z ^ k s j =^ xsin s + ^ ycos s (2.2e) ^ v s = ^ h s ^ k s = ^ xcos s cos s + ^ ysin s cos s ^ zsin s : (2.2f) On the other hand, in the BSA convention, which is typically used for backscatter measurements rather than bistatic setups, the wave-orientation is with respect to the transmitting and receiving radar antennae. The BSA transmitting coordinate system is identical to that of the FSA incident coordinates ( ^ k i ; ^ v i ; ^ h i ) = ( ^ k t ; ^ v t ; ^ h t ). The scattered elds, however, are transformed to received elds and converted as ( ^ k s ; ^ v s ; ^ h s ) = ( ^ k r ; ^ v r ; ^ h r ). The p-polarized scattered wave, reradiated from an object in the far-eld at a distance r is generally a spherical wave with the functional form of e (jk r) r S pq . S pq is the scattering amplitude function from the q-polarized incident wave to the p- polarized scattered wave and e (jk r) r is the Scalar Green's Function. Combining both E i and E s in matrix format, their relation can be expanded as E h E v s = e jk r r S hh S vh S hv S vv E h E v i : (2.3) or alternatively E s = e jk^ r ^ r ~ S E i : (2.4) The complex scattering matrix ~ S describes the scattering behavior of an object and relates the incident and scatter elds. The individual scattering terms are functions of 13 the objects geometry, electrical properties and the illumination and scattering angles ( i , i ) and ( s , s ). The individual elements, in the far-eld, take the following form S pq = lim ^ r!1 ^ re jk^ r E s p E i q : (2.5) It is within the complex denition of the Scattering Matrix, and its elements, where the geophysical properties of remote sensing targets are manifested. Whereas the Scattering Matrix relates the incident and scattered elds, the Stokes Matrix relates the incident and scattered Stokes vectors which dene the specic intensities of the incident and scattered waves. These Stokes vectors are dened as I i = 2 6 4 I i v I i h U i V i 3 7 5 = 2 6 4 jE i v j 2 jE i h j 2 2<e(E i v E i h ) 2=m(E i v E i h ) 3 7 5 = (2.6) for the incident elds and I s = 2 6 4 I s v I s h U s V s 3 7 5 = 2 6 4 jE s v j 2 jE s h j 2 2<e(E s v E s h ) 2=m(E s v E s h ) 3 7 5 = (2.7) for the scattered eld and are related, as mentioned, by the Stokes matrix I s = 1 R 2 r M I i (2.8) where M, known as the modied Stokes Matrix, is written as M = 2 6 4 jS vv j 2 jS vh j 2 <e(S vv S vh ) =m(S vv S vh ) jS hv j 2 jS hh j 2 <e(S hv S hh ) =m(S hv S hh ) 2<e(S vv S hv ) 2<e(S vh S hh ) <e(S vv S hh +S vh S hv ) =m(S vv S hh +S vh S hv ) 2=m(S vv S hv ) 2=m(S vh S hh ) =m(S vv S hh +S vh S hv ) <e(S vv S hh +S vh S hv ) 3 7 5 : (2.9) The power density, in units of [W=m 2 ], due an incident wave E i q , at a target R t 14 away form an transmitting antenna with gain G t and power P t q is S i q = P t q 4R 2 t G t : (2.10) Portion of this power is scattered (reradiated) and the rest absorbed at the tar- get. Reradiation of scattered energy is, in general, in all directions however, only the fraction of this power in the direction of the receiving antenna is of interest. Furthermore, the scattered wave has both q and p polarization states of which only one at a time is measured at the receiving antenna. Therefore, the p-polarized power reradiated from the target, towards the receiver is P rec p = pq S i q (2.11) where pq is dened as the Scattering Cross-section (RCS) of the target. pq is in general a function of the target's Electromagnetic properties. As a function of the transmitted power P t q the reradiated power is now P rec p = P t q G t 4R 2 t pq (2.12) which, when received at the receiving antenna yields the Power Density S s p = P rec 4R 2 t = P t q G t (4R t R r ) 2 pq : (2.13) The eciency of antennae in absorbing incidence power over an eective aperture A r is r which when combined with the above equation and the fact thatA r = 2 4 D r and G r = r D r results in the well known Radar Equation P r p = r A r S s q = G t G r 2 P t q (4) 3 R 2 t R 2 r pq : (2.14) Equation 2.14 is the Bistatic Radar Equation, and is applicable for cases when the 15 transmitting and receiving antennae are at dierent spatial locations or ( i ; i )6= ( s ; s ). Typically, for remote sensing scenarios, both transmitter and receiver are located at the same position, such that R t = R r = R and G t = G r . Thus, the Monostatic Radar Equation is P r p = G 2 2 P t q (4) 3 R 4 pq : (2.15) Due to the two-way wave propagation, the received power now decreases at 1=R 4 . The expression of pq in Eqs. 2.14 and 2.15 indicates a target small enough to be considered as a Point Target. Such targets are dened such that their subtended Solid Angle is smaller than the radar antenna beam illuminating the object [24]. The total radar backscattering cross-section of a target pq is determined by the ratio of the incident and scattered power densities as pq = lim Rr!1 4R 2 r S s p S i q (2.16) where the q-polarized power densityS i q is incident on the object and the p-polarized scattered or re ected power densityS s p is measured by the receiver. From the Poynting Theorem of Electromagnetic Waves, for a plane wave, it can easily be shown that the Power Density of a q-polarized wave isS q =jE q j 2 = 0 . Thus we haveS s p =jE s p j 2 = for the scattered eld andS i p =jE i p j 2 = for the incidence eld. Therefore, in view of Eqs. 2.5 and the above expressions, the backscatter strength of an object is pq = 4jS pq j 2 : (2.17) jS pq j is the pq element of the Scattering Matrix. For distributed Targets, extending beyond the radar's beamwidth, the radar equa- tions in 2.14 and 2.15 must be integrated over the target area such as 16 P r p () = x A P T q G 2 ( a ; a ) 2 (4) 3 R 4 a 0 pq dA: (2.18) ( a ; a ) are the integrand variables extending within the target area. Note that a dierent RCS is dened as 0 pq which is now the Normalized Radar Cross-section of the a target with are A. Whereas pq was in units of m 2 , 0 pq is unit less, i.e., 0 pq = 0 =A. The generalized denition of 0 pq is the average value of radar backscatter cross-section of distributed targets within an illuminated are A, normalized to the area itself. It is important to note that radars operating on moving platforms, such as air- crafts or satellites, are in Synthetic Aperture Radar (Synthetic Aperture Radar (SAR)) mode to synthesize an eectively larger aperture to enhance spatial reso- lution. Extensive signal processing is performed in SARs prior to obtaining 0 mea- surements. The scale and depth of SAR processing is beyond the scope of this work. Therefore, at the current state of radar remote sensing, 0 pq measurements are ob- tained and are input into geophysical parameter estimation schemes thus segregating SAR processing from the physics of Electromagnetic scattering. 2.1.2 Electromagnetic Radar Scattering Model The counterpart to radar remote sensing observations and data are the radar wave scattering models wherein a mathematical description of a target under ob- servation is dened and then using Maxwell's Equations the amount of expected radar backscatter is calculated. An extensive body of literature exists which have developed, evaluated and derived numerous Electromagnetic scattering models un- der many dierent condition. The focus of this section is on a particular model [15] due to its simple form and applicability to soil moisture remote sensing. Heuristically, the total backscatter from a vegetated target, Fig. 2.2, such as 17 Ground Layer Vegetation Layer Figure 2.2: Schematic Representation of Remote Sensing Targets including a surface or ground layer, and a vegetation layer. forested areas or cropland is written as 0 pq = surf pq e 2pq + veg pq + vegsurf pq (2.19) where the subscripts pq indicate the polarization state of the incident and scattered waves (HH,VV,HV, or VH). The total backscatter 0 is, in general, the sum of (1) backscatter contribution from rough surfaces modied by the wave's two-way attenu- ation through the vegetation or canopy layer surf e 2pq (2) backscatter contribution from the vegetation layer by itself veg and (3) the mutual scattering interaction between the surface and vegetation layer vegsurf . The attenuation factor pq is polarization dependent as well as being dependent on the vegetation layer's geome- try. A more complete description of surface scattering will be presented later in this section. Target features or constituents, such as trunks, branches, etc., are viewed as - nite length dielectric cylinders distributed by size and orientation based on certain probability density functions PDF and grouped into dierent categories: small or large branches, trunks, leaves or needles, etc. Trunks, for example, are near vertical cylinders with specic average diameters and heights. Graphically, such a discretiza- tion of the medium can be seen in Fig. 2.3, were the vegetation layer consists of a trunk/stem layer and canopy or branch layer. 18 Ground Layer Trunk/Stalk Layer Branch & Leaf Layer Figure 2.3: Target Structural Discretization: a general vegetated target is discretized in to groups of individual constituents. In Fig. 2.4, typical vegetation features and structures are shown. Modeling and accounting for each individual part of a canopy is impractical. To over comes this limitation, through extensive eld studies and data collection tasks for dierent land cover types statistics on trunk/stem densities, trunk diameters, branch lengths and diameters and many more are collected and an \average" tree is dened. Thus, when modeling the radar response of a given region, the model output refers to the average or most common constituent. Table 2.1 lists the major target categories and parameters. Figure 2.4: Dierent vegetation shapes and structures. Such an approach is taken in order to better dene and calculate dierent scat- tering mechanisms which are detected by a radar. As outlined in Eqs. 2.16 the pq polarized cross-sections are related to terms within the Stoke Matrix M of the target. Therefore, the task now is to determine the Stokes Matrix based on the discretized 19 Table 2.1 Radar Model Parameters Category Parameters Branch Small Radius/diameter Large Length Trunk (Stalks) Density Leaf Disk Orientation Needle Permittivity r Surface Moisture & Roughness view shown in Fig. 2.3. The Discrete Scatterer Approach, extensively outlined in [15], states that the total Stokes Matrix M total is found by summing individual scattering mechanisms' Stokes matrices M i . Mathematically, this is written as M total = n X i=1 T i M i T i (2.20) where n is the number of layers, typically 3, as depicted in Fig. 2.3 and T i is the Transmission or attenuation matrix of the ith layer which is also a function of the layer's Scattering Matrix Elements, but in the forward scattering direction only. Once M total is determined, 0 pq or pq can be readily found. For a typical remote sensing target the break down of the four individual scattering contributions can be seen in Fig. 2.5 and are described below. 1. Direct Ground Scatterig (G): Rough surface scattering studies have a long and diverse history with endless existing models and approaches. Focusing on L-band remote sensing, in general the surface Permittivity (or dielectric) prole r and surface roughness prole dictate the amount of radar scattering. The dielectric layer is assumed a half-space in contact with air with an eective r determined from Clay, Sand, Silt and Water fractions within a given volume of soil [25]. In 2D, the surface roughness is described as height variations in the z direction as z =f(x;y); f can be a random function or periodic with a 20 Ground Layer Trunk/Stem Layer Branch & Leaf Layer Radar Antenna G B BG TG Figure 2.5: Dominant Scattering Mechanisms: Ground, volume and interaction scat- tering mechanisms all contribute to the total measured backscatter. surface height distribution of p f (z). Most commonly, p f (z) takes a zero-mean Gaussian form as p f (z) = 1 s p 2 e 1 2 z 2 s 2 (2.21) with a single parameters as the height root mean squared surface standard de- viation fromz = 0 plane. Most commonly, the surface RMS height is expressed in terms of the wave number k as ks. A wide range of surface scattering models such as the Small Perturbation Model (SPM), Kirchho Model, Geometrical and Physical Optics models [24, 26, 27] exists which can be used to determine radar backscatter from surfaces. Since Stokes Matrix denition of scattering is of interest, the direct ground contribution is denoted as M g . 2. Branch Volume Scatterig (B): The total volume scattering withing the branch or canopy layer is the sum-total of individual wave scatterings from dierent components, especially arbitrarily oriented nite dielectric cylinders such that a new Scattering Matrix element S( i ; i ; s ; s ; ;) can be calcu- 21 lated resulting in the branch layer's Stokes Matrix M b . The angles ( ;) dene tilts and orientations within and out-o the plane of incidence. With prior knowledge, extensive eld campaign, data collection of dierent land cover types, typically an \average" branch of class large or small is determined. Such an \average" branch is parameterized as having a specic length, diam- eter, permittivity, and orientation with respect to the trunk. The following steps are then taken: (a) Calculate Electromagnetic scattering from an \average" branch assuming vertical cylindrical geometry with the parameterization given above, i.e., nd S( i ; i ; s ; s ; ;). (b) Dene and calculate orientation-averaged Stokes Matrix over a distribu- tions of of angles and orientations exists p( ;) within the canopy layer as M av = x M( ;)p( ;) d : (2.22) (c) Evaluate the total attenuation within the canopy layer of height h as M = h Z 0 T (z)M av T (z)dz: (2.23) T (z) is just the Stokes Matrix due to the Scattering matrix in the forward scattering direction. 3. Branch-Ground (BG) and Trunk-Ground Scattering (TG): Lastly, wave scattering also includes interactions between the trunk or branch layer with the ground, as shown in Fig. 2.5. Since we are predominantly interested in radar backscatter, BG and TG mechanisms can be broken down as two consecutive actions: (a) Specular scattering from the trunk or branch layer towards the ground. In the case of trunks, we can assume this is just the specular scattering 22 from vertical dielectric cylinders from the direction of the incident wave towards the ground. (b) Specular scattering from the ground towards the receiving antenna due to the incident waves which was just scattered o the trunks (or branches). Due to Reciprocity Theorem, the converse also holds true and is equal to the above; that is, rst the wave rst specularly scatters from the ground the o the trunks and branches(TG = GT and BG = GB). For brevity, only the Scattering Matrix for TG or BG is given: S double = 2r h S hh (r h +r v )S hv (r h +r v ) 2r v S vv (2.24) S double is the either the TG or BG Scattering Matrix. r h and r v are the h or v-polarized Fresnel Equation for rough surfaces describing the specular ground scattering of the incident scattering matrix elements S hh ; S vv ; S hv and S vh . Therefore, the total Stokes Matrix for G, TG and BG scattering mechanisms is M total = M branch +T b T t M bg T t T b +T b T t M g T t T b +T b T t M t T t T b (2.25) Here, for completeness, the trunk T t and branch T b attenuation layers have been separated. Fig. 2.6 shows a simple model simulation of scattering from a single trunk over a rough surface as well as individual mechanisms within the geometry. This is a typical scenario for most land covers and clearly shows the inter relatedness of dierent parts of the canopy. 23 z Specular Scattering Trunk-Ground Double Bounce TG-DB GND GND+Atten. Total E i −100 −50 0 50 100 −40 −30 −20 −10 0 10 20 30 θ s [deg] σ 0 vv [dB] TG−DB GND GND+Atten. Total Figure 2.6: Simulation Example showing Total, Ground and Trunk Ground scatter- ing due to a single trunk over a rough surface both in the backscatter and forward scattering direction. 24 ( ϕ , δ ) S S M cyl i i s s cyl cyl i i s s ( , ; , ; , ) ( , ; , ; , ) θ φ θ φ ϕ δ θ φ θ φ ϕ δ → S S cyl cyl i i s s = ( , ; , ) θ φ θ φ E E i i i i = ( , ) θ φ 2- The Stokes Matrix (M cyl ) of tilted cylinders is then determine. 1- Scattering Matrix (S cyl ) of !nite length vertical cylinders must be known M T z M T z dz T z e h ave i i s s h z i = ⋅ ⋅ ∝ ∫ − ⋅ ( ) ( , ; , ) ( ) ( ) /cos( ) θ φ θ φ τ θ 0 0 h 5- Interaction & “Double-Bounce” scattering terms are calculated. (For Branch-Ground & Trunk-Ground) 6- Total Stokes Matrix is the sum of di"erent Mechanisms G B BG TG M M T T M T T T T M T T T T M T T total brn b t gb t b b t g t b b t tg t b = + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ S r S r r S r r S r S S M cyl gnd hh h v hv h v vh v vv cyl gnd t h − − = + +       → 2 2 ( ) ( ) g g 3- Orientation Averaging for Random Distribution of Cylinders is performed M M d d ave cyl = ∫∫ ∑ ( , ) ( , ) ϕ δ ρ ϕ δ ϕ δ α α = categorty types 4- Attenutaion through the Canopy is calculted. Figure 2.7: Steps in calculating individual scattering mechanisms from a vegetated target (1) Scattering from a nite dielectric cylinder (2) tilt and orientation correction (3) tilt and orientation distribution (4) attenuation calculation through canopy (5) Trunk-Ground interaction (6) Cascading of individual contributions. 25 From a practical perspective, within an optimization scheme, constant evaluation for the DSA model can be computational costly. Therefore Van zyl et.al. [28, 29], etc., have proposed to evaluate these deterministic models prior to use and create look-up tables or Datacubes. These datacubes are precomputed and are functions of the three main parameters aecting total backscatter (1) soil moisture, or soil permittivity (2) surface roughness, and (3) the total overlaying amount of vegetation, or Vegetation Water Content (VWC). For each classied land cover, a specic datacube can be dened. All geometric constituents of the trunk and canopy layers can be condensed in to an eective vegetation layer represented by a single VWC value. Extensive empirical studies have shown that VWC is a function of vegetation density, water density within the volume as well as the water fraction within the volume. 26 2.2 Radiometer Remote Sensing In contrast to radar remote sensing, were geophysical targets are actively illumi- nated by an incident wave, radiometer remote sensing, or Radiometry, measures the natural incoherent radiant Electromagnetic energy of observed targets. It is a well known fact that objects, partly due to their thermal temperature, emit a spectrum of energy. This section will thus discuss the fundamentals of Radiometry from Planck's Law and Radiative Transfer. In closing an overview of current geophysical emissions models is presented which will be used extensively to estimate and retrieve surface soil moisture. 2.2.1 Black Body Radiation and Brightness Temperature Radiometry stems from the application of Black Body Radiation to natural world sources. From a Quantum Mechanic view points, a Black Body is dened as: 1. Idealized and perfectly opaque object which absorbs all incident radiation. 2. A perfect emitter of Electromagnetic energy at Thermodynamic Equilibrium. Accordingly, Planck's Law of radiation shows that a Black Body uniformly radi- ates energy in all directions. The spectral energy of radiation, or spectral Intensity or Brightness, in units of Wm 2 st 1 Hz 1 , is I f = 2 h f 3 c 2 1 e hf=kT 1 (2.26) h is Planck's Constant, f is frequency in Hz, c is the speed of light in vacuum, k is the Boltzmann Constant and T is the objects physical temperature in units of Kelvin. All quantities in Eqs. 2.26 other than f and T are constant, therefore as both frequency and temperature vary, the amount of Brightness, or spectral intensity varies. 27 UV VISIBLE INFRARED 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 Wavelength (μm) 5000 K 4000 K 3000 K Classical theory (5000 K) 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 Spectral Radiance Wavelength (μm) 5000 K 4000 K 3000 K Classical theory (5000 K) Figure 2.8: Spectral Radiance as a function of wavelength By interchanging frequency with wavelength, Planck's Law can also be expressed as I = 2 h c 2 5 1 e hc= kT 1 (2.27) A simple plot spectral radiance as a function of wave length for dierent temper- atures is shown in Fig. 2.8. Analogous to the Radar Scattering Cross-section, a blackbody source has an area ofA s which emits with intensityI f detected at a receiving antenna a distanceR away. The antenna's eective aperture, similar to the Radar case, isA r and subtends a solid angle of r . The received power P f per frequency [WHz 1 ] is simply P f =I f A s r : (2.28) By interchanging the eective source aperture A s and receiving solid angle r , with the receiving aperture A r and source solid angle s , the received power becomes P f =I f A r s (2.29) 28 which when considering only a limited bandwidth of frequency becomes P = f2 Z f 1 P f df =A r s f2 Z f1 I f df: (2.30) In the low-frequency limit of Planck's Law, which also includes the Microwave region, Eqs. 2.26 can be approximated as I f 2 kT 2 (2.31) This approximation is commonly known as the Rayleigh-Jeans Law which considers the case when hf=KT 1. For almost all remote sensing of Earth scenarios, this approximation holds true and the actual deviation from Planck's law is negligible. Black Bodies are idealized objects which, when at Thermodynamic Equilibrium, radiate as much energy as any other object at the same temperature. Natural world targets are not black bodies, but rather gray bodies which emit and absorb less energy than actual Black Bodies. In the Microwave region of the Electromagnetic spectrum, and within a limited bandwidth B, the Brightness intensity of a Black Body I bb is I bb = kT 2 B (2.32) which is the same expression as the Rayleigh-Jeans Law, Eqs.2.31, modied by a factor of 1/2. This is because at any instance of time, antennae can only measure a single polarization of incident intensity. Since emission is unpolarizaed, on average half of the energy is in one orthogonal polarization to the other, hence the 1/2 factor. Considering a gray body target with direction dependent intensity I(;) at a physical temperature T, the equivalent Black Body Temperature for an actual black 29 body which yields the same intensity is T B (;). Thus I(;) = k 2 T B (;) B: (2.33) The quantityT B is commonly known as the Brightness Temperature of the object as is a key data product in Radiometer Remote Sensing. The ratio of Black Body Intensity I bb to gray body intensity I(;) is dened as emissivity e(;). Further- more, since I(;) < I bb emissivity is constraint 0 e(;) 1; thus an object's Brightness Temperature is equal to or smaller than its Physical Temperature. It is important to note that the measured spectral intensity, or henceforth Bright- ness Temperature, is due to a cascade and accumulation of multiple sources of emis- sion, Fig.. 2.9. Not only ground and vegetation emission is observed, but also contri- butions from the atmosphere and cosmic radiation is also measured. Furthermore, the antenna which itself has a Brightness Temperature, detects incident radiation through both its main lobe and side lobes. Therefore, similar to SAR processing for Radar Remote Sensing, signicant radiometric processing is performed prior to delivering surface and vegetation TB. 2.2.2 Radiative Transfer The ow and propagation of energy and intensity is governed by Radiative Trans- fer Theory which denes and describes the extinction and emission processes between Electromagnetic waves. Extinction is the loss of energy due to either absorption or scattering of waves and emission is the source of new energy added to the medium. The extinction coecient e is a measure of power attenuation and is the sum of the absorption coecient a and the scattering coecient s , i.e., e = a + s . The Radiative Transfer equation for intensity is dI d +I =J (2.34) 30 Cosmic Radiation TB Atmosphere Brightness Temperature Source Distribution Upwelling Ground+Vegetaion Emission Upwelling Atmoshpheric Emission Downwelling and Re!ected Atmoshperic Emission Total Incident Emission Figure 2.9: The total measured Brightness Temperature is due to the accumulation from ground, vegetation, atmospheric and Cosmic sources. such thatd denes the optical depth of the medium, or in other words, the extinction per unit length d = e dR. J is the total source function of the medium given as J(R; ^ R) = (1a)J a (R; ^ R) +a J s (R; ^ R) (2.35) a is the single scattering albedo dened as the radio of the scattering to extinction coecients a = s = e . Here, J a and J s describe the total absorption and scattering of waves from direction R to ^ R. The direct relationship between intensityI and Brightness Temperature TB was given in Eqs. 2.32 and 2.33. Thus, similar to Eqs. 2.34, the radiative Brightness Temperature equation is dTB d +TB = (1a)T +a T VS (2.36) here the new quantityT VS descries the volumes scattering of Brightness Temperature as the wave propagates through a medium; analogous to J s . 31 A key insights which Radiative Transfer oers is the direct linkage between Scat- tering of waves and the concept of emission. Within Eqs. 2.34 and 2.35 J s describes wave scattering as J s (R; ^ R) = 1 s x 4 ( ^ R; R 0 )I(R; ^ R)d 0 (2.37) Such that the scattering Phase Function (R; ^ R) is fundamentally in uenced by the medium's size (with respect to ), orientation, distribution and electrical properties. A generalized solution for Eqs. 2.36 can be derived by assuming a stratied medium and incrementally solving the Radiative Transfer Equation [24]. For a medium with vertical thickness H and an observing antenna at an angle , the total observed Brightness Temperature is TB(;H) =TB(0;H)e (0;H) +sec H Z 0 [(1a)T (z 0 ) +a T VS (;z 0 )]e (z 0 ;H) e dH: (2.38) The integration over the height H accounts for multiple scattering and absorption events throughout the medium while accounting for the incremental one-way atten- uation e (z 0 ;H) . 2.2.3 Geophysical Emission Model A specic solution to Eqs. 2.36 when applied to geophysical media such as a ground-vegetation scenario with the assumption of limited to no scattering within the vegetation layer (T vs or J s = 0 ) is the commonly referred to as the -! emis- sion model [11]. With this assumption, the vegetation layer eectively becomes an attenuation later, which reduces the upwelling ground emission. Due to the thermal temperature of the vegetation layer, it also contributes a certain amount of emission. 32 For this scenario, the p-polarized vegetation Transmissivity in the direction is T p () =e psec =e ep d sec : (2.39) Derivation details for -! are numerous in literature [30][11][24] thus only the nal form is presented here: TB p () =T s e p e p sec +T c (1! p ) (1e p sec ) (1 +r p e p sec ): (2.40) each term is dened as follows: 1. T s & T v are the surface and vegetation canopy physical temperatures in [K]. Most often, they are assumed to be equal. 2. e p is the ground layer emissivity and thus the re ectivity is r p = 1e p . Re- ectivity in turn is related to the Fresnel Equations. 3. ! is the single scattering albedo previously dened as the radio of the scattering to extinction coecient ratios. Emphasis is placed on the surface emissivity and re ectivity terms which are directly related to the surface temperature and complex permittivity. Natural Soil is a complex mixture of Sand, Clay, Silt and Water. It is this water within the soil which aects the mixtures eective permittivity, thus emission. Ultimately this is the goal of soil moisture estimation: to work backwards from the measured TB, to surface emissivity to surface permittivity. Once permittivity is determined, a whole range of soil mixture models can be used to determined the moisture content [25]. Plots of both surface emission and scattering response to soil moisture are shown in Fig.. 2.11. As a direct consequence of Snell's Law at a two-media boundary as well as prin- ciples of Boundary Conditions in Electromagnetic , the Fresnel equations are 33 r v = 2 cos 1 1 cos 2 2 cos 1 + 1 cos 2 (2.41a) r h = 2 cos 2 1 cos 1 2 cos 2 + 1 cos 1 (2.41b) where and are the intrinsic impedance and incidence angles of the rst or second media. For the case of Normal Incidence angles, the Fresnel equations take on the exact Transmission Line re ection coecient expression. For the general case of Lossy media with complex relative permittivity of r , = q 0 r 0 . Therefore, without presentation of deviation detail, in its most simplis- tic form, the p-polarized Brightness Temperature TB, of a dielectric medium with physical temperature T in contact with air is TB p () = (1jr p (; r )j 2 ) T =e p (; r ) T: (2.42) This form of TB is directly applicable to emission from smooth unvegetated areas such as deserts. 2.3 Radar vs. Radiometer With the theoretical and fundamental basis of Radar and Radiometer Remote Sensing presented in Sections 2.1 and 2.2, this section will focus on the more practical and applied aspects of these two complementary techniques; especially in the context of combined soil moisture retrieval. Two main features are discussed (1) Dierences in measurements spatial resolution (2) Dierences in sensitivity to soil moisture and complementarity. 34 2.3.1 Measurement Spatial Resolution On almost all moving platforms, especially spacecrafts, Radars operate in a Syn- thetic Aperture mode resulting in sub-kilometer scale backscatter cross-section mea- surements 1 at the cost of smaller swath widths. On the other hand, Real Aperture Radiometers yield kilometer scale, or larger, Brightness Temperature data products covering a very large swath. To compare and contrast this spatial resolution disparity a non-exhaustive list of select SAR and RAD spacecraft and satellite missions as well as their approximate spatial resolutions is given in Table 2.2. A schematic representation of this resolution dierence is also shown in Fig. 2.10.b such that within a typical Radiometer pixel, multiple SAR measurements can exists. For example, in the case of SMAP, Earth- gridded L-band Radiometer Brightness Temperature measurements are reported at a 36 km resolution whereas SAR acquisitions are obtained at 250 m scale, reprocessed and then reported at 1-3 km resolution. 1 It is important to note that soil moisture products derived from spaceborne SAR are typically reported at a coarser scale than the native radar resolution. This is due to SAR processing and down-scaling to overcome speckle noise, etc. 35 Table 2.2: SAR and Radiometer Satellite Missions Satellite or Instrument Years of Operation Instrument Frequency Nominal Resolution SMOS 2009 - present Radiometer L-Band 35 km Aquarius 2011 - 2015 Radiometer L-Band 40 km AMSR-E 2002 - 2011 Radiometer Multi-Band 5-50 km SMAP 2015 - present Radiometer L-Band 36 km SMAP 2015 - present SAR L-Band 300 m Radarsat-1/2 1995 - 2013 2007 - present SAR C-Band 3-8 m Sentinal-1 2014 - present SAR C-Band 5 m PALSAR 2006 - present SAR L-Band 7-100 m ERS-1/2 1991 - 2011 SAR C-Band 30 m 36 Radar View Radiometer View Radar & Radiometer Platform (a) Spacecraft Observatory Radar View Radiometer View Ob (b) Figure 2.10: (a) Single Resolution Radar-Radiometer measurement schematic (b) Multi-resolution Radar-Radiometer measurement schematic from a spacecraft. It is assumed that Earth-gridded Radar measurements are nested within a single Ra- diometer measurement. Schematic (b) also represents a SMAP like scenario. Clearly, high resolution soil moisture estimates obtained from Radar-only mea- surements are preferred over the coarser resolution Radiometer-only estimates and are potentially more suitable for high level science applications. However, in terms of estimation accuracy, the performance of radar-based and radiometer-based retrievals must be examined as well. 2.3.2 Soil Moisture Sensitivity & Estimation Accuracy To estimate geophysical parameters of interest, using microwave remote sensing measurements which have the most sensitivity and least ambiguity across a wide range of soil moisture and vegetations conditions, will yield the most accurate esti- mate of that parameters. Extensive prior studies [31] have examined and evaluated 0 and TB sensitivity to changes in Land Surface Parameters (LSP), land cover types, frequency bands (L, C, and X-band) as well as for dierent incidence angels. In short, for L-band soil moisture remote sensing, (a) for low to moderate amounts of Vegetation Water Content (VWC) (<5kg/m 2 ), L-band radiometry shows signif- 37 Volumetric Soil Moisture [cm 3 /cm 3 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Brightness Temep. [K] 120 140 160 180 200 220 240 260 280 300 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Backscatter σ 0 [dB] -35 -30 -25 -20 -15 -10 Volumetric Soil Moisture [cm 3 /cm 3 ] V-pol H-pol HH VV (a) (b) Rough s = 1 cm l = 15 cm Smooth s = 0.5 cm l = 15 cm Rough s = 1 cm l = 15 cm Smooth s = 0.5 cm l = 15 cm Figure 2.11: (a) Variation in TB vs. soil moisture as the surface roughens RMS height (s) doubles at 300 K physical temperature. (b) Variation in Radar backscat- ter vs. soil moisture for the same amount of roughness change. Radar backscatter shows signicant sensitivity to roughness whereas emission is slightly aected. Ap- proximately backscatter changes 5dB compared to3K at 0.02m 3 /m 3 . Solid lines represent smooth surface (s = 0.5 cm & l = 15 cm); Dashed lines are for rougher surface (s = 1 cm & l = 15cm.); H-pol are in blue and V-pol are in red. The surface roughness prole is Gaussian with a correlation length (l). icant response to changes in soil moisture, at times up to90 K from dry to wet conditions [32] (b) radar backscatter is highly sensitive to surface roughness as seen in Fig. 2.11, and (c) both TB and 0 observations are aected with increasing VWC, however vegetation scattering is more signicant. The later has also been a moti- vation to use SAR for land cover, crop, and wetlands classication applications. A thorough analysis and assessment on soil moisture products derived from AMSR-E (passive) and ASCAT (active) by Brocca et. al. [33] indicated a noticeable degrada- tion in soil moisture estimation with increasing vegetation optical depth, i.e., more vegetation, which arm the above emission and scattering sensitivity responses. Although the physical processes of emission and scattering are interrelated, their individual responses and sensitivities to variations in LSP, especially soil moisture, are dissimilar. Scatter plots, Fig. 2.12, of actual measured TB and 0 [34] across a range of soil moisture conditions clearly show this dierence. As the surface be- comes wetter, backscatter increases and typically tends to saturate. For the case 38 Brightness Temep. TBV [K] Backscatter σ 0 vv [dB] Brightness Temeperature TBV [K] Volumetric Soil Moisture [cm 3 /cm 3 ] (a) (b) Radar Backscatter σ 0 VV [dB] -30 -25 -20 -15 -10 -5 260 265 270 275 280 285 290 295 300 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -30 -25 -20 -15 -10 -5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 260 270 280 290 300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Radar Radiometer Volumetric Soil Moisture [cm 3 /cm 3 ] Figure 2.12: (a) Radar backscatter vv [dB] (top) and Brightness Temperature TB V [K] (bottom) plots vs. Soil Moisture [m 3 /m 3 ] (b) Radar vs. Radiometer general rela- tionship indicates a negative correlation. The data is a subset of the PALS-SMEX02 campaign collocated with in situ sampling locations. H-polarized measurements have similar behaviors. Size's of circles indicate the amount of VWC with a maximum of 6kg/m 2 , i.e., large VWC is large circle & vice versa. of Fig. 2.12.a-top vv varies from -25 dB to -10 dB due to a 0.3 m 3 /m 3 increase in soil moisture. On the other hand, Brightness Temperature across the geophysical soil moisture ranges of interest, generally exhibits a linear decrease and as seen in Fig. 2.12.a-bottom exhibits30 K change in TB at a rate of10 K/(m 3 /m 3 ). Simultaneously examining variations in both backscatter and emission, as shown in Fig. 2.12.b, highlights the negative correlation between the two phenomena, such that with increasing radar backscatter, i.e., increasing soil moisture, the amount of emission, for the case presented here, decreases with an approximate slope of 1.6 K/dB and a correlation coecient of -0.76. Earliest known attempts at examining synergistic use of both Active and Passive soil moisture observations are from Ulaby et. al. [35] where reportedly an absolute estimation error accuracy of30% in soil moisture over Corn and bare soil was determined when considering both TB and 0 . This outcome stemmed from observ- ing the dierent sensitivities of TB and 0 to soil moisture and then balancing and 39 averaging independent active and passive over- and under-estimated predictions. 2.3.3 Information Theoretic Perspective From an Information Theoretic view point, it can be shown that radar and ra- diometer measurements include a certain amount of Mutual Information (MI) re- garding the target's soil moisture or even vegetation. Konings et.al. [36], using the concepts of Entropy and Mutual Information, determined the number of independent parameters between as set of observations, or Degrees of Freedom (DOF) between TB and 0 measurements. In general, and as expected for N measurements (N typi- cally being 2) N-1 DOF< N. That is between a single TB and single 0 observation at least 1 parameter can be independently estimated. Given the limited amount of suitable single-resolution TB and 0 data which cover a wide range of soil moisture and vegetation conditions, numerical simulations are instead used to determine MI between Radar and Radiometer Observations. Fundamentally, MI is the degree of interdependence between two random vari- ables X and Y each with probability distributionp(x) andp(y). In other words, MI is the degree to which knowledge of one random variable conveys information about another random variable. If, for example, two variables are completely independent, they have no MI. On the other hand, if one is a deterministic function of the other, knowledge of the rst variable and its associated uncertainty is enough to infer in- formation about the other. In the context of Probability and Information theory, a random variable's uncertainty is knowns as Entropy H(x) = P p(x) log(p(x)). For discrete random variables MI is I(X; Y) = X y2Y X x2X p(x;y) log p(x;y) p(x)p(y) (2.43) p(x;y) is the joint probability distribution between the random variables. Expansion 40 of Eqs.2.43 in terms of Entropy yields I(X; Y) =H(X)H(XjY) (2.44a) =H(Y)H(YjX) (2.44b) =H(X) +H(Y)H(XjY): (2.44c) similar to p(x;y), H(XjY) is the Joint Entropy between X and Y. In the context of radar and radiometer remote sensing active and passive ob- servations 0 and TB can be considered random variables, with latent or hidden variables being soil moisture or vegetation. To understand how the interrelatedness between TB and 0 change, a series of numerical simulations are performed to rst draw probability distributions for either term and then calculate the amount of MI under various conditions. Of specic interest is how MI and the relationship between TB and 0 changes with respect to surface soil moisture. The underlying rational is essentially because these two data types are to be considered simultaneously to estimate soil moisture, thus the extent to which they convey useful information is of importance. The following steps are taken: 1. The range of soil permittivity ( r 2 [3 30] or 4% sm40%) is considered and binned into step sizes of r = 2.5 (or approximately 4% moisture steps). Also, the entire range of land cover dependent VWC is considered. For the discussions in this section, a single surface roughness values is considered ks =0.15. 2. For each r bin, TB and 0 model simulations are performed. 3. TB and 0 are normalized to ranges between [0 1]. 4. Histogram based Probability Density Functions (PDF) are calculated and then the Entropy, Joint Entropy and MI are calculated. In Fig. 2.13, via numerical simulations, probability density distributions for 0 41 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.03 0.04 0.05 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Normalized Sigma 0 Normalized TB Prob(Sigma 0 ) Prob(TB) Increase in Soil Moisture (a) Normalized Sigma 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Prob(Sigma) 0 0 0.05 0.1 0.15 0.2 Normalized TB -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Prob(TB) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 (b) Figure 2.13: Probability Distributions for Normalized Radar Backscatter and Bright- ness Temperature. (a) PDFs for Woody Savanna (b) PDFs for Grass. Dashed (`{`) lines are HH pol and Solid lines are V-pol. and TB can be seen. Each PDF curve covers a specic permittivity bin and a wide range of VWC. In general, the shapes of the PDFs across the range of permittivity are similar however the skewness becomes more negative (i.e., longer left hand tail). Also, in order to have a consistent basis of comparison, both TB and 0 are normalized; this can be an underlying reason why the shape of the distributions are similar since model response are generally smooth, just shifted up or down, i.e., Fig. 2.11. Furthermore, in the gure, two dierent land covers have been considered; Fig. 2.13a is for a Woody Savanna and Fig. 2.13b is for a Grassland. At rst look the PDFs for 0 between Grassland and Woody Savannas are noticeably dierent and are due to the underlying Electromagnetic scattering response to media of dierent structures. On the other hand, PDFs for TB are very similar and almost on the same order. This is because in the -! model, vegetation structure is not properly accounted for; hence similar distributions. This is an important consideration which must be addressed in future research and possibly considering more complete modeling of the emission process. Using Eqs. 2.43 and the PDFs determined in Fig. 2.13, MI can now be calculated for radar-only, radiometer-only and co-pol combinations between radar and radiome- 42 ter, i.e., I( 0 vv ; 0 hh ); I(TBV;TBH) ; I( 0 vv ;TBV ); I( 0 hh ;TBH). The results are seen in Fig .2.14. For both Woody Savanna and Grassland, I( 0 vv ; 0 hh ) shows very little variation, especially for the grass scenario. This is most likely due to the fact that across the entire range of soil moisture and vegetation condition, the radar re- sponse for HH and VV-pols are similar. That is, as seen in Fig. 2.11, although the amount of backscatter is dierent, the trends of the curves are the same hence an almost steady MI. Soil Permittivity Bins (3,5.5) (5.5,8) (8,10.5) (10.5,13) (13,15.5) (15.5,18) (18,20.5) (20.5,23.5) (23.5,25) (25.5,28) (3,5.5) (5.5,8) (8,10.5) Mutual Information 0 0.5 1 1.5 2 2.5 MI(Sig HH ,Sig VV ) MI(TB H ,TB v ) MI(Sig vv ,TB V ) MI(Sig HH ,TB H ) (a) (3,5.5) (5.5,8) (8,10.5) (10.5,13) (13,15.5) (15.5,18) (18,20.5) (20.5,23.5) (23.5,25) (25.5,28) (3,5.5) Mutual Information 0 0.5 1 1.5 2 2.5 MI(Sig HH ,Sig VV ) MI(TB H ,TB v ) MI(Sig vv ,TB V ) MI(Sig HH ,TB H ) Soil Permittivity Bins (b) Figure 2.14: Mutual Information for (a) Woody Savanna and (b) Grasslands. With increasing soil moisture all coupled MI also increase but then gradually tend to level-o. This indicates that as the soil gradually becomes wetter, complementary TB or 0 observation do not provide more unique information and there is a limit on how one observation informs about the other. Overall, although some degree of mutual information is observed, more exten- sive studies are required to fully capture the relationship between radar backscatter and radiometer Brightness Temperature from an Information Theoretic view point; especially with respect changes in vegetation and surface roughness. 43 2.4 Chapter Summary To encapsulate the concept soil moisture radar and radiometer remote sensing a comprehensive overview is shown in Fig. 2.15. 44 Cosmic Radiation TB Atmosphere Brightness Temperature Source Distribution Upwelling Ground+Vegetaion Emission Upwelling Atmoshpheric Emission Downwelling and Re!ected Atmoshperic Emission Total Incident Emission Ground E i E s Incident Plane Wave Scattered Wave Front Due to Ground, Volume and Interaction Scattering Radar and Radiometer Remote Sensing Data Figure 2.15: Overview of Radar and Radiometer Remote Sensing Process 45 CHAPTER III Single-Resolution Combined Radar-Radiometer Soil Moisture Retrieval The discussion on C-AP soil moisture estimation is divided into two categories (1) Single-Resolution (SR) perspective and (2) Multi-Resolution (MR) scenario. SR- CAP address soil moisture estimation when the complementary set of radar and radiometer observations represent the same spatial resolution such as instruments mounted on towers or airborne platforms. The natural evolution of SR-CAP is for space-based observatories, where as mentioned in Chapter II, due to the dierent eective antenna aperture and wave coherency, measurements yield very dierent spatial resolutions. Therefore, a robust and complete cross-platform methodology which is suitable and applicable to all observation geometries is highly desirable and the topic of discussion in this and the following chapter. 3.1 Introduction Inherently, from a physical perspective, emission and scattering from a geophys- ical target are related to each other and are jointly aected by the target's Electro- magnetic properties and conguration. In the context of soil moisture remote sensing 46 these properties are soil moisture, or the soil permittivity, surface roughness and the amount and geometry of the overlaying vegetation. Thus, active and passive measurements acquired simultaneously from a target, for example a forest or cropland, convey overlapping information about the target, which when used synergistically within a proper estimation framework to predict surface soil moisture will outperform any radar-only or radiometer-only technique. The key here, however, is take proper advantage of the share parameter kernel, here dened as x, between the emission process and scattering process. Since at unit resolution both SAR and RAD observe the same area their corresponding measure- ments are due to this so called parameter kernel x a bicriterian Objective, or Cost, function, L ap ( x) can be dened such that SAR and RAD contributions are tied and constraint together. A generic mathematical description of L ap ( x) is L ap ( x) =L a ( x) + L p ( x) (3.1) where L a ( x) represents the radar contributions and L p ( x) the corresponding ra- diometer contributions. The subscripts a and p refer to Active and Passive processes. is a regularization term, yet to be dened. In physics based retrieval and optimiza- tion methods, typically model parameters are modied based on certain optimization rules until the dierence, usually an L 2 norm between model predictions and obser- vations, is minimized. Generalizing radar backscatter models as 0 ( x) and emission models as TB( x) and taking a Chi-squared form of the L 2 norm, the full form of L ap ( x) can be written as L( x) = X pp=vv;hh 0 pp 0 pp ( x) kp 2 + X p=v;h TB p TB p ( x) T 2 (3.2a) L( x) = X pp=vv;hh 0 pp 0 pp ( x) 2 + kp T 2 X p=v;h jTB p TB p ( x)j 2 (3.2b) 47 Table 3.1 Active-Passive Soil Moisture Retrieval Simulation Model Parameters Symbol Quantity Value Certainty f a Radar Frequency 1.26 GHz NA f p Radiometer Frequency 1.41 GHz NA i Incidence Angle 40 deg NA T s Surface Temperature 300 K 5% T c Vegetation Temperature 300 K 5% ! Scattering Albedo 0.05 5% b Vegetation Parameter 0.1 5% ks Electromagnetic Roughness 0.1-0.3 5% h Emission Roughness Parameter 0.1 5% vsm Volumetric Soil Moisture 0.01-0.4 m 3 /m 3 Target vwc Vegetation Water Content 0-5 kg/m 2 10% The rst term in 3.2a is the radar-only contribution withinL ap ( x) and the second term is the radiometer contribution. Each term is the normalized dierence between model predictions ( 0 pp ( x) or TB p ( x)) and measurements were the normalization factor is the uncertainty associated with each measurement, i.e., k p for radar and T for radiometer. The form of L ap ( x) shows summation over all polarimetric observations (HH and VV for 0 and H- and V-pol for TB). By rearranging the terms in 3.2a with the assumption that the uncertainty, or noise standard deviation, for each channel are the same, a regularization term can be dened, i.e., kp T . The continuation of this chapter will, in detail, assess the performance of this pro- posed joint radar-radiometer objective function via extensive numerical simulations as well as tests on real eld data. Unless specied otherwise, base model parameters use are given in Table 3.1. 48 3.2 Soil Moisture Estimation: Numerical Simulations To understand the performance of Eqs. 3.2 the shape and form of each contribu- tion must be examined. This is done by plotting L ap ( x) at soil moisture and VWC extremes, based on parameter values given in Table 3.1. That is, the normalized value of the radar-only, radiometer-only and radar-radiometer cost function within the vicinity of the cost function's minimum are plotted and shown in Fig. 3.1. For radar-only and radiometer-only, observe that the shape of the cost function is either relatively at or relatively steep at the extremes. In Fig. 3.1.a, for dry soil conditions and low VWC both L a ( x) and L p ( x) exhibit reasonable response to changes in soil moisture however the passive response is comparatively smoother i.e., less sensitive. Furthermore, in Fig. 3.1.b and 3.1.d for high VWC valuesL p ( x) shows almost no response to changes in soil moisture which is indicative of the vegetation emission contribution dominating the total expected emission. In this case radar-only con- tributions are dominant. For at cost function cross sections, small changes in the target parameter x result in small changes in the cost function hence re ecting a lack of sensitivity to the parameter of interest. On the other hand, changes in x for very sharp cost functions yield larger sensitivity to x with the possibility to cause the optimizer to overshoot the minimum and oscillate around it. The applicability of the proposed joint cost function can be clearly seen in this context. As observed in Fig. 3.1.a-d, the active-passive cost function at the extremes encapsulates the be- havior of both radar and radiometer terms. The dominance of one term over the other can be addressed by the regularization term kp T and is discussed in later sections in detail. To test the performance of the proposed radar-radiometer objective function, initially, closed-loop numerical simulations are performed. This is done via Monte- Carlo simulations were synthetic noisy \true" data are generated using applicable forward models across the parameter space of interest. Then, the objective function is minimized to estimate the desired parameters of interests. For three independent scenarios representing Corn, Soybean, and Grass land 49 0.04 0.06 0.08 0.1 0.12 x 10 -3 0 2 4 6 8 0.04 0.06 0.08 0.1 0.12 |Model - Data| 2 0 0.005 0.01 0.015 0.02 0.32 0.34 0.36 0.38 0.4 x 10 -3 0 0.5 1 1.5 0.32 0.34 0.36 0.38 0.4 x 10 -3 0 0.2 0.4 0.6 0.8 1 1.2 (a) Dry Soil - Low VWC (c) Wet Soil- Low VWC (b) Dry Soil - High VWC (d) Wet Soil - High VWC Active Passive Active-Passive soil moisture [m 3 /m 3 ] |Model - Data| 2 |Model - Data| 2 |Model - Data| 2 soil moisture [m 3 /m 3 ] soil moisture [m 3 /m 3 ] soil moisture [m 3 /m 3 ] Figure 3.1: Normalized cost function value around local minimum at VSM and VWC extremes. (a) Dry soil and low VWC (0.5 kg/m 2 ). (b) Dry soil and high VWC (4.5 kg/m 2 ). (c) Wet soil and low VWC (0.5 kg/m 2 ). (d) Wet soil and high VWC (4.5 kg/m 2 ). covers types, three sets of soil moisture retrievals are performed: 1) radar-only, 2) radiometer-only, and 3) joint radar-radiometer. The objective is to present the per- formance of the rst two methods separately as a baseline comparison to the radar- radiometer joint approach. In Fig.3.2 for each vegetation type considered, the mean RMS retrieval error over the entire range of soil moisture for given a VWC value is shown for active-only, passive-only, and the active-passive methods. In all cases, using soil dielectric mixing models and knowledge of soil texture, the estimated soil dielectric constant or permittivity (real part only) is converted to soil moisture. The Mironov soil dielectric model [25] is used in this work. Results discussed here only account for noise eects on radar and radiometer measurements, as well as the algo- 50 Figure 3.2: Average soil moisture retrieval error for (a) Corn, (b) Grass, and (c) Soybean, over the range of VWC, for active (\ dashed lines), passive (\-- dashed-dot lines), and joint active-passive (solid lines). Note the improvement of retrieval errors for the joint active-passive optimization method. rithm performance. Discussions on the inherent dielectric-soil moisture conversion errors are beyond the scope of this work and are not accounted for presently. As shown in Fig. 3.2 there is a noticeable decrease in the retrieval error for the active-passive approach. The overall trend of increasing retrieval error with in- creasing VWC is observed, even in the joint active-passive optimization. This is expected since, with increased VWC and for wetter soil conditions, radar backscat- ter loses sensitivity and saturates. Similarly, for high VWC values, the vegetation emission contribution increases, and a loss of sensitivity to the ground emission is observed. The eectiveness of the joint optimization, in comparison with radar-only or radiometer-only retrievals, is in the overall reduction of errors across the entire range of soil moisture and VWC. For example, in Fig. 3.2.a, for very high VWC, the maximum error observed drops from 0.045 to 0.032 m 3 /m 3 . Similarly, a 0.035 m 3 /m 3 error reduction is also observed, from 0.08 m 3 /m 3 for radar-only retrieval to 0.045 m 3 /m 3 , as shown in Fig. 3.2.b, for Grass. These retrieval errors are based on noisy simulated data and input parameter uncertainty. Table 3.2 shows maximum incurred retrieval RMS errors for two measurement scenarios: low noise and high noise. The errors reported are the maximum possible er- rors under the given conditions covering the range of 0 VWC5 kg/m 2 . For low to 51 Table 3.2 Maximum RMS Error [m 3 /m 3 ] for Low and High Noise Measurement Scenarios Noise Level Method Corn Soybean Grass Low Noise k p = 0.5 dB; T = 1.5 K Active 0.050 0.070 0.080 Passive 0.045 0.035 0.055 Combined 0.035 0.032 0.048 High Noise k p = 0.7 dB; T = 3 K Active 0.066 0.08 0.090 Passive 0.053 0.038 0.057 Combined 0.044 0.038 0.059 moderate measurement errors, the active-passive retrieval technique outperforms the other two methods. The performance, however, suers for higher levels of measure- ment noise, but is still comparable to or better than radar-only or radiometer-only methods. For all cases, higher errors occur for higher values of VWC, as expected. To further examine the CAP algorithm performance in the presence of input pa- rameter uncertainty, another set of numerical simulations are performed, throughout which only one parameter at a time is varied and assumed uncertain while all other model input parameters are kept the same with no error. For each parameter, the retrieved volumetric RMS error over the range of soil moisture and VWC of interest is shown in Table 3.3 for the three dierent land covers under examination. Each of the input parameters of interest is listed along with their assigned errors and un- certainties. The total root-sum-square (RSS) error, which captures the total error contribution from all parameters, is also calculated. Across the range of conditions considered, the total error accounting for each pa- rameter is signicantly below the SMAP soil moisture retrieval accuracy requirement of 0.04 m 3 /m 3 and indicates acceptable algorithm performance within the tested limit and robustness with respect to parameter error. Thus far, in the analysis presented here, the regularization term within the ob- jective function was set to 1 giving unity and equal weight to radar and radiometer 52 Table 3.3 Active-Passive Algorithm Parameter Sensitivity Analysis Parameter Uncertainty Corn Soybean Grass VWC 5% 0.0023 0.0034 0.0041 10% 0.0045 0.007 0.0075 Roughness ks & h 5% 0.0036 0.0046 0.005 Temperature T s =T c 5% 0.0065 0.0180 0.0230 b & ! 5% 0.0011 0.0016 0.0027 Total RSS Error * [m 3 /m 3 ] 0.0088 0.0190 0.0240 * RSS: Root Sum Squared. measurements and models. However, forward emission and scattering models, in general, have limitations and do not always fully capture the underlying physics of scattering and emission. Furthermore, for soil moisture and VWC values of interest, it is desirable to have comparable contribution from radar and radiometer mea- surements within the cost function such that no one term dominates the retrieval process. To overcome these limitations and to nd an optimum balance between the contributing weights ofL a ( x) andL p ( x), the regularization is varied across a wide range of values (0.001-100). The addition of this regularization term within the cost function enables shifting the balance and cost function in uence between L a ( x) and L p ( x). In practice, this gives the optimization scheme the exibility to give relatively more weight to the forward model that best represents the expected backscattering cross section or emission of the pixel under consideration. At this stage, the noise ration p T is assumed to be unity. To demonstrate this exibility, similar to before, joint active-passive optimiza- tion and Monte-Carlo simulations are performed for various land-cover types, while sweeping over the range of values. For each value, over the range of soil moisture and VWC given in Table 3.1, noisy radar and radiometer measurements are gener- ated and soil moisture estimation performed. Average RMS errors across the range 53 of true soil moisture and VWC are then reported. In Fig. 3.3, retrieval errors for Corn, Soybean, and Grass are shown, where each point on the plots represents the average error. For =1, radar and radiometer measurements have equal contribu- tion to the overall cost function, and the cost function is balanced. When 1, more weight is given to radar measurements. Similarly, when 1 radiometer measurements and the -! model are given more weight in the cost function. Thus, essentially, is used to nd the best, or optimum, contribution between active and passive measurements. Closer examination of Fig. 3.3 indicates that for 1 5, i.e., for slightly more weight to passive measurements, overall estimation errors can be reduced. For example, for Grass, between the two extremes of active-only and passive-only, minimum error is achieved when is approximately 2, corresponding to 0.032 m 3 /m 3 volumetric error. Fig. 3.3 is summarized in Table 3.4 by examining the single measurement retrieval methods (the two extremes in Fig. 3.3) and the joint method. Figure 3.3: Eect of parameter in reducing overall retrieval errors for Corn, Soy- bean, and Grass. When = 1, radar and radiometer contribute equally. For 1, more weight is given to radar, and for 1, more weight is given to radiometer measurements. For 1 5, minimum RMS errors are achieved. 54 Table 3.4 Error Reduction due to Regularization Retrieval Method RMSE [m 3 /m 3 ] Grass Corn Soybean Active-Only 0.055 0.025 0.041 Passive-Only 0.040 0.030 0.021 Combined 2-4 0.032 0.018 0.019 Therefore, it can be seen that with proper regularization and weighting between radar observation and radiometer observations, obtaining comparable and even su- perior soil moisture estimates is possible. 3.3 Soil Moisture Estimation: Application to Field Mea- surements Simulation based assessment of CAP soil moisture estimation was demonstrated and outlined in the previous section. It was shown that by simultaneous and ecient merging of radar and radiometer measurements improved soil moisture estimation is possible. In the face of actual eld data and real measurement, a range of factors in uence and degrade the performance any estimation scheme. Most noticeably, measure- ment error, data processing, and model performance aect retrieval outcomes. This section will thus demonstrate CAP with two real datasets and will highlight some key lessons learned which must be address to further improve soil moisture remote sensing techniques. 55 3.3.1 PALS SMEX02 Dataset The same CAP algorithm discusses in the previous section is applied to data obtained during the 2002 Soil Moisture Experiment (SMEX02) [34, 37], in Walnut Creek, Iowa, USA. The campaign took place between June 25 and July 12, 2002, throughout which the airborne PALS instrument was used to obtain radar backscat- ter and radiometer TB measurements over various land-cover conditions, predom- inantly Corn and Soybean elds. PALSs polarimetric radars operate at 1.26 and 3.15 GHz and the non-scanning microwave radiometers at 1.41 and 2.69 GHz. Both instruments yield the same spatial resolution of approximately 400 m, at 1-km ight altitude, with a 45 o incidence angle. Thirty-one eld locations were selected for soil moisture sampling throughout the Walnut Creek Watershed. To account for moisture variations and dierent soil types, 16 dierent sampling points were distributed within each eld. During the campaign experiment, the PALS instrument ew on eight dierent days (day of year (DOY) 176, 178, 182, 183, 186, 187, 188, and 189) during which soil moisture measurements of the top 5 cm were recorded from each site and averaged to represent the eld's soil moisture content. For soil probe calibration purposes and to characterize soil texture as well as bulk density, various gravimetric soil samples were collected during each sampling round. Handheld infrared thermometers and temperature probes were also used to determine the surface, 1-cm, 5-cm, and 10-cm soil temperature. VWC val- ues for Soybean elds reached a maximum of 1.5 kg/m 2 , and between 2 kg/m 2 to 5 kg/m 2 for Cornelds. In situ VWC measurements were taken three times throughout the campaign experiment. A complete description of the eld campaign is available in [37]. In addition to the airborne measurements, all the ground in situ soil mois- ture, texture, and VWC measurements collected were used to validate both forward emission and scattering models as well as the retrieval algorithms. The soil moisture retrieval algorithm highlighted earlier is applied to SMEX02 PALS data, such that radar backscattering cross-section and TB measurements repre- senting a unit pixel, roughly 400 m400 m, were collocated with in situ soil moisture 56 Figure 3.4: Single resolution soil moisture retrieval for PALS SMEX02 with param- eter varied. Minimum eld-averaged retrieval errors for = 6 are achieved. Corn (\ dashed lines), Soy (\-- dashed-dot lines), and Field-averaged (solid lines). and VWC measurements. Field-averaged RMS retrieval errors, as shown in Fig. 3.4, are calculated and reported for all in situ locations (31 sites) over the duration of the campaign. Field-averaged errors are calculated using both Corn and Soybean sites. Errors associated with either crop type are also shown separately in Fig. 3.4. Similar to Fig. 3.2, overall retrieval errors are reduced when slightly more weight is given to the radiometer contribution. For = 6 , optimum retrieval errors for the entire SMEX02 campaign including the large variations in VWC and soil moisture conditions are achieved. The eld-averaged (combination of Corn and Soybean) errors are, however, relatively high for 2 and are due to the high radar retrieval errors for the pixels. From a forward scattering and emission model perspective, this is expected. As shown in Fig. 3.5, the expected radar backscattering cross-section values extracted 57 from the Soybean datacubes, with knowledge of ancillary collocated parameters, compare relatively poorly with PALS backscattering cross-section measurements. For Soybean, the comparison yields higher RMS errors when compared to Corn. For vv and hh , the associated errors are 4 dB and 3.6 dB, respectively. This is, on average, 2 dB higher compared to Corn model-data mismatches. Therefore, it is expected that any soil moisture retrieval attempted on Soybean data using the current forward model will yield higher retrieval errors. This is the result re ected in Fig. 3.4. The applicability and usefulness of this retrieval method can now be explained as follows: at unit resolution with available radar and radiometer measurements, even with relatively decient forward models (both radar scattering and emission models), it is still possible to achieve minimum overall errors by nding the right balance between radar backscatter contributions and radiometer TB contributions. 3.3.2 ComRad Dataset As a further test of the proposed active-passive soil moisture estimation technique, radar and radiometer measurements obtained at the U.S. Department of Agriculture- Agricultural Research Service (USDA-ARS) research center during the summer of 2012 [37], using the truck-mounted ComRad [38]instrument, are examined. Field- averaged and collocated radar and radiometer measurements, representing the same eld of view, over a Corneld and a Soybean eld are available, as well as in situ soil moisture, temperature, and VWC measurements. The campaign duration ranged from June to October 2012. Numerous radar and radiometer observations were made over Corn and Soybean crops throughout these dates, and in-situ soil moisture and VWC measurements were collected at regular intervals. Measurements from a subset of dates were selected, i.e., from July 16th to August 28th, 2012, and active- passive soil moisture retrievals were performed. These dates were selected based on the availability of radar and radiometer data before and after rain events as well as dry down periods. The same cost function form as in Eqs.3.2 and the same 58 Figure 3.5: Simulated and PALS observed radar backscatter comparison for Corn (top) and Soybeans (bottom). RMSE for Corn (top gure) is 2 dB and 2.8 dB, for vv and hh , respectively. Soybean errors are 4 dB and 3.6 dB, respectively. optimization technique are employed to estimate soil moisture. A plot of the eld- averaged soil moisture for each of the crop plots is shown in Fig.3.6. The available soil moisture values range from 0.03 to 0.25 m 3 /m 3 , indicating a relatively dry season. Furthermore, Corn VWC, within the selected dates, ranges from 0.3 to 2 kg/m 2 , and Soy VWC ranges from 0.3 to 0.4 kg/m 2 . In order to obtain an acceptable match between ComRad measurements and model predictions throughout the selected campaign dates, model parameters were empirically chosen. For the radar datacubes, soil moisture and VWC information were available from in situ measurements. However, no information of surface rough- ness was available. With the assumption of near-constant surface roughness through- out the selected measurement dates, a single surface roughness value, which mini- mizes the error between model predictions and ComRad radar measurements, was 59 0 5 10 15 20 25 30 Soil Moisture Samplind Dates Field Average VSM [%] 7−16 7−17 7−30 7−31 8−2 8−3 8−4 8−5 8−8 8−9 8−11 8−12 8−13 8−14 8−15 8−16 8−17 8−21 8−22 8−23 8−24 8−27 8−28 Corn Soy Figure 3.6: Corn and Soy plot of eld-averaged soil moisture between July 16th and August 28th.H indicates available radar and radiometer measurement dates. determined. For each polarization (HH and VV) and for each crop type (Corn and Soy), this surface roughness value was determined via two approaches. First, the entire data set was used to nd the optimum roughness ks f , which yields the best match, i.e., minimum RMS error, between model predictions and ComRad measure- ments. Second, at random, at least 50% of data points were chosen as a training set to obtain the optimum surface roughness value and repeated 50 times to determine, on average, the best value for surface roughness (ks p ). Note that model predictions are based on the available Corn and Soy radar backscatter datacubes. The results of this analysis are shown in Table 3.5, where the mean and standard deviation of ks f and ks p are shown. As observed in Table 3.5, for all polarizations and crop types, the optimum sur- face roughness values determined using the partial data set (ks p ) are comparable to the parameters determined using the full data set (ks f ). Furthermore, due to the datacube axis discretization, the mean roughness values ks p map to the same 60 Table 3.5 Datacube Roughness Parameter Fitting. Crop Polarization Partial Dataset Full Dataset Mean ks p Stndev. Index Mean ks f Index Corn HH 1.1048 0.0131 117 1.1055 117 VV 1.2694 0.0243 134 1.2637 134 Soybean HH 0.868 0.0278 91 0.863 91 VV 0.0454 0.0119 3 0.420 3 datacube index as those of the full dataset, therefore extracting the same numeri- cal value from the datacubes. For the Corn plots, the best roughness values that minimize the RMS error between model and data are 1.10 for HH and 1.26 for VV. Regardless of the polarization of the incident wave, the physical surface roughness does not change. Therefore, a single surface roughness value must be used, which in this case isks = 1.10. The increased error for VV, by selecting this value, is on the order of 0.01 dB; therefore, a signicant deviation between model predictions and data is not expected for VV backscatter. For Soy plots, on the other hand, the best ks values for HH and VV are signicantly dierent. For HH, a much higher surface roughness value compared to VV provides the best t. The underlying reason for this deviation is still under investigation, but is possibly due to Soybean datacube inaccuracies, or due to surface features such as periodicity. Therefore, in the following analysis for Soybeans, VV radar backscatter are excluded, and only HH backscatter are used along with its determined surface roughness, which is more consistent with the Corneld data. It is important to note that the structure of the cost function permits using multiple polarizations and combinations to estimate soil moisture. This exibility and innovative feature allows, for instance, in this case, the exclusions of the VV backscatter data. The estimation of soil moisture for Soybeans, therefore, will include only backscatter as well as H- and V-pol TBs. Similar to the datacube model calibration, radiometer measurements were also 61 Table 3.6 Radiometer -!Optimum Model Parameters For ComRad. Polarization Parameter Corn Soybean Partial Full Partial Full H-pol ! 0.010 0.010 0.010 0.010 b 0.110 0.100 0.350 0.350 h 0.190 0.200 0.200 0.200 V-pol ! 0.010 0.100 0.020 0.010 b 0.020 0.010 0.014 0.010 h 0.010 0.010 0.170 0.180 used to determine the optimum , b, and h values. Surface and vegetation physical temperatures, as well as VWC and soil moisture information are available. The same approach as earlier was performed for the Corn and Soybean areas and for both H and V polarizations. In Table 3.6, similar to Table 3.5, partial data set and full data set optimum parameters are shown. Note that these parameters are numerically very close. Therefore, for the following analysis, within the radar forward model datacubes and the radiometer emission model, parameters determined via the full data set are used. Using these parameters in addition to other in situ information, comparisons between the radar datacubes and ComRad radar measurements are shown in Fig. 3.7. Similarly, radiometer measurements in comparison with the -! forward model sim- ulations are shown in Fig. 3.8. Model-data discrepancies, such as RMS error and biases, are summarized in Table 3.7 and, similar to the discussion with respect to the PALS data, have important implications in the inversion process. Corn and Soy RMS errors range from 1.2 to 1.6 dB, and biases are approximately 0.02 dB, except for VV backscatter for Corn with a 0.5-dB bias. Brightness temperature RMS errors are approximately 4 K for V-pol and 7-8 K for H-pol. Biases for either polarization or crop type are no more than 1.2 K. Using the model parameters in Tables 3.5 and 3.6 soil moisture estimation in the proposed active-passive framework is performed on the ComRad data and eld- 62 Figure 3.7: Datacube and ComRad radar backscatter model comparison. (a) Dat- acube 0 Corn. (b) Datacube 0 ; Soy. averaged soil moisture RMS errors calculated and presented in Fig. 3.9. Similar to Figs. 3.3 and 3.4, as the regularization parameter is varied, soil moisture estimates due to radar-only, radiometer-only, or a combination are produced. Note that, unlike Fig. 3.4, no minimum error exists between the two extremes, and best estimates are possible when more weight is given to radiometer measurements, i.e., 10. This response clearly re ects the forward model performance, as shown in Fig. 3.7 and 3.8, as well as the summary in Table 3.7: relative to radar measurements and models predictions, the observed model-data match for TB is better, and discrepancies are comparable to system noise (1.5-3 K); therefore, better soil moisture estimation is possible. 63 The increased radar-only soil moisture estimation errors for Corn, i.e., 0.046 m 3 /m 3 , relative to the Soybean radar-only estimates, i.e., 0.035 m 3 /m 3 , are due to the fact that, within the optimization process for Corn, both co-pol radar backscatter mea- surements were used, one of which presented signicant mismatch with model com- parisons. Examining the shape of the cost function cross-section using ComRad data over the Corneld gives valuable insight into the behavior of the algorithm. In Fig. 3.10, the cost function in Eqs. 3.2, for = 1 is plotted for soil moisture and VWC ex- tremes. These soil and vegetation values are based on available in situ data, and model parameterization is as outlined earlier. Emission and scattering predictions TB; Corn TB; Soybean Figure 3.8: -!model and ComRad emission comparison. (a) TB; Corn. (b) TB; Soy. 64 Table 3.7 ComRad Data and Forward Model Prediction Comparisons for Corn and Soybean Metric Polarization Corn Soybean RMSE 0 hh 1.25 [dB] 1.57 [dB] 0 vv 1.26 [dB] NA TB H 6.8 [K] 7.8 [K] TB V 4 [K] 3.8 [K] Bias 0 hh 0.020 [dB] 0.03 [dB] 0 vv 0.5 [dB] NA TB H 1.1 [K] -1.08 [K] TB V -0.6 [K] 0.170 [K] Figure 3.9: Average retrieval RMS errors for Corn (solid) and Soy (dotted). Estima- tion absolute biases shown as dashed lines. are simulated based on available parameters, except that (1) soil moisture is swept from 0.01 to 0.4 m 3 /m 3 and (2) VWC values are inputted based on the four test 65 Figure 3.10: ComRad normalized cost function values around local minimum at VSM and VWC extremes. (a) Dry soil and low VWC (0.03m 3 /m 3 , 0.8 kg/m 2 ). (b) Dry soil and high VWC (0.05 m 3 /m 3 , 2.06 kg/m 2 ). (c) Wet soil and low VWC (0.13m 3 /m 3 , 0.52 kg/m 2 ). (d)Wet soil and high VWC(0.14 m 3 /m 3 , 1.4 kg/m 2 ). Passive-onlyL p ( x)(dashed lines \). Active-onlyL a ( x)(dashed-dot lines \--). Active- Passive (dashed-circle lines \-o-). conditions. ComRad 0 and TB measurements coinciding with the test cases are ex- tracted from the data. Then, similar to Fig. 3.2, the individual radar and radiometer contributions to the cost functions, as well as the combined outcome, are shown. Although 0 and TB are due to the same target conditions, i.e., soil moisture and VWC, the cost function minima forL a ( x) andL p ( x) are at dierent locations. This behavior is mostly due to forward model deciencies in predicting the measurements, as well as independent instrument uncertainties. This is very dierent than the case in Fig. 3.2, where the simulation test cases were closed loop, yielding the same global minimum for all parts of the cost function. Furthermore, as outlined in Fig. 3.7 and Table 3.6, given the best model parameterization available, 0 predictions from the datacubes show a signicant deviation (in terms of RMS error) from actual 66 measurements, i.e., on the order of 1.5 dB for Corn. The eect of this is shown in the plots in Fig. 3.10, where, for the case ofL a ( x), the magnitude of the cost function at the minimum is larger compared to the radiometer case, particularly for high soil moisture values. For example, referring to Fig. 3.10.d, at their associated minima, the magnitude ofL a ( x) is 0.04, andL p ( x) is approximately 8:30 10 5. Given that = 1, the resulting combined active-passive cost function will mostly be dominated by radar contributions. The implications of this so-called \double-minimum is such that within the op- timization scheme, unless is large enough to give enough weight to the radiometer contributions within L ap ( x), radar-only estimates will be both poor and retrieval errors almost constant over a range of . This is exactly observed in Fig. 3.9, such that, for 10 3 10 1 radar contributions are dominant; then, onward radiome- ter contributions become more signicant, and the algorithm delivers more accurate estimates, i.e., lower RMS errors. Another feature to examine is the existence of biases between model outcomes and data. As noted in Table 3.7, the VV radar backscattering cross section, as com- pared to the HH backscattering, exhibits a higher model-data bias, i.e., 0.5 dB and 0.02 dB, respectively. It is expected that, by removing these biases, estimation errors will improve. The radiometer-only estimate errors are re ective of the radiometer forward model behavior, with respect to Corn and Soybean. From a forward model perspective, Soybean model and data RMS errors are comparable to Corn, but bi- ases are notably smaller, hence an overall better Soybean model-data match. As a further attempt to reduce average errors, model and data mean dierences (biases) are removed from the data before any soil moisture estimation is attempted. Soil moisture estimation is then performed and retrieval errors shown in Fig. 3.11. Note, that by removing model-data biases before optimization, both estimation errors and estimation biases are reduced. Most noticeably, for example, Corn radar-only errors reduce to 0.038 m 3 /m 3 , from 0.046 m 3 /m 3 . This outcome is predominantly due to the removal of the 0.5-dB mean dierence between model prediction and data in the VV radar backscattering channel. 67 Figure 3.11: Average retrieval RMS errors for Corn (solid) and Soy (dotted) with model-data biases removed. Estimation absolute biases shown as dashed lines. This approach can be employed for long-duration campaigns, such as SMAP, where, over time, existing model and data biases will be removed before any soil moisture retrieval is attempted. 68 3.4 Multi-parameter Estimation and Self-regularization In the previous section Combined Radar-Radiometer soil moisture estimation, at unit resolution, was demonstrated and discussed in detail. To place focus on the C-AP framework, a few key assumptions and simplications were initially made: 1. The amount of surface roughness was either assumed known or the emis- sion/scattering roughness parameter was optimized to yield best match be- tween model predictions and data. 2. Measurement noise and uncertainty was ignored and not accounted for within the objective function L ap ( x). 3. The regularization term was tuned, i.e., empirically varied, until best soil moisture predictions were obtained. Henceforth, in this section, none of the aforementioned assumptions are taken to be true and a more robust and complete presentation of C-AP is given. An important consideration for a global and widely applicable C-AP algorithm is how the process and methodology accounts for surface roughness. In addition to soil moisture and vegetation, the amount of surface roughness greatly aects measured radar backscatter and radiometer TB. Typically both backscatter and emission increase with increasing roughness, but at dierent rates. In Chapter II the eects of surface roughness on emission and scattering was presented in detail and an equivalent Electromagnetic roughness parameter ks was dened. More importantly, on regional, let alone global scales, knowledge of surface rough- ness is very limited and dicult to measure. Typically two approaches are taken to overcome this issue (a) assuming a land cover dependent value for surface roughness such as [15,16] for radar-only methods or [17] in radiometer-only methods and (b) time-series approaches, such that within a short window of time, the amount of sur- face roughness is assumed constant and a two-step optimization is performed; rst for roughness then for soil moisture [18]. 69 From a scene or target under observation more information can be inferred from simultaneous use of both radar backscatter and radiometer TB measurements. Nom- inally, in a resolution independent Active-Passive soil moisture retrieval framework, two co-pol radar (HH and VV) as well as two co-pol radiometer measurements (H- and V-pol) exist. Each process, emission and scattering, independently exhibit dier- ent and unique sensitivities to the underlying land surface conditions and collectively convey overlapping information from the target cell. Therefore, by taking advantage of this added so called mutual information be- tween backscatter and emission, it is possible to assume, along with soil moisture, surface roughness as an unknown parameter. Validation of retrieved surface rough- ness values however are generally not possible, thus within the retrieval framework they should be considered as free parameters and more emphasis placed on validation and examination of retrieved soil moisture. The continuing discussion will rst highlight forward model ambiguities with respect to multiple parameter (soil moisture and roughness) then will demonstrate multi-parameter estimation in the presence of measurement noise. Similar to the previous section, outcomes are rst examined through numerical simulation then application to real data. 3.4.1 Forward Model Ambiguity In forward model-centric estimation and retrieval methods, understanding both model and cost function behavior is key. Forward scattering and emission models can be, and are in this case, non-unique with respect to model parameters and have ambiguities. Specically, model predictions based on various soil moisture, surface roughness and VWC combinations can yield similar backscatter or Brightness Tem- perature values. This feature greatly complicates the inversion process, especially in the presence of measurement noise. Fig. 3.12 schematically shows this issue. Under nominal conditions, a single set of model parameters X 1 produces a single observed radar backscatter or radiometer emission value D. Another set of parameters X 2 70 Parameter Space Objective Space x 1 D x 2 Figure 3.12: Schematic representation of Model parameter and Data (Or Objective) space non-uniqueness. Both set of model parameters X 1 or X 2 can generate the same model prediction value D . Within the inversion process, it becomes unclear what set of parameters caused D, shown as the shaded region. can also generate the same observed value. Ambiguities further arise in the inverse process were, at rst glance, it is unclear whether D is due to X 1 or X 2 and in the presence of noise due to a range of possible parameters, shown as the grey shaded area in Fig. 3.12. In the case of estimating multiple unknown parameters, model ambiguities are one of the limiting factors which aect the retrieval performance. In a joint radar- radiometer framework, such limitations can be mitigated or even eliminated by proper utilization of the complimentary information provide by 0 and TB mea- surements. To highlight the eects of model ambiguity within the inversion process and how simultaneously combining radar backscatter and radiometer emission measurements can improve soil moisture retrievals, plots of the cost function hyper-planes, Eqs.3.2, are examined with respect to two parameters, rather than one such as in Fig. 3.1. For simplication, the hyper-plans are thresholded such that only model predictions within a certain range of true test points, D t , are shown. That is, the range of x which makejD t FM( x)j 2 k 2 p or T 2 hold true are plotted. Using radar datacube as forward models, the space of all possible x = [ r ;k s ] values is a 28030 matrix 71 such that r ranges from 3 to 30 and roughness, scaled by the wavenumber k, is limited between 0 and 0.3. FM( x) is either the radar backscatter model, in this case the datacubes, or the-! emission model. k p and T are the expected measurement noise standard deviations which are squared for consistency in units. For simplicity and demonstration, D t is initially assumed noiseless. Measurement and observation noise eects are investigated and presented in detail in the next section. It is important to note within the C-AP framework here, both models share the same key parameter kernel, i.e., x = [ r ;k s ], although the underlying theoretical development of these models is signicantly dierent. In Fig. 3.13, an example hyper-plane for the radar-only cost function, L a ( x), can be seen. The example here is specic to Corn with VWC of 2.5 kg/m 2 and without noise. Therefore, variations with respect to soil permittivity and surface roughness are initially considered. VV (top-panel) and HH (middle-panel) responses have also been separated since scattering polarization behaviors are also dierent. The shaded regions in Fig. 3.13 indicate the space of all possible model parameters which produce a model prediction withink 2 p of the actual measurement. Ask p is gradually reduced from 1.5 dB to 0.5 dB, the eective model parameter search space is reduced, thus showing a gradual convergence towards the true set of model parameters x true (Red- squares in Fig. 3.13). Observe that due to a very ambiguous model response a large range of soil permit- tivity and roughness combinations are acceptable; from very dry and rough surfaces to wet smooth surfaces. When both VV and HH radar backscatter coecients are accounted for simultaneously the parameter search space and therefore the ambi- guity are eectively reduced (from 25% to 3.5% of the entire range). The range of possible soil permittivity values, however, is still very large (10 r 25). On the other hand, possible values ofk s are approximately 0.15. Therefore, from a retrieval perspective it is initially unclear which set of [ r ;k s ] parameter values to select. Con- sequently, due to high model ambiguity attempting to estimate soil moisture using a single set of co-pol radar measurements in snapshot mode is prone to higher errors. 72 5 10 15 20 25 30 0.1 0.2 0.3 5 10 15 20 25 30 0.1 0.2 0.3 5 10 15 20 25 30 0.1 0.2 0.3 2.25 1 0.25 k p 2 [dB] Soil Permittivity ε r Surface Roughness (ks) |σ VV -σ VV (x)| 2 ≤ k p 2 |σ HH -σ HH (x)| 2 ≤ k p 2 |σ VV -σ VV (x)| 2 + |σ HH -σ HH (x)| 2 ≤ k p 2 Figure 3.13: 2D Radar-only Cost Function Hyper-plane. Shaded regions indicate j 0 true 0 ( x)j 2 k 2 p ; k p = [0.5, 1, 1.5] dB. Red square is the true test point. Top panel VV-only; middle panel HH-only; bottom panel VV and HH; Fig. 3.14 is analogous to Fig. 3.12, but for the radiometer case. Again, a large possible search space exits which spans a wider range of surface roughness than soil permittivity. Observe that, unlike the radar scenario, the space of possible solutions is limited to a smaller range of soil permittivity values (12.5 r 17.5). Inclusion of both TBV and TBH, similar to the radar scenario, reduces the eective search space signicantly and the span of possible soil permittivity values is much less than single polarization TB (12.5 r 25). The power of combined active-passive estimation can clearly be seen when both radar and radiometer contributions are tied together and jointly accounted for within 73 5 10 15 20 25 30 0.1 0.2 0.3 5 10 15 20 25 30 0.1 0.2 0.3 5 10 15 20 25 30 0.1 0.2 0.3 9 6.25 2.25 ∆T 2 [K] Soil Permittivity ε r Surface Roughness (ks) |TB V -TB V (x)| 2 ≤ ∆T 2 |TB H -TB H (x)| 2 ≤ ∆T 2 |TB V -TB V (x)| 2 +|TB H -TB H (x)| 2 ≤ 2"∆T 2 Figure 3.14: 2D Radiometer-only Cost Function Hyper-plane. Shaded regions indi- catejTB true TB( x)j 2 T 2 ; T = [1.5, 2, 3] K. Red square is the true test point. Top panel TBV-only; middle panel TBH-only; bottom panel TBV and TBH; the objective function. As seen in Fig. 3.15, when L a ( x) and L p ( x) along with an appropriate regularization term are simultaneously evaluated the model parameter search space is signicantly reduced. The resulting space is essentially a weighted overlap between radar and radiometer contributions, as indicated by the yellow region in Fig. 3.15. Since possible model parameters are limited to a smaller region around the true point, more accurate soil moisture retrievals are therefore possible. Furthermore, by varying the contributions of passive data toL ap ( x), via changing , an optimum weight between L a ( x) and L p ( x) can be determined which further 74 improves nal soil moisture estimates. In the presence of noise, however, the true set of model parameters may fall outside the search space. Under this condition, a set of parameters x opt which minimize L ap ( x) must be determined and report. Soil Dielectric 5 10 15 20 25 30 Soil Roughness k· s 0.05 0.1 0.15 0.2 0.25 0.3 2.25 1 0.5 k p 2 ΔT 2 9 6.25 2.25 Combined σ-TB L p (x) ≤ Δ T 2 L a (x) ≤ k p 2 L a (x) + α · L p (x) ≤ 2· k p 2 α = (k p /Δ T) 2 Figure 3.15: Overlay of Radar-only (L a ( x)k 2 p ; blue shades) and Radiometer-only (L p ( x) T 2 ; green shades) parameter search spaces. Combined Radar-Radiometer region is the overlap region (yellow shades). The true point is the red square. Variations in the amount of VWC also increases or decreases the eective model ambiguity and the ability to predict soil moisture. In general, as VWC increases, soil moisture estimation, both from a radar-only and radiometer-only perspective, becomes more erroneous [25, 26] since scattering or emission contributions due to the vegetation gradually dominate the surface response. This feature therefore also aects combined radar-radiometer retrieval approaches. To demonstrate this be- 75 havior, the parameter search space based on L a ( x)+L p ( x) k 2 p for dierent soil permittivity, roughness and VWC conditions is evaluated and shown in Fig. 3.16. Here k p =0.5-1 dB and T =1.5-3K. For a given set of r and ks values, as VWC increases, in general, the parameter search space also increases. However, unlike radar-only or radiometer-only scenarios in Fig. 3.12 and Fig. 3.13, the span of pos- sible model parameters is much smaller. For drier soil conditions and changing VWC values, model ambiguity with respect to variation of surface roughness are larger compared to changes in soil permittivity. This can clearly be observed in the smaller search regions in Fig. 3.16. With in- creasing soil permittivity and roughness C-AP ambiguity increases such that as the surface becomes wetter and wetter, both permittivity and VWC ambiguity become prominent. 5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 Soil Permittivity ε r Soil Permittivity ε r Surface Roughness (ks) Surface Roughness (ks) 4 3 2 1 0 5 VWC [kg/m 2 ] (a) α= (0.5/1.5) 2 (b) α= (1.5/3) 2 Figure 3.16: Combined Radar-Radiometer search spaces gradually increase as VWC is increased from 1 to 5 kg/m 2 . Red Squares indicate dierent soil and conditions. 76 Table 3.8 Simulations Model Parameters for Soil Moisture and Roughness Estimation Parameter Value Unit Land Cover Corn, Soybean and Grass NA Soil Permittivity ( r ) 3-30 NA Surface Roughness s 0.1-1 cm VWC 0-5 kg/m 2 Radar Noise k p 0.5-0.7 dB Radiometer Noise T 1.5-3 K 3.4.2 Simultaneous Soil Moisture and Surface Roughness Estimation Concurrent 0 and TB observations enable simultaneous estimation of soil mois- ture and surface roughness. This can be demonstrated through a series of Monte- Carlo like numerical simulations. In the presence of measurement and observation noise, numerical simulations are performed on three distinct land cover types of Corn, Soybean, and Grass. Noisy radar and radiometer measurements are generated based on model parameter listed in Table 3.8. Within a Monte Carlo simulation scheme, while varying the complete regularization term in Eqs.3.2, soil permittivity and roughness values are retrieved for pairwise sets of measurements. In the analysis presented in the previous section, the regularization term varied within the optimization to obtained the least retrieval errors and it did not include information regarding the observation's noise. For completeness, Eqs. 3.2 is presented here again, L ap ( x) = X pp=vv;hh 0 pp 0 pp ( x) 2 + kp T 2 X p=v;h jTB p TB p ( x)j 2 (3.3) The ratio kp T is the ratio of uncertainty associated to the corresponding radar or 77 Table 3.9 Active-Passive Noise Standard Deviation Ratios and Regularization parameter ranges Parameter Low-Low* High-Low Low-High High-High kp T 2 0.11 0.05 0.03 0.22 min ( = 10 3 ) 10 4 10 4 10 4 10 4 max ( = 10 2 ) 11 5 3 22 * Radar noise (k p ) is mentioned rst, then radiometer T radiometer measurements. In other words, they are the expected noise standard deviation of the measurement instrument. The objective function's eectiveness to estimate soil moisture, in the from writ- ten in Eqs. 3.3, can now be explained: radar and radiometer measurements are tied and constrained to each other but their relative weights are modied based on mea- surement noise and an additional regularization term . Assuming for the moment = 1, if the ratio of kp T increases, which is indicative of reduced radiometer noise, or increased radar noise, within L ap ( x) more weight is given to radiometer data. Conversely, if the ratio decreases, e.g., reduced radar noise or increased radiometer noise, less weight is given to the radiometer data. In Table 3.9 various noise ratio combinations and the resulting minimum and maximum values of kp T based on varying are shown. Lowk p values are considered to be 0.5 dB and higher values 0.7 dB [18]. Similarly, T values range from 1.5 to 3 K. These same set of parameters are also used in the following numerical simulations to demonstrate the performance of this method when attempting to retrieve soil moisture. For the high radiometer noise scenarios, High-High and Low-High, such that T 3K , the ratio kp T 2 is small thus automatically reducing the contribution of radiometer measurements. Under these scenarios, it is expected that the minimum of the curve of least retrieval errors with respect to changes in the regularization 78 term, similar to Fig. 3.3, tends more towards radar data. When T is lower, such as in the Low-Low and High-Low scenarios, kp T 2 is larger than before, by factor of 3-5, thus naturally adding more weight to radiometer data compared to the previous scenario. Therefore, the location of minimum error shifts towards more radiometer contributions. This self-regularizing feature of the cost function greatly improves its robustness with respect to measurement noise when compared to previous discussions were no distinction was made between the regularization term and the noise terms. Furthermore, in the context of SMAP, were TB measurements at the radar res- olution do not exist and are produced via a disaggregation scheme [9], T can be interpreted as the uncertainty associated with the disaggregation process and soil moisture estimation performed at the native radar resolution (3km). Given the compact form of the -! emission model and the fact that no calcula- tions are needed to nd 0 from the datacues, a global optimization scheme known as Simulated Annealing, SA, is used [27]. Optimum values of soil permittivity and roughness which minimize in Eqn. 3.3 are reported as the retrieved parameters of interest. The Root Mean Squared Error (RMSE) over the entire range of simulated parameters is then calculated and reported as a function of . In Fig. 3.17, plots of the RMS error for both soil permittivity and roughness are shown. Panels (a)-(b) are for Corn, (c)-(d) Grass, and (e)-(f) for Soybean. By varying the regularization term , through sweeping and dierent kp T ratios, the contributing weights of radar 0 and radiometer TB measurements can be changed such that soil moisture estimates with the least retrieval errors are obtainable. This feature is clearly seen in the dips, or minima of the curves in Fig. 3.17. At the extremes, the optimization process utilizes more radar data ( is small) or more radiometer data ( is larger). In between, active and passive measurements are weighted such that further reduction in retrieval errors is possible. As expected, for all cases, under the high noise scenario, minimum retrieval errors are larger than all other test scenarios. Furthermore, for the Low-High case, mini- 79 10 -4 10 -2 1 1.5 2 2.5 3 3.5 4 0.15 0.1 0.25 0.2 0.35 0.4 1 10 -4 10 -2 1 RMSE Soil Permittivity ε r RMSE Roughness [cm] α = γ � ( k p /� T) 2 α = γ � ( k p /� T) 2 7 0.4 1 2 3 4 5 6 0.2 0.25 0.3 0.35 10 -4 10 -2 1 10 -4 10 -2 1 α = γ � ( k p /� T) 2 α = γ � ( k p /� T) 2 RMSE Soil Permittivity ε r RMSE Roughness [cm] 3.5 4 4.5 5 0.4 0.45 0.5 10 -4 10 -2 1 10 -4 10 -2 1 α = γ � ( k p /� T) 2 α = γ � ( k p /� T) 2 1 1.5 2 2.5 3 0.2 0.25 0.3 0.35 RMSE Soil Permittivity ε r RMSE Roughness [cm] (a) RMSE in ε r for Corn (c) RMSE in ε r for Grass (e) RMSE in ε r for Soybean (b) RMSE in ks for Corn (d) RMSE in ks for Grass (f) RMSE in ks for Soybean Low Noise High Noise HighLow LowHigh Figure 3.17: Plots of Soil Permittivity RMS error (left column) and surface rms height (right column). Panels (a)-(b) are Corn, (c)-(d) Grass, and (e)-(f) Soybean. All four noise scenarios listed in Table 3.8 are included. 80 Table 3.10 Soil Permittivity Retrieval RMS Errors and Optimum Regularization Term Values. Land Cover Corn Soybean Grass Noise Scenario RMSE r opt RMSE r opt RMSE r opt High-Low 1.71 0.32 1.91 0.32 2.16 0.32 High-High 2.06 0.08 2.16 0.05 2.45 0.05 Low-High 1.71 0.04 1.91 0.04 2.16 0.04 Low-Low 1.47 0.13 1.40 0.10 1.73 0.16 mum retrieval errors are shifted more towards the radar measurements, indicative of a smaller ratio and discounting TB data. Conversely, for the High-Low scenario, the minimum is shifted more towards radiometer contributions. Table 3.10 summarizes Fig. 3.17 by listing the minimum achieved RMS errors for soil permittivity and the optimum regularization term for each noise scenario. Similarly, Table 3.11 summarizes RMS errors for surface roughness. Across the entire range of soil moisture, roughness and vegetation parameters, the average RMS error for estimated soil permittivity is at most 2.5 and 0.25 cm for surface roughness. Observe that the shape and location of error minima for permittivity and roughness are dissimilar. This is expected since the parameters are disjoint and forward model responses and sensitivities to these two parameters are very dierent. Given that independent validation of surface roughness, on any scale, is dicult if not impossible, emphasis is placed on validation and assessment of the retrieved soil permittivity. Although the minimum reported errors for surface roughness values, as seen in Figs. 3.17.b,c d and Table 3.11, are particularly small and are on the order of a few millimeters, it is considered a free parameter within the optimization framework. In practice, sweeping over or even to nd their optimum values, which yield the retrieval with minimum error, is impractical and time consuming. Therefore, for 81 Table 3.11 Surface Roughness Retrieval Errors and Optimum Regularization Term Values. Land Cover Type Corn Soybean Grass Noise Scenario RMSE opt RMSE opt RMSE opt High-Low 0.22 0.16 0.25 0.20 0.23 0.16 High-High 0.24 0.05 0.28 0.06 0.27 0.06 Low-High 0.22 0.02 0.25 0.03 0.23 0.02 Low-Low 0.17 0.05 0.21 0.16 0.21 0.13 each land cover type, under the High-High noise scenario, an optimum regularization term is selected, hh opt . The eect of selecting a single regularization on the estimation accuracy for all other cases is examined and shown in Table 3.12. That is, for the Low-Low, High-Low and Low-High cases the actual minimum RMS error, RMS error at hh opt , and the incurred relative error by selecting hh opt are evaluated. Table 3.12 Worst Case RMS Errors in Soil Permittivity. Land Cover Type Parameter Low-Low High-Low Low-High Corn Min RMSE 0.47 1.71 1.71 Error at hh opt 11.5 1.71 1.71 Relative Error (%) 1.71 0.01 0.01 Soybean Min RMSE 1.40 1.91 1.91 Error at hh opt 1.40 1.92 1.92 Relative Error (%) 0.01 0.12 0.12 Grass Min RMSE 1.73 2.16 2.16 Error at hh opt 1.73 2.17 2.17 Relative Error (%) 0.47 0.35 0.35 Referring to Tables 3.10 and 3.11, it is observed that if the optimum regularization term value is selected from the high noise scenario and applied to all other cases, 82 the resulting relative error is negligible and at most 1.7%. Therefore, a single set of regularization terms can be used for Corn, Soybean, and Grass ( corn opt = 0:08; soy opt = 0:05; grass opt = 0:05). For added exibility and to ne-tune soil moisture retrievals, can be manually varied between 0.9-1.5. To assess the quality of retrievals a pair-wise comparison between true test param- eters ( r ;s) and their corresponding mean estimates (h ^ r i;h^ si) is performed. That is, for discrete pairs of surface roughness and permittivity parameters, covering the ranges as listed in Table 3.9, the mean of the retrieved parameters, due to the out- come of the Monte-Carlo simulations is calculated and compared to the true test conditions. This comparison can be seen in the scatter plot in Fig. 3.18 where the true and estimated pairs of ( r ;s) are shown. The gures show outcomes at the min- imum of the worst-case scenario, High-High noise, thus can be viewed as the upper bound of the algorithm's error performance. 10 15 20 25 30 Soil Surface RMS Height s [cm] 0 0.2 0.4 0.6 0.8 1 1.2 Soil Permittivity ε r 5 0 Corn Grass Soybean “True” Test Point Figure 3.18: Scatter plot of pair-wise Roughness-Dielectric test points and mean estimates for Corn (Blue Triangles), Soybean (Yellow Diamonds), and Grass (Green Squared). Red Circles indicate true test points. Estimation Error increases for rougher and wetter soils 83 Under perfect retrieval conditions, i.e., no retrieval errors, all mean estimates would align on top of the test case pairs (red-circles). However, in the presence of noise, mean estimates acquire error, also known as bias. This error is particularly strong for very rough and very wet surfaces. As observed in Fig. 3.18, as the surface roughness and soil moisture increase, mean estimates of surface roughness, across the entire test range of interest, degrades. Accordingly, errors in predicting soil dielectric constant also increase, however their increase is not as severe as surface roughness. This is due to selecting optimization results where estimates of surface roughness are not minimum, but rather soil permittivity estimates are minimum. To capture the error performance of the comparisons in Fig. 3.18, the RMS errors between true and mean estimates of r across the whole range of surface roughness are calculated. In the top panel of Fig. 3.19 this error is shown with a maximum error about 1.4 for Grass. Components of the Mean Squared Error (MSE), i.e., variance and bias-squared, are also shown in the bottom panel of the gure. For all three land cover types, as soil permittivity increases both the variance and bias increase. This is due to lack of forward as model sensitivity with increasing soil moisture, especially for radar scattering models. The larger bias in surface roughness estimates can clearly be seen in Fig. 3.18 and increase for increasing roughness amounts. The MSE for s, similar to Fig. 3.19, is calculated and shown in Fig. 3.20. The majority of the contributing error to the Total MSE is due to the existing bias. As mentioned previously, surface roughness outcomes are selected from where r is minimum therefore higher errors fors are ex- pected. Given that validation of surface roughness, in practice, is almost impossible, this parameter is viewed as a free parameter allowing the optimization scheme to compensate for measurement and observation noise. An important metric when evaluating retrieval algorithms is the error perfor- mance with respect to changes in VWC. With increasing VWC, vegetation emission and scattering contributions begin to dominate the total measured 0 or TB, thus masking surface contributions. This eect was seen in Fig. 3.16 where the eective search space increased as VWC increased. To evaluate the upper error bound, for 84 Figure 3.19: Worst Case average RMS Error for r for all thee vegetation types (top panel). Mean Squared Error (MSE) for r shown as Variance and Bias 2 components of the error. each VWC value the RMS error in r across the range of permittivity and roughness is calculated and shown in Fig. 3.21. Similarly, the error in surface roughness is also calculated. Errors in both parameters increase as VWC increases which is commonly observed in many retrieval methods. In Section 3.3.2, prior to estimating soil moisture, surface roughness was numer- ically optimized to yield best match between model predictions and observations. This optimum roughness parameter was then using within a C-AP approach, based on Eqs.3.2b, to retrieve for soil moisture. Here, no a prior roughness estimation is performed but rather assumed unknown in conjunction with soil moisture. Furthermore, little to no roughness information is available from the experiment campaign, thus validation of soil moisture is only emphasized. As a function of the regularization term , the RMS error between in situ Com- 85 Surface Roughness (cm) 0 0.2 0.4 0.6 0.8 1 1.2 Surface Roughness RMSE 0 0.2 0.4 0.6 0.8 1 Corn Soybean Grass Surface Roughness (cm) 0.04 0.2 0.3 0.5 0.7 0.9 1 Total Squared Error ǫ r 0 0.2 0.4 0.6 0.8 1 Corn Variance Corn Bias 2 Soy Variance Soy Bias 2 Grass Variance Grass Bias 2 Figure 3.20: Worst Case average RMS Error for s for all thee vegetation types (top panel). Mean Squared Error (MSE) for s shown as Variance and Bias 2 components of the error. RAD soil moisture and C-AP predicted values, for both crop types is shown in Fig. 3.22. When is small, radiometer data are weighted less, thus radar backscat- ter measurements dominate the cost function. The radar-only RMS error for Corn is 0.22 m 3 /m 3 and 0.12 m 3 /m 3 for Soybean, both of which are substantially higher than the acceptance criteria of 0.04 m 3 /m 3 volumetric accuracy. As increases, the error for Corn and Soybean reduce, such that when the regularization term is largest, 11.1 [dB/K] 2 , radiometer-only inversions yield the least errors. This outcome is consistent with previous Fig.3.19 where comparisons between forward model pre- dictions and radar measurements showed noticeable error and bias. As the relative weight on L a ( x) within L ap ( x) increases radar induced model-data mismatch eects are reduced and the retrieval errors improve. In the simulation analysis presented earlier, an optimum regularization parameter was selected for each land cover type of interest ( corn opt = 0:08; soy opt = 0:05; grass opt = 0:05). RMS errors for each crop, at the selected optimum regularization term value 86 VWC [kg/m 2 ] 0 1 2 3 4 5 RMS Error ǫ r 0 1 2 3 4 Average Error in ǫ r Corn Soy Grass VWC [kg/m 2 ] 0 1 2 3 4 5 RMS Roughness [cm] 0 0.1 0.2 0.3 0.4 0.5 Average RMS Error in Roughness (k·s) Corn Soy Grass Figure 3.21: Average RMS error for r with increasing VWC for Corn, Soybean and Grass (top panel). Bottom panel shown RMS error for roughness for the same three vegetation types. Soybean maximum VWC is 3 kg/m 2 . are 0.07m 3 /m 3 and 0.05m 3 /m 3 for Corn and Soybean elds respectively. A convenient way to compare the performance of various estimation models with respect to true observations is the use of Taylor diagrams [29] where the three metrics of unbiased RMSE (or Centered RMSE), Correlation Coecient and Standard De- viation are summarized and presented simultaneously. These statistics for ComRad retrievals are calculated and shown in the Taylor diagrams in Fig. 3.23. Radar-only (SAR), radiometer-only (RAD) and combined radar-radiometer at opt statistics are presented as well as a series of other combined active-passive combinations to show the progression of statistics as the regularization term changes. These values are also summarized in Table 3.13. 87 10 -4 10 -3 10 -2 10 -1 1 10 0 0.05 0.1 0.15 0.2 0.25 α = γ � ( k p /� T) 2 RMS Error in Soil Moisture [cm 3 /cm 3 ] Corn Soybean α opt Corn α opt Soy Figure 3.22: ComRad Soil Moisture Retrieval RMS error for Corn and Soybean. Error values at opt are show by the squares and are 0.06 and 0.053 m 3 /m 3 for Corn and Soybean respectively 88 Standard Deviation 0.05 0.1 0.15 0.05 0.1 0 0.15 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Co r r e l a t i on Coe f f i c i en t True SAR opt RAD 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Co r r e l a t i on Coe f f i c i en t True SAR opt RAD Standard Deviation (a) Taylor Plot for Corn (b) Taylor Plot for Soybean Figure 3.23: Taylor diagrams for (a) Corn and (b) Soybean. Radar-only (SAR), Radiometer-only (RAD) and combined radar-radiometer at opr are shown. Smaller red circles are the statistics for a series of combined eorts due to dierent regularization term values prior to opt . ComRad in situ data are marked as True with standard deviation of 0.028 m 3 /m 3 . 89 Table 3.13 Worst Case RMS Errors in Soil Permittivity. Land Cover Metric Radar-only Radiometer-only C-AP at opt Corn ubRMSE 0.113 0.025 0.310 R 2 0.880 0.805 0.923 Stndev. 0.137 0.042 0.055 Soybean ubRMSE 0.067 0.017 0.018 R 2 0.870 0.830 0.900 Stndev. 0.090 0.030 0.040 With respect to in situ eld observations, Radiometer-only retrievals have the least unbiased RMS error (0.025 and 0.017 m 3 /m 3 for Corn and Soy), and comparable standard deviation of 0.03 and 0.04 for Corn and Soy respectively. Although radar- only estimates show a large correlation with respect to in eld measurements, their retrieval RMS errors and variations are much larger than Radiometer-only outcomes. Statistics calculated at opt are comparable to radiometer-only values with slightly higher correlations. Both methods, however, do meet the SMAP unbiased RMS error criteria of 0.04 m 3 /m 3 volumetric soil moisture content. Individual Radar-only, Radiometer-only, and C-AP retrieved soil moisture values are also shown in the scatter plots of Figs. 3.24 and 3.25 for Corn and Soy respectively. For Radar-only Corn soil moisture estimates, a large portion of the retrievals are capped at approximately 0.44 m 3 /m 3 , i.e., the upper bound of soil moisture within the optimization algorithm. This behavior is the SA optimization's response to forward model and parameter constraints. In the case of radar backscatter datacubes, surface roughness eects are derived from analytical Electromagnetic models with a range of validity up to 5 cm. On the other hand in the -! model form used here, surface roughness eects on emission are modeled as an exponential modication to the p-polarized Fresnel Equation, i.e., r 0 = r p e (2ks) 2 . The upper theoretical limit for this model is 90 Figure 3.24: ComRAD C-AP Soil Moisture Scatter Plot for Corn. Radar-only (di- amonds), Radiometer-only (squares) and C-AP at opt (circles) soil moisture esti- mates; respective unbiased RMS errors are 0.113, 0.025, 0.031 m 3 /m 3 . typically when ks 3 or 1 cm. Beyond this value, incoherent surface re ectivity and emission overtake the coherent component and thus are not captured properly. Parameters must also be bounded within the Simulated Annealing process such that the algorithm does not produce non-physical results. For the analysis here, the smaller roughness value was selected as the upper bound on s (1 cm). For the Radar-only retrievals in Fig. 3.24, which are capped at large soil moisture values, the SA algorithm is attempting to produce higher radar backscatter values for smaller soil moisture values. Since surface roughness is limited to about 1 cm, the algorithm responds by increasing soil moisture to produce higher backscatter until L ap ( x) and L a ( x) is minimum. The result is a set of over-estimated soil moisture points, which mathematically minimize the objective function, but are not consistent with in situ measurements. This outcome highlights the importance of consistent forward emission and scat- tering modeling when performing combined active-passive soil moisture estimation. 91 Figure 3.25: ComRAD C-AP Soil Moisture Scatter Plot for Soybean. Radar-only (diamonds), Radiometer-only (squares) and C-AP at opt (circles) soil moisture es- timates; respective unbiased RMS errors are 0.067, 0.017, 0.018 m 3 /m 3 . Since the measured backscatter and TB are dependent on the same set of physical properties of the scene, both models must consistency capture the underlying phys- ical phenomena. Therefore, it is hypothesized that the accuracy of soil moisture retrievals, within an C-AP framework, will signicantly increase with a uniform the- oretical development of a forward model which concurrently predicts the amount of emission and scattering, while using a single parameter kernel valid for both physical processes. 3.5 Chapter Conclusion A joint Radar-Radiometer surface soil moisture retrieval algorithm was presented in this chapter, wherein both active and passive microwave remote sensing measure- ments, at the same spatial resolution, are simultaneously combined to achieve optimal soil moisture estimates with minimum retrieval error. Same- resolution co-pol radar backscatter and H- and V-pol TB in- formation are constrained to each other within 92 a joint cost function, such that microwave emission and scattering forward models share the same parameter space. More specically, soil moisture (dielectric constant), surface roughness, and VWC are the shared parameters. Within the optimization, the soil moisture values that globally minimize the cost function are reported as the retrieved soil moisture values of interest By constraining active and passive measurements, as written in Eqs. 3.2, backscat- ter and TB sensitivities to soil moisture and vegetation can be captured in a single framework when performing soil moisture estimation. The analysis presented ear- lier, for both PALS SMEX02 data and ComRad data, and the numerical simulations highlight important features of this new proposed estimation method as summarized in the following 1. Using the adaptive regularization parameter , and utilizing radar and ra- diometer data at the same time, optimal soil moisture estimates with minimum overall error are possible. When = 1, both radar and radiometer measure- ments contribute equally. When 1 radar measurements are weighted more, and when 1, radiometer measurements are valued more within the cost function. 2. The regularization parameter within the cost function enables the optimization process to compensate for model deciencies and inaccuracies by giving more weight to the forward model that best ts the data 3. The numerical simulations presented show that, for the current forward models (radar datacubes and -! emission model), best estimates over a large range of soil moisture and VWC (up to 5 kg/m 2 ) are possible for slightly larger than 1. 4. The larger data space (4 data points: 2 radar and 2 radiometer) proves ad- vantageous, such that the optimization process is not limited to using specic input polarization combinations, and certain questionable data points can be excluded. This unique approach of combining dierent scattering and emission 93 polarization responses to estimate soil moisture will enable the algorithm to overcome or adapt to various issues related to erroneous data or forward model limitation. 5. The analysis presented, with respect to Fig. 3.10 and the \double-minimum cost function cases, highlights potential challenges in physics-based optimiza- tion and retrieval algorithms. Soil moisture estimates would be most accurate when associated forward models are well parameterized and accurate enough to capture the underlying physics. This point further motivates the devel- opment of joint-physics and consistent combined active-passive models, along with proper regularization, in order to drive the optimization toward delivering the best estimates. 6. With the increase information space and degrees of freedom provided by using multiple measurements of dierent polarizations, more than one unknown pa- rameter can be assumed. In the over-determined C-AP methodology outline in this chapter, surface roughness in addition to soil moisture is an assumed unknown target parameter. Since validation of retrieved surface roughness is impractical it is considered a free parameter thus allowing the Objective func- tion to compensate for model-data mismatched and deciencies. This methods, and concept serves as a foundation for a physics-based joint radar- radiometer retrieval algorithm for the case of multi-resolution data obtained from spacecraft data such as the SMAP mission. The next logical step in further developing and enhancing this algorithm is to utilize physically consistent forward mod-els, such that the expected emission from the footprint under consideration and the scattering cross section are fundamentally related from rst electromagnetic principles. Using the current models, the expected radar cross section and TB are determined independently of each other. Utilization of higher order solutions of the radiative transfer equation (to incorporate scattering) and Peakes emissivity relationships [39] will enable the algorithm to fully capture the 94 complementary behaviors and sensitivities of emission and scattering and yield more accurate surface soil moisture estimates. 95 CHAPTER IV Multi-Resolution Combined Radar-Radiometer Soil Moisture Retrieval This chapter will discuss the Multi-resolution scenario of Combined Radar-Radiometer Soil Moisture estimation by placing emphasis on model and data driven physics-based optimization techniques. Analysis on satellite data from the NASA SMAP mission will be performed and outcomes will be compared to the mission's baseline product. 4.1 Introduction For the case of spacecraft instruments and remote sensing observations, radar and radiometer products are reported at signicantly dierent spatial resolutions. On almost all moving platforms, especially spacecrafts, radars operate in a Synthetic Aperture mode resulting in sub-kilometer scale backscatter cross-section measure- ments 1 typically at the cost of smaller swath widths. On the other hand, Real Aperture radiometers yield kilometer scale, or larger, Brightness Temperature data products covering a very large swath. To compare and contrast this spatial resolu- tion disparity a non-exhaustive list of select SAR and RAD spacecraft and satellite 1 It is important to note that soil moisture products derived from spaceborne SAR are typically reported at a coarser scale than the native radar resolution. This is due to SAR processing and down-scaling to overcome speckle noise, etc. 96 Table 4.1 SAR and Radiometer Satellite Missions Satellite Years of Operation Instrument Frequency Nominal Resolution SMOS 2009 - present Radiometer L-Band 35 km Aquarius 2011 - 2015 Radiometer L-Band 40 km AMSR-E 2002 - 2011 Radiometer Multi-Band 5-50 km SMAP 2015 - present Radiometer L-Band 36 km SMAP 2015 - present SAR L-Band 300 m Radarsat-1/2 1995 - 2013 2007 - present SAR C-Band 3-8 m Sentinal-1 2014 - present SAR C-Band 5 m PALSAR 2006 - present SAR L-Band 7-100 m ERS-1/2 1991 - 2011 SAR C-Band 30 m missions as well as their approximate spatial resolutions is given in Table 4.1. A schematic representation of this resolution dierence is also shown in Fig. 4.1 such that within a typical radiometer pixel, multiple SAR measurements can exists. For example, in the case of SMAP, Earth-gridded L-band radiometer Brightness Tem- perature measurements are reported at a 36 km resolution whereas SAR acquisitions are obtained at 250 m scale, reprocessed and then reported at 1-3 km resolution. Clearly, high resolution soil moisture estimates obtained from Radar-only mea- surements are preferred over the coarser resolution Radiometer-only estimates and are potentially more suitable for high level science applications. However, in terms of estimation accuracy, the performance of radar-based and radiometer-based retrievals must be examined as well. Multi-resolution Combined Active-Passive soil moisture estimation is therefore viewed as a balance and trade-o between (a) high resolution yet, lower accuracy radar-only soil moisture estimation and (b) low spatial resolution, but more accurate radiometer-only estimation. Such a trade-o in estimation abilities can be seen in the 97 curve in Fig. 4.2. The ideal outcome is both high resolution, sub-kilometer, and high accuracy soil moisture estimation, indicated by the red circle. However, given current remote sensing technologies and techniques, this outcome is unachievable. Therefore, to meet soil moisture science criteria and mission requirements, in the case of SMAP for example, the compromise is to balance and utilize the complementary strengths of each technique; thus a fully C-AP approach. 4.2 NASA SMAP Mission In Chapter I the context and setting of this work was related to the NASA Soil Moisture Active-Passive (SMAP) mission. Launched in January 2015 the mission aims to provide the science community with global surface soil moisture estimates to address many of the pressing and current climate dynamics questions and its eorts will yield unprecedented high resolution global soil moisture estimates at 9 km spatial resolution. Essentially, the goal of SMAP is to address the high spatial and temporal Radar View Radiometer View Radar & Radiometer Platform (a) Spacecraft Observatory Radar View Radiometer View Ob (b) Figure 4.1: (a) Single Resolution Radar-Radiometer measurement schematic (b) Multi-resolution Radar-Radiometer measurement schematic from a spacecraft. It is assumed that Earth-gridded Radar measurements are nested within a single Ra- diometer measurement. Schematic (b) also represents a SMAP like scenario. 98 High Resolution Lower Accuracy Low Resolution Higher Accuracy Radar-only Estimation Radiometer-only Estimation Optimum Solution? Possibly. Unachievable Solution Figure 4.2: Combined Radar-Radiometer Soil Moisture Estimation Trade-o Curve. resolution global soil moisture science requirements. Spanning the Northern Hemisphere spring and summer months (04/25/2015- 07/07/2015) global active SAR and passive Brightness Temperature data are avail- able for analysis and algorithm development. These datasets are readily accessible from the National Snow and Ice Data Center (NSIDC) 2 and Alaska Satellite Fa- cility (ASF) 3 . The Beta Release Version R12170 [38] are used for analysis in this work. A high resolution L-band (1.26 GHz) SAR along with an L-band (1.41 GHz) radiometer coexist within SMAPs observational platform. The unique conically scanning antenna, shared between the SAR and RAD instruments, provides a near- constant incidence angle of about 40 in a near-polar sun-synchronous orbit with an eight-day exact revisit cycle. Given the satellite's orbital conguration and processing schemes, Level-1 (L1) geolocated and Earth gridded radar backscatter (L1_S0_HiRes) and radiometer Brightness Temperature (L1C_TB) will be available, within a three-day global coverage temporal window, as inputs to any retrieval and estimation algorithm. 2 https://nsidc.org 3 https://www.asf.alaska.edu/ 99 9km Cell 3km Cell 36km Cell Figure 4.3: SMAP EASE 2.0 Nested Griding at 36, 9 and 3km Resolutions. Within a 36km TB cell, 144 3km Radar cells can exists. EASE grids center's row and column indices for the 36km TB can be transformed to all 3km radar cell indices within that grid. The SMAP mission follows a unique data processing scheme were all L2 soil moisture products are geolocated and gridded base on the Equal-Area Scalable Earth (EASE) Grid 2.0 [39, 40]. This nested and scalable griding scheme is shown in Fig. 4.3. High-resolution backscatter information is provided at a 3 km resolution covering the outer 70% of the swath and Earth gridded Brightness Temperature at 36 km resolution. The primary data product of the mission, i.e., L2_SM_AP, is the high accuracy and high resolution surface soil moisture estimates obtained through the mission's baseline C-AP [41] algorithm, explained as follows: Three main geophysical features aect both the radar and radiometer signals: soil dielectric constant (synonyms to soil moisture), surface roughness, and vegetation at- tenuation and scattering. Even with multiple polarized measurements, the number of parameters exceeds the number of observations. Correlation between polariza- tions further reduces the information content of measurements and prohibits robust retrieval of all the parameters of interest that aect the observations [36]. There is, however, a separation of scales in the variability of these parameters which opens the 100 opportunity to use multi-temporal information to estimate the parameters. Surface soil moisture (and hence soil dielectric constant) varies at daily and sub-daily time scales in response to precipitation, evaporation and drainage. Vegetation structure and water content as well as soil roughness however vary on times scales of vegetation growth which is typically weeks to months. Therefore, over short periods of time, such that vegetation growth and surface roughness conditions remain relatively stable, TB and 0 are correlated with re- spect to variations in soil moisture, and as seen in Fig. 2.12b are in fact negatively correlated. Thus, a regression based linear mapping between TB and 0 can be es- tablished, TB/ +. Das et.al. [41][20] extended this understanding to the SMAP multi-resolution scenario such that, prior to actual soil moisture retrieval, high resolution co- and cross-pol SMAP active data are used to predict an intermediate Brightness Temperature product at medium resolution (9 km). This step is typi- cally known as \Brightness Temperature Disaggregation." To retrieve soil moisture, after the disaggregation step, a well known Radiometer-only retrieval algorithm, the Single Channel Algorithm (SCA) [11], is then used. The medium-resolution choice is to conform to SMAP Hydrometeorological science requirements calling for 0.04m 3 /m 3 volumetric accuracy at10 km or less. This two-step Disaggregation-Estimation approach is formally the SMAP baseline C-AP algorithm [42]. Furthermore, the process is specic for every individual TB pixel such that time series linear mapping is applied to obtain unique regression parameters. The TB Disaggregation step is formulated as TB V (M) =TB V (C) +(C)f 0 vv (M) 0 vv (C) + 0 hv (C) 0 hv (M) g (4.1) were TB V (C) is the SMAP coarse resolution V-pol Passive measurement. (C), in units of [K/dB] is the data-driven time-series regression slope between TB and 0 , i.e. TB V (C) = (C) 0 vv (C) +(C). The parameter projects variations in the soil moisture contributions to the co-pol backscatter onto the Brightness Tempera- ture space. The choice of V-pol observations stems from the observed higher linear 101 Time-series Regression and Disaggregation SCA Soil Moisture Estimation Radiometer TB at 36km Disaggregated TB Medium Resolution (9km) Soil Moisture Medium Resolution (9km) 0 36 18 km Radar (SAR) At 3, 9 & 36 km Ancillary Information Vegetation (VWC), Temperature, etc. Figure 4.4: Schematic overview of the SMAP L2SM AP Algorithm correlation between TB V - vv compared to TB H - hh . The third term in Eq. 4.1, [ 0 hv (C) 0 hv (M)], is a strong indicator of heterogene- ity within a given pixel (C) which is then projected to the co-pol radar space through the factor . Based on co- and cross-pol 0 statistical regression, specic to a given grid cell, can be predicted. In other words, this third term is the soil contribu- tion to the co-pol backscatter such that volume scattering component is isolated in the cross-pol measurement. then represents a proportionality factor that depends on the architecture (distribution of the shape and orientation of the lossy dielectric elements in the vegetation canopy) within the pixel under consideration. Disaggregation is performed for every medium-resolution pixel within TB(C) such that grid averaged disaggregated TB(M) should be close to TB(C) (TB(C) 1 N P N i=1 TB(M i )). A simple schematic overview of this algorithm is shown in Fig. 4.4. A 3 day global composite of 9 km surface soil moisture from the SMAP L2SM AP products is shown in Fig.4.5. Although in the SMAP approach radar observations are not directly used within the soil moisture estimation step, the derived linear relationship between TB and 0 based on their fundamental physical inter-relatedness is utilized to related one to the other. The availability of disaggregated TB also have additional science benets. 102 Figure 4.5: 3 Day Global Composite of SMAP 9 km Combined Active-Passive Surface Soil Moisture. 4.3 Multi-Resolution Soil Moisture Estimation Unlike the SMAP baseline approach, for the Multi-resolution optimization frame- work no assumptions will be made to relate radar backscatter and Brightness Tem- perature, other than their model behavior with respect to soil moisture. Further- more, the greater emphasis of the methodology is on forward physical models, their performance, and ways to improve them in future studies. By placing these stringent constraint, the only available sources of information are: 1. SMAP Level-1 3 and 9 km Radar backscatter information (L1C_S0_HiRes). 2. SMAP Level-2 Active-Passive and Passive-only data products, L2_SM_AP, and P (Passive-only) [38, 43], as the source of 3, 9, and 36 km reference soil moisture data and ancillary model parameters such as vegetation, surface temperature, etc. 3. SMAP Radar Datacubes [29] as proxies to radar scattering models and the 103 -! model as the emission model. 4.3.1 Perspective and Approach The multi-resolution Radar-Radiometer concept of Fig. 4.2 is analogous to solving multi-objective optimization problems with the following basic formulation: min or max x f m (x); m = 1; 2;:::;M: subject to g i (x) 0; i = 1; 2;:::;I: h k (x) = 0; k = 1; 2;:::;K: x L j x i x U j ; j = 1; 2;:::;J: (4.2) where a set of M objective functions f m (x) are to be minimized, maximized, or a combinations, subject to equality or non-equality constraints, h k (x) and g i (x), either of which can be linear or non-linear in nature. Here, the last condition in Eqs. 4.2 enforces the parameters x i to be valid within certain Lower (L) and Upper (U) bounds. In contrast to single-objective optimization problems were only one cost function is considered, multi-objective optimization involve higher dimensional spaces with, at times, con icting objective functions. Therefore, there is no single \optimum" solution which outperforms all other solution. By con icting objective functions, for example, it is meant that the optimum solution x which minimizes f 1 (x) does not necessarily yield the minimum for f 2 (x). Thus, a trade-o must occur between f 1 and f 2 and set of solutions and outcomes reported. This set of solutions is typically referred to as the Pareto Front, or Pareto optimality, and dominate all other options. If, for a given solution x (1) with respect to x (2) , the following two conditions are met, x (1) is said to be dominant, or non-dominated and lies on the Pareto Front. These conditions hold only for the case of minimization of all objective function. 104 1. x (1) is no worse that x (2) in all objective functions. That is, the solutions are compared and evaluated based on their corresponding objective function value and if f m (x (1) )f m (x (2) ) for all m, this condition is met. 2. x (1) is strictly better than x (1) in at least one of the object functions i.e., f m (x (1) ) < f m (x (2) ) for at least on m. In other words, once the multi-objective optimization trade-o occurs, and a set of solutions are obtained, those solutions which meet the above conditions, when compared to all other solutions, are non-dominated and meeting the problem's con- straints, hence lie on the Pareto Front. For the case of minimization of two objective functions, Fig. 4.6 highlights the Pareto Front and non-dominance features. A set of model parameters x from the Parameter Space map in to the Objective Space, i.e., space of all f m (x) values. The thick-black portion of the Objective Space is the Pareto Front such that all the model parameters which map to this curve are non-dominated. In the example here, x 2 is the only non-dominated and desired solution. All other x values yield objective functions which do not meet the conditions set above. More specically, for example, both f 1 (x 2 ) and f 2 (x 2 ) are strictly better, less than, f 1 (x 1 ) and f 2 (x 1 ). Furthermore, note that x 3 and x 4 map to the same objective function values, thus indicative of a non-unique, or ambiguous model. This is exactly the scenario discussed in Section 3.4 where both radar and radiometer forward models exhibit no one-to-one relationship with respect to soil moisture and surface roughness. This is a realistic scenario and attempts will be made to overcome these short comings. The resulting Pareto Front of solutions, in this context, can thus be viewed as a set of local minima spanning the Parameter Space of Soil Moisture and Roughness. More elegant and complete descriptions of Multi-objective optimization can be found in numerous literature such as [44]. An alternate perspective on MR C-AP is a Bias-Variance trade-o between spa- tially distributed soil moisture. Considering a specic 36 km Radiometer pixel, passive-only soil moisture estimation will yield a coarse grade soil moisture predic- 105 Parameter Space Objective Space x 1 x 2 f 1 (x) f 2 (x) x 1 (f 1 (x),f 2 (x)) Pareto Front x 2 x 3 x 4 Figure 4.6: Parameter and Objective Function Space; point x 2 is considered domi- nant and lies on the Pareto Front. tion which is in fact an aggregate of many ner scale values; such high resolution values can be derived from active-only retrievals. Thus, radiometer controls the so called Bias, or in this context the mean value of soil moisture, while radar controls the spatial variance or sub-pixel heterogeneity and variability. The total residual error between an estimate and truth can be decomposed in to the sum of Bias-squared and Variance, or the variability, of predictions. It is a well known fact in error minimization [45] the expected Bias in the error and the Variance cannot be simultaneously minimized and there is always a trade-o. This scenario is, again, analogous to the Radar vs. Radiometer trade-o. Application of Evolutionary Algorithms (EA), specically Genetic Algorithms (GA) to solve the multi-objective at hand are ideal and powerful methods. Many of the details of Multi-objective Evolutionary Algorithms (MOEA) can be found in [44]. The basic principles, however, are to 1. Find a set of optimal solution which lie on the Pareto Front. 2. Find a diverse enough set of solutions covering the entire Pareto Front, or as 106 much as possible. MOEA are \population" based, gradient-free, methods such that within a single iteration, multiple objective function evaluations occur in order to determine opti- mum solutions. At every stage the methodology must consider (a) overcoming local minima (b) at regions of the objective function (c) parameter bounds (d) infeasible regions, etc. A population here is considered as a set of model parameters x, whilst an individual is an element of the population x i 2 x. Furthermore, progression of MOEA through the optimization process are based on stochastic operators, rather than deterministic, gradient-based, or xed-transition rule techniques. Therefore, the solutions converge towards giving higher probabilities of acceptance for more favorable outcomes. Beginning with an initial population, at each iteration MOEA generates a new population, or children, using the previous iteration's population (parents). Typi- cally, three or four actions occur to dene a new population: 1. Crossover: The process of selecting two or more parent solutions to create a new solution (child). 2. Mutation: Perturbation of child solutions within its vicinity. 3. Elite-section: Process of keeping and maintaining superiors solution from one iteration to another. Via the above operations, and throughout the optimization process, the algorithm can gradually demonstrate non-degrading performance by emphasizing better and superiors solutions. Furthermore, the objective functions are constantly evaluated and ranked based on the best set of individuals and selections are made to proceed to the next iteration. Typical algorithm termination criteria are (a) number of iterations (b) time limit (c) objective function value and/or tolerance limits. Currently, implementation of MOEA for Active-Passive soil moisture retrieval is done in MATLAB 2015b using thegamultiobj function with upper and lower bounds 107 for soil moisture and roughness as the only constraints. 4.3.2 Multi-Objective Active-Passive Formulation The trade-o concept between high resolution radar and coarse resolution ra- diometer is similar to the aggregated objective function in Eqs. 3.2,L a ( x) + L p ( x). For the case presented in Section 3.1, the trade-o between radar and radiometer data, at the same resolution, was controlled through the regularization term . To proceed, similar to Eqs. 3.2, radar-only and radiometer-only objective func- tions must be dened while keeping the spatial resolution dierence in mind. The only control and link between TB and 0 here is indirectly through surface soil mois- ture and roughness. VWC is assumed known and available through other sources. The following outline is the problem formulation for a single 36 km radiometer pixel with multiple nested 9 km radar pixels. For any given coarse scale pixel, the aggregated contributions from all N nested radar pixels, each dening an objective function L sub ( 0 pp;i ;x i ), can be written as: L a (x) = 1 N N X i=1 L sub ( 0 pp;i ;x i ) (4.3a) L a (x) = 1 N N X i=1 X pp=hh;vv 0 pp;i 0 pp (x i ) 0 pp;i 2 (4.3b) where 0 pp;i is the pp-polarized radar backscatter for the ith pixel with the corre- sponding model prediction 0 pp (x i ) due to x i = [sm i ;ks i ]. For brevity, all other model parameters are suppressed in the notation. x is dened as the 2N 1 vector of soil moisture and surface roughness parameters; that is, two unknowns per pixel. For the case of SMAP, within every 36 km pixel nominally sixteen 9 km radar pixels exist. Thus, x is 321. Due to model performance and data calibration uncertainty cross-pol backscatter hv and vh are neglected without loss of generality or negative eects. 108 On the other hand, the denition of the Radiometer-only objective function takes a dierent form since actual TB data at the radar resolution do not exits. However, at the ner scale it can be assumed that emission from adjacent pixels are (1) indepen- dent of each other (2) their aggregate yields the actual coarse resolution observation TB c . In this case higher resolution emission information TB f are produced by evalu- ating the forward models with knowledge, or some understanding of, the underlying parameters. For the case of SMAP at 9 km nearly all other model parameters p, ex- cept soil moisture are known. Thus, for a given 36 km pixel and measured p-polarized emission TBp c we can state that TBp c 1 N N X i=1 TBp f (x i ; p) (4.4) which when written as an objective function takes the following form L p (x; p) = 1 2 X p=H;V TBp c hTBp f (x i )i TBp c 2 (4.5) wherehi indicate the arithmetic average over N values. Clearly stated, Eqs. 4.5 says that the dierence between mean TB f s due to the predicted values of soil moisture x i , and TB c can be minimized. Since for every ith pixel, -! model parameters are known, producing TB f is possible within the optimization cycle. This is exactly anal- ogous to the disaggregation process within the SMAP baseline algorithm discussed in the previous section and Eqs. 4.1. Alternatively, the averaging operation can be substituted with more physically representative operators such as the antenna Point Spread Function or other Bright- ness Temperature up-scaling methods; in actuality, the total measured TB is a con- volution of all constituents within the antenna pattern. The multi-objective active-passive problem therefore is stated as trade-o be- 109 tween L a (x) from Eqs. 4.3 and L p (x) from Eqs. 4.5: 1 N N X i=1 X pp=hh;vv 0 pp;i 0 pp (x i ) 0 pp;i 2 vs: 1 2 X p=H;V TBp c hTBp f (x i )i TBp c 2 which for every pixel examined will yield the best approximation to the true Pareto Front with a solution set of X p . The solution set is K 2N large. Note that this process is deemed a trade-o hence a range of possible, and math- ematically acceptable, outcomes exist. The next logical step, some times referred to as the \Decision Making" step, is to identify and select the best, or most acceptable, individual from the Pareto set. The Decision Making step, or in this context \Sub-pixel" selection, is performed as follows 1. For every 9 km pixel, and for all x k 2 X p ; k = 1; 2;:::K L sub ( 0 pp;k ;x k ) is evaluated. 2. L sub ( 0 pp;k ;x k ) are then ranked, or scored, in ascending order based on the correspondingx k . That is, solutionsx k with the least norm are ranked highest. 3. Highest ranking solutions are reported and recored as the 9 km retrieved soil moisture and roughness. In other words, across the Pareto set, L a (x) from Eqs. 4.3 is segregated and L sub ( 0 pp;k ;x k ) evaluated on an individual pixel basis. Solutions with the least norm are then selected. A simplied and high level owchart for this entire process is shown in Fig. 4.7. Prior to optimization, for every pixel rigorous preprocessing is performed to check data and parameter qualities, land covers and exclusion regions. SMAP data prod- ucts include binary ags indicating various states of data. A high level preprocessing owchart is given in Appendix A. 110 MOEA: Multi-objective Evolutionary Algorithm SMAP 36 km TB & 9 km Sigma 0 Radar Forward Model Radiometer Forward Model Figure 4.7: High Level MOEA Flowchart. 1. 36 km: Water body corrected TBV and TBH data quality ags are checked rst, then Snow, Water, Ice or Frozen (SWIF) surface ags. Via an \AND" operation, if both are acceptable, then the preprocessors goes to the next step. 2. 3 & 9 km: SWIF ags are checked at 3 km rst. By majority rule, if most 3 km pixels within a 9 km box are agged, that 9 km pixels is ignored. 3. Pixels where no radar scattering forward model exists are agged and excluded. These included urban areas, and mixed crop-forested lands. 4. Pixels identied during precipitation or rain events are agged post-optimization and discarded. Thus far, the discussion has focused only on obtaining best soil moisture re- trievals. Surface roughness is also an unknown parameter within MOEA, but since on any scale inferring any knowledge on its value is impractical, expect for deserts and barren land, this parameter is viewed as a free parameter. To enforce physically meaningful roughness values they are only bounded within the process. In future studies, other approaches can be implemented to mitigate this issue: 111 1. Increase the Information Content of the problem by introducing complementary data layers. 2. Apply time-series approach wherein a short time frame surface roughness is assumed constant and two step retrieval is performed, rst for roughness then for soil moisture; this is similar to the current SMAP L2SM Active (radar-only) baseline algorithm. Furthermore, within a 36 km coarse pixel, the spatial variation and distribution of soil moisture is assumed smooth and unimodal, such that no outliers or extreme values exist. Thus, by properly bounding soil moisture within each 36 km box, surface roughness can be assumed a free parameter to compensate for model-data mismatches. This feature was discussed extensively in Section 3.4. 4.4 Soil Moisture Estimation Using SMAP Data 4.4.1 Global Comparison In this section, the multi-resolution MOEA active-passive approach detailed in the previous section is applied to global SMAP data. Currently, due to the extensive computational time, only one full global coverage, which is approximately 3 days of data, will be discussed. A more detailed case study, covering the full time span of available data will be presented in the next section along with comparisons with ground truth. On a global scale a large range of surface and land cover states exists which in turn due to their soil conditions and vegetation structure, will aect measured radar backscatter and radiometer Brightness Temperature. The International Geosphere- Biosphere Programme (IGBP) [46] Land Cover classication scheme, currently em- ployed by the SMAP mission, is a comprehensive approach to identify and classify regions on earth based on their representative and specic land cover and overlaying vegetation; there are a total of 17 distinct categories shown in Fig. 4.8. The gure 112 Water Evergreen Needle e Evergreen Broad Deciduous Needle Deciduous Broad Mixed Forest Closed Shrublands Open Shrublands Woody Savannas Savannas Grasslands Wetlands Croplands Urban Cropland SnowIce Barren Figure 4.8: Global IGBP Land Cover Classication (EASE-grid 9km). shows IGPB classes at a 9 km resolution based on the EASE-gird projection. For any pixel under consideration and in order to mimic the expected TB or 0 a specic model must be used. To be consistent with the SMAP processing schemes, the same radar datacube, -! emission model, and parameterization are used here. Note that for Water, Snow, Ice, Mountains, Frozen ground, and Urban Areas no scattering or emission process will be dened. Thus, such pixels are identied and ignored within the processing scheme. Global 3-day composites of SMAP Active and Passive data are shown in Fig. 4.9- 4.12 (respectively, TBH, TBV, hh , vv ) with temporal span of 04/24/2015 - 04/26/2015. Distinct patterns can be observed from this data. For example: 1. Signicantly low backscatter values over the African Sahara Desert compared to forested areas such as Central Africa or the Amazon can be seen. This is due to the fact the deserts are relativity smoother therefore have more forward or specular scattering than backscatter. On the other hand vegetation volume scattering and especially the double bounce scattering mechanism begin to dominate the total backscatter. Therefore for regions with high vegetation, 113 soil moisture estimation becomes more challenging. 2. V-pol Brightness Temperature on average is higher than H-pol TB. 3. The negative correlation between TB and 0 can clearly be seen such that for regions with lower backscatter, such as the African Sahara, the measured TB is higher compared to other locations. 4. Gaps in the data, especially for 0 , are either due to lack of coverage, or exclusion of data. The later is particularity true for mountainous regions such as the Himalayas or very high Northern Latitudes which may be Frozen. A global map of MOEA and SMAP combined radar-radiometer soil moisture outputs can now be seen in Fig. 4.13 and Fig. 4.14. Overall, there is a strong agreement in capturing the spatial distribution of soil moisture. Most noticeably, dry and semi-arid regions such as the African Sahara and central Australia yield, as expected, low soil moisture predictions. Northern latitudes (N50 o ), such as the state of Alaska and Siberia are currently agged as frozen ground. A more detailed comparison between MOEA and SMAP can been seen in Figs. 4.15- 4.26 where soil moisture scatter plots for dierent IGBP land covers can be seen. Furthermore, in each gure the marker size and color indicates the amount of VWC and overlaying vegetation. The insert histograms show, expect for a few case, overall agreement in soil moisture distribution. 114 Figure 4.9: SMAP 3 day TBH [K] Global Composite (04/24/2015- 04/25/2015) Figure 4.10: SMAP 3 day TBV [K] Global Composite (04/24/2015- 04/25/2015) 115 Figure 4.11: SMAP 3 day Sigma-HH [dB] Global Composite (04/24/2015- 04/25/2015) Figure 4.12: SMAP 3 day Sigma-VV [dB] Global Composite (04/24/2015- 04/25/2015) 116 Figure 4.13: MOEA 3-day global soil moisture composite(04/24/2015- 04/25/2015). Figure 4.14: SMAP Active-Passive 3-day global soil moisture composite(04/24/2015- 04/25/2015). 117 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: Bare 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.15: SMAP vs. MOEA soil moisture for bare and sparsely vegetated land areas. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: Grasslands 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.16: SMAP vs. MOEA soil moisture for grassland. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 118 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: Croplands 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.17: SMAP vs. MOEA soil moisture for croplands. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: OpenShrublands 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.18: SMAP vs. MOEA soil moisture for open shurblands. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 119 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: Savannas 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.19: SMAP vs. MOEA soil moisture for savannas. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: WoodySavannas 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.20: SMAP vs. MOEA soil moisture for woody savannas. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 120 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: PermanentWetlands 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.21: SMAP vs. MOEA soil moisture for permanent wetlands. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: EvergreenBroadleaf 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.22: SMAP vs. MOEA soil moisture for Evergreen broad-leaf forests. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 121 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: ClosedShrublands 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.23: SMAP vs. MOEA soil moisture for closed shrublands. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: MixedForests 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.24: SMAP vs. MOEA soil moisture for mixed forests. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 122 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: DeciduousNeedleleaf 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.25: SMAP vs. MOEA soil moisture for deciduous needle Leaf forests. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 0.1 0.2 0.3 0.4 0.5 0.6 SMAP L2SM AP [cm 3 /cm 3 ] 0.1 0.2 0.3 0.4 0.5 0.6 MOEA SM [cm 3 /cm 3 ] Land Cover Type: DeciduousBroadleaf 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vegetation Water Content [kg/m 2 ] SMAP MOEA Figure 4.26: SMAP vs. MOEA soil moisture for deciduous broad-leaf forests. Marker sizes and colors indicate the amount of VWC (kg/m 2 ). 123 Bare OpenShrublands Grasslands Croplands ClosedShrublands Savannas PermanentWetlands DeciduousNeedleleaf WoodySavannas EvergreenNeedleleaf MixedForests DeciduousBroadleaf EvergreenBroadleaf 0 0.05 0.1 0.15 0.2 Error Statistics [m 3 /m 3 ] 11.8% 15.2% 13.0% 8.0% 0.1% 6.4% 0.7% 1.2% 6.7% 2.9% 7.4% 0.8% 8.8% |Bias| ubRMSD Figure 4.27: Plot of RMSD for dierent land cover classications. Land-cover type and class percentage is shown above each error bar. To capture and summarize the entire performance the MOEA, RMS dierence area calculated for each dierent land cover type. The following pixels, for obvious reasons are excluded (a) nadir region pixels (b) regions with VWC > 5kg/m 2 (c) upper and lower hard boundary points. Overall, these plots and gures are essential in identifying sources of mismatch and error. 124 4.4.2 Case Study: Tonzi Ranch Study Site Global comparisons of MOEA and SMAP L2SM AP outcomes show an overall acceptable agreement. Here, in this section, a more detailed case study and compar- isons with ground truth is presented. That is the, MOEA algorithm is applied to a specic study site over a longer time period and 9 km soil moisture predictions are compared to in situ ground sensors. For the following analysis SMAP 36 km EASE-grid TB and 0 observation over the Tonzi Ranch Study site will be considered. Located in North-Central California, Tonzi Ranch is a Savanna region with a cluster of 18 in situ soil moisture sensor proles (5,10, 15cm) acquiring data at 20 min intervals. The general geographical location of this region can be seen in Fig. 4.28 where the red boundary box is the 36 km extent, the nested blue boxes are the 9 km grids and the small Green box is representative of a 3 km pixel. The cluster of in situ probes are located in the North-East section of the green 3 km pixel. A temporal plot of the daily averaged soil moisture within the site is shown in Fig. 4.29. Two rain-events and a prolonged dry period provide quality soil moisture dynamics to assess the MOEA estimated soil moisture values. However, two important facts must be cautioned prior to evaluation of MOEA soil moisture estimates: 1. Soil Moisture Representativeness: compared to the study domain and area (3 or 9 km) the extent of the in situ probes is substantially less. Therefore a simple arithmetic average may not fully capture the large scale actual soil moisture. 2. Sensor Calibration: although signicant eort has been devoted to in situ sensor calibration, the reported values should be considered with some caution since residual calibration error can still aect the outcomes. Nevertheless, the availability of this sensor network, although limited in global coverage, is extremely signicant and benecial to understanding algorithm, model, and data performance. 125 Tonzi Ranch 36 km 9 km Tonzi 3 km Figure 4.28: EASE-grid Nesting over the greater Tonzi Ranch Area. Each color box is representative of a specic spatial resolution: Red = 36 km, Blue = 9 km, Green = 3 km. The Yellow box is the designated Tonzi Ranch box and the in situ sensors are located within the 3 km Green box. As previously mentioned, the MOEA method outputs a set of Pareto optimum X p , or near-optimum, soil moisture and roughness values for every pixel considered. Then, Sub-pixel selection is performed to identify the most suitable individual from X p by evaluatingL sub from Eqs. 4.3. To shed more details on this process, in Fig.4.30 the collection of Pareto sets for Tonzi Ranch are shown. These sets cover the entire duration of the study (04/24/2015-07/07/2015). The shape of the Pareto sets in Fig. 4.30 are essentially the same as the Front discussed in Fig. 4.6. The dierences between each set of due to dierence in the Objective Space, which in turn are due to dierence in observations and data. How- ever, for all cases, across the entire Front, all plausible C-AP scenarios exists. The next step is to select an individual form each of these sets. In Fig. 4.31, the Parameter space of surface permittivity (analogous to soil mois- ture) and roughness ( r ;ks) is shown along withL sub (x k ) for each individual within 126 04/26 05/06 05/16 05/26 06/05 06/15 06/25 07/05 Date 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Soil Moisture [cm 3 /cm 3 ] Tonzi Ranch Daily Mean Soil Moisture (04/26/2015-07/07/2015) Rain Average Stn.Dev. Figure 4.29: Temporal Plot of Tonzi Ranch Soil Moisture with daily average (Red- line) and Standard deviation of multiple sensors readings in the area (Pink shaded region) the Pareto set such that the size and colors of the markers are proportional to the objective function's value, i.e., smaller and darker makers have lower values. The red thick line indicates the daily mean Permittivity of Tonzi Ranch. Since the actual roughness within the site is unknown, the line spans the entire range of allowable surface roughness. The plots can be interpreted as such: the closer the selectedx k is to the red-line, the closer the reported soil moisture is to the in situ aggregate soil moisture. Two specic markers are also shown (1) red marker indicating the smallest L sub (x k ) and (2) green star which is indicative of the median value and used for Bias correction. Fig. 4.31a is for a typical dry scenario and as seen the range and diversity of solutions is very small, thus indicative of convergence towards an acceptable minimum. On the contrary, Fig. 4.31b re ects an example from the beginning of the study cycle during a rain event. Surface dynamics during rain are unstable, thus soil moisture predictions are typically agged and used with caution. The larger spread of the Pareto set and higher values of L sub re ect this fact. The temporal soil moisture plot of Fig. 4.29 can now be overlaid with C-AP 127 0 0.1 0.2 0.3 0.4 0.5 0.6 Objective-1 (Radar-only) 0 0.005 0.01 0.015 0.02 0.025 0.03 Objective-2 (Radiometer-Only) Pareto Fronts For Tonzi Ranch (04/25/2015 - 07/07/2015) Figure 4.30: MOEA Pareto Fronts for Tonzi Ranch; all study days are shown. estimates from both MOEA and SMAP L2SM AP, Fig. 4.32. The inset gure is a scatter plot of the two retrieval outcomes with respect to the validation data. A few important features can conclusions can be drawn from these outcomes. 1. Similar to the in situ probes MOEA tracks the soil moisture's evolution over time. This is a very signicant and important feature. If, after the optimiza- tion, the estimates showed no to little change compared to validation data, most likely (a) the method has not converged correctly (b) temporal changes in data weren't aecting the methods. The resulting biases, or o-sets, are either correctable or explainable with further investigation. Their underlying reasons are typically due to (1) ancillary data or parameterization issues (2) 128 (a) (b) Figure 4.31: Parameter-space ( r ;ks) for L sub values evaluated on the Pareto set for Tonzi Ranch. Red-line is mean Tonzi Ranch Permittivity; Color markers refer to value of L sub ; Red circle is minimum value and Green start is median value.(a) Dry period (b) During rain event. system and observation noise (3) model-data discrepancies. 2. Both MOEA and SMAP underestimate soil moisture. This can be due to the representativeness of scale of the validation data. 3. The green star markers are 0 data points which fall within the nadir region which are generally agged due to SAR processing considerations and should be ignored. Error statistics for these retrievals can be seen in Table 4.2. Note that the smaller unbiased RMSD for SMAP is due to removing the substantially large bias in the estimates. It is important to reiterate that the SMAP data products (R12170) are still in Beta version, and thus should be viewed with caution. To proceed, more detailed insight in to the nature of actual SMAP radar data is needed. vv and hh observations can be seen in Fig. 4.33 such that for each data point a distinction based on the radar-cell's location within the SMAP swath is also made, i.e., dierent size markers. 129 04/26 05/06 05/16 05/26 06/05 06/15 06/25 07/05 Date 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Soil Moisture [cm 3 /cm 3 ] 0 0.1 0.2 0.3 Tonzi Ranch SM 0 0.1 0.2 0.3 Active-Passive 9 km C-AP Soil Moisture Estimation (Tonzi Ranch) (04/26/2015-07/07/2015) MOEA SMAP Figure 4.32: Temporal Plot of Tonzi Ranch mean soil moisture along with 9 km C-AP Retrieval; purple diamond are MOEA; green squares are SMAP L2SM AP (R12170). Stars indicate data points which fall within the Nadir region. Immediate observations from vv and hh are 1. Towards the end of the study duration (06/06-07/07) 0 observations show little to no variations although in situ sensors decrease from 0.1 m 3 /m 3 to 0.06m 3 /m 3 . The underlying reasons can be traced to two possibilities (a) lack of representativeness of soil moisture sensors at 9 km scale (b) measurement or instrument related issues. The later point is beyond the scope of analysis. 2. With respect to all other observation those 0 data points falling within the satellite's Nadir region clearly show a dierent pattern and are noticeably lower and at times more than 1dB. These data points should be generally ignored. This feature, i.e., lack of radar response, is clearly re ected in the inversion out- comes as well where for the same period of time, soil moisture estimates are near 130 Table 4.2 C-AP Soil Moisture Retrieval Errors over Tonzi Ranch [m 3 /m 3 ]. C-AP Method RMSD Bias ubRMSD R 2 MOEA vs. TZ 0.034 0.014 0.031 0.7 SMAP vs. TZ 0.094 -0.09 0.026 0.7 constant. Even though this will negatively aect the error statistics, the outcomes are clearly explainable and acceptable. Over time, SMAP measurements over a xed point on Earth, are acquired from multiple observation geometries, such that the eective incidence and azimuthal an- gles can vary; this is manifested as the radar cell's location varying with respect to the satellites Nadir Track, i.e., Cross-track distance. A more complete discussion of Cross-track distance will be given in the next Chapter when considering radar-only soil moisture retrieval and model performance. However, for the discussions following it is important to emphasize that due to varying eective observation angles, the actual measured 0 , at any scale is slight dierent. For example, for stable and near constant soil moisture conditions, a 0 observation from a xed radar cell at distance d 1 is dierent than d 2 , i.e., 0 (d 1 ) 6= 0 (d 2 ). Such a case can be seen in Fig. 4.33 and towards the end fo the study such that consecutive radar observations, but with dierent Cross-track distances, are 0.1-0.3 dB dierent. This residual dierence will thus manifest as slight dierences in retrieved soil moistures. At the current state of radar scattering models and implementations, none of the above geometry eects are accounted for, thus this can be considered the current lower limit the in algorithm's capabilities to estimating soil moisture. Finally, over the duration of this study, any changes in measured 0 are due to (1) soil moisture variations (2) observation geometry; vegetation dynamic and their eects are negligible since minimal vegetation change was observed. Moderate Reso- lution Imaging Spectroradiometer (MODIS) derived VWC estimates indicate a slight 131 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 -15 -14 -13 -12 -11 -10 -9 Tonzi Ranch Mean Soil Moisture [cm 3 /cm 3 ] SMAP Sigma 0 [dB] Sigma 0 -VV Sigma 0 -HH In Nadir Gap 04/26 05/06 05/16 05/26 06/05 06/15 06/25 07/05 Date -15 -14 -13 -12 -11 -10 -9 SMAP Sigma 0 [dB] Sigma 0 -VV Sigma 0 -HH In Nadir Gap Figure 4.33: Plots of SMAP 9 km 0 over Tonzi Ranch area. Top panel shows 0 vs. soil moisture variations. Bottom panel shows temporal evolution of 0 observations. D indicate 0 values within the Nadir Gap of SMAP. decline from 2.6 kg/m 2 to 2 kg/m 2 . To demonstrate this fact, in Fig. 4.34 scatter plots of MOEA and SMAP 9 km soil moisture verses Tonzi Ranch soil moisture is shown. Each marker's color is indicative of the amount of VWC and the plots shows that soil moisture estimates were not in uenced signicantly by vegetation for this case study. 132 0 0.05 0.1 0.15 0.2 0.25 0.3 Tonzi Ranch Soil Moisure [cm 3 /cm 3 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 Active-Passive Soil Moisure [cm 3 /cm 3 ] 0 0.5 1 1.5 2 2.5 3 3.5 4 VWC [kg/m 2 ] MOEA SMAP Figure 4.34: Vegetation Eects on Tonzi Ranch Soil Moisture Estimation. are MOEA and are SMAP L2SM AP retrievals. Marker colors refer to the amount of VWC which for this study ranges fro 2kg/m 2 to 2.6kg/m 2 133 CHAPTER V Forward Emission and Scattering Modeling Considerations The performance of model-driven soil moisture estimation methods is highly de- pendent on the accuracy and completeness of the physical forward models under consideration. In general, both emission and scattering models are strong in track- ing and capturing trends of soil moisture or vegetation dynamics. However, in ab- solute accuracy terms, in the context of soil moisture retrieval, signicant room for improvement exists. This chapter will therefore highlight and present some key issues and considera- tions which must be addressed in order to achieve optimum model-data match-ups. Furthermore, in the context of C-AP soil moisture retrieval, joint and cohesive emis- sion and scattering responses, which stem of a unied theoretical development is key. An opening discussion on this topic is also presented. 5.1 Observation Geometry Eects In Chapter II the Discrete Scatter Approximation model (e.g., Radar Datacubes) and the -! emission model were presented. In the remote sensing community and especially the SMAP mission, both models are widely used. 134 Although both methods are popular they suer from two important drawbacks due to the following underlying assumptions in their theoretical development: 1. Flat Earth: Rough surface scatter and emission contributions are derived with the assumption that the ground is at and has no aspect or slope with respect to the transmitting and receiving antennae. In other words, no topographical information about the targets exist in current models. 2. Azimuthally Symmetric Targets: Natural world targets, with the exception of deserts and bare surfaces, are generally asymmetric especially when viewed at dierent observation points. Although remote sensing instruments on board moving platforms have an \almost" constant incidence angle, their azimuth an- gles can be dierent. On the contrary, in current modeling techniques targets are assumed azimuthally symmetric. Therefore, with a xed and determined set of input parameters, model outputs are independent of the azimuthal per- spective of the instrument and aspect with respect to the antenna. The following section will therefore present a more detailed case study on how these assumption will aect soil moisture retrieval using SMAP satellite data. 5.1.1 Spacecraft Observation View Angles The conically scanning SMAP antenna on the surface of Earth maps out a helical pattern with a swatch width of approximately 1000 km. The radiometer's charac- teristic and native resolution is dened by the 3 dB real aperture beamwidth of the antenna with a post-processed and Earth gridded spatial resolution of 36 km. The radar resolution, on the other hand, is dictated by Synthetic Aperture processing such that for a given ground location, or cell, depending on the cross-track distance, a spatial resolution of 1 km (far away from nadir track) to 30 km (close to nadir track) is possible. To avoid low resolution radar data a so called nadir gap of 150 km is dened around the nadir ground track and data which fall within this region will 135 are excluded. It is important to emphasize that SMAP products are projected based on the Equal-Area Scalable Earth version 2.0 (EASE 2.0) grid projection [39, 40] Focusing on SMAP SAR processed data, in general, for each radar cell four unique observation geometries per half-orbit, Descending (D) or Ascending (A), can be dened: 1. Forward-looking, fore-look, with positive Cross Track (CRTK) distance. 2. After-look, or aft-look, with positive CRTK distance. 3. Fore-look with negative CRTK distance 4. Aft-look with negative CRTK distance. Positive and Negative cross-track distances are dened as the location of the radar cell being on the right hand side of the spacecraft nadir track when viewed in the spacecraft velocity direction, Fig. 5.1.a, and negative cross-track as when the radar cell location is to the left of the nadir track, Fig. 5.1.b. These orbital and observation congurations apply to both Descending and Ascending half-orbits. 136 Spacecraft Velocity Direction Cross-track Distance Aft-look Fore-look (a) (b) Figure 5.1: SMAP Descending half-orbit SAR geometry (a) Descending Fore- and Aft-Looks with positive cross-track distance (b) Descending half-orbit Fore- and Aft-looks with Negative cross-track distance. For both case, the location of the red radar cell is unchanged. A similar conguration also exists for Ascending Orbits. 137 USGS Topo Map Google Earth Satellite Tonzi Ranch Figure 5.2: Tonzi Ranch study site in North-Central California. Cluster of red squares is the in situ soil moisture sensors. It can be seen that for each D or A half-orbit, two radar observations per site exist, fore- and aft-looking, and for each observation HH, VV, and HV radar backscatter measurements are acquired. Although the local incidence angle is always approxi- mately 40 degrees, unless a given target cell shows complete azimuthal symmetry, fore- and aft-look observation will yield dierent measurement values. This dier- ence is further enhance for regions having pronounced topography, such as hilly areas where the slopes and aspects can face towards or away from the antenna. Under these circumstances it becomes clear that scattering from a given radar cell is captured from eight dierent geometry and observation viewpoints. For the following analysis SMAP 3 km EASE-grid 0 observation over the Tonzi Ranch Study site will be considered. Located in North-Central California, Tonzi Ranch is a Savanna region with a cluster of 18 in situ soil moisture sensor proles (5,10, 15cm) acquiring data at 20 min intervals. Fig. 5.2 shows both a USGS Topo- graphical map of the region as well as satellite image from Google Earth. The blue bounding box is the extend of the 3 km radar EASE-grid cell. It is clear that this region shows noticeable topographical variation as well as diversity in cover. 138 HH and VV 1 km SMAP 0 observations which fall within the Tonzi Ranch radar cell are shown in Fig. 5.3. The plot clearly shows a noticeable dierence between all fore-looking and aft-looking observations with a few outliers. Related comparison statistics are summarized in Table 5.1. The underlying reasons can be limited to: 1. SAR processing of radar Cells. 2. Enhanced target asymmetry due to dierent incident azimuth angles. 3. Regional topography. To mitigate the eects of the latter two cases, in the future, within the radar scattering models more information about the target radar cell can be incorporated. Typically to overcome radar speckle noise eects ner resolution measurements are aggregated to a slightly coarse scale [24]. Furthermore, since radar-only soil moisture estimation for SMAP is performed at 3 km, the following analysis will SMAP σ 0 Fore Look [dB] -30 -25 -20 -15 -10 -5 0 SMAP σ 0 Aft Look [dB] -30 -25 -20 -15 -10 -5 0 σ 0 hh σ 0 vv Figure 5.3: Comparison of SMAP 1 km Fore and Aft-looking 0 over the Tonzi Ranch study site. 139 Table 5.1 Fore vs. Aft 1 & 3 km SMAP Radar 0 measurements over Tonzi Ranch Resolution Polarization RMSD [dB] Bias [dB] R 2 1km HH 2.8 0.15 0.32 VV 2.6 0.17 0.31 3km HH 1.05 0.74 0.43 VV 0.75 0.20 0.70 focused on the 1 km aggregate radar measurements. Similar to Fig. 5.3, fore vs aft looks are plotted in Fig. 5.4, but for 3 km data and similarly summarized in Table 5.1. 140 SMAP σ 0 Fore-look HH [dB] -16 -15 -14 -13 -12 -11 -10 -9 -8 SMAP σ 0 Aft-look HH [dB] -16 -15 -14 -13 -12 -11 -10 -9 -8 SMAP σ 0 Fore-look VV [dB] -16 -15 -14 -13 -12 -11 -10 -9 -8 SMAP σ 0 Aft-look VV [dB] -16 -15 -14 -13 -12 -11 -10 -9 -8 Left Side of Nadir Track (CrossTrack < 0 ) Right Side of Nadir Track (CrossTrack > 0 ) Size is Proportional to Cross Track Distance (a) (b) Figure 5.4: Comparison of SMAP 3k m Fore and Aft-looking 0 over the Tonzi Ranch Study Site (a) HH, and (b) VV-pol. Marker sizes are proportional to the Cross Track Dristance and the color inidicates the site location with respect to the satellite nadir track (left or right side). 141 Date 04/06 04/16 04/26 05/06 05/16 05/26 06/05 06/15 06/25 07/05 07/15 SMAP Sigma 0 [dB] (3km) -15 -13 -11 -9 -7 -5 Soil Moisture [cm 3 /cm 3 ] 0 0.1 0.2 0.3 Soil Moisture VV HH Figure 5.5: Temporal plot of Tonzi Ranch daily mean soil moisture (green line) and 3 km SMAP 0 . Red error bars indicate the Fore vs. Aft range. It is obvious that the look discrepancies at 1 km also transfer to 3 km 0 . Also, note that 0 measurements with larger Cross Track distances (larger circles in Fig. 5.4) are better correlated than radar cells closer to the Nadir Track. This is due to the fact that the closer the target is, the more pronounced the Fore vs. Aft-looks' azimuth angle dierence is. Therefore the antenna is eectively observing a very dierent scene! Figure 5.5 shows the temporal evolution the daily mean soil moisture at Tonzi Ranch as well as the corresponding mean radar 0 measurements; averaged radar data are the arithmetic mean of the Fore and Aft-looks. The red error bars around each radar point is the Fore vs. Aft range. Two rain events at 04/26 and 05/16 are clearly captured by the in situ sensors and radar measurements show the expected variation with soil moisture. It is important to note, that soil moisture retrievals concurrent with rain events generally include a cautionary ag since surface con- ditions become very dynamic and unstable, thus obtaining accurate estimates are 142 questionable. The daily Fore-Aft range in 0 increases for radar cells closer to the Nadir Track, (i.e., smaller markers in Fig. 5.5) which is re ective of the same discrepancy observed in Fig. 5.4. In order to perform soil moisture estimation over this study site, the question is raise about which SMAP radar look is most representative of the scene and how well does the radar scattering model match-up with actual data. 5.1.2 Forward Model Comparison with SMAP Radar Details of the radar scattering model were presented in Chapter II. In this section, the model performance is evaluated and compared to 3 km SMAP 0 measurements at the Tonzi Ranch Study site. In Table 5.3 a complete list of model parameters is shown. For each day and each SMAP Descending half-orbit, the closest-time in situ soil moisture observation is extracted from the database and used. With respect to each 3 km measurement a model predictions is simulated for both HH and VV- polarization. The parameters given in Table 5.3 were collected during an extensive eld cam- paign in November 2014. Information on trunk and total tree height, diameters, estimates of branch lengths and orientations, as well as Trunk Permittivity measure- ments were gathered and processed to determined the \average" tree within the area. The dominant Tree type is Oak and based on the IGBP Land Cover Classication, the region is a Savanna class. MODIS based VWC estimates indicate minimal vegetation change over the study duration, reduction of VWC from 2.6 kg/m 2 to 2.2 kg/m 2 . However, the extent of the ground sampling study site covered regions with relatively less tree density than the Southwestern part of the site. Therefore, trunk destines were optimally modied to achieve best model-data match. 143 Tonzi Ranch Mean SM [cm 3 /cm 3 ] 0.05 0.1 0.15 0.2 0.25 Radar Sigma 0 HH [dB] -14 -13 -12 -11 -10 -9 -8 -7 Tonzi Ranch Mean SM [cm 3 /cm 3 ] 0.05 0.1 0.15 0.2 0.25 -14 -13 -12 -11 -10 -9 -8 -7 Left Side of Nadir Track (CrossTrack < 0 ) Right Side of Nadir Track (CrossTrack > 0 ) Size is Proportional to Cross Track Distance Radar Sigma 0 VV [dB] Model SMAP Model SMAP Figure 5.6: Forward model comparison with SMAP 3 km 0 ; top panel is hh and bottom panel is for vv . Figure 5.6 shows plots of SMAP Radar measurements with respect to changes in soil moisture. Model predictions and outputs are also overlaid (thick black lines) for both HH and VV-polarizaitons. As expected, both model and data show an overall increase in radar backscatter with increasing soil moisture. The dynamic range of moisture variations is indicative of a dry period (0.05-0.25 m 3 /m 3 ) whilst 0 varies from -13 to -8 dB. The red error bars are, again, the range of Fore vs. Aft 0 and are comparability larger for smaller cross track distances as noted before. Aft-look comparisons show a larger RMSD with respect to model predictions than Fore or mean-looks. However, overall no solid conclusion can be made on which 0 look 144 Table 5.2 Comparison Between Model Predictions and SMAP 3 km 0 for Fore, Aft and Mean Observations. Polarization Look RMSD [dB] Bias [dB] ubRMSD [dB] HH Fore 1.07 0.34 1.02 Aft 1.3 0.55 1.18 Mean 1.05 0.41 0.97 VV Fore 1.06 0.35 0.99 Aft 1.34 -0.47 1.25 Mean 1.11 0.43 1.01 matches the best with model outputs. 145 Table 5.3 Radar Scattering Model Parameters for Savanna Land Cover Class. Constituent Parameter Value Unit Large Branch Permittivity (25,10) NA Length 2.2 m Radius 0.0153 m Density 0.2 #/m 2 Orientation 45 Deg Small Branch Permittivity (25,10) NA Length 0.68 m Radius 0.007 m Density 0.924 #/m 2 Orientation 45 Deg Leaf Permittivity (30,8) NA Thickness 0.5 mm Radius 0.01 m Density 462 #/m 2 Trunks Permittivity (28.0,8.0) NA Height 2.95 m Canopy Height 4.2 m Radius 0.09 m Density 0.09 #/m 2 Orientation 45 Deg Surface RMS Height 0.02 m Clay Fraction 21 % 146 5.1.3 Radar-only Soil Moisture Estimation Similar to all prior soil moisture retrieval methods presented in Chapter III Radar- only retrieval is performed for the collection of SMAP radar data obtained over Tonzi Ranch. The objective function to be minimized is: L a (sm) = 1 2 X pp=hh;vv 0 pp 0 pp (sm) 0 pp 2 : (5.1) Note that unlike prior case studies, here, surface roughness is assumed known and corrected for prior to optimization. Therefore, the only unknown parameter is soil moisture (sm). For each Descending Half-orbit, soil moisture estimation is performed using (a) Fore-look 0 only, (b) Aft-look 0 only, and (c) Mean of Fore- and Aft-looks. RMS Errors, Biases and other statistics are calculated for comparison. In Fig. 5.7 a temporal plot of Tonzi Ranch mean soil moisture along with retrieved values is shown; results using the mean 0 are only presented here. A scatter plot comparison between retrievals and mean soil moisture is also shown in Fig. 5.8a and indicates an overall strong correlation between estimates and in situ measurements. However, for larger soil moisture values, the comparison degrades. 147 Date 04/06 04/16 04/26 05/06 05/16 05/26 06/05 06/15 06/25 07/05 07/15 Soil Moisture [cm 3 /cm 3 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 Left Side of Nadir Track (CrossTrack < 0 ) Right Side of Nadir Track (CrossTrack > 0 ) Size is Proportional to Cross Track Distance Tonzi Ranch Mean SM Radar-only Estimate Figure 5.7: Temporal Plot of Tonzi Ranch Soil Moisture and Radar-only Estimates 148 Tonzi Mean Soil Moisture [cm 3 /cm 3 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 Estimated Soil Moiture [cm 3 /cm 3 ] 0 0.05 0.1 0.15 0.2 0.25 0.3 Left Side of Nadir Track (CrossTrack < 0 ) Right Side of Nadir Track (CrossTrack > 0 ) Size is Proportional to Cross Track Distance (a) Standard Deviation 0.02 0.04 0.02 0.04 0 0.06 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Cor re l at i on Coef f i ci ent True Mean Fore Aft Standard Deviation 0.06 RMSE (b) Figure 5.8: (a) Scatter plot of true vs. retrieved soil moistures (b)Taylor plot comparing the three soil moisture estimation scenarios. in situ moisture Standard Deviation is shown as a red square labeled \True." 149 Table 5.4 Radar-only Soil Moisture Retrieval Error Statistics in [m 3 /m 3 ]. Radar Look RMSD Bias ubRMSD Fore-only 0.034 0.023 0.024 Aft-only 0.034 0.014 0.031 Mean 0.030 0.017 0.024 The Taylor Plot of Fig. 5.8b compares the statistics of unbiased RMSD, Standard Deviation and Correlation Coecient between the three radar estimation scenarios, which are also summarized in Table 5.4. The red square labeled \True" is the in situ soil moisture Standard Deviation over the study period. The plot can be interpreted as such: the closer a method is to the reference true, the higher the likelihood of it being a more accurate methodology therefore having a higher correlations, similar standard deviation and less error compared to other techniques. Fore-only and Mean 0 show an overall better match with respect to ground truth, and are comparable to each other. Although a short and concise case study was presented here, a few key issues are raised which address the discrepancies between true and estimated soil moistures values. Furthermore, these points are viewed as open topics and opportunities to address in future Electromagneticand remote sensing related studies. 1. Representativeness of Soil Moisture: in situ probes are point measurements covering a signicantly smaller area than 3 km. Simple arithmetic averaging of moisture values may not be sucient enough to capture the wider scale soil moisture distribution, especially since 0 is an aggregated and eective value at 3 km. Utilization of properly \upscaled" soil moisture values can be signicantly benecial to improving current scattering models. 2. Model-data mismatches: although the DSA model was corrected to achieve best match, with least error, with respect to 3 km SMAP 0 , the correction applied accounted for all data points at once. On the other hand, estimation was 150 performed point-wise, thus some residual error is expected. This can be seen in Figs. 5.6 and 5.8a for the moisture value of 0.23 m 3 /m 3 . There is a signicant model-data deviation (2 dB) at 0.23 m 3 /m 3 which in turn results in a relative error in estimation of 20%! The over-estimation of soil moisture towards the end of the study period is due to the 1.5 dB o-set between vv measurements. This is also clearly seen in the lower panel in Fig. 5.6. 5.2 Joint Modeling of Emission and Scattering From a physical point of view, it can be shown that the amount of emission and scattering from an object are fundamentally related to each other. A simple example was show in Chapter II where plots of 0 vs. TB indicated a negative correlation. Studies such as [20] have built on this near-linear relationship to develop simple multi-temporal C-AP schemes. From an Electromagnetic forward modeling perspective, Peake [47] showed that under Thermodynamic Equilibrium, the amount of bistatic scattering from an object can be related to its total emission. Kirchho's Law of Radiation under the equilib- rium conditions states that an object's p-polarized emission is equal to its absorption coecient, e p ( i ; i ) =a p ( i ; i ). Furthermore, the fraction of power incident on an object which is not absorbed, must be scattered, i.e.,a p ( p ; p ) = 1! p ( i ; i ). Thus emission is 1! p ( i ; i ). The scattering albedo is ! p ( i ; i ) = Fraction of Power Scattered Total Power Incident (5.2a) ! p ( i ; i ) = 1 4 X q=h;v =2 Z 0 2 Z 0 qp ( i ; i ; s ; s ) sin s d s d s : (5.2b) where qp is the qp-polarized Bistatic Scattering Coecient from the incident di- rection ( i ; i ) to the scattering direction ( s ; s ) and includes both coherent and 151 incoherent scattering terms: qp ( i ; i ; s ; s ) = lim ^ r!1 4 r 2 jE s p j 2 jE i q j 2 : (5.3) The limits on integration in Eqs. 5.2 refer to a hemispherical integration with the assumption of limited to no energy transmission through the object. This holds true for L-band soil moisture remote sensing, since the wave penetration in to the soil is limited to 5-10 cm therefore the soil medium can be considered as a dielectric half-space. By combing Eqs. 5.2 with the expression of emission, Peake's statement can be written as e p ( i ; i ) = 1 1 4 X q=h;v =2 Z 0 2 Z 0 qp ( i ; i ; s ; s ) sin s d s d s : (5.4) This simple expression uniquely relates Radar scattering to Radiometer Bright- ness Temperature such that if, for any geophysical target, the total Bistatic scattering coecient can be calculated, in the backscatter direction it will yield 0 and if in- tegrated over the entire upper hemisphere, emissivity can be determined. This so called \Joint-physics" approach uses a single parameter kernel to predict both pro- cesses and has the potential to out perform the DSA and -! in CAP. A conceptual diagram of this process is show in in Fig. 5.9 5.2.1 Eects of Surface Roughness To highlight the importance and signicance of Joint-physics modeling on soil moisture estimation, a series of numerical simulations are performed and error statis- tics calculated and compared. Two scenarios for rough surface scattering and emis- sion are considered (a) smooth surface (b) very rough surface; in both cases soil moisture varies from very dry to very wet (in terms of soil permittivity r ). In the discussion in Chapter II it was stated that a variety of advanced surface 152 Bistatic Radar Scattering Model TB e T Radiometer response p i i p i i phys ( , ) ( , ) θ φ θ φ = ⋅ = γ θ φ θ φ γ θ φ θ φ pq i i s s pq i i i i Radar response ( , ; , ) ( , ; , ) = = Emission In Backscatter Direction e d p i i qp i i s s q h v ( , ) ( , ; , ) , θ φ π γ θ φ θ φ = − ∫∫ ∑ = 1 1 4 Ω Figure 5.9: Joint-physics emission and scattering concept. scattering models exists which can be used to calculate the amount of rough surface scattering. On the other hand, within the-! model roughness eects on emission are accounted for by an exponential modication to the bare surface Fresnel Equations, sometimes known as the Kirchho Approximation [26]. The p-polarized re ectivity is therefore expressed as r p;rough =r p0 e h =r p0 e (2ks) 2 =cos i (5.5) wherer p0 is the p-polarized Fresnel Re ectivity equation. At times, to correct model performance, an empirical and land cover dependent value for the h parameter is used rather than the actual surfaces RMS height s. Although the last expression on the right hand side of Eqs. 5.5 is theoretically correct, it has an upper bound on validity. In general, as the amount of surface roughness increases, so does incoherent, or diused, wave scattering. Studies have shown that as the surface RMS height approaches 2 cm and beyond, the Kirchho Approximation for rough surface emission is no longer valid. This drawback can be mitigated by using a more complete rough surface emis- sion model. In the case presented here initially the bistatic scattering coecient pq ( i ; i ; s ; s ) for rough surfaces is calculated, which does include all coherent, in- coherent and specular scattering components. Then, using Eqs. 5.4 the total surface emissivity is determined. It is expected that as the surface roughness increases, the 153 latter approach, compared to the Kirchho Approximation performs better. Therefore, for the Radar-Radiometer soil moisture retrieval analysis which fol- lows, two model setups are used 1. Traditional Approach: Kirchho Approximation Model is used for Emission along with SPM scattering model for 0 . 2. Joint-physics Approach: SPM bistatic scattering is used in Eqs. 5.4 to calcu- lated emission, the same bistatic scattering, in the backscatter direction will yield 0 . Clearly, it can be seen that for the second case, both emission and scattering are tightly related through their theoretical basis. To proceed, a Monte-Carlo like nu- merical simulation and optimization is performed such that for 3 r 30 and ks2 [0.1 0.3] Radar-Radiometer soil moisture retrieval is attempted. To avoid the so called \Inverse Crime" the IEM method proposed by Tsang et.al. [48] is used as the Forward Simulator to generate \true" emission and scattering values. By doing so, inversion models are segregated from forward models. Comparison plots for these models are also shown in Fig. 5.10. It is clear that for rougher surfaces, the Kirchho model performs poorly with respect to the reference truth, i.e., IEM model. There- fore, signicant error in soil permittivity retrieval is expected. Gaussian noise with standard deviation of 0.5 dB for radar and 1.5 K for radiometer are assumed. Fur- thermore, the regularization term in Eqs. 3.2, balancing 0 and TB contributions, is varied to obtain best outcomes. In Fig. 5.11 Error maps of the entire optimization and simulation scenario is shown. Every pixel in Fig. 5.11 represents the RMS error in retrieving soil permit- tivity for a given \true" r (y-axis) as a function of the regularization term (x-axis). The thick black line is the average RMS across all r at a given value. When is small, more weight is given to Radiometer data, and when is large, more emphasis is placed on radar data. The following features are discussed: 154 0 5 10 15 20 25 30 Soil Permittivity (Real Part) 120 140 160 180 200 220 240 260 280 300 TB [K] V-ie m H-iem V-sp m H-spm V-kirch H-kirch Surface Roughness ks= 0.1 (a) 0 5 10 15 20 25 30 Soil Permittivity (Real Part) 120 140 160 180 200 220 240 260 280 300 TB [K] V-ie m H-iem V-sp m H-spm V-kirch H-kirch Surface Roughness ks= 0.3 (b) Figure 5.10: IEM, SPM and Kirchho emission model comparison (assuming Phys- ical Temperature of 300 K) vs. soil permittivity for (a) smooth surface (ks = 0.1); maximum deviation between IEM and Kirchho is 2 K and (b) rough surface (ks=0.3); maximum deviation between IEM and Kirchho is 10 K. For both case, IEM and SPM are comparable. 1. From the gure it can be seen that for almost all case, for very wet surfaces, Radar-only retrieval incurs noticeable error. This is a well understood fact and is due to the reduced radar backscatter sensitivity to wetter soils. 2. For the Traditional Kirchho Approximation approach, as the surface rough- ness increases, a signicant error is reported for Radiometer-only outputs Fig. 5.11.b indicative of model performance degradation. 3. In the Joint-physics approach, Fig. 5.11.c and d (a) radar response is as ex- pected (b) for rough surfaces with respect to the Kirchho Approximation the amount of RMSE in r is insignicant 4. Although in Fig.5.11.b a noticeable error exists at either extremes, it is still possible to obtain acceptable soil permittivity predictions via proper regular- ization. This is seen in the dip in the average error curve in the same gure or Fig. 5.12. When 5, the average error across all the soil moisture scenarios considered is smaller than all other regularized conditions. Approximately, it 155 Figure 5.11: Error maps and RMSE in soil permittivity for (a) smooth surface (ks = 0.1) using Kirchho roughness model (b) rough surface (ks = 0.3)using Kirchho roughness model (c) smooth surface (ks = 0.1) using Joint-physics roughness model (b) rough surface (ks = 0.3) using Joint-physics roughness model. is 1/2 of Radar-only and Radiometer-only scenarios. The last statement in the list above is a strong and powerful feature of C-AP soil moisture estimation: when faced with decient forward models, proper utiliza- tion and regularization of complementary information, i.e., TB and 0 , will still outperform either radar or radiometer-only methods! A natural extension of these examples is to apply the same analysis to vegetated scenarios and natural targets. The current hypothesis is that with Joint-physics model, across a whole range of surface and land cover conditions obtaining improved and acceptable soil moisture retrievals are possible. 156 (a) (b) Figure 5.12: (a) Soil Permittivity retrieval RMS Error smooth surface (b)Soil Permittivity retrieval RMS Error for rough surface. In both case, the Joint-physics C-AP outperforms traditional Kirchho method. 157 5.3 Chapter Conclusion The discussions and case studies presents in this chapter are viewed as an opening discussion and way forward to address current modeling issues in Active and Passive remote sensing. Brie y, 1. Incorporation of a target's (a) azimuthal behavior and (b) topographical infor- mation within forward scattering models will signicantly improve mode-data comparisons and match-ups, thus improving soil moisture estimation capabili- ties. 2. In the context of C-AP soil moisture retrieval, consistent and coherent theoret- ical development of emission and scattering from a target using a single shared parameter kernel will increase the robustness of C-AP method. The numerical analysis presented in the previous section demonstrated, using rough surfaces, how a single model can yield both TB and 0 and acceptable soil permittivity retrievals. 3. In the face of models which perform poorly with respect to actual data, proper use of complementary information can assist in reducing errors. This was shown in the example using Kirchho Approximation for very rough surfaces. 158 APPENDICES 159 APPENDIX A Active-Passive Data Processing Flow Chart 160 Figure A.1: SMAP Active-Passive Data Processing Flow Chart. SWIF: Snow, Water, Ice and Frozen. 161 APPENDIX B Equations for Scattering from Finite Dielectric Cylinder The following equations for scattering from nite length dielectric cylinders are at the heart of the Radar scattering model discussed in Chapter II. First, scattering from an innite cylinder is determined, then then tapered based on the length of the cylinder. For an innite length cylinder, with incident eld E i and incident angle i , the h and v components of the scattered eld are E h E v s =e i3=4 r 2 kr sin i e ik(r sin i zcos i ) A hh A hv A vh A vv E h E v i (B.1) The reader is refers to [49] for exact details and derivations of the elements of the scattering matrix which are typically series expansions of Bessel and Hankel Function. They are, however, strong functions of the cylinder's permittivity, radius, incident angles ( i ; i ) and scattering angles ( s ; s ). To account for the nite length of the cylinder, the elements of the scattering matrix are tapered using a Sinc function proportional to the length of the cylinder, 162 which is generally valid with the length of the cylinder is larger than the wave length, i.e., L>. Eectively, this method assumed a small aperture placed on top of the innite cylinder, hence the Sinc function. Thus, the pq-polarized element of the nite cylinder scattering matrix is: S pq ( i ; i ; s ; s ) = ikL sin s sin i sinkL(cos i + cos s )=2 kL(cos i + cos s )=2 A pq ( i ; i ; s ; s ): (B.2) For an arbitrary oriented cylinder, the scattering matrix must be multiplied by two rotation matrices, once in-to and the other out-o the cylinder's local coordinate system ASB. The tilt of the cylinder in and out of the Plane of incidence of (plane made by k and z vectors) are and . The elements of the 22 rotation matrices A and B are A 11 = cos 1i (B.3a) A 12 = cos i2 sin i1 sin i2 sin cos i1 (B.3b) A 21 = cos cos i1 sin i1 + sin i1 sin (B.3c) A 22 = cos i2 cos i1 cos i1 sin i2 cos i1 sin sin i1 + sin i2 sin i1 cos (B.3d) 163 B 11 = sin s1 sin s + cos s1 cos s cos (B.4a) B 12 = cos s1 cos s1 sin s + cos s1 sin s1 cos s cos + sin s1 cos s sin (B.4b) B 21 = sin s1 cos s cos s2 + cos s2 cos s1 sin s cos cos s1 sins2 sin (B.4c) B 22 = cos s2 cos s1 cos s cos s1 + cos s1 cos s2 cos sin s sin s1 cos s2 sin s2 sin sin s1 + sin s2 sin s1 cos: (B.4d) where we have i2 = i + (B.5a) s2;back = s + (B.5b) s2;forward = s (B.5c) i1 = arccos(cos cos i2 ) (B.5d) s1 = arccos(cos cos s2 ) (B.5e) i1 = arctan sin tan i2 cos i + cos i cos (B.5f) s1 = arctan sin tan s2 cos s + cos s cos : (B.5g) 164 LIST OF ABBREVIATIONS AirMOSS Airborne Microwave Observatory of Subcanopy and Subsurface ASF Alaska Satellite Facility C-AP Combined Active-Passive ESA European Space Agency EASE Equal-Area Scalable Earth GCMs Global Climate Models IGBP International Geosphere-Biosphere Programme MODIS Moderate Resolution Imaging Spectroradiometer NWP Numerical Weather Prediction NSIDC National Snow and Ice Data Center PALS Passive and Active L- and S-band Sensor RAD Radiometer SAR Synthetic Aperture Radar SCA Single Channel Algorithm SMAP Soil Moisture Active Passive SMOS Soil Moisture Ocean Salinity 165 UAVSAR Uninhabited Aerial Vehicle Synthetic Aperture Radar VWC Vegetation Water Content 166 BIBLIOGRAPHY 167 BIBLIOGRAPHY [1] NASA Workshop Report, Soil Moisture Active/Passive Mission, Report of NASA HQ Community Workshop, 2007. 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Abstract (if available)

Abstract This dissertation presents a detailed investigation of combined and synergistic utilization of active radar and passive radiometer microwave remote sensing techniques to estimate surface soil moisture across multiple measurement platforms, i.e., tower-mounted, airborne and Spacecraft data. Chapter I provides a comprehensive introduction to Combined Active-Passive soil moisture estimation as well as listing the objectives and contributions of this dissertation. The theoretical basis of emission and scattering, along with radar and radiometer comparisons are discussed in Chapter III. Thereafter, Chapters III and IV present in detail Combined Radar-Radiometer soil moisture retrieval for multiple different measurement scenarios and instruments. Lessons learned and future consideration are then elaborated in the final Chapter V.

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Creator Akbar, Ruzbeh (author)

Core Title Active and passive microwave remote sensing synergy for geophysical parameter estimation

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 02/22/2016

Defense Date 01/13/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag applied electromagnetics,Earth Sciences,Electrical Engineering,estimation,OAI-PMH Harvest,optimization,Radar,radiometer,soil moisture

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Moghaddam, Mahta (committee chair), Hashemi, Hossein (committee member), Jafarpour, Behnam (committee member), Narayanan, Shrikanth S. (committee member), Povinelli, Michelle L. (committee member)

Creator Email rakbar@mit.edu,rakbar@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-213076

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Legacy Identifier etd-AkbarRuzbe-4138.pdf

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