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Solution of inverse scattering problems via hybrid global and local optimization

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Solution of inverse scattering problems via hybrid global and local optimization

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Content Solution of Inverse Scattering Problems via Hybrid Global and Local Optimization By Aslan Etminan A dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL In partial fulfillment of the requirements for the degree of Doctor of Philosophy (ELECTRICAL ENGINEERING) in the University of Southern California December 2019 Doctoral Committee: Professor Mahta Moghaddam, Chair Professor Andreas Molisch Associate Professor Behnam Jafarpour Research Assistant Professor Alireza Tabatabaeenejad To my parents, for their love and endless support... ACKNOWLEDGEMENTS I would like to extend thanks to all the people who so generously contributed to the work presented in this thesis. First and foremost, I would like to express my sincere gratitude to my advisor, Professor Mahta Moghaddam for the continuous support of my Ph.D. studies and research, and for her patience, enthusiasm, motivation and immense knowledge. I feel extremely fortunate and blessed to have the opportunity of having her as my advisor and mentor. Beside my Ph.D. advisor, my profound gratitude goes to the rest of my committee members: Prof. Behnam Jafarpour, Prof. Andreas Molisch, and Prof. Alireza Tabatabaeenejad for their time, suggestions and helpful com- ments. I would like to acknowledge my colleagues and friends in Microwave and Systems, Sensors and Imaging Laboratory (MiXIL). Special thanks to Ruzbeh Akbar, Majid Albahkali, Amir Azemati, Kazem Bakian-Dogaheh, James Camp- bell, Guanbo Chen, Richard Chen, Negar Golestani, Samuel Prager, Pratik Shah, John Stang, Alireza Tabatabaeenejad, and Jane Whitcomb. I would also like to thank the entire MIXiL group for the fun, hours-long discussions, and friendship over the years. Last but not least, I would like to express my deepest gratitude to my mother Parvaneh Basirpour and my father Nader Etminan for their unbelievable support 2 at every step of my life and my special thanks to Parisa Pouya for her sacrifice, emotional support, and patience. Without their unconditional love and support, I would not have gone this far in pursuing my dreams and exploring the beauty of life. 3 Contents ACKNOWLEDGEMENTS 2 LIST OF FIGURES 7 LIST OF TABLES 15 ABSTRACT 17 1 Introduction 19 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . 20 1.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.1 Microwave Imaging of Dielectric Objects . . . . . . . . . 24 1.2.2 Inversion of Subsurface Properties of Layered Dielectric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3 Objectives and Research Goals . . . . . . . . . . . . . . . . . . . 27 1.4 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Inverse-Scattering Problem 30 2.1 Theory and Background . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Multi-Directional Search Based Simulated Annealing . . . . . . . 31 4 2.3 Inversion Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Benchmark Function Tests . . . . . . . . . . . . . . . . . . . . . 43 2.5 Effect of Algorithm Parameters on Optimization Results . . . . . 45 2.6 Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . 47 3 Microwave Imaging of Dielectric Regions 48 3.1 Background and Theory . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Complexity of the Algorithm . . . . . . . . . . . . . . . . . . . . 59 3.4 Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . 63 4 Radar Remote Sensing Retrieval of Subsurface Properties of Layered Dielectric Structures 64 4.1 Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Retrieving the Model Parameters of a Two-Layer Dielectric Struc- ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.1 Background and Theory . . . . . . . . . . . . . . . . . . 66 4.2.2 Numerical Experiments . . . . . . . . . . . . . . . . . . 68 4.3 Retrieving Subsurface Soil Moisture Profiles . . . . . . . . . . . . 82 4.3.1 Introduction and Background . . . . . . . . . . . . . . . 82 4.3.2 Inversion scheme . . . . . . . . . . . . . . . . . . . . . . 83 4.3.3 Soil Moisture Retrieval Using Synthetic Data . . . . . . . 85 4.3.4 Retrieving RZSM Using AirMOSS Data . . . . . . . . . 92 4.3.5 A New Soil Moisture Model for Retrieving RZSM . . . . 101 4.4 Conclusion and Summary . . . . . . . . . . . . . . . . . . . . . . 103 5 Conclusion and Future Work 108 5 Appendices 112 A Overview of the Simulated Annealing Method 113 B Multi-Directional Search Algorithm 117 List of Abbreviations 120 Bibliography 122 6 LIST OF FIGURES 1.1 Using microwave imaging to improve the accuracy of tumour de- tection. (a) Microwave imaging system setup system in our lab. (b) Detection of a cancer tumor, which is not visible in x-ray (im- age courtesy: https://newsnetwork.mayoclinic.org). . . . . . . . . 21 1.2 SMAP radar image acquired from data from March 31 to April 3, 2015. Weaker radar signals (blues) reflect low soil moisture or lack of vegetation, such as in deserts. Strong radar signals (reds) are seen in forests. Credit: NASA/JPL-Caltech/GSFC. . . . . . . 23 1.3 The radar backscatter over the AirMOSS site in Arizona from Oc- tober 23, 2012 (image reprinted from [14]). . . . . . . . . . . . . 24 2.1 General configuration of 2-D inverse-scattering problem in which we illuminate the target and collect the scattered fields. . . . . . . 32 2.2 Multi-directional search with non-deterministic reflection, expan- sion, and contraction steps. . . . . . . . . . . . . . . . . . . . . . 36 2.3 Flowchart of the conventional form of the simulated annealing method using the sequential perturbation scheme. . . . . . . . . . 37 2.4 Flowchart of the multi-directional-search based simulated annealing. 38 7 2.5 Two dimensional Rosenbrock test function. . . . . . . . . . . . . 44 2.6 Root-mean-square error (RMSE) in the optimization of 2 dimen- sional Rosenbrock function for different values ofN t andN s . . . . 46 3.1 (a) Permittivity distribution of the investigation domain (b) Re- construction of the target for noise-free measurements and (c) the corresponding cost function. . . . . . . . . . . . . . . . . . . . . 51 3.2 (a) Reconstruction of the target in the presence of noise (30 dB signal-to-noise ratio) and (b) the corresponding cost function. (c) Re- construction of the target in the presence of noise (20 dB signal- to-noise ratio) and (d) the corresponding cost function. (e) Recon- struction of the target in the presence of noise (10 dB signal-to- noise ratio) and (f) the corresponding cost function. . . . . . . . . 52 3.3 (a) Reconstruction of the target for noise-free measurements by using the standard simulated annealing method and (b) the corre- sponding cost function. . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 (a) Permittivity distribution of the investigation domain (b) Re- construction of the target for noise-free measurements and (c) the corresponding cost function. . . . . . . . . . . . . . . . . . . . . 54 3.5 (a) Reconstruction of the target in the presence of noise (30 dB signal-to-noise ratio) and (b) the corresponding cost function. (c) Re- construction of the target in the presence of noise (20 dB signal- to-noise ratio) and (d) the corresponding cost function. (e) Recon- struction of the target in the presence of noise (10 dB signal-to- noise ratio) and (f) the corresponding cost function. . . . . . . . . 55 8 3.6 (a) Permittivity distribution of the investigation domain (b) Re- construction of the target for noise-free measurements and (c) the corresponding cost function. . . . . . . . . . . . . . . . . . . . . 57 3.7 (a) Reconstruction of the target in the presence of noise (30 dB signal-to-noise ratio) and (b) the corresponding cost function. (c) Re- construction of the target in the presence of noise (20 dB signal- to-noise ratio) and (d) the corresponding cost function. (e) Recon- struction of the target in the presence of noise (10 dB signal-to- noise ratio) and (f) the corresponding cost function. . . . . . . . . 58 3.8 (a) Permittivity distribution of the investigation domain (b) Re- construction of the target for noise-free measurements and (c) the corresponding cost function. . . . . . . . . . . . . . . . . . . . . 60 3.9 (a) Reconstruction of the target in the presence of noise (30 dB signal-to-noise ratio) and (b) the corresponding cost function. (c) Re- construction of the target in the presence of noise (20 dB signal- to-noise ratio) and (d) the corresponding cost function. (e) Recon- struction of the target in the presence of noise (10 dB signal-to- noise ratio) and (f) the corresponding cost function. . . . . . . . . 61 3.10 Required number of function evaluations for the convergence of the proposed optimization algorithm for different number of func- tion variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9 4.1 Examples of geometries of the problems for retrieving the sub- surface properties of dielectric structures. (a) The structure could contain multiple rough interfaces and the complex dielectric con- stant is continuous for each layer. (b) The structure has rough up- per surface and smooth interfaces below with a continuous depth- varying depth profile. . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 A 3-D two-layer isotropic dielectric structure with complex per- mittivities. The boundaries are zero-mean stationary random pro- cesses. The layers mean separation is denoted asd. . . . . . . . . 68 4.3 The value of the cost function for determining the properties of the layered structure in the first experiment. . . . . . . . . . . . . 70 4.4 Changes of the model parameters during the iterations of the first experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 The value of the cost function for determining the properties of the layered structure in the second experiment. . . . . . . . . . . . 72 4.6 Changes of the model parameters during the iterations of the sec- ond experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.7 Cost function variation with respect to each of the model parame- ters of Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.8 Cost function variation with respect to each of the model parame- ters of Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.9 Relative error distribution between the actual and retrieved values of the model parameters of Case 1. We applied a uniform noise distribution with +/- 0:25 dB bounds. . . . . . . . . . . . . . . . . 78 10 4.10 Relative error distribution between the actual and retrieved values of the model parameters of Case 2. We applied a uniform noise distribution with +/- 0:25 dB bounds. . . . . . . . . . . . . . . . . 79 4.11 Relative error distribution between the actual and retrieved values of the model parameters of Case 1. We applied a uniform noise distribution with +/- 0:50 dB bounds. . . . . . . . . . . . . . . . . 80 4.12 Relative error distribution between the actual and retrieved values of the model parameters of Case 2. We applied a uniform noise distribution with +/- 0:50 dB bounds. . . . . . . . . . . . . . . . . 81 4.13 Flowchart of the multi-directional-search based simulated anneal- ing adapted for the inversion of soil moisture profile. . . . . . . . 84 4.14 Geometry of the forward problem for retrieving soil moisture pro- files: A three-dimensional dielectric structure with complex per- mittivities. Boundaries are zero-mean stationary random processes. 86 4.15 Two nominal Examples of soil moisture profiles as second-order polynomials: increasing and decreasing moisture with respect to depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.16 Distribution of RMSE between the actual and retrieved soil mois- ture profiles for 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the hybrid multi-directional search based simulated annealing algorithm for Profile 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 11 4.17 Distribution of RMSE between the actual and retrieved soil mois- ture profiles for 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the hybrid multi-directional search based simulated annealing algorithm for Profile 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.18 Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using both polarizations as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . . . . . 94 4.19 Comparison between the retrieved and measured profiles on Au- gust 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the second-order polyno- mial model (using both polarizations as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . 95 4.20 Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using both polarizations as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . . . . . 96 4.21 Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the second-order polyno- mial model (using the VV polarization as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . 98 12 4.22 Comparison between the retrieved and measured profiles on Au- gust 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the second-order polyno- mial model (using the VV polarization as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . 99 4.23 Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the second-order polyno- mial model (using the VV polarization as our measurement). The RMSE is calculated based on depths of down to 80 cm. . . . . . . 100 4.24 Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.25 Comparison between the retrieved and measured profiles on Au- gust 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.26 Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Wal- nut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 13 B.1 Reflection, expansion, and contraction steps of the multi-directional search algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 119 14 LIST OF TABLES 2.1 A comparison between the proposed optimization scheme and simulated annealing method on Rosenbrock functions in 2 and 4 dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Root-mean-square error (RMSE) and CPU-time (seconds) of the reconstructed images in the numerical experiments. . . . . . . . . 60 4.1 Actual values of the model parameters in the cases used for nu- merical simulations. All of the unites are in SI. . . . . . . . . . . 69 4.2 Constraints on the unknown variables in the numerical simula- tions. All of the unites are in SI. . . . . . . . . . . . . . . . . . . 69 4.3 Actual values, initial guesses, and retrieved values of the first nu- merical experiment. All of the unites are in SI. . . . . . . . . . . . 70 4.4 Actual values, initial guesses, and retrieved values of the second numerical experiment. All of the unites are in SI. . . . . . . . . . 73 4.5 Average error of estimating the model parameters for 100 realiza- tions of the perturbed data modeled by Eq. 4.3 with +/- 0:25 dB bounds All of the unites are in SI. . . . . . . . . . . . . . . . . . 77 15 4.6 Average error of estimating the model parameters for 100 realiza- tions of the perturbed data modeled by Eq. 4.3 with +/- 0:50 dB bounds. All of the unites are in SI. . . . . . . . . . . . . . . . . . 77 4.7 Numerical results of 100 realizations of the perturbed data mod- eled with Eq. 4.5 using the simulated annealing method and the proposed hybrid optimization technique for estimating the soil moisture profile 1 in Fig. 4.15. The numbers reported in this table are the average of all realizations for each inversion method. . . . 89 4.8 Numerical results of 100 realizations of the perturbed data mod- eled with Eq. 4.5 using the simulated annealing method and the proposed hybrid optimization technique for estimating the soil moisture profile 2 in Fig. 4.15. The numbers reported in this table are the average of all realizations for each inversion method. . . . 89 4.9 Breakdown of RMSE for Walnut Gulch Sites Using the Second- Order Polynomial Model with Observations of HH and VV Polar- izations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.10 Breakdown of RMSE for Walnut Gulch Sites Using the Second- Order Polynomial Model with Observations of HH and VV Polar- izations (Hybrid Optimization scheme) . . . . . . . . . . . . . . . 97 4.11 Breakdown of RMSE for Walnut Gulch Sites Using the Second- Order Polynomial Model with Observations of HH and VV Polar- izations (Simulated Annealing Method) . . . . . . . . . . . . . . 97 4.12 Breakdown of RMSE for Walnut Gulch Sites Using the Soil Mois- ture Profile Model in Eq. 4.6 . . . . . . . . . . . . . . . . . . . . 103 16 ABSTRACT This dissertation presents a detailed investigation of solving inverse-scattering problems using hybrid global and local optimization. The main goal in the inverse- scattering problem is to characterize the electrical and geometric scattering prop- erties of a target from the scattered fields. Inverse-scattering problems are usually modelled as an optimization problem in which we try to minimize the mismatch between the scattered fields collected from the target and our estimation. Al- though local optimization methods have substantially higher speed of convergence compared to global search methods, they cannot generally provide the desired so- lution of these highly nonlinear inverse problems. On the other hand, heuristic techniques have significantly better performance in obtaining the global solution with the cost of substantially higher amount of computation. To address these is- sues, the first part of this dissertation presents a global optimization scheme that is a novel combination of the simulated annealing method and the multi-directional search algorithm. Numerical results of optimizing some test functions is presented to develop and benchmark the proposed inversion scheme. In the next part of the thesis, the application of the presented inversion is inves- tigated for a microwave-imaging system to obtain the electrical properties of ob- jects. The proposed global optimizer significantly improves the performance and speed of the simulated annealing method by utilizing a nonlinear simplex search, 17 starting from an initial guess, and taking effective steps in obtaining the global solution of the minimization problem. Due to the efficient performance of the proposed global optimization method, we are able to obtain the shape, location, and material properties of the target without considering anyapriori information about them. The accuracy and applicability of the proposed imaging method is demonstrated with some numerical results in which two-dimensional images of multiple objects are successfully reconstructed. In the last part of this dissertation, retrieving the subsurface properties of lay- ered dielectric structures is investigated for two different models of these struc- tures. In the first one, the unknown variables are the modelling parameters of the structure, which are complex dielectric constants (permittivity and conductivity), thickness of layers, and statistical properties of the boundaries. Using some nu- merical experiments, successful retrieval results of this class of inverse problems are demonstrated. Thereafter, an inversion model for retrieving the root zone soil moisture profile using the P-band radar observations collected during the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) mission is pre- sented. We present numerical results to demonstrate significant improvements in the inversion results and computational speed compared to the previously used AirMOSS baseline inversion algorithm. The presented method is validated within situ soil moisture measurements. A physics-based soil moisture profile model de- rived from Richards’ equation is employed and its greater performance in RZSM retrieval compared to the previous model is demonstrated. 18 Chapter 1 Introduction Given a complete description of a physical system, we can predict the outcome of some measurements by using physical theories linking the parameters of the model to the parameters being measured. This problem of predicting the result of measurements is often called the forward problem. The inverse problem con- sists of using the actual result of some measurements to infer the values of the parameters that characterize the system. Inverse problems are difficult to solve because they do not usually have a unique solution (contrary to the forward prob- lem, which has a unique solution in deterministic physics) and obtaining the de- sired solution requires the exploration of a large parameter space. The solution to the inverse problem is useful because it generally provides information about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied topics in science. An electromag- netic inverse problem typically estimates the distribution of electrical properties of a region, such as permittivity and conductivity, based on the measured samples of the scattered field. 19 1.1 Background and Motivation Due to the non-contact penetration property, non-destructive, and non-ionizing nature of electromagnetic (EM) waves in the microwave regime, EM inverse prob- lems are of immense interest in a number of areas such as biomedical applications, remote sensing, nondestructive evaluation, and buried object detection. In the biomedical area, many imaging modalities have been used for screen- ing, diagnosis, and monitoring applications. These include X-ray imaging, ultra- sound, and magnetic resonance imaging. Microwave imaging is a newer alternate (or complement) to these established modalities, and is important because of its non-ionizing nature, specificity, and low-cost. Application of microwave imaging is being considered, for example, in breast cancer detection. The most significant action in fighting against breast cancer is perhaps to diagnose the shape and type of tumors at an early stage. There is, therefore, a growing and urgent need for early stage detection and treatments of this cancer. Many research studies have been focused on imaging techniques to find accurate, specific, and non-invasive screening methods[1, 2, 3]. Many efficient technologies are already utilized, such as X-ray mammography, ultrasound, and Magnetic Resonance Imaging (MRI). However, these imaging systems have roughly 70% overall accuracy in detecting breast cancer [2], which means either they fail to detect tumors (false negative) or diagnose a tumor in a healthy tissue (false positive), or both, a significant frac- tion of the time. In addition, X-ray mammography, which is the most commonly used breast cancer screening method, is not recommended to be used as a regular screening tool because of its harmful ionizing nature [4]. While ultrasound imag- ing is safe and provides millimeter-scale resolution, high rates of false positive and false negative diagnoses [5, 6] and severe image artifacts like shadowing, rever- 20 (a) (b) Figure 1.1: Using microwave imaging to improve the accuracy of tumour de- tection. (a) Microwave imaging system setup system in our lab. (b) De- tection of a cancer tumor, which is not visible in x-ray (image courtesy: https://newsnetwork.mayoclinic.org). beration, and speckles are the shortcomings of this imaging technique. Although breast MRI has the highest resolution among the three screening methods, it has a higher false-positive rate than mammography [7]. In addition, MRI is expensive, which makes it inaccessible to large portions of population. These limitations and concerns have motivated research in alternative biomedical imaging techniques, and in particular, microwave imaging. Based on the expected sensitivity (hence specificity), low cost, and non-ionizing nature of microwaves, there has been substantial interest in developing microwave imaging systems for breast cancer detection, which can help improve the accuracy of tumor detection (Fig. 1.1), leading potentially to better patient outcomes. Another application of inverse-scattering problems is to use electromagnetic waves in through-the-wall radar imaging to detect obstructed targets. The main 21 objective in this case is to obtain physical properties of a scene from the targets through-the-wall microwave scattering. Through-the-wall radar imaging can be employed in different areas, such as military and rescue security applications [8, 9]. One of the major challenges in this area is that available methods for the full problem are not fast enough. A recent review article, [10], summaries the different imaging schemes and highlights that the solving a full problem is needed to have better focused images. Furthermore, in the other class of inverse-scattering problems, determining the properties of layered rough surface structures from scattering data is currently of special interest in geoscience and remote sensing applications, because these structures are representative models for many naturally occurring structures, such as layered soil, rivers, and lakes. In the inverse problem associated with lay- ered dielectric structures with rough boundaries, we use radars to collect the scat- tered fields and retrieve their subsurface properties, such as complex dielectric constants (permittivity and conductivity), thickness of layers, and statistical prop- erties of the boundaries. Since, in these remote sensing problems, the incident and scattered fields are usually in the microwave regime, they fall into the cate- gory of microwave inverse problems. An important application of this category is mapping of the global soil moisture, which can improve our understanding of the global water, climate, and carbon cycles. Due to the importance of mapping global soil moisture, several missions such as the Soil Moisture and Ocean Salin- ity (SMOS) [11], NASA Soil Moisture Active Passive (SMAP) [12] mission, and Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) [13] have been developed to map regional and global distributions of soil moisture. Figure 1.2, which is the radar image obtained during the SMAP mission, presents 22 Figure 1.2: SMAP radar image acquired from data from March 31 to April 3, 2015. Weaker radar signals (blues) reflect low soil moisture or lack of vegeta- tion, such as in deserts. Strong radar signals (reds) are seen in forests. Credit: NASA/JPL-Caltech/GSFC. the map of backscattering coefficient for HH polarization on specific dates. Fig- ure 1.3 is another example of the actual radar data, which demonstrates the radar backscattering coefficient over the AirMOSS site in Arizona [14]. 1.2 Previous Works There has been a substantial work done in the electromagnetic inverse problems. Here, we present a short review of the previous works in this topic. This review is focused on two major categories of the inverse-scattering problems, which are the investigated in this dissertation. 23 Figure 1.3: The radar backscatter over the AirMOSS site in Arizona from October 23, 2012 (image reprinted from [14]). 1.2.1 Microwave Imaging of Dielectric Objects In the microwave imaging problem, the main goal is to characterize the electri- cal and geometric scattering properties of a target from the scattered fields. For this purpose, we usually model the inverse-scattering problem as an optimization problem in which we try to minimize the mismatch between the scattered fields collected from the target and our estimation. Based on the optimization tech- niques utilized to solve the inverse-scattering problems, they can be classified as two major groups, namely, local and global optimization methods. Due to the substantially higher speed of convergence in local optimization algorithms [15], they have been more popular in solving the inverse problems with large number of unknowns. Born iterative method, for example, is widely used as an effec- tive imaging method to obtain the conductivity and permittivity distribution of 2-D [16] and 3-D investigation domains [17], as is the distorted Born iterative method [18, 19]. In addition, Guass-Newton algorithm has been applied to the microwvae imaging problem for reconstructing 3-D conducting [20] and dielec- tric [21, 22] targets. Due to the nonlinearity of the inverse-scattering problem, the main disadvantage of utilizing local optimization algorithms is the convergence to 24 one of the multiple local minima of the cost function, which may not necessarily be the desired solution (global solution). To overcome this drawback in local op- timization algorithms, one solution is to perform a global search over the entire unknown domain of the inverse-scattering problem. Depending on the formation of inverse-scattering problem, appropriate global multi-agent optimization meth- ods, such as the genetic algorithm (GA) [23], particle swarm optimization (PSO) [24], and differntial evolution (DE) [25], single-agent techniques such as simu- lated annealing method [26, 27], and hybrid algorithms such as GA/CG method [28], memetic algorithm [29, 30], hybrid DE [31], hybrid GA [32], hybrid PSO [33], and the hybrid optimizer presented in this thesis [34, 35] have been applied to solve the inverse-scattering problem. While meta-heuristics, such as GA, PSO, and SA, have acceptable performance for small numbers of unknowns, they do not usually scale well with complexity. As a result, simplifications are often made to reduce the computational complexity of reaching the solution. In general, the the microwave imaging system is under-determined as the number of measurements are less than the number of unknowns. Therefore, the solution of the problem is not necessarily unique. Different regularization approaches have been utilized to obtain a meaningful and desirable solution by considering a constraint orapriori information in the problem. L2 norm based regularizations are commonly used to produce smooth solutions and blur the edges of the objects [36, 37, 38]. On the other hand, due to the immense popularity of the compressive sensing approaches [39], L1 norm based methods [40, 41] and total variation based approaches [42] have been applied to the microwave imaging problem as the regularization terms to improve the image reconstruction. 25 1.2.2 Inversion of Subsurface Properties of Layered Dielectric Structures In an inverse scattering problem associated with a layered dielectric structure with rough boundaries, the unknown variables are complex dielectric constants (per- mittivity and conductivity), thickness of layers, and statistical properties of the boundaries. There has been a substantial work in this category of electromagnetic inverse problems focusing on the estimation of the soil moisture, which can be translated as the complex dielectric constant of the soil based on its texture. Soil moisture is a key variable in the earth system and is subject of much research in environmental studies, including hydrology, agriculture, flood predic- tion, drought analysis, and weather forecasting. Soil moisture plays an important role in understanding the processes connecting the interface between the land sur- face and the atmosphere, as well as partitioning the elements of the water cycle such as the precipitation, surface runoff, and ground water storage [43, 44]. Mon- itoring the global soil moisture is therefore crucial in understanding the hydro- logical and ecological processes responsible for climate change [45, 11]. Several missions such as the Soil Moisture and Ocean Salinity (SMOS) [11], NASA Soil Moisture Active Passive (SMAP) [12] mission, and Airborne Microwave Obser- vatory of Subcanopy and Subsurface (AirMOSS) [13] have been developed to map regional and global distributions of soil moisture. The problem of retrieving soil moisture from the scattered electromagnetic waves falls within the broad category of inverse scattering problems. If the for- ward model, which estimates the scattered EM waves, has an invertible form the inversion process is straightforward [46]. Otherwise, the inverse-scattering prob- lem is modelled as an optimization problem for minimizing the mismatch be- 26 tween the scattered fields collected from the target and an estimation of those fields from a numerical model. To solve the optimization problem, several inver- sion algorithms, such as neural network techniques [47, 48], statistical inversion approaches[49, 50], and Bayesian estimation methods [51], have been developed. In addition, due to the nonlinearity of inverse-scattering problems and presence of multiple local minima, utilizing global optimizers, such as genetic algorithm [52] and simulated annealing method [14], have been preferred in some related works, despite their heavy computational load. 1.3 Objectives and Research Goals The primary research goal in this thesis is to develop an Inversion algorithm, which can not only improve the performance (accuracy and speed) of inverse- scattering problems and increase their applicability range, but also to enable in- version solutions in scenarios where existing methods are not able to generate any feasible solutions. To avoid the aforementioned drawbacks of using local optimizers for solving the inverse-scattering problems, the proposed optimiza- tion technique is a hybrid heuristic algorithm based on the simulated annealing method and the multi-directional search method. The main reason for using the combination of these methods is to achieve the best of both worlds: enhance the performance and speed of the simulated annealing method by utilizing a nonlin- ear simplex search to take effective steps in obtaining the global solution of the minimization problem. Once the method is developed and tested, we apply the inversion algorithm to two major classes of inverse-scattering problems, which are (1) the microwave 27 imaging problem and (2) the radar remote sensing problem of determining the Subsurface properties of layered rough surface structures from monostatic po- larimetric scattering data. For the microwave imaging problem, we are aiming for reconstructing investigation domains with multiple targets having different dielec- tric constants. For the remote sensing problem, the goal is to develop an inversion model for estimating subsurface properties of dielectric structures, such as com- plex dielectric constant, statistical and geometrical properties, and soil moisture profile, in a more accurate and faster way. Extensive sensitivity analyses will be performed and the inversion uncertainties will be investigated in each class of problems. 1.4 Thesis Overview This dissertation consists of five chapters. In Chapter 2, a general framework for solving the inverse-scattering problems will be presented. A global optimization method, which is a novel combination of the simulated annealing method and the multi-directional search algorithm will be proposed to solve the inverse scattering problems. To benchmark our method, we have used some test functions to in- vestigate the performance of the proposed heuristic and compare the optimization results with other methods. In Chapter 3 a microwave imaging method based on the proposed heuristic scheme will be presented for reconstructing 2-D investigation regions. We then provide some numerical results of reconstructing 2-D dielectric regions containing multiple targets with different dielectric constants. To investigate the uncertain- ties of the inversion results in the presence of the noise, an extensive sensitivity 28 analyses has been performed by adding different noise levels to the measurements. In Chapter 4, the presented optimization technique will be adapted to solve an important class of inverse-scattering problem, which is the remote sensing prob- lem for inversion of subsurface properties of layered dielectric structures such as soils, from polarimetric radar data. We will mostly focus on estimating the Root Zone Soil Moisture (RZSM) profile in which we consider second-order polynomi- als for modelling the soil moisture profile and parametrize these quadratic poly- nomials with soil moisture values at three different depths, which are the top, the middle, and the bottom of the investigation domain. A new physically-based soil moisture profile model will be employed and its effect on the retrieval results will be investigated. By providing some numerical results, we demonstrated that the proposed inversion method has significantly better computational efficiency than the standard simulated annealing method used previously. We validate the retrieval method using the measured radar data collected during the AirMOSS mission flights and in situ soil moisture measurements. Finally, the conclusions and future work of this dissertation are presented and discussed in the last chapter. 29 Chapter 2 Inverse-Scattering Problem 2.1 Theory and Background The general framework in solving the inverse-scattering problem in which we reconstruct the desired properties of the target, is to illuminate the investigation domain with electromagnetic waves from different directions and collect the scat- tered fields. Figure 2.1 demonstrates general configuration of this framework for two-dimensional (2-D) inverse-scattering problem. In this approach, we usually begin with an initial guess of the target’s desired characteristics and minimize the mismatch between the measured scattered fields of the object (measured data) and those of the initial guess (simulated data). As a consequence of this approach, one of the major mathematical challenges is to utilize an appropriate optimiza- tion algorithm to solve the inverse-scattering problem. It is worth mentioning that the inverse-scattering problem, which is usually formulated using a scattered field volume integral equation, is highly non-linear. Mathematically, we model the inverse-scattering problem as a nonlinear optimization problem in which we 30 minimize a cost function defined as C(x) = M X i=1 ke i (x)k 2 km i k 2 = M X i=1 kS i (x)m i k 2 km i k 2 ; (2.1) wherex denotes the vector of unknowns that model the distribution of the electri- cal properties of the investigation domain. The residual vectore(x) =S(x)m represents the mismatch between the simulated (or predicted) measured dataS(x) in the reconstructed domain and the measured datam in the original domain. We have normalized the mismatches in the measurements byk m i k 2 to have bal- anced contribution of each term in the cost function. The index i denotes the measurement index andM is the number of measurements. In the next section, we present the proposed global search algorithm to find the optimum solution of the cost function defined in (2.1). 2.2 Multi-Directional Search Based Simulated An- nealing To solve the optimization problem described in the previous section, a novel global search algorithm that takes advantage of the speed of local optimization while en- suring convergence to a global minimum will be proposed. More specifically, this is done by modifying the perturbation scheme in the simulated annealing method using a non-linear simplex search algorithm. This modification will enhance the performance of global search by choosing judicious perturbation directions. The conventional form of the simulated annealing method for optimization of func- tions with continuous variables, presented in [60], is based on a heuristic opti- 31 Figure 2.1: General configuration of 2-D inverse-scattering problem in which we illuminate the target and collect the scattered fields. 32 mization scheme with adaptive moves along coordinate directions in the domain of unknowns. The procedure starts with an initial solution (initial guess) of the unknownx, sets a parameter called temperatureT 0 , and follows by an iteratively decreasing series of temperature and random perturbations of the initial solution to reach the global optimum of the problem. According to the Corana algorithm [60], these random perturbations are obtained by sequentially moving the current solution point along the coordinate directions in the domain of variables. In other words, each perturbation is performed on one of the variables to obtain a new can- didate solutionx 0 . The candidate solutionx 0 will be accepted and replaced byx based on the Metropolis Criterion [61], that: if C(x) 0 then Accept the new candidate solutionx 0 . else Accept the new candidate solutionx 0 by the probabilitye C(x)=T k . end if whereT k is the temperature parameter at the current iterationk and C(x) de- notes C(x 0 )C(x). At each temperature, the process of perturbing currently accepted solution and replacing the perturbed solution will be repeated until the function values of the sequence solution points reach a stable value. At ‘high’ temperatures most of the solutions would be accepted as the criterion is usually satisfied, while at ‘low’ temperatures solutions that reduce the cost function would be accepted as the probability of accepting worse solutions significantly decreases. It has been shown that by reducing the temperature sufficiently slowly and keeping the number of perturbations at each temperature sufficiently high, the simulated 33 annealing algorithm always converges to a global minimum of the cost function [62]. A detailed discription of the simulated annealing method’s algorithm is pre- sented in Appendix A. While the simulated annealing method in [60] has good performance in the optimization problems with small number of variables, such as the inversion scheme presented in [26], increasing the number of unknowns extends the duration of algorithm’s convergence to a global solution significantly. Furthermore, as a result of the sequential perturbation scheme proposed in [60], convergence of the method to a satisfactory solution may not be achievable in a reasonable amount of time. Therefore, employing an efficient scheme to move the current solution along the directions in which we expect significant reduction in the cost function will enhance the performance of the algorithm in optimizing functions with large number of continuous variables, such as the cost function in an inverse-scattering problem. For this purpose, we combine the multi-directional search scheme proposed in [63] with the simulated annealing method to accelerate the convergence and im- prove the performance of this global search method. The multidirectional search algorithm is presented in Appendix B. We start with an initial guess of the un- known variables and produce a non-degenerate simplex--in which no lower di- mensional hyperspace can be found that contains all vertices of the simplex-- by randomly perturbing the current solution along coordinate directions of the un- knowns. Similar to the simulated annealing scheme in [60], we will generate new perturbed points based on the currently accepted solution, but instead of sequen- tially perturbing the accepted solution along the coordinates, we obtain the per- turbed solutions using the vertices of the evolving simplex in the multi-directional search algorithm. At the lth iteration of the multi-directional search algorithm, 34 for a given simplex with verticesfv l 0 ;v l 1 ;:::;v l n g, we update the solution points, which are the vertices of this simplex, by reflecting, expanding, and contracting the simplex based on the following scheme: Perform reflection step: v l+1 i =v l 0 +r (v l 0 v l i ) i = 1;:::;n - Calculatef(v l+1 i ). if minff(v l+1 i ); i = 1;:::;ng<f(v l 0 ) then Perform expansion step: v l e i =v l 0 + 2r (v l i v l 0 ) i = 1;:::;n - Calculatef(v l e i ). if min i ff(v l e i )g< min i ff(v l+1 i )g then Replacev l e i withv l e i fori = 1;:::;n. end if else Perform contraction step: v l+1 i =v l 0 +r=2 (v l 0 v l i ) i = 1;:::;n - Calculatef(v l+1 i ). end if As it is expressed in this scheme, instead of applying deterministic movements of the simplex vertices in the given directions of the reflection, expansion, and contraction steps, we generate random perturbations in these directions to develop our heuristic scheme ensuring that the process does not stop at a local minimum. Figure 2.2 is an example of evolving a 2-D simplex using multi-directional search in a non-deterministic manner. 35 Figure 2.2: Multi-directional search with non-deterministic reflection, expansion, and contraction steps. 36 Figure 2.3: Flowchart of the conventional form of the simulated annealing method using the sequential perturbation scheme. As illustrated in Figures 2.3 and 2.4, the main difference between simulated annealing and the proposed optimization algorithm is the perturbation scheme, in which we obtain more effective updates to the solution, manifested in an accel- erated rate of cost function reduction. A detailed description of the algorithm is described in the next section. The flow charts in Figures 2.3 and 2.4 illustrate the similarities and differences between the classical simulated annealing algorithm and the proposed hybrid method. 37 Figure 2.4: Flowchart of the multi-directional-search based simulated annealing. 38 2.3 Inversion Algorithm In this section, we present a formal statement of the algorithm to minimize the cost functionf(x), which is a function ofn variables. Initialization Step: - Set the initial temperature parameterT 0 and the rate of temperature reduction r T . - Set the parameters related to internal cycles of algorithmN t ,N s , andN mul . - Choose an initial-guess pointx 0 and initialize the size of step lengthss i for i = 1; 2;:::;n. - Evaluate the cost function atx 0 ,f 0 = f(x 0 ), and initialize the accepted and optimum points: x accept =x 0 ; f accept =f 0 x opt =x 0 ; f opt =f 0 Iterations: form = 1; 2;::: do - Check the stopping criteria of the algorithm. forj = 1;:::;N t do - Setn s i = 0; i = 1; 2;:::;n. fork = 1;:::;N s do - Setl=1. - Setv l 0 =x accept . 39 - Randomly generate the vertices < v l 0 ;:::;v 1 n > to provide a non- degenerate simplex using the following process: v 1 i =v 0 +rs i e i i = 1; 2;:::;n; where r is a random number generated in [0,1] ande i denotes the vector with a 1 in theith coordinate and 0’s elsewhere. forl = 1; 2;::: do - Check the stopping criteria of the random multi-directional search. Perform random reflection step: fori = 1;:::;n do v l+1 i =v l 0 +r (v l 0 v l i ), where r is a random number generated in [0,1]. - Calculatef(v l+1 i ). - Accept or reject the candidate pointv l+1 i according to the Metropo- lis Criterion. if l=1 then ifv l+1 i is accepted, then Add 1 ton s i end if end if end for if minff(v l+1 i ); i = 1;:::;ng<f(v l 0 ) then Perform random expansion step: fori = 1;:::;n do 40 v l e i =v l 0 + 2r (v l i v l 0 ), where r is a random number generated in [0,1]. - Calculatef(v l e i ). - Accept or reject the candidate pointv l e i according to the Metropo- lis Criterion. end for if min i ff(v l e i )g< min i ff(v l+1 i )g then Replacev l e i withv l e i fori = 1;:::;n. end if else Perform random contraction step: fori = 1;:::;n do v l+1 i =v l 0 +r=2 (v l 0 v l i ), where r is a random number generated in [0,1]. - Calculatef(v l+1 i ). - Accept or reject the candidate pointv l+1 i according to the Metropo- lis Criterion. end for end if end for 41 Update the step length (presented in [60]): s i = 8 > > > > > > < > > > > > > : s i 1 + 2 ns i Ns 0:6 0:4 ; ifn s i > 0:6N s s i 1 1+2 0:4 ns i Ns 0:4 ! ; ifn s i < 0:4N s s i ; otherwise end for end for - Reduce the temperature for the next iteration,T m+1 =r T T m - Reset the accepted point and its cost function by currently obtained opti- mum solution: x accept =x opt ; f accept =f opt end for The parameter N t denotes the number of step length adjustments at each it- eration. In addition, since we initialize the multi-directional search algorithm by randomly perturbing the current solution along coordinate directions of the unknowns (sequential perturbations), we employ the random multi-directional searchN s times to investigate the individual effect of each function variable sep- arately. This is done by tracing the first iteration of the multi-directional search, which only includes sequential perturbations of the variables. After obtaining the ratio of the accepted solutions to the total perturbations of each variable, we utilize the step-length adjustment approach similar to the one provided in [60]. We consider two stopping criteria for the random multi-directional search al- gorithm. As the first stopping criterion, we consider the following test proposed 42 by Woods[64] atkth iteration 1 max 1in kv k i v k 0 k<; (2.2) where = max(1;kv k 0 k), is a preset tolerance, andkk indicates the second norm. In addition to the stopping test shown in Eq. 2.2, which measures the size of the simplex and how far it can move in the next iteration, we consider another stopping criterion, namely one that restricts the maximum number of iterations for this random search algorithm. 2.4 Benchmark Function Tests In this section, we have used some test functions to investigate the performance of the proposed heuristic scheme and compare the optimization results with the standard simulated annealing method. For this purpose, we consider 2 and 4 di- mensional Rosenbrock test functions [65], which are f 2 (x 1 ;x 2 ) = 100(x 2 x 2 1 ) 2 + (1x 1 ) 2 ; f 4 (x 1 ;x 2 ;x 3 ;x 4 ) = 3 X i=1 100(x i+1 x 2 i ) 2 + (1x i ) 2 : (2.3) The global minimum of then-dimensional Rosenbrock function is atx n = [1; ; 1], which is located inside a long, narrow, parabolic shaped flat valley, which makes it difficult to be found. Figure 2.5 illustrates the two dimensional Rosenbrock test function. To provide a fair comparison with the standard SA, we have used same algorithm parameters suggested in [60]. Therefore, we set the initial temperature 43 Figure 2.5: Two dimensional Rosenbrock test function. toT 0 = 10 and apply the exponential cooling schedule ofT k = (0:85) k T 0 for the kth iteration. In addition,N t is chosen to be 10 and 20 for the 2 and 4 dimensional test functions, respectively, andN s is chosen to be 20 for both cases. Moreover, we restrict the maximum number of iterations in multi-directional search algorithm, N mul , to be 100 and set the parameter = 0:01 in Eq. 2.2. Table 2.1 provides the optimization results of test functions using the proposed optimizer and simu- lated annealing method. The optimization process is repeated for different starting points. As provided in Table 2.1, the number of function evaluation is significantly lower using our optimization method, which demonstrates its advantage over the simulated annealing method. 44 Table 2.1: A comparison between the proposed optimization scheme and simu- lated annealing method on Rosenbrock functions in 2 and 4 dimensions. Simulated annealing Proposed Optimizer Starting point Final function value Number of function evaluations Final function value Number of function evaluations 2-D 1001,1001 1.8E-10 500001 1.5E-11 31080 1001,-999 2.6E-9 508001 2.5E-12 66226 1443,1 1.5E-8 492001 7.5E-9 90828 4-D 101,101, 101,101 5E-5 1288001 1.7E-7 256662 201,0, 0,0 7.5E-7 1288001 2.2E-7 89454 -99,-99, -99,-99 3.3E-7 1304001 2.1E-7 358205 2.5 Effect of Algorithm Parameters on Optimiza- tion Results In this section, we investigate dependence of the proposed heuristic on two algo- rithm parametersN t andN s . For this purpose, we apply our optimization method to find the global solution of 2-dimensional Rosenbrock function for different val- ues ofN t andN s . Figure 2.6 shows the Root-mean-square error (RMSE) in the obtained solution after 10 iterations for each of the N t and N s values changing from 1 to 30. The initial temperature and cooling schedule are same as in the previous experiment and the number of iterations for the multi-directional search algorithm is fixed at 20. Since we are dealing with a heuristic method with non- deterministic steps, we have repeated the optimization process 10 different times, 45 Figure 2.6: Root-mean-square error (RMSE) in the optimization of 2 dimensional Rosenbrock function for different values ofN t andN s . presenting here the average RMSE for each set ofN t andN s . As evident in Fig. 2.6, forN t andN s values larger than or equal to 10, the er- ror in the obtained solutions drops significantly and the algorithm converges to the global solution. Although these results can provide reasonable ranges for choos- ingN t andN s values and obtaining the global solution of this problem, applying the presented heuristic to other optimization problems may lead to other feasible ranges ofN t andN s values. Therefore, we choose these parameters empirically for each specific problem. 46 2.6 Conclusion and Summary In this chapter, we presented an inversion scheme using a multi-directional ran- dom search based on the simulated annealing method to solve inverse scattering problems. As we perform random searches in multiple effective directions instead of sequential perturbations in the standard simulated annealing method, the pro- posed optimization approach works more successfully compared to the classical simulated annealing. We provided some numerical results of optimizing some benchmark functions and demonstrated its effective performance. We also dis- cussed the effect of algorithm parameters on the performance of the optimization algorithm. In the next two chapters, the presented optimization scheme is employed to solve the underlying inverse-scattering problem of the microwave imaging sys- tems and inversion of subsurface properties of layered structures. Although both of these problems fall into the broad category of inverse scattering problems, the corresponding forward model and the number of unknowns of these problems are quite different, and therefore, their inversion schemes are entirely distinct in that sense. 47 Chapter 3 Microwave Imaging of Dielectric Regions Microwave imaging is based on retrieving dielectric properties of materials, such as their permittivity and conductivity. Over the last few decades, microwave imag- ing has attracted increasing interests in biomedical applications (in particular, for breast cancer screening and therapy monitoring), mainly due to the significant di- electric property contrast between normal and malignant breast tissue [69, 70]. Compared to other conventional imaging modalities, such as X-ray mammogra- phy, microwave imaging is attractive in several important aspects, namely its non- ionizing and non-compressive nature, in addition to relatively low-cost associated with the hardware system. 48 3.1 Background and Theory In the microwave imaging, the scattering problem is governed by the following electric field volume integral equation, expressed for a heterogeneous, isotropic, non-magnetic medium in a regionD, with measurement points on a surfaceS that enclosesD, as [68], E s (r) =k 2 b Z D G(r;r 0 )(r 0 )E(r 0 )dv 0 ; r2s;r 0 2D (3.1) whereE s (r) = E(r)E i (r) is the scattered electric field in terms of the total field, E(r), and the incident field, E i (r), G(r;r 0 ) is the Green’s function, k b is the wavenumber of the background medium with lossless permittivity, b is the lossless relative permittivity of a background medium,(r 0 ) = [ r (r 0 )= b 1] j[((r 0 ) b )=( b 0 !)] is the dielectric contrast in terms of the permittivity and conductivity contrast, where the subscriptr denotes the relative permittivity, and b is the background conductivity. The imaging problem deals with determining the dielectric contrast,(r),r2 D, of an unknown medium, given some observations of the scattered field,E s (r), r2S. Due to the presence of the electric field within the integral, which itself is a function of the dielectric contrast, Eq. (3.1) is nonlinear and generally, the inverse problem is ill-posed. If the contrast of a medium is known, the electric field is estimated using a standard field estimation method such as Finite Difference Time Domain (FDTD), Method of Moments (MOM), Finite Element Method (FEM) and their variants. 49 3.2 Numerical Experiments We performed several numerical simulations to validate the proposed heuristic in the inverse scattering problems. In the first numerical experiment, we retrieve the permittivity distribution of the investigation domainD shown in Fig. 3.1 (a), which is a region discretized by=7=7 pixels, where is the free space wavelength, and clearly, the number of pixels is equal to the number of unknown variables in all experiments. To illuminate the target and collect the scattered fields, we have 7 transmitting/receiving antennas that are symmetrically located on a circle of 4 diameter, which provides a total number of 49 measurements. For solving the forward electromagnetic scattering problem, we have utilized a two-dimensional method of moment-based formulation [66, 67]. Fig. 3.1 (a) illustrates the investigation domainD in which we have 2 different objects with permittivities of 2 and 7. We set the initial temperature toT 0 = 10 and apply the exponential cooling schedule ofT k = (0:85) k T 0 for thekth itera- tion for all of the simulations. In addition, the values of N t and N s parameters are chosen to be 10 and 2, respectively. In addition, we restrict the maximum number of iterations in multi-directional search algorithm, N mul , to be 100 and set the parameter = 0:01 in Eq. 2.2. We have chosen these values so that the number of function evaluations at each iteration of the proposed algorithm would be comparable to the classical simulated annealing method. Moreover, based on the empirical observations, the presented values for the algorithm parameters pro- vide the most efficient performance of the proposed method. As it is evident in Fig. 3.1 (b)-(c), for noise-free measurements, the permittivity distribution of the investigation domain is reconstructed nearly perfectly and the cost function converges to zero, which are the indications of reaching the global solution. To 50 (a) (b) (c) Figure 3.1: (a) Permittivity distribution of the investigation domain (b) Recon- struction of the target for noise-free measurements and (c) the corresponding cost function. investigate the noise effect, white Gaussian noise was then added to the measured data to get 10 dB, 20 dB, and 30 dB signal-to-noise ratio in the measurements. Figure 3.2 illustrates the imaging results obtained when the white Gaussian noise contaminates the measurements. While we cannot perfectly reconstruct the per- mittivity distribution in the presence of noise in the measurements, the retrieved image provides the precise information about the shape and location of the objects in addition of obtaining the permittivity values close to the original distribution. Figures 3.3 (a)-(b) demonstrate the reconstruction results of the same exper- 51 (a) (b) (c) (d) (e) (f) Figure 3.2: (a) Reconstruction of the target in the presence of noise (30 dB signal-to-noise ratio) and (b) the corresponding cost function. (c) Reconstruction of the target in the presence of noise (20 dB signal-to-noise ratio) and (d) the corresponding cost function. (e) Reconstruction of the target in the presence of noise (10 dB signal-to-noise ratio) and (f) the corresponding cost function. 52 (a) (b) Figure 3.3: (a) Reconstruction of the target for noise-free measurements by using the standard simulated annealing method and (b) the corresponding cost function. iment using the standard simulated annealing method. In addition, the values of N t and N s parameters are chosen to be 10 and 20, respectively. The simulated annealing method has reconstructed the investigation domain in 65 iterations and 2258 seconds, while the proposed global optimization method has reached the same solution with the same accuracy in five iterations and 552 seconds, which illustrates the effectiveness of the presented method in searching for the global solution. In the second experiment, we present the results of reconstructing the permit- tivity distribution of a more complicated target. As shown in Fig. 3.4 (a), we have 3 different objects with the permittivity distribution range between 2 to 7. The configuration of antennas, the size of investigation domain, the values of the al- gorithm parameters (N t ,N s , andN mul ), and the resolution of the pixels are same as the first experiment. As shown in Fig. 3.4 (b)-(c), for noise-free measurements, the cost function converges to zero and the permittivity distribution of the investi- 53 (a) (b) (c) Figure 3.4: (a) Permittivity distribution of the investigation domain (b) Recon- struction of the target for noise-free measurements and (c) the corresponding cost function. gation domain is perfectly reconstructed. Figure 3.2 illustrates the imaging results in the presence of noise in the measurements. Although we do not have an ex- act reconstruction of the permittivity distribution, it is observed that the retrieved image provides the shape and location of the target precisely in addition of the permittivity values close to the original distribution. In the third simulation, we present the results of reconstructing the permittivity distribution of the region shown in Fig. 3.6 (a). The resolution of the image is 54 (a) (b) (c) (d) (e) (f) Figure 3.5: (a) Reconstruction of the target in the presence of noise (30 dB signal- to-noise ratio) and (b) the corresponding cost function. (c) Reconstruction of the target in the presence of noise (20 dB signal-to-noise ratio) and (d) the correspond- ing cost function. (e) Reconstruction of the target in the presence of noise (10 dB signal-to-noise ratio) and (f) the corresponding cost function. 55 increased by using =10=10 pixels and 10 transmitting/receiving antennas have been utilized to obtain the measurements. In addition, we choose the values ofN t andN s parameters to be 20 and 2, respectively, and restrict the maximum number of iterations in multi-directional search algorithm, N mul , to be 200. As shown in Fig. 3.6 (b)-(c), for noise-free measurements, the cost function converges to zero and the permittivity distribution of the investigation domain is perfectly reconstructed. Figure 3.2 demonstrates the impact of adding white Gaussian noise to the measured data. In the last numerical experiment, we try to reconstruct the permittivity distri- bution of a investigation domain shown in Fig. 3.8 (a) in which we have two objects with different permittivity values. The resolution of the image is increased by using=15=15 pixels. In addition, we set the parameters of the algorithm byN t = 30,N s = 15, andN mul = 200. To collect the scattered fields, we have assumed 15 transmitting/receiving antennas to keep the number of unknowns and measurements equal. Although we present the results of reconstructing the do- main by using 15 transmitting/receiving antennas, we obtain quite similar and ac- ceptable reconstruction results by reducing the number of transmitting/receiving antennas to 7. Figures 3.8 (b)-(c) demonstrate the reconstruction results for noise-free mea- surements, in which the algorithm successfully converges to the global optimum solution. The solution correctly reflects the shape and the location of the target. The existing error in the results is due to the significant reduction in the perfor- mance of the simulated annealing method in obtaining new candidate solution points in low temperatures. Figure 3.2 illustrates the sensitivity of this method to noise, where we add white Gaussian noise to the measurements. As a conse- 56 (a) (b) (c) Figure 3.6: (a) Permittivity distribution of the investigation domain (b) Recon- struction of the target for noise-free measurements and (c) the corresponding cost function. 57 (a) (b) (c) (d) (e) (f) Figure 3.7: (a) Reconstruction of the target in the presence of noise (30 dB signal- to-noise ratio) and (b) the corresponding cost function. (c) Reconstruction of the target in the presence of noise (20 dB signal-to-noise ratio) and (d) the correspond- ing cost function. (e) Reconstruction of the target in the presence of noise (10 dB signal-to-noise ratio) and (f) the corresponding cost function. 58 quence of adding noise to the measurements, the accuracy of the reconstructed contrast decreases, yet it provides information about the shape and location of the objects in the investigation domain. In the last three simulations, applying the standard simulated annealing method does not lead to a successful reconstruction of the investigation domain. The main reason for this is inability of the sequential perturbation scheme to generate can- didate solutions close to the global solution. As a consequence, the searching domain for finding new candidate solutions is not large enough to encompass the global optimum solution. Therefore, by reducing the temperature, the algorithm is forced to get trapped in a solution that is not necessarily the desired solution of the inverse problem. To sum up the presented inversion results, Table 3.1 shows the CPU-time needed for the convergence of each simulation and the root-mean-square error (RMSE) between the reconstructed permittivity domain and the original target us- ing pixel-by-pixel values of the regions. As evident in Table 3.1, by increasing the resolution of the image, the RMSE of the reconstructed images would be higher compared to the low resolution images. In addition, by increasing the noise level in the measurements, the error in the pixels of reconstructed domain will increase accordingly. 3.3 Complexity of the Algorithm Here we briefly investigate the performance of the proposed method with respect to the number of unknown variables using a numerical experiment. For this pur- pose, we consider the same configuration of antennas, investigation domain ( 59 (a) (b) (c) Figure 3.8: (a) Permittivity distribution of the investigation domain (b) Recon- struction of the target for noise-free measurements and (c) the corresponding cost function. Table 3.1: Root-mean-square error (RMSE) and CPU-time (seconds) of the re- constructed images in the numerical experiments. Noise free 30 dB 20 dB 10 dB CPU-time Root-mean-square error Experiment 1 552 0.014 0.068 0.082 0.088 Experiment 2 986 0.025 0.076 0.294 2.135 Experiment 3 34239 0.021 0.094 0.183 0.396 Experiment 4 180510 0.259 0.489 0.608 0.636 60 (a) (b) (c) (d) (e) (f) Figure 3.9: (a) Reconstruction of the target in the presence of noise (30 dB signal- to-noise ratio) and (b) the corresponding cost function. (c) Reconstruction of the target in the presence of noise (20 dB signal-to-noise ratio) and (d) the correspond- ing cost function. (e) Reconstruction of the target in the presence of noise (10 dB signal-to-noise ratio) and (f) the corresponding cost function. 61 Figure 3.10: Required number of function evaluations for the convergence of the proposed optimization algorithm for different number of function variables. region discretized by=7=7 pixels), and parameters for the global optimizer as the first experiment. However, we assume that only some of the pixels in the investigation domain are unknown variables of the optimization problem. Our goal is to obtain the number of function evaluations required to solve the inverse problem for different number of variables. Figure 3.10 demonstrates the number of function evaluations required for con- vergence of the algorithm for different number of unknown variables. It is ob- served that we have an almost linear increase as the number of variables in the optimization problem is raised. Since the proposed method is non-deterministic, we have repeated each case for 100 times and taken their average in Figure 3.10. 62 3.4 Conclusion and Summary In this chapter, we presented an inverse model for reconstructing permittivity dis- tribution of the dielectric investigation domains. By combining the global and local optimization methods, we are able to reconstruct investigation domains with multiple objects with different permittivity values. The numerical results verify the capability of this method in obtaining the permittivity distribution without con- sidering any specific constraint on the initial guess or a priori information about the size, location, or permittivity values of the target. Although a higher resolu- tion (via reducing the pixel size) will result in slower convergence, applying the proposed imaging technique leads to the successful reconstruction of investigation domain. The proposed method can apply equally well to other imaging modalities that use inverse scattering, such as acoustic/ultrasound imaging. The difference is the employed forward model in these imaging techniques, which provide different types of measurements from the target. 63 Chapter 4 Radar Remote Sensing Retrieval of Subsurface Properties of Layered Dielectric Structures In this chapter, we investigate the problem of obtaining the subsurface properties of layered dielectric structures with rough surfaces from remotely sensed radar scattering data, which arises in many areas of science and engineering. Since these structures are representative models for many naturally occurring structures, such as layered soil, rivers, and lakes, determining the properties of layered rough surface structures from scattering data is currently of special interest in geoscience and remote sensing applications. In an inverse scattering problem associated with a layered dielectric structure with rough boundaries, the unknown variables are complex dielectric constants (permittivity and conductivity), thickness of layers, and statistical properties of the boundaries. In this chapter, we will first introduce the forward model that we use for calculating the scattered fields from dielectric 64 (a) (b) Figure 4.1: Examples of geometries of the problems for retrieving the subsur- face properties of dielectric structures. (a) The structure could contain multiple rough interfaces and the complex dielectric constant is continuous for each layer. (b) The structure has rough upper surface and smooth interfaces below with a continuous depth-varying depth profile. layered structures. We then consider two configurations for the inverse problem. In the first one, the unknown variables are the modelling parameters of the struc- ture, such as the permittivity, roughness and thickness of layers. In the second configuration, we will present an inversion scheme for retrieving soil moisture profiles from radar data. 4.1 Forward Model To solve the inverse problem of retrieving the subsurface properties of a layered structure, we need to employ a proper forward model for estimating the scattered electromagnetic waves during the inversion process. For this purpose, we have employed the forward model in [53], which uses the small perturbation method to calculate the scattered electromagnetic fields from multi-layer rough surfaces such as the ones shown in Fig. 4.1. The radar observations are the bistatic scattering 65 coefficients of the layered structures. It can be shown that the bistatic scattering coefficients for layered structures can be calculated as o pq = 4k 2 0 v cos 2 s 2 j f1 pq (k s ? )j 2 W f1 (k s ? k i ? ) +j f2 pq (k s ? )j 2 W f2 (k s ? k i ? ) ; (4.1) where i and s are incident and scattering angles, respectively, and vectors k i ? andk s ? denote their corresponding directions. The quantitiesW f1 (k s ? k i ? ) and W f2 (k s ? k i ? ) are the spectral densities of the rough surfaces, which are assumed to be independent. The reader is referred to [53] for details of calculating the co- efficients f1 pq and f2 pq , where p;q2fh;vg. As is illustrated in Figure 4.1, the rough surface boundaries are assumed to be zero-mean stationary random pro- cesses with known statistical properties denoted byf(x;y). The structure could contain multiple rough interfaces (Figure 4.1(a)) or a continuous depth-varying profile (Figure 4.1(b)). In the latter case, which is shown in Fig. 4.1(b), the profile is modelled as a finely stratified medium with a rough upper surface and smooth interfaces below. 4.2 Retrieving the Model Parameters of a Two-Layer Dielectric Structure 4.2.1 Background and Theory In this part, we investigate the problem of estimating the model parameters of layered dielectric structures using backscattering coefficients for different polar- izations, observation angles, and frequencies. As shown in Fig. 4.2, we have a 66 two-layer dielectric structure with infinite random rough surfaces. Conductivities and dielectric constants of the layers are denoted by ( 0 , 0 ), ( 1 , 1 ), and ( 2 , 2 ), the permeability of all of the layers is assumed to be 0 , and the boundaries are denoted byf 1 (x;y) andd +f 2 (x;y), which are assumed to be zero-mean sta- tionary random processes with known statistical properties. The top and bottom layers are half-spaces regions. To characterize the layered structure demonstrated in Fig. 4.2, we consider nine parameters, which are the permittivity and conductivity of the first layer ( 1 and 1 ), the permittivity and conductivity of the second layer ( 2 and 2 ), the standard deviation and correlation length of each interface ( f 1 ,l f 1 , f 2 , andl f 2 ), and the mean separation between the two rough interfaces referred to as layer thickness (d). In this problem, we consider multiple measurements at different frequencies and observation angles, so we can rewrite Eq. 2.1 as C(x) = n f X i=1 n X j=1 kS hh ij (x)m hh ij k 2 km hh ij k 2 + kS vv ij (x)m vv ij k 2 km vv ij k 2 ; (4.2) where n f is the number of frequencies and n is the number of observation an- gles at each frequency to collect the scattered fields. x denotes the vector of unknowns that model the layered structure in Figure 4.2. The residual vector e(x) =S(x)m represents the mismatch between the simulated (or predicted) measured dataS(x) of the reconstructed model and the measured datam of the target. In this problem, our measurements are the backscattering coefficient of the layered structure at different frequencies and different observation angles and the superscriptshh andvv show their polarizations. We have employed the forward model presented in [53] to obtain the scattered field measurements from the rough 67 Figure 4.2: A 3-D two-layer isotropic dielectric structure with complex permit- tivities. The boundaries are zero-mean stationary random processes. The layers mean separation is denoted asd. surface structure shown in Figure 4.2. 4.2.2 Numerical Experiments Here we provide some numerical results to demonstrate the performance of the proposed algorithm in determining the properties of layered rough surface struc- tures from scattering data. In this section, we consider two different cases for the simulations with the model parameters presented in Table 4.1. The measure- ment angles are 1 = 30 o and 2 = 45 o at the frequencies f 2 = 435 MHz and f 2 = 1:2 GHz. The values ofN t andN s parameters are chosen to be 10 and 20, respectively. In addition, we restrict the maximum number of iterations in multi- 68 Table 4.1: Actual values of the model parameters in the cases used for numerical simulations. All of the unites are in SI. d 1 1 2 2 l f1 f1 l f2 f2 Case 1 0.4 4 0.02 12 0.1 0.1 0.02 0.2 0.01 Case 2 0.2 5 0.05 10 0.1 0.1 0.02 0.2 0.01 Table 4.2: Constraints on the unknown variables in the numerical simulations. All of the unites are in SI. d 1 1 2 2 Lower Bound 0 0 0 0 0 Upper Bound 1.0 15 0.2 15 0.2 directional search algorithm, N mul , to be 50 and set the parameter = 0:001 in Eq. 2.2. In these experiments, we consider a five-parameter problem, where only 1 , 1 , 2 , 2 , andd are unknown variables of the optimization problem, as we are usually most interested in these parameters. The lower and upper bounds considered for each variable is presented in Table 4.2. In the first experiment, the true values are 1 = 4, 1 = 2 10 2 , 1 = 12, 1 = 1 10 1 , d = 0:4, l f 1 = 0:5l f 2 = 0:1, and f 1 = 2 f 2 = 0:02 (same as Case 1 in [26]). As evident in Figure 4.3 the cost function reduces significantly in the first iterations and converges to zero in 10 iterations as we reach a nearly exact global solution of the problem. As shown in Table 4.3, the true values of the layered structure are successfully retrieved and the number of forward model evaluations are 856060. Figure 4.4 demonstrates how unknown variables evolve during the iterations. As evident in this figure, the thickness, permittivity, and conductivity of the first layer are retrieved in the first iteration, which shows that the variations in these variables significantly affect the cost function. For the second experiment, we changed the the true values to 1 = 5, 1 = 69 Figure 4.3: The value of the cost function for determining the properties of the layered structure in the first experiment. Table 4.3: Actual values, initial guesses, and retrieved values of the first numerical experiment. All of the unites are in SI. Experiment 1 Actual Value Initial Guess Retrieved Value 1 4 11.25 4 1 0.02 0.15 0.02 2 12 11.25 12 2 0.1 0.15 0.1 d 0.4 0.75 0.4 70 Figure 4.4: Changes of the model parameters during the iterations of the first experiment. 71 Figure 4.5: The value of the cost function for determining the properties of the layered structure in the second experiment. 5 10 2 , 1 = 10, 1 = 1 10 1 , d = 0:2, l f 1 = 0:5l f 2 = 0:1, and f 1 = 2 f 2 = 0:02 (same as Case 2 in [26]). Similar to the previous experiment, the true values of the layered structure are successfully retrieved as shown in Table 4.4. Moreover, Figure 4.5 demonstrates that the cost function converges to zero in 12 iterations as we reach the global solution of the problem. The number of forward model evaluations in this case is 1120436 as it takes more iterations for the inversion algorithm to converge to the solution. The variations in the model parameters during the iterations are presented in Figure 4.6. In this experiment, it takes a few more iterations to retrieve each unknown variable comparing to the first experiment. 72 Table 4.4: Actual values, initial guesses, and retrieved values of the second nu- merical experiment. All of the unites are in SI. Experiment 2 Actual Value Initial Guess Retrieved Value 1 5 11.25 5 1 0.05 0.15 0.05 2 10 11.25 10 2 0.1 0.15 0.1 d 0.2 0.75 0.2 Figure 4.6: Changes of the model parameters during the iterations of the second experiment. 73 4.2.2.1 Variation of the Cost Function With Respect to Unknown Variables In this part, we briefly investigate how the cost function changes with respect to the variations of each unknown variable in the optimization problem. For this purpose, we consider the cases presented in Table 4.1 for the model parameters and change each parameter between its lower and upper bounds while the other parameters remain the same. Figures 4.7-4.8 demonstrate the variation of the cost function to unknown variables for both cases. As evident in these figures, there is a wide range of oscillations in the cost function as we change the thickness of the first layer. In addition, the permittivity and conductivity of the first layer have more impact on the cost function than the ones of the second layer, which is intuitively perceivable as the lower layers usually have less impact on the scattered fields. Moreover, the cost function does not have drastic changes with respect to the permittivity of the second layer, which makes it a challenging parameter to be obtained during the inversion process. 4.2.2.2 Sensitivity Analysis Here we investigate the effect of noise in retrieval results. Therefore, we repeat both experiments by adding noise to the backscattering coefficients using the fol- lowing model: n pq = pq +r; (4.3) where n pq and pq represent the measured and noise-free radar signal (in deci- bels), respectively, andr is a random number with the uniform distribution, which models the noise in measurements. For each of the cases presented in Table 4.1, we have applied the inversion 74 Figure 4.7: Cost function variation with respect to each of the model parameters of Case 1. 75 Figure 4.8: Cost function variation with respect to each of the model parameters of Case 2. 76 Table 4.5: Average error of estimating the model parameters for 100 realizations of the perturbed data modeled by Eq. 4.3 with +/- 0:25 dB bounds All of the unites are in SI. d 1 1 2 2 Case 1 0.069 0.276 0.022 3.318 0.091 Case 2 0.021 0.126 0.042 1.041 0.092 Table 4.6: Average error of estimating the model parameters for 100 realizations of the perturbed data modeled by Eq. 4.3 with +/- 0:50 dB bounds. All of the unites are in SI. d 1 1 2 2 Case 1 0.103 0.362 0.027 3.756 0.096 Case 2 0.064 0.262 0.070 2.116 0.090 algorithm to 100 realizations of the perturbed data modeled with 4.3. We have performed this experiment for two different noise distribution bounds of +/- 0:25 dB and +/- 0:50 dB. Figures 4.9-4.10 and 4.11-4.12 show the distribution of rel- ative error between the actual and retrieved values of the model parameters of both cases for +/- 0:25 dB and +/- 0:50 dB noise distributions, respectively. In the presence of noise in the measurements, the error in estimating the thickness, per- mittivity, and conductivity of the first layer is significantly small for both cases in comparison to the permittivity and conductivity of the second layer. Table 4.5-4.6 summarize the results of estimating model parameters in the presence of noise in measurements. The values reported in these tables are the average of 100 realiza- tions for each case. 77 Figure 4.9: Relative error distribution between the actual and retrieved values of the model parameters of Case 1. We applied a uniform noise distribution with +/- 0:25 dB bounds. 78 Figure 4.10: Relative error distribution between the actual and retrieved values of the model parameters of Case 2. We applied a uniform noise distribution with +/- 0:25 dB bounds. 79 Figure 4.11: Relative error distribution between the actual and retrieved values of the model parameters of Case 1. We applied a uniform noise distribution with +/- 0:50 dB bounds. 80 Figure 4.12: Relative error distribution between the actual and retrieved values of the model parameters of Case 2. We applied a uniform noise distribution with +/- 0:50 dB bounds. 81 4.3 Retrieving Subsurface Soil Moisture Profiles 4.3.1 Introduction and Background In this section, we present an inversion scheme for estimating root zone soil mois- ture (RZSM). The operating frequency in this work is in P-band frequency range (0:42-0:44 GHz), which allows the electromagnetic waves to penetrate the soil profile down to the root zone, which has been shown to predominantly reside in the 30-100 cm range below the ground surface [14]. The P-band frequency choice provides up to a meter of sensing depth, which is the depth at which a sensing system is no longer sensitive to variations of soil moisture. This is determined based on the system noise floor, signal calibration accuracy, and the physical variation of radar scattering cross section as a function of the variable soil moisture profile [55]. For the sites considered in the AirMOSS mission, the sensing depth ranged from 30 cm to about a meter [14]. The forward model used to develop and test the new inversion algorithm is the multi-layered small perturbation method of [53] , which is used for calculating the scattering cross sections for bare surfaces. Vegetated- surface models [76] are not considered in this paper, as our purpose in this work is to benchmark the proposed inversion method with the previously used Air- MOSS baseline inversion algorithm. We then present some numerical results and demonstrate that the new inversion algorithm provides more accurate results in a significantly shorter time compared to the standard simulated annealing method, which is utilized in the current AirMOSS retrieval algorithm. The inversion algo- rithm is validated by retrieving the surface to root-zone profiles of soil moisture using data collected by the P-band synthetic aperture radar during the AirMOSS 82 mission. Following the baseline AirMOSS retrieval algorithm [14], we consider a second-order polynomial form as a function of subsurface depth for modelling the soil moisture profile. The quadratic model is parametrized with three soil mois- ture values at different depths as its unknowns instead of the three coefficients of the polynomial. Furthermore, we employ a physically-based soil moisture pro- file model derived from Richards’ equation [54] and demonstrate its improved performance in RZSM retrieval compared to the previous quadratic model. 4.3.2 Inversion scheme The framework for inversion model is similar to the previous problem. It con- sists of illuminating the region of interest with electromagnetic waves from one direction, measuring the scattered fields from the same direction (backscattering direction), comparing the measured data with an estimation of the scattered field obtained from forward model simulations, and finally, minimizing the mismatch between the measured and estimated data by properly updating the unknown vari- ables. Here, the cost function is defined as C(x) = k hh (x)m hh k 2 km hh k 2 + k vv (x)m vv k 2 km vv k 2 ; (4.4) where pq (x) andm pq are the estimated and measured backscattering coefficients forpq polarization (pq2fhh;vv;hvg), respectively. However, We only consider HH and VV polarizations in this work, because of unknown calibration uncertain- ties in the HV channel of the AirMOSS system [77]. The vector x consists of the unknowns of the optimization problem, which model the subsurface proper- ties of the dielectric structure. Figure 4.13 illustrates the flowchart of the multi- 83 Figure 4.13: Flowchart of the multi-directional-search based simulated annealing adapted for the inversion of soil moisture profile. directional-search based simulated annealing method adapted for the inversion of soil moisture profile. The reader is referred to [34] for the details of this hybrid optimization technique. Fig. 4.14 illustrates the configuration of this inverse-scattering problem with a rough upper surface and smooth interfaces below, where the rough surface bound- ary is assumed to be zero-mean stationary random process with known statistical properties denoted by f(x;y). As mentioned earlier, the number of radar ob- servations available in this work is limited to two backscattering coefficients for HH and VV polarizations. As a result, the number of free parameters for math- 84 ematical modelling of the soil moisture profile should not significantly exceed the number of observations. In this work, we model the subsurface soil moisture profiles with two different models. In the next two sections that we test and vali- date our retrieval model, we assume that the unknown soil moisture profile has a second-order polynomial form, which is shown in [14, 56] to be a reasonable fit for several cases of the soil moisture profiles in AirMOSS sites. Then, we employ a new model, which is a closed-form analytical solution to Richards’ equation, containing three free parameters to enhance the retrieval results. To parametrize each of these two models, we consider the soil moisture values at three different subsurface depths, which are the top, the middle, and the bottom of the investiga- tion domain as the modelling parameters and the elements of vectorx. To convert between the dielectric constant and soil moisture values, we use the soil mixture model presented by Peplinskietal. [57] to calculate the dielectric constant profile corresponding to the soil moisture profile. 4.3.3 Soil Moisture Retrieval Using Synthetic Data To demonstrate the applicability of the proposed inversion algorithm and compare its performance with the simulated annealing method, we perform a numerical experiment for estimating the soil moisture profile of a 50 cm deep subsurface re- gion with rough interfaces using synthetic data. To calculate the forward solution, we consider an incident angle of 40 , two polarizations (HH and VV), and the AirMOSS center frequency of 430 MHz. Moreover, to discretize the subsurface region and soil moisture profile, we consider 11 layers with the thickness of 5 cm 85 Figure 4.14: Geometry of the forward problem for retrieving soil moisture profiles: A three-dimensional dielectric structure with complex permittivities. Boundaries are zero-mean stationary random processes. for each layer. The radar data are assumed to have the following noise model n pq = pq +r; (4.5) where n pq and pq represent the measured and noise-free radar signals (in deci- bels), respectively, andr is a random number with a uniform distribution between 0:25 and 0:25, which models the expected noise in measurements [77]. As the values are considered in dB, the noise model in Eq. 4.5 corresponds to multiplica- tive noise. The specific value of noise distribution depends on the instrument used to make the backscattering observations. In this case, since our target observations are those from the P-band AirMOSS radar, which has a calibration uncertainty of 0:5 dB, we have used +/- 0:25 dB as the bounds of the noise distribution. Figure 4.15 shows the two nominal quadratic soil moisture profiles, which will 86 be reconstructed using the standard simulated annealing method and our proposed hybrid optimization method. For both of the methods, we set the initial temper- ature to T 0 = 10, apply the exponential cooling schedule of T k = (0:85) k T 0 for the kth iteration, and perform 100 step length adjustments at each iteration (N t = 100). For each step length adjustment, we perturb each unknown variable 20 times (N s = 20) in the simulated annealing method, whereas in our hybrid optimizer, we perform the multi-directional search twice (N s = 2) and restrict the maximum number of iterations in the multi-directional search algorithm to 100 (N mul = 100). For each soil moisture profile shown in Fig. 4.15, we have run both inversion algorithms 100 realizations of the perturbed data modeled with Eq. 4.5. The unknown soil moisture values are constrained to be between completely dry (0m 3 =m 3 ) and saturated (0:5m 3 =m 3 ) limits. Tables 4.7-4.8 summarize the results of estimating soil moisture profiles shown in 4.15. The values reported in this table are the averages of 100 realizations for each inversion method. In estimating both soil moisture profiles, the root mean square errors (RMSEs) and the final values of the cost function are smaller for the proposed inversion scheme compared to the standard simulated annealing method. In addition, these results are obtained in a much shorter time and significantly less number of function evaluations using the hybrid optimization technique. Fig- ures 4.16-4.17 show the distribution of RMSE between the actual and retrieved soil moisture profiles. Similar results are obtained for other profile examples, but will not be included here for brevity. 87 Figure 4.15: Two nominal Examples of soil moisture profiles as second-order polynomials: increasing and decreasing moisture with respect to depth. 88 Table 4.7: Numerical results of 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the proposed hybrid op- timization technique for estimating the soil moisture profile 1 in Fig. 4.15. The numbers reported in this table are the average of all realizations for each inversion method. Profile 1 Simulated Annealing Method Hybrid Optimization Method Number of Function Evaluations 412651 211454 Final Value of Cost Function 2.87e-6 6.83e-7 Root Mean Square Error 7.22e-2 6.98e-2 Computation Time (Seconds) 777 441 Table 4.8: Numerical results of 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the proposed hybrid op- timization technique for estimating the soil moisture profile 2 in Fig. 4.15. The numbers reported in this table are the average of all realizations for each inversion method. Profile 2 Simulated Annealing Method Hybrid Optimization Method Number of Function Evaluations 411942 214851 Final Value of Cost Function 2.77e-6 6.03e-7 Root Mean Square Error 0.11 9.86e-2 Computation Time (Seconds) 792 453 89 Figure 4.16: Distribution of RMSE between the actual and retrieved soil moisture profiles for 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the hybrid multi-directional search based simu- lated annealing algorithm for Profile 1. 90 Figure 4.17: Distribution of RMSE between the actual and retrieved soil moisture profiles for 100 realizations of the perturbed data modeled with Eq. 4.5 using the simulated annealing method and the hybrid multi-directional search based simu- lated annealing algorithm for Profile 2. 91 4.3.4 Retrieving RZSM Using AirMOSS Data Next, we use the P-band radar observations collected during the AirMOSS mis- sion andinsitu soil moisture measurements to validate the proposed hybrid inver- sion algorithm. For this purpose, we present the inversion results for the AirMOSS flight over the Kendall Grassland and Lucky Hills Shrubland in the Walnut Gulch Experimental Watershed, Arizona. This site was chosen among the AirMOSS sites because it is the closest to a bare surface among all of its sites. The radar observations for each AirMOSS site cover areas of approximately 25 km by 100 km and including hundreds of thousands of 3 arcsec pixels. However, since all of the several in situ soil moisture profile measurements are only available for one pixel at each site, we present the inversion results of that pixel on three different dates for each site (July 12, 2014, August 13, and September 1, 2015). The parameters for the inversion algorithm are the same as the ones that we choose in the previous section. We consider an 80 cm retrieval depth and 17 layers each with the thickness of 5 cm to discretize the subsurface region and soil moisture profile. Similar to [14], we have utilized the Soil Survey Geographic (SSURGO) database to obtain the soil texture of pixels. The incidence angles are around 33 to 34 and the frequency is 430 MHz. In this experiment, we consider both HH and VV polarizations of the backscattering coefficients as our measurements. Figures 4.18-4.20 show the comparison between the retrieved soil moisture profiles and measured soil moisture profiles. The RMSE values reported in this figure have been calculated over the depths down to 80 cm where in-situ data were available. The average RMSE values on July 12, 2014 are 0:096 m 3 =m 3 and 0:101 m 3 =m 3 , on August 13, 2015 are 0:069 m 3 =m 3 and 0:089 m 3 =m 3 , and on September 1, 2015 are 0:082 m 3 =m 3 and 0:087 m 3 =m 3 in the Kendall Grassland 92 Table 4.9: Breakdown of RMSE for Walnut Gulch Sites Using the Second-Order Polynomial Model with Observations of HH and VV Polarizations Date Lucky Hills Shrubland Kendall Grasslands Both Sites July 12, 2014 0.101 0.096 0.099 August 13, 2015 0.089 0.069 0.082 September 1, 2015 0.087 0.082 0.085 All Dates 0.092 0.082 0.089 and Lucky Hills Shrubland, respectively. The average RMSE over all profiles, all dates, and both sites is 0:089 m 3 =m 3 . Table 4.12 presents the breakdown of RMSE for Walnut Gulch sites. As mentioned earlier, we assume bare surfaces in our forward model. For the Kendall and Lucky Hills sites in Walnut Gulch, the surfaces near the validation pixels include some grass and shrub cover. Although at P-band the effects of such vegetation are not expected to dominate, nevertheless, to rule out significant er- rors due to vegetation, which have more impact on the HH channel than the VV one. Therefore, one way to improve the retrieval results is to consider only the VV polarization of the backscattering coefficient in retrievals. In the future, we plan to use the fully vegetated model [76] for retrievals with the new algorithm. Figures 4.21-4.23 show the comparison between the retrieved soil moisture pro- files and measured soil moisture profiles based on using the VV polarization as our measurement. The RMSE values reported in this figure have been calculated over the depths down to 80 cm where in-situ data were available. The average RMSE values on July 12, 2014 are 0:056 m 3 =m 3 and 0:055 m 3 =m 3 , on August 13, 2015 are 0:096 m 3 =m 3 and 0:060 m 3 =m 3 , and on September 1, 2015 are 0:042 m 3 =m 3 and 0:063 m 3 =m 3 in the Kendall Grassland and Lucky Hills Shrub- 93 (a) (b) Figure 4.18: Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using both polarizations as our measurement). The RMSE is calculated based on depths of down to 80 cm. 94 (a) (b) Figure 4.19: Comparison between the retrieved and measured profiles on August 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using both polarizations as our measurement). The RMSE is calculated based on depths of down to 80 cm. 95 (a) (b) Figure 4.20: Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using both polariza- tions as our measurement). The RMSE is calculated based on depths of down to 80 cm. 96 Table 4.10: Breakdown of RMSE for Walnut Gulch Sites Using the Second-Order Polynomial Model with Observations of HH and VV Polarizations (Hybrid Opti- mization scheme) Date Lucky Hills Shrubland Kendall Grasslands Both Sites July 12, 2014 0.055 0.056 0.055 August 13, 2015 0.060 0.096 0.072 September 1, 2015 0.063 0.042 0.055 All Dates 0.059 0.063 0.061 Table 4.11: Breakdown of RMSE for Walnut Gulch Sites Using the Second-Order Polynomial Model with Observations of HH and VV Polarizations (Simulated Annealing Method) Date Lucky Hills Shrubland Kendall Grasslands Both Sites July 12, 2014 0.067 0.059 0.064 August 13, 2015 0.069 0.084 0.074 September 1, 2015 0.059 0.052 0.056 All Dates 0.065 0.064 0.065 land, respectively. The average RMSE over all profiles, all dates, and both sites is 0:061 m 3 =m 3 . Table 4.10 presents the breakdown of RMSE for Walnut Gulch sites, which shows the significant reduction in RMSE of the retrieved soil mois- ture profiles compared to the previous results based on using both polarizations as our measurements. In addition, Table 4.11 presents the breakdown of RMSE for same dates and sites employing previously used AirMOSS baseline inversion algorithm, which is based on the simulated annealing method. Comparison of RMSE values in Tables 4.10 and 4.11 demonstrates improvements in retrieving soil moisture profiles for these sites. 97 (a) (b) Figure 4.21: Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using the VV polarization as our measurement). The RMSE is calculated based on depths of down to 80 cm. 98 (a) (b) Figure 4.22: Comparison between the retrieved and measured profiles on August 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using the VV polarization as our measurement). The RMSE is calculated based on depths of down to 80 cm. 99 (a) (b) Figure 4.23: Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the second-order polynomial model (using the VV polar- ization as our measurement). The RMSE is calculated based on depths of down to 80 cm. 100 4.3.5 A New Soil Moisture Model for Retrieving RZSM In this section, a new soil moisture profile model is employed in the inversion algorithm to replace the quadratic profile model and to demonstrate the greater flexibility for fitting the measured soil moisture data. This new model, which is derived in [78], is a closed-form analytical solution to the nonlinear Richards’ Equation (RE) [54] and contains three free parameters: m v (z) = c 1 z +c 2 exp(z=h cM ) +c 3 1=P ; (4.6) wherem v (z) is the soil moisture function with respect to depth andc 1 ,c 2 , andc 3 are the free parameters, or unknowns, of the model. The parameterP is an em- pirical parameter related to the soil pore size distribution andh cM is the effective capillary drive introduced in [79]. The soil hydraulic parameters (h cM andP ) can be approximated knowing the texture of the soil [78]. It can be shown that the sensitivity ofm v with respect toc 1 ,c 2 , andc 3 is quite significant [78], which is a good attribute in that the variability of soil moisture profile can be captured. At the same time, the greater sensitivity makes the direct retrieval of these parameters, which are unknowns more difficult. To overcome this issue, instead of retrieving c 1 , c 2 , andc 3 , we retrieve the soil moisture values at three different depthsm v1 , m v2 , andm v3 corresponding to depthsz 1 ,z 2 , andz 3 , respectively. Then, we cal- culatec 1 ,c 2 , andc 3 directly from these soil moisture values using the following equations: c 1 = m P v3 m P v1 A m P v2 m P v1 z 3 z 1 A z 2 z 1 (4.7) 101 c 2 = m P v2 m P v1 c 1 z 2 z 1 exp(z 2 =h cM ) exp(z 1 =h cM ) (4.8) c 3 =m P v1 c 1 z 1 c 2 exp(z 1 =h cM ); (4.9) where A = exp(z 3 =h cM ) exp(z 1 =h cM ) exp(z 2 =h cM ) exp(z 1 =h cM ) : (4.10) From this point onward, we proceed to apply the inversion algorithm for the AirMOSS data in the previous section using the RE-based model in Eq. 4.6 and demonstrate its flexibility for fitting the measured soil moisture data in compari- son with the quadratic model. Based on the soil texture in Lucky Hills Shrubland and Kendall Grassland, we classify them as sandy and clay loam using the United States Department of Agriculture classification scheme [80]. Using Table 1 in [78], we choose the hydraulic parameters as P = 6:73 and h cM = 5:7 cm for Lucky Hills Shrubland andP = 13:05 andh cM = 100:39 cm for Kendall Grass- land. Other parameters for the inversion algorithm are the same as the ones that we chose in the previous section. Figure 4.24-4.26 illustrates the in situ measured soil moisture profiles and the retrieved soil moisture profiles using the model in Eq. 4.6. The RMSE values reported in this figure, have been calculated over the depths down to 80 cm. The average RMSE values for Lucky Hills Shrubland are 0:068 m 3 =m 3 , 0:054 m 3 =m 3 , and 0:049 m 3 =m 3 and for Kendall Grassland are 0:043 m 3 =m 3 , 0:036 m 3 =m 3 , and 0:044 m 3 =m 3 on July 12, 2014, August 13,2015, and September 1, 2015, respectively. Table 4.12 Summarizes the breakdown of RMSE for both Walnut Gulch sites for different dates, which shows the reduction of RMSEs in both sites 102 Table 4.12: Breakdown of RMSE for Walnut Gulch Sites Using the Soil Moisture Profile Model in Eq. 4.6 Date Lucky Hills Shrubland Kendall Grasslands Both Sites July 12, 2014 0.068 0.043 0.059 August 13, 2015 0.054 0.036 0.048 September 1, 2015 0.049 0.044 0.047 All Dates 0.057 0.041 0.051 for all dates (except July 12, 2014 in Lucky Hills Shrubland). The average RMSE over all profiles, all dates, and both sites is reduced to 0:051 m 3 =m 3 . 4.4 Conclusion and Summary In this chapter, we presented an inversion scheme for estimating subsurface prop- erties of layered dielectric structures. We applied the method two different con- figuration of the problems. In the first configuration, we retrieved the model pa- rameters of dielectric structures with rough surfaces. In this problem the unknown variables to be retrieved are permittivity and conductivity of each layer, the stan- dard deviation and correlation length of each interface, and the mean separation between the two rough interfaces referred to as layer thickness. For the second configuration in which we estimate the RZSM profile, we considered second- order polynomials for modelling the soil moisture profile and we parametrized these quadratic polynomials with soil moisture values at three different depths, which are the top, the middle, and the bottom of the investigation domain. By providing some numerical results, we demonstrated that the proposed inversion method has significantly better computational efficiency than the standard simu- 103 (a) (b) Figure 4.24: Comparison between the retrieved and measured profiles on July 12, 2014 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. 104 (a) (b) Figure 4.25: Comparison between the retrieved and measured profiles on August 13, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. 105 (a) (b) Figure 4.26: Comparison between the retrieved and measured profiles on Septem- ber 1, 2015 in (a) Walnut Gulch Kendall Grassland and (b) Walnut Gulch Lucky Hills Shrubland using the soil moisture profile model in Eq. 4.6. The RMSE is calculated based on depths of down to 80 cm. 106 lated annealing method used previously. We validate the retrieval method using the measured radar data collected during the AirMOSS mission flights andinsitu soil moisture measurements. Finally, we utilized a closed-form analytical solution to Richards’ equation and demonstrate significant improvements in soil moisture retrieval results. 107 Chapter 5 Conclusion and Future Work In this work, a general framework for solving inverse-scattering problems has been presented. To improve both the computation time and capability of finding the desired solution, a novel combination of local and global optimization meth- ods is proposed for solving the inverse-scattering problem,which is often modelled as an optimization model. By performing random searches in multiple effective directions instead of sequential perturbations in the standard simulated annealing method, the proposed optimization approach works more successfully compared to the classical simulated annealing. The inversion algorithm is developed and benchmarked with numerical simulations. Moreover, the effect of algorithm pa- rameters on the inversion results has been studied. Although the choice of algo- rithm parameters depends on the problem, suggested values for each parameter have been provided for the presented inverse scattering problems and test func- tions. To demonstrate the applicability of the presented inversion scheme, two major classes of electromagnetic inverse problems have been investigated: 108 A microwave imaging system based on the proposed inversion scheme is pre- sented which is able of reconstructing investigation regions with multiple targets and different dielectric constants. The numerical results of reconstructing the elec- tromagnetic properties of two-dimensional dielectric targets verify the capability of this method in obtaining the permittivity distribution without considering any specific constraint on the initial guess or a priori information about the size, lo- cation, or permittivity values of the target. The effect of the resolution of the reconstructed images on the inversion results has been investigated. Although a higher resolution (via reducing the pixel size) will result in slower convergence, applying the proposed imaging technique leads to the successful reconstruction of investigation domain. To investigate the uncertainties of the reconstruction results in the presence of the noise, an extensive sensitivity analyses has been performed by adding different noise levels to the measurements. In the second class of inverse problems, the problem of determining the prop- erties of layered rough surface structures from radar data has been investigated. Two different scenarios are considered for this problem. The more basic sce- nario is retrieving the model parameters of layered structures, which are complex dielectric constants (permittivity and conductivity), thickness of layers, and statis- tical properties of the boundaries. The inversion results for different simulations cases in addition of the sensitivity analysis are presented to demonstrate the per- formance of the method. In the second scenario, estimating the RZSM profile using the P-band radar data collected during the AirMOSS mission is thoroughly studied. To compare with the previously used AirMOSS baseline inversion algo- rithm, some numerical simulations in the presence of noise in the measurements are presented to demonstrate significant improvements in both accuracy and com- 109 putation time of the inversion results. The the retrieval method is then validated using the measured radar data collected during the AirMOSS mission flights and in situ soil moisture measurements. Moreover, a new soil moisture profile model derived from Richards’ equation is applied, which shows significantly better per- formance in estimating RZSM profile compared to the previous model. It is worth mentioning that the proposed inversion method can apply equally well to other imaging modalities that use inverse scattering, such as acoustic/ultrasound imaging. The difference is the employed forward model in these imaging tech- niques, which provide different types of measurements from the target. A direction that can be pursued regarding the microwave imaging work pre- sented in this dissertation is to use parallel computing techniques in the inver- sion scheme for accelerating the convergence speed. This opportunity is provided because perturbations required for obtaining new candidate solutions at each it- eration of the multi-directional search algorithm are independent of each other. As a result, the corresponding forward model evaluation of each perturbation can be done independently. Therefore, depending on the available processors, the computation time for the image reconstruction can be reduced significantly. By reducing the computation time, it will be then possible to increase the number of unknowns, and consequently, the image resolution. As it is described earlier, the forward model calculates the scattered fields from layered dielectric structures assuming bare surfaces. Neglecting the vegeta- tion of the surfaces may lead to errors in the calculated backscattering coefficients, particularly for the HH polarization. As a result, another extension of the work presented in this dissertation is to apply the proposed inversion model to vegetated surfaces. Since all of the AirMOSS sites include vegetated areas, a more compli- 110 cated forward model should be employed to model the scattered electromagnetic fields from these sites more precisely by incorporating the effect of vegetation in the scattered waves [75, 76]. This can significantly improve the retrieval results of RZSM profiles. 111 Appendices 112 Appendix A Overview of the Simulated Annealing Method In this appendix, we present a detailed description of the algorithm of simulated annealing method for minimizingf(x), which is a function ofn continuous vari- ables: Initialization Step: - Choose an initial-guess pointx 0 and initialize the size of step lengthss i for i = 1; 2;:::;n. - Set the initial temperature parameterT 0 and the rate of temperature reduction r T . - Set the parameters related to internal cycles of algorithmN t andN s . 113 - Evaluate the cost function atx 0 ,f 0 =f(x 0 ), and initialize the optimum solu- tion: x opt =x 0 f opt =f 0 Iterations: form = 1; 2;::: do - Check the stopping criteria of the algorithm. forj = 1;:::N t do - Setn s i = 0; i = 1; 2;:::;n. fork = 1;:::;N s do - Setl=1. forh = 1;:::;n do - Generate new candidate solution by perturbing current solutionx m along coordinate directionh x 0 m =x m +rs i e i i = 1; 2;:::;n; where r is a random number generated in [0,1] and e i denotes the vector with a 1 in theith coordinate and 0’s elsewhere. where r is a random number generated in [0,1]. - Calculatef(x 0 m ). iff(x 0 m )<f(x m ) then x m =x 0 m ; f(x m ) =f(x 0 m ) 114 x opt =x 0 m ; f opt =f(x 0 m ) Add 1 ton s i else - Generate a pseudorandom numberp in [0,1]. ifp< exp( f(xm)f(x 0 m) Tm ) then x m =x 0 m ; f(x m ) =f(x 0 m ) - Add 1 ton s i end if end if end for Update the step length using: s i = 8 > > > > > > < > > > > > > : s i 1 +c i ns i Ns 0:6 0:4 ; ifn s i > 0:6N s s i 1 1+c i 0:4 ns i Ns 0:4 ! ; ifn s i < 0:4N s s i ; otherwise end for end for - Reduce the temperature for the next iteration,T m+1 =r T T m - Reset the accepted point and its cost function by currently obtained opti- mum solution: x m =x opt ; f(x m ) =f opt end for Here, s i denotes the step length size for the i-th variable of the function x i . 115 The parameterN t is the number of step length adjustments at each iteration and N s denotes the number of perturbations of each unknown for the each step length adjustment. After obtaining the ratio of the accepted solutions to the total pertur- bations of each variable, we adjust the step-lengths of the variables based on their ratio. The parameter T is called the temperature, which determines the rate of accepting new candidate solutions at each iteration. We reduce the temperature at each iteration by the reduction rater T . Here are the suggested values for some of these parameters: N s = 20 N t = maxf100; 5ng c i = 2 i = 1; 2;:::;n r T = 0:85 116 Appendix B Multi-Directional Search Algorithm Here, we present the description of multi-directional search algorithm: Initialization - Start with an initial simplexS 0 with verticesfv 0 0 ;:::;v 0 n g - Set the expansion factor in (1; +1) range. - Set the contraction factor in (0; 1) range. - Calculatef(v 0 i ); i = 1;:::;n. min argminff(v 0 i ); i = 1;:::;ng. - Swapv 0 0 andv min 0 forl = 0; 1;::: do - Check the stopping criteria. Perform the reflection step: v l+1 i =v l 0 + (v l 0 v l i ) i = 1;:::;n - Calculatef(v l+1 i ). if minff(v l+1 i ); i = 1;:::;ng<f(v l 0 ) then 117 Perform the expansion step: v l e i =v l 0 +(v l i v l 0 ) i = 1;:::;n - Calculatef(v l e i ). if min i ff(v l e i )g< min i ff(v l+1 i )g then Replacev l e i withv l e i fori = 1;:::;n. end if else Perform the contraction step: v l+1 i =v l 0 +(v l 0 v l i ) i = 1;:::;n - Calculatef(v l+1 i ). end if min argminff(v l+1 i ); i = 1;:::;ng. - Swapv l+1 0 andv l+1 min end for Figure B.1 illustrates the reflection, expansion, and contraction steps of the multi-directional search algorithm for a 2-D simplex. The values of the expansion and contraction factors are usually chosen to be = 2 and = 0:5, respectively. 118 Figure B.1: Reflection, expansion, and contraction steps of the multi-directional search algorithm. 119 List of Abbreviations AirMOSS . . . . . Airborne Microwave Observatory of Subcanopy and Subsurface CG . . . . . . . . . . . . Conjugate Gradient DE . . . . . . . . . . . . Differential Evolution EM . . . . . . . . . . . Electromagnetic FDTD . . . . . . . . . Finite Difference Time Domain FEM . . . . . . . . . . Finite Element Method GA . . . . . . . . . . . . Genetic Algorithm MOM . . . . . . . . . Method of Moments MRI . . . . . . . . . . Magnetic Resonance Imaging NASA . . . . . . . . . National Aeronautics and Space Administration PSO . . . . . . . . . . . Particle Swarm Optimization RE . . . . . . . . . . . . Richards’ Equation RMSE . . . . . . . . . Root Mean Square Error 120 RZSM . . . . . . . . . Root Zone Soil Moisture SA . . . . . . . . . . . . Simulated Annealing SMAP . . . . . . . . . Soil Moisture Active Passive SMOS . . . . . . . . . Soil Moisture and Ocean Salinity SPM . . . . . . . . . . Small Perturbation Method 121 Bibliography [1] E. Fear, S. Hagness, P. M. Okoniewski, and M. Stuchly, ”Enhancing breast tumor detection with near-field imaging,” IEEE Microw. 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Abstract (if available)

Abstract This dissertation presents a detailed investigation of solving inverse-scattering problems using hybrid global and local optimization. The main goal in the inverse-scattering problem is to characterize the electrical and geometric scattering properties of a target from the scattered fields. Inverse-scattering problems are usually modelled as an optimization problem in which we try to minimize the mismatch between the scattered fields collected from the target and our estimation. Although local optimization methods have substantially higher speed of convergence compared to global search methods, they cannot generally provide the desired solution of these highly nonlinear inverse problems. On the other hand, heuristic techniques have significantly better performance in obtaining the global solution with the cost of substantially higher amount of computation. To address these issues, the first part of this dissertation presents a global optimization scheme that is a novel combination of the simulated annealing method and the multi-directional search algorithm. Numerical results of optimizing some test functions is presented to develop and benchmark the proposed inversion scheme. ❧ In the next part of the thesis, the application of the presented inversion is investigated for a microwave-imaging system to obtain the electrical properties of objects. The proposed global optimizer significantly improves the performance and speed of the simulated annealing method by utilizing a nonlinear simplex search, starting from an initial guess, and taking effective steps in obtaining the global solution of the minimization problem. Due to the efficient performance of the proposed global optimization method, we are able to obtain the shape, location, and material properties of the target without considering any a priori information about them. The accuracy and applicability of the proposed imaging method is demonstrated with some numerical results in which two-dimensional images of multiple objects are successfully reconstructed. ❧ In the last part of this dissertation, retrieving the subsurface properties of layered dielectric structures is investigated for two different models of these structures. In the first one, the unknown variables are the modelling parameters of the structure, which are complex dielectric constants (permittivity and conductivity), thickness of layers, and statistical properties of the boundaries. Using some numerical experiments, successful retrieval results of this class of inverse problems are demonstrated. Thereafter, an inversion model for retrieving the root zone soil moisture profile using the P-band radar observations collected during the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) mission is presented. We present numerical results to demonstrate significant improvements in the inversion results and computational speed compared to the previously used AirMOSS baseline inversion algorithm. The presented method is validated with in situ soil moisture measurements. A physics-based soil moisture profile model derived from Richards’ equation is employed and its greater performance in RZSM retrieval compared to the previous model is demonstrated.

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Asset Metadata

Creator Etminan, Aslan (author)

Core Title Solution of inverse scattering problems via hybrid global and local optimization

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 09/18/2019

Defense Date 07/19/2019

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

Tag global optimization,inverse problems,microwave imaging,OAI-PMH Harvest,remote sensing

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Moghaddam, Mahta (committee chair), Jafarpour, Behnam (committee member), Molisch, Andreas (committee member), Tabatabaeenejad, Alireza (committee member)

Creator Email aslanetminan@gmail.com,etminan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-218732

Unique identifier UC11674154

Identifier etd-EtminanAsl-7820.pdf (filename),usctheses-c89-218732 (legacy record id)

Legacy Identifier etd-EtminanAsl-7820.pdf

Dmrecord 218732

Document Type Dissertation

Rights Etminan, Aslan

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA