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A Kalman filter approach for ionospheric data analysis

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A Kalman filter approach for ionospheric data analysis

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Content A KALMAN FILTER APPROACH FOR IONOSPHERIC DATA ANALYSIS by Michael Christian Lehn A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (APPLIED MATHEMATICS) May 2001 Copyright 2001 Michael Christian Lehn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1414825 UMI UMI Microform 1414825 Copyright 2003 by ProQ uest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United S tates Code. ProQ uest Information and Learning Company 300 North Z eeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA The Graduate School U niversity Park LOS ANGELES, CALIFORNIA 90089-1695 This thesis, w ritten b y h t t U a t l C ltn 'd u tw Under th e direction o f h.!.h ... Thesis Com m ittee, an d approved b y a ll its members, has been p resen ted to an d accepted b y The Graduate School, in p a rtia l fulfillm ent o f requirem ents fo r th e degree o f o f -S c iV h c c Apjfh ect f i d r V — ............................ jf........................ -........................................ Dean o f Graduate Studies D ate May. I I , 2001 THESIS COM M ITTB Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to Jenny and my parents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgment s I would like to thank my adviser Chunming Wang for his support, guidance and priceless help during my time at USC. My education and research experience received invaluable contribution through my work with Dr. Wang. I’m also thankful to George Hajj for putting so much enthusiasm into our discus sions and for his encouragement during my research. And I would also like to thank Gary Rosen for his support and help. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Dedication ii Acknowledgments iii List of Figures vi List of Tables vii Abstract viii 1 Introduction 1 2 M athem atical M odel for Earth’s Ionosphere 3 2.1 Basic Physical M odel................................................................................... 4 2.1.1 Conservation of Mass ..................................................................... 4 2.1.2 Conservation of M om entum ............................................................ 5 2.1.3 Geo-Magnetic Coordinates ......................... 7 2.2 Equations for the Ionosphere Model and Numerical Solution................ 9 2.2.1 Differential E q u a tio n s..................................................................... 9 2.2.2 Numerical Solution........................................................................... 12 3 M easurements of the Atmosphere using GPS 17 3.1 GPS Radio Occultation C o n c e p t............................................................. 17 3.2 GPS Observables......................................................................................... 18 3.3 Ionospheric O bservables............................................................................. 20 3.3.1 Relative TEC from LI and L2 (TECr,p) ...................................... 20 3.3.2 Absolute TEC from PI and P2 (TECa,n) ................................... 21 3.3.3 Relative TEC from LI and PI (TECr,n) ...................................... 21 3.3.4 Relative TEC from LI Only (TECr!P tLi) ...................................... 22 3.3.5 LI and L2 amplitude m easurem ents............................................ 22 3.4 Mathematical Model For O bservations.................................................... 23 4 The Kalman Filter 25 4.1 Recursive Estimation P ro cess................................................................... 26 4.2 Statistical Concepts ................................................................................... 27 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.1 Hilbert Space of Random Variables............................................... 28 4.2.2 Projection T h e o re m ........................................................................ 29 4.2.3 Minimum Variance E s tim a te ......................................................... 30 4.3 Recursive Filtering A lgorithm ..................................................................... 32 4.3.1 Forecast S te p .................................................................................... 32 4.3.2 Update Step .................................................................................... 35 5 Com putational Results 38 5.1 Structure of the Simulation Process............................................................ 38 5.2 Theoretical Results for Comparison............................................................ 41 5.3 Evaluation of the Simulation R u n s ............................................................ 42 6 A ppendix 44 Bibliography 45 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 2.1 Sketch of magnetic field lines............................................................... 8 2.2 The ionosphere in geo-m agnetic coordinates.................................. 9 3.1 Occultation geom etry defining a, r, a and the tangent point and showing the split of the LI and L2 signals due to the dispersive ionosphere................................................................................ 18 3.2 A 2-dimensional sketch of an observation link.............................. 23 4.1 The projection theorem .......................................................................... 30 4.2 The working concept o f the Kalman filter...................................... 33 5.1 Flowchart for the Observation System Simulation Experim ent (O SSE)........................................................................................................... 39 5.2 D ata links created by 5 ground receiver and 4 GPS transm it ters in 2 hours............................................................................................. 40 5.3 The theoretical error compared to the experim ental error. . . 42 5.4 The theoretical norm of the innovation vector compared to the experim ental value.................................................................................... 43 v i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract There are several reasons why a deterministic approach to model a physical system does not provide sufficient means of performing a system analysis. In this work we consider a stochastical model for earth’s ionosphere as well as a stochastical model for the process of sensing the ionosphere. Based on those models a Kalman filter is used to estimate the distribution of the electron density of the ionosphere at a given time. In a Observation System Simulation Experiment (OSSE) we show that the filter provides optimal estimates according to our criterion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Given a physical system an engineer first attempts to develop a mathematical model that adequately represents some aspects of the behavior of that system. With such a mathematical model he is able to investigate the system structure. In order to observe the actual system behavior measurement devices provide data about certain variables of interest. There are several reasons why a deterministic approach does not provide sufficient means of performing a system analysis. First of all, no mathematical system model is perfect. The objective of the model is to represent only the dominant characteris tics. This is essentially in order to generate computational feasible algorithms. But even the effects, which are modeled are necessarily approximated by a mathematical model. Thus there are many sources of uncertainty in any mathematical model of a system. A second shortcoming of deterministic systems is that dynamic systems are partially driven by disturbances which we can not model deterministically at all. A final shortcoming is that sensors do not provide perfect and complete data about a system. A Kalman filter incorporates all information that can be provided to it. It processes all available measurements, regardless of there precision, to estimate the current value 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the variables of interest, with use of knowledge of the system and measurement device dynamics, the statistical description of the system noise, measurement errors and uncertainty in the dynamics models. It can be shown that the Kalman filter is optimal with respect to virtually any criterion that makes sense. In chapter 2 we introduce our mathematical model for the ionosphere. The success of ionospheric modeling depends mainly on accurate knowledge of the forces which enter into the hydrodynamic plasma equations for the ionosphere. These include solar EUV and UV radiation, magnetospheric electric fields, particle precipitation, dynamo electric fields thermospheric neutral densities, temperatures and wind. Therefore, ideally, one needs a coupled magnetospheric-thermospheric-ionospheric model that assimilates global and continuous data in all these regions. However, a first step toward reaching that goal is to learn to do data assimilation optimal in each region independently. Therefore, we only investigate a simplified system model for earth’s ionosphere in this work. Chapter 3 gives a short overview how global GPS networks are used to monitor the ionosphere. We will further derive a mathematical model for the process of sensing the ionosphere. In chapter 4 we introduce the Kalman filter. We will start by giving an overview of the statistical tools it is based on and we will then go on to derive its equations. In the context of an Observation System Simulation Experiment (OSSE) the Kalman filter is used based on the ionospheric model and the GPS-TEC data obtained from ground and space. Through the OSSE we will show in chapter 5 that our imple mentation of the Kalman filter is correct. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 M athem atical M odel for Earth’s Ionosphere In this chapter we introduce a mathematical model for earth’s ionosphere. In general such a model should explain the dynamics of the ions and electrons in the ionosphere and electro-magnetic fields. Therefore the density distribution, velocity and temper ature of all particles as well as the vector electrical and magnetic fields characterize the state of the ionosphere. However, in order to keep the model tractable and to make it possible to solve the equations numerically we will make several simplifying assumptions: 1. The o+ ions are the only dominant ion species in the ionosphere. 2. The meridianal component of the ion drift v$ is equal to zero. This has as a result that the density distribution in all meridianal planes are identical for the same local time. 3. The ions and electrons have the same temperature as the neutral species and t e m p e r a t u r e fields a r e g i v e n by a n e m p i r i c a l f o r m u l a . 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1 Basic Physical M odel The general goal of an ionosphere model is as mentioned above to derive equations for the density distribution. However the exact distribution can only be approximated. We therefore will divide the ionosphere into regions and assume that each single region has a homogeneous density. By this modeling the dynamics of the system means modeling how the densities in this regions change. 2.1.1 C onservation o f M ass What the law of conservation of mass simply states is that the net flux of particles into or out of a volume must be equal to the rate of increase or decrease of total number of particles inside the volume. This however is not true for the ions in the ionosphere. Ion and electron pairs may be produced by impact of energetic particle or lost through recombination between opposite charged particles. We will use the terms P and L to denote the production rate and loss rate distri bution, respectively. This quantities are given by empirical formulas, which we won’t further investigate in this work, see e.g. [1]. Considering production and loss of ions the law of conservation of mass for a particular region of interest Q , can mathematically expressed by j t J n(t, x)dV = - J nVffdS + J {P - L)dV (2.1) V S V where S is the surface of V and dV, dS represent Lebesgue measure of the volume and surface area in Euclidean space and ff is the outward normal vector on the surface 5. The vector field V is the vector field for the velocity of the ions. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The ion flux across boundary points can be classified by the physical phenomenon, which is causing it: 1. The flux crossing the boundary at any given point is called diffusion flux if it depends on the relative density between inside and outside of the boundary. This simply has the effect that if a region with high density lies next to a region with low densities there will be a flux down the density gradient. Obviously we need to know for each boundary the densities of the two affected regions in order to get the density gradient. 2. If the flux across a boundary is caused by the migration of ions carried by the flow of gases is said to be a transportation flux. In this case we further can distinguish between the out-flow case if the velocity V is in the positive direction and the in-flow case for negative directions of V. For the out-flow case we only need information about the densities inside the region fi. On the other hand for the in-flow case we only need information about the density outside the boundary. 2.1.2 C onservation o f M om en tu m The flux vector nV in (2.1) is given by the law of conservation of momentum. The principle of conversation of momentum relates the forces acting on the fluid to the fluid velocity. It requires that the change of momentum per unit time within a volume be equal to the pressure gradient force and the total external force field F acting on the material inside the volume plus the momentum flux carried across the surface bounding the volume by a viscosity, advection or wave flux. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We will use the steady-state form of the law of conservation of momentum (see [2]), which is given by - V{nkBT ) + nMg + cn{E+ V x B) - W = 0, (2.2) where ks is Boltzmann’s constant, c the elemental charge, M is the molecular mass of the o+ ion, g is the gravitational field and E, B are the electrical and magnetic fields, respectively. The vector field W represents the change in momentum due to fictional forces caused by collision with other ion species and neutral particles. However we made the assumption the o+ ions are the dominating ion species and therefore we get W in the simple form of W = nMvn{V - U) (2.3) where U represents the velocity of the neutral particles and the coefficient vn is the collision frequency between o+ ions and the neutral particles. Similar to the ions we can apply the law to electrons. As we only consider o+ ions, the electron density ne is equal to the ion density n. So we only need to replace M in (2.2) by the mass of the electron and change the sign for the n(E + V x B) term, as it depends on whether a particle is positive or negative charged. - V(nekBT) + neMeg - ene(E + Ve x B) - neMeuejn(Ve - U) - 0 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We will further simplify the equations by neglecting the mass of the electron. Therefore we get 0 = -V (n k BT) -en {E + V x B) & E = _ V (nfca T ) - V e X B (2.4) en 2.1.3 G eo-M agnetic Coordinates In the previous section we didn’t refer to a particular coordinate system. However in order to make use of the law of conservation of mass we need to specify a region of interest ft. Choosing the geo-magnetic coordinate system will simplify the equations enormously. The earth’s magnetic field B is a dipole, which points down toward the surface of the earth in the northern hemisphere and away from it in the southern hemisphere. The magnetic field at a given point has the form „ r%cos(0) B = - B 0Vq, q = ----- 2— ' where ro is the earth’s radius, r the distance to the dipole center and theta the angle between the dipole axis and the vector linking the the dipole center to the specific point. A dipole field is sketched in Figure(2.1). We consider the center of a spherical coordinate system to be centered at the dipole center and with co-latitude angle 6 and longitude angle < f> defined with respect to the dipole axis and to a specified zero meridianal. If we fix < j> an arbitrary point has a unique position on a particular field line. This principle allows us to build the geo-magnetic coordinate system. Coordinates (p , q, < f> ) in this system refer to the geo-magnetic field line, geo-magnetic potential line and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. north geomagnetic d ip o le m o m e n t - ph Figure 2.1: Sketch of m agnetic field lines. longitude. Geo-magnetic potential lines are perpendicular to the geo-magnetic field lines. For a point on a field line the direction where the magnetic intensity changes most is the direction along the geo-magnetic potential line. Graphically this means that two points on the same field line have the same p coordinate and two point on the same geo-potential line have the same q coordinate. We will denote the unit vectors for orthonormal basis of this coordinate system by up, uq and u^. For a given scalar field / the gradient of f is given in geo-magnetic coordinates by V / = hpdp(f)up + hqdq(f)uq + h't> d(j)(f)u( p, (2.5) where the values hp, hq, and h < f, correspond to the scaling between the metrics in p, q and < f> and the metric given by the distance. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.2: The ionosphere in geo-m agnetic coordinates. The main result of this is that we can create a grid for the ionosphere or in other words we can discretize our region of interest. In Figure (2.2) we show the grid of the ionosphere used for the numerical computations in this work. 2.2 Equations for the Ionosphere M odel and Numerical Solution 2.2.1 Differential Equations If the vector V is written as a linear combinations of the unit vectors {up, uq, u^} of the form V = V p U p + V q U q + V j U j , , 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the components vp and are assumed to be known and independent of the ion density and ion temperature. We can take advantage of the fact that V x B and Ve x B are perpendicular to Uq and obtain by using (2.5) for (2.2) and (2.4) nvg = - ^ - ( 1 + e^cjdginksT) + n(Uq + g ^ 1). (2.6) Now we can rewrite the law of conservation of mass in a volume A V given by A V - {(p, q ,< f> ) : po < p < Po + Ap,q0 < q < qo + Aq, < / > 0 < < /> < < f> 0 + A< f> } . (2.7) If Ap, Aq and A (f> is small enough A V can be approximated by a rectangular box with edges of length Ap/hp, Aq/hq and A< f> /h^. The assumption that vphi = 0 will simplify the surface integral, as we only need to consider the boundaries of A y, which are perpendicular to u^, those are A S P j o = {(P >g, 4 > ) :p-- = Po, qo < q < qo + Aq, < < t > < h + A0} A V = {(P >Q , 4 > ) :p--= po + Ap, q0 < q < q0 + Aq, 4)o < < t > < + A <j>} A V = {(P >q, < t > ) ■P0 < p < po + A p, q = qo, 4 > o < < t > < 4 > o + A4> } A V = {(P >q, 4 > ) ■ Po < p < po + Ap, q = go + Aq, < j ) Q < (j) < 4 > o + A<j>} 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then equation (2.1) becomes go+Ag go+Ag 0o+A0 d f f f n II I h l h f {t'!’' qA)dpdqd4‘ di J J J hphqhfj^ g o p o < t > o 0 O + A 0 g o + A g = - J J (j^ {t,P 0 +&p,q,< i> ) - -pj^(t,Po,q,$))dpd(j> < j> o g o <j>o+A<j>po+Ap ~ f f ( l T h ~ ( t , p , q ° + $ ) ~ - r ~ r - ( t , P , < 1 0 i 4 > ))d p d < f> J J n q n ^ n q n $ 0 0 P o go+Aq go+Ag 0o+A0 + f f f h~Th (P ~ (*’P’ 4> )dpdqd< j> . J J J n p h g h tf, g o p o < j> o We define 0 O + A 0 n v 00 0O +A 0 \p (2 .8) (t.Jb?) = J fifty ^ 2 '9^ P(t,P,Q) = J ■j^(P(t,P,q,< j> ) ~ L(t,p,q,(f>))d(l). (2.10) 00 By dividing both sides of equation (2.8) by ApAq and taking the limit as Ap and Aq approaches zero, we obtain dtn{t,p,q, < j))d (j) = hPh s ( l + ce ) dq( - ^ - d q(nkBT)) - h qhpdq(Uq+,9qn^-n) (2 .11) tip - hphqdp(^-n) + P{t,p, q) tlq l i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.2 N um erical S olu tion For solving the partial differential equations we will use a numerical approximation. In order to do this we will use two steps: 1. Space discretization The region of interest for the solution of the model is a subset of the 4-dimensional time-space domain S = IR+ x fi. The space discretization consists of selecting a finite dimensional family of functions to represent the approximation of any function defined on S. This is also referred to as grid generation in the case of finite difference or finite element methods. 2. Approximation of differential operators The approximation of differential opera tors consists of replacing the partial differential equations by a system of algebraic equations that the approximate solution in the function family selected in the space discretization must satisfy. In this subsection we will introduce our discretization and the resulting approximation for the ionosphere model equations. The used method is generally knows as finite vol ume method. For the discretization of time we simply choose time intervals (tm,tm+1] with length given by At. For the space discretization the region fl will be approxi mated by a collection of small regions of the form A V given in (2.7). We assume that a rectangular grid in the geo-magnetic coordinate system is used, which is given by three sets of values { q i } ^ Q and Therefore we can specify a volume-element y i j , k b y Vi,j,k = {(p, q, 4> ) : P i-i <P<Pi, q i-i < q < qi, <t>i-1 < < / > < 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We assume that the ion density inside such a volume-element is homogeneous and changes linear in time. That means the ion density function n(...) has the form n(t,p,q,4) = ■ * ro+" " T n(m,i,j,k) + —^ + £m+1 ^m+1 for t E (tm,tm+1], (p,q,<p) G Vi,j,k- The approximation of the differential operator consists of of translating the partial differential equations for the ionosphere model into algebraic equations in n(m, i,j, k). The approximation of the law of conservation of mass(2.8) therefore is given by n(m + l,i,j,k) =n(m,i,j,k) + At(P(m,i,j, k) - L(m,i,j,k)) ~ 7 7 } \{\Sij,k;i+l,j,k\fp(.m i h j ) ® T + 7 7}— - iJ> M ,j,fc)) (2.12) \V i,j,k\ ~ 7 7 }— + 1, k)) \vi,j,k\ T 7 7 } ~ 1> ® j j i & )) \vi,j,k | where is the volume of the volume element and j j ^ is the surface between the volume elements Vy,fc and with representing the surface area of j The production rate P and the loss rate L correspond to the values in the center of Vijtk- The quantities fp and f q correspond to the approximated ion 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. flux in the directions up and uq. As we assumed that vp is given by an empirical function we can define fp by n{m, i,j, k)vp(tm,pi, qcj, (j)c,i) vp{tmiPiiQc,ii( l> c,i) > 0) 0 vpij'miPi')( lc,ii4> c,i) = 0, 71(771,2 + 1 1 j, k)vp{tm,pi, qC ti, (f)C ti) vp(tm,pi, qcj, ( f> c > i) < 0, (2.13) where qc > i, are the coordinates in q and < fr of the point at the center of the surface Si,j,k\i,j,fe+i- This corresponds to the transportation flux we discussed in section 2.1.1. If the velocity vector has the direction of an out-flow then the amount of ions leaving the volume element is determined by the inside density. On the other hand for the in flow case it is determined by the outside density. For the quantity f q however we also need to consider the diffusion flux. In equation (2.6) we determined the ion flux vq, but in order to determine the sign we would have to solve a system of linear equations. But as the diffusion process is the main driving force for the flux in the uq direction, the smoothing effect of the diffusion allows us to approximate (2.6) where we use an implicit/explicit hybrid schema. Therefore we get fq(i,j,k-,i,j + 1, k) =nu9(pC j,gj,</>cj) _ hq (pc,j, qj, (1 + e 1c)A:sA(nT) Mvn (2.14) + nvq(pc,j, qj, 4> c,j) {Uq + gqvn ) 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where _ _ n(m,i,j,k) +n(m ,i,j + 1 ,k) n - - and A(nT) = [n(m + l,i, j + 1, k) — n(m + 1, i, j, k)]T(m + 1,*, j + 1, k) + n(m + l,i,j,k)[T(m + 1, i, j + l,k) - T(m + l,i,j,k)\. Now we can write (2.12) in a matrix form. Therefore we line up the n(m,i,j,k) and denote the density distribution as vector nm. This is possible as we are only interested in finite set of volume elements. As a result (2.12) in matrix notation will be of the form flm + l ~ n m "t" A n m +1 "h M p n m + M qTljn + P m where the form of matrix A mainly depends on our approximation of A(nT), Mq mainly on vq and Mp on fp. Pm is obtained by lining up the terms At(P(m, i,j , k) — L(i,j,k)). We can solve for nm+i and get n m + 1 — (I ~ A) 1(I + Mp + Mq)nm + Pm = M nm + Pm. It can be shown that (I — A)-1 is a symmetric invertable matrix for a sufficient small At. For our Kalman filter approach however we would like to have a homogeneous 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. form. This can be achieved by defining a sequence of vectors n * m nm+1 M nm + Pm. Now we can consider a sequence given by Obviously xm can be recursively defined by Xm+1 = M xm with X m ;; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 M easurem ents of the A tm osphere using GPS In this chapter we want to provide an overview of how space measurements are used for ionospheric sensing. Space measurements are obtained using the Global Position System (GPS). GPS observables can be used to measure integrated electron density (known as total electron content - TEC ) along the transmitter-receiver line-of-sight. For the purpose of this work we are interested in how precise this measurements are. Furthermore we want to know, how to represent the process of obtaining measurements in our Kalman filter. For a more comprehensive and detailed introduction we refer to [5]. 3.1 G PS Radio O ccultation Concept The Global Position System (GPS) consists of 24 satellites in six orbital planes about the globe. Each satellite orbit is circular, with an inclination of about 55°, a period of 12 hours and an altitude of 20,200 km. A schematic representation of atmospheric pro filing by GPS radio occultations, using a low earth orbiter (LEO) satellite as receiver, is given in Figure 3.1. An occultation takes place when the GPS satellite sets or rises behind the Earth’s ionosphere or lower neutral atmosphere as seen by a receiver in 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G PS (3) * f i / i T an g en t Point L2 Signal G P S (2) G S (1 ) L1 Signal LEO (4) Earth Figure 3.1: Occultation geom etry defining a, r , a and the tangent point and showing the split of the LI and L2 signals due to the dispersive ionosphere. LEO. Each occultation therefore consists of a set of links with tangent points ranging from the LEO satellite height to the surface. One occultation takes nearly 4 to 10 minutes depending on the relative geometry of the LEO and GPS satellites. When the signal passes through the Earth media, it experiences delay, bending and ampli tude change. These changes can be used to extract information such as profiles of electron density in the ionosphere, or temperature, pressure and water vapor in the lower neutral atmosphere (0-50km), in the vicinity of the tangent point. 3.2 GPS Observables Each GPS satellite continuously broadcasts two signals (LI, L2) at L-band, with fre quencies / i = 1575.42 MHz and f?, — 1227.60 MHz. A receiver will detect amplitude, pseudo-range and phase measurements for both frequencies. Pseudo-range is an abso lute measurement of group delay between the time a signal is transmitted and received. It is the sum of the actual range between the transmitter and the receiver, atmospheric 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and ionospheric delays and transmitter and receiver clock offsets. Range and phase measurements can be modeled as: rp Tpfrtj P* = pij + 6ii + d— ^ + Ci + Ci + vk (3.1) h r T FC^ 4 = - f * ' i = 4 4 + + + + Ci = e„ (3.2) where k equals 1 for L 1-frequency or 2 for L2-frequency; P^ is the pseudo-range between transmitter i to receiver j; is the range corresponding to the light travel time in vacuum between the transmitter and the receiver; 5% is the extra delay due to neutral atmosphere; d is a constant corresponding to the square of the plasma frequency divided by the electron density; TEC^ is the TEC along the ray path; C \ Ci transmitter and receiver clock errors, respectively (including constant bias and time dependent terms); uk is the measurement error including the receiver’s thermal noise and local multipath; Ll £ is the recorded phase between transmitter i and receiver j; is the recorded phase in cycles; c is the speed of light; N l kJ is an integer which is constant over a connected arc; Xk is the operating wavelength and ek the phase measurement noise. The neutral ionosphere is non-dispersive at radio frequencies; however because the LI and L2 split in the ionosphere, at a given time they sense slightly different parts of the neutral atmosphere (see Figure 3.1). This split is the reason for having subscripts on and TEC% J in equations (3.1) and (3.2). 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Ionospheric Observables There are several methods for extracting the ionospheric delay along the LEO-GPS line-of-sight. The selection of a specific method depends on many factors which include the data noise, desired retrieval accuracy and resolution. For completeness we sum marize the different methods below. 3.3.1 R ela tiv e T E C from LI and L2 (TECr> p) By ignoring the split of the LI and L2 signals, we can calculate the relative TECrjP (subscript r refers to relative, p to precise), along the LEO-GPS line-of-sight by taking the linear combination T E C r,P = -L 2 ) + B + v (3.3) Otfl J2J where B is an undetermined bias which is constant for a connected arc, and u is the TEC noise. The constant multiplying LI — L2 is equal to 9.52 x 1016 in International Standard (IS) units. Therefore, each 1 meter of differential delay between LI and L2 corresponds roughly to 10 TECU (1 TECU — 101 6e/m 3). The TEC noise is related to the LI and L2 noise by = (3,4) 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.2 A b so lu te T E C from PI and P 2 (TECa,n) Absolute TECa,^i (a for absolute, n for noisy) can be derived from PI and P 2 by taking the linear combination TEC™ = m r j f ) iP' - P i ) + c + ( where C is the sum of the transmitter and receiver differential code bias (DCB’s), which experience little variation and are usually assumed constant. The TE C I — a noise C is determined by the P I and P2 noise in a relation similar to (3.4). Since range measurements tend to be about 1 0 0 times noisier than those of the phase, a standard procedure is to solve for B in (3.3) by a least-square minimization of ^2(TECa,n — TECr,p)2, where the sum is over all measurements in a connected arc. This way, we obtain an absolute and precise TECa,p with a bias which is of order 1 - 3 TECU. 3.3.3 R ela tiv e T E C from LI and P I {TECr,n) Thus fax, it has been assumed that the GPS receivers tracks both LI and L2 signals. In many applications, the GPS receiver can be a single frequency receiver tracking LI only. In this case the TECr,n can be derived by taking TECr,n = ~ ( P l ~ L l ) + B + C (3.5) 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where B is an undetermined bias which is constant over a connected arc. We note that TECrtn is a relative TEC measurement with noise (, which is dominated by the P I noise. 3.3.4 R ela tiv e T E C from LI O nly (TECrtP tLi) If the range and clock terms (<W,C'*,C'J) in (3.2) are known, then the sum of the atmospheric and ionospheric delays contribute to a phase measurement (< W — d x TEE"J ) can be determined up to a constant bias. The range term can be determined from Precise Orbit Determination (POD) of the LEO and GPS satellites. GPS POD is routinely available by use of a ground network of receivers. LEO satellite POD is possible by use of GPS satellites in view. In addition, the clock terms can be solved relatively to an accurate clock on the ground by a process similar to ‘double differencing’ where four links (the occulting link and the dashed links in Figure 3.1) are differenced such that all clocks cancel out. When the tangent point of the occulted signal is above 80 km, the neutral atmospheric delay is less than 2 mm and can be, ignored, therefore a relative and precise TEC can be obtained. 3.3.5 LI and L2 am p litud e m easurem ents In addition to phase and range measurements, the receiver measures the voltage signal- to-noise ratio (VSNR) for both LI and L2 from which amplitude scintillation can be detached. Scintillation measurements, from which ionospheric irregularity maps can be obtained, are of vital importance to space weather because of their potentially detrimental effects on satellite and ground communications. Amplitude measurements can also be combined with the phase measurements to improve occultation retrievals 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. OPS MtdUle (it limit JO ,200 km) 50% 0 0 3000 2000 1000 0 *1000 -2000 -3000 -4000 0 0 - 1 0 0 0 x (Km, m agnetic) Figure 3.2: A 2-dimensional sketch of an observation link. in various ways including detecting and correcting for atmospheric multipath effects, accounting for diffraction for the purpose of improving the horizontal resolution and possibly localizing electron density irregularities along the GPS-LEO ray path. 3.4 M athem atical M odel For Observations Mathematically the TEC value of an observation gained by a receiver R and a sender S can be expressed as space discretization introduced in chapter 2 and the simplifying assumption that the density is constant in each volume element we can easily approximate this integral. grid. The TEC value is determined by the densities of the volume elements intersecting TEC r,s L where n is the electron density and L the link between sender and receiver. With the Figure 3.2 shows a link between a satellite and a ground receiver through the space 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the link and the corresponding lengths of the link inside the elements. Thus the observation can be written as (3.6) hit where n(i,j, k) is the electron density in the volume element and k) is the length of the link inside this volume. For our Kalman filter approach we have to deal with many simultaneous observa tions. As we are only interested in the densities on a bounded region, say the densities k) were 0 < * < iV), 0 < j < Nj and 0 < k < Nf. we can also represent density distribution as iV-dimensional vector n, were N — NiNjN^. W ith the same concept we can construct a corresponding observation matrix H. The rows Hi of H correspond to the values of /(.,.,.). As a result m simultaneous observations can be modeled as y° = Hn (3.7) were the observation matrix H is a m x N matrix and the elements of y° correspond to the individual observation measurements. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 The Kalman Filter In this chapter we introduce the Kalman filter, which provides a optimal recursive data processing algorithm. There are many ways of defining optimal depending on the performance measure chosen as criterion. However we will show that under the assumptions made in the next sections the Kalman filter is optimal with respect to virtually any criterion making sense. The word recursive in the above description means that the Kalman filter unlike some other assimilation concepts does not require that all previous measurements have to be kept in memory. This is an important issue for the practicality of filter implementations. The typical situation in which a Kalman filter can be used advantageously is a system of some sort driven by known controls, and measuring devices provide the value of certain pertinent quantities. The need for a filter becomes apparent as the variables of interest cannot measured directly, and some means of inferring these values from the available data must be generated. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Recursive Estim ation Process First we want to formalize mathematically the estimation process for which we want to use the Kalman filter. A dynamic model of a random process consists of tree parts 1. A discrete model for the evolution of some system from time U to U+i is governed by the equation x‘(fi+1) = + rji (4.1) where x*(-) and Mj are the system’s state vector and its corresponding dynamics operator respectively. The state vector x4(-) has dimension n. In this work we assume that the dynamics operator M, is linear and given by a n x n matrix. 77* is a noise process with zero mean and covariance matrix Efar/J) = Q?(%. In chapter 2 we gave an overview on how a system model for the ionosphere can be derived and showed with equation (2.15) that it indeed can be represented in the above form. 2. An initial random vector x*(to) describing the true state of the system at time to together with an initial estimate x-^(to)- In this work we assume that the initial estimate is the true system state: x / (t0) = x*(io) (4.2) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Observations y° at time fj are available in the form of y °(ti) = H + ei (4.3) where Hj is an observation operator, and e is a noise process. The observa tion vector y°(ti) has dimension mj. Like with the dynamics operator we only consider linear operators and assume that H is given as a mj x n matrix. The noise process ej has zero mean and its covariance matrix is denoted by Rj. We further require that R(.) is positive definite for all tk and satisfying E[eieJ] = Rj<5jj In chapter 3 we showed how this observations are obtained and how an obser vation matrix H is constructed. Equation (3.7) represents exact observation measurements, here however we also have to take care of measurement errors. In addition we assume that the random vectors x 4(fo), r](ti) and e(tj) are uncorrelated for i > 0, j > 0. 4.2 Statistical Concepts In this section we will put together the tools, which we need later to derive the equa tions for the Kalman filter. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.1 H ilb ert Space o f R andom V ariables For a given sample space with probability distribution P we define a Hilbert space of random variables. Let {yi, 2/2 , • • ■ iVm} be a finite collection of random variables {yi,y2, ■ ■ ■ , V m } with E(yf) < 0 0 for each i, then the Hilbert Space H consists of all random variables that are linear combinations of the j/j’s. The inner product of two elements x, y in H is given by Obviously he resulting Hilbert space is finite-dimensional and dim(H) < m. We want to generalize this for the case where the y^s are random vectors rather than random variables. For a collection of n-dimensional random vectors {yii V 2 i • • •, ym} we require, that all components have a finite second moment (E(yfJ) < 0 0 ). We define the Hilbert space H of n-dimensional random vectors as consisting of all vectors whose components are linear combinations of the components of the yi s. Thus an arbitrary element y in this space can be written as where all Ki s are real n x m matrices. We define the inner product as the expectation value of the n-dimensional inner product 00 < x ,y > = E(xy) = / xydP 00 y = Kiyi + K 2y2 H 1 - Knyn < x,y >= E(xTy) = trace(E(xyT)) 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The norm of an element x in H can be written as ||aj|| = V< x,x > = y E ( x Tx) 4.2.2 P ro jectio n T heorem The reason to introduce a Hilbert space for random variables and random vectors is the advantage that we can use the following famous projection theorem for our statistical problems. The Classical Projection Theorem Let H be a Hilbert space and M a closed subspace of H. Corresponding to any vector x G H, there is a unique vector mo G M such that ||x — m0\\ < ||a; — m|| for all m G M. Furthermore, a necessary and sufficient condition that m0 6 M be the unique minimization vector is that x — m 0 be orthogonal to M. The 2-dimensional case is illustrated in Figure 4.1. In this trivial case H = IR2 and the subspace M is a line through (0,0). We intuitively agree that the best choice for mo is such that x — mo is perpendicular to the line. The big advantage mentioned above is that problems of minimizing norms become as simple as this in any Hilbert space. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.1: The projection theorem . 4.2.3 M inim um V ariance E stim ate Let x and y be random variables of dimension n and m respectively. Based on x we want to find an estimate y for y. In particular we seek a linear estimate y of the form y = Kx where K is a constant m x n matrix. A natural criterion for optimality for this kind of problem is minimization of the norm of the error. Using the Euclidean m-space norm this cost function of the mini mization problem can be written as m m E{\\y - yll2) = E (% 2(yj - Vj)2) = E (yi ~ y t f j =i j =i 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore the problem can be decomposed into m sub-problems, which separately minimize E(yj — yj)2 for j = 1,... ,m. If we denote the j-th row of K with Kj this can be express in terms of yj and x by minimize E(yj — Kjx )2. (4-4) Now we finally brought it into a form were the projection theorem can be applied. The solution to the problem (4.4) is choosing Kj in such a way, that yj — KjX is orthogonal to the subspace spanned by the components x i ,...,x n of x. Using the the inner product for the Hilbert space of random variables we can find the sufficient and necessary condition for Kj yj — KjX _ L span-faq,... ,xn} <yj - KjX, Xi >— 0 i = 1,..., n < f> - < yj, X{ >—< KjX, Xi > i = 1,..., n ■&E(yj,Xi) — E(KjX,Xi) i - l , . . . , n Writing this normal equations for all m sub-problems in matrix form we get E(yxT) = K E ( xxt ) 4^ K = E(yxT)[E(xxT)]~1 (4.5) 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and we call y = Ky = E(yxT)E(xxT) lx the linear minimum variance estimate of y based on x. The corresponding error covariance matrix can be easily computed, as E{{y - y){y - y)T) = E(yyT) - E(yyT) = E(yyT) - E(yxT)[E(xxT)]~lE(xyT) (4.6) 4.3 Recursive Filtering Algorithm Our goal is to estimate the system’s true state x*(-) at times to, ii, £2 • • • • Whereby an estimate x at time fj is considered to be optimal, if £7(||x*(ij) — x||2) is minimal. If we know the true system state at time ti or if we have an optimal estimate x, we will show that an optimal estimate for fj+i can be computed using the system model given by (4.1): x^ (fj+i) = MjX. This is called the forecast estimator and we will use the superscript (-)^ to denote it. After obtaining observations y° at time U we can adjust or update the a priori estimate xf . This new estimate is denoted by the superscript (•)“ for analysis and depends on (y°,xf) The computations for x-f and xa for time ti are called forecast step and update step respectively and constitute the recursive Kalman algorithm. How the filter works is shown in Figure 4.2. 4.3.1 Forecast Step In the forecast step we assume that for a time ti we already have a linear minimum variance estimate Kxi of the true system state x\ = x1^ ) . 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [Observations] [Forecast] [Update] Ionospheric Model Initial state Optimization Ionosphere Figure 4.2: T he w orking concept of th e K alm an filter. For the case i = 0 we can assume x = a:*(to) the initial state of the system, which we required to be known. Otherwise we will take the estimate gained from the previous forecast or update step (if measurements were available). The true system state at time ti+i is given by the model equation xi+1 = x (*»+i) = M(ffc)x (fj) + rj(ti) (4.7) and we want to find a linear minimum variance estimate based on x. By (4.5) we know that K was chosen in order to minimize E{\\x\ — K x ||) and therefore is given by K = E{x\xt )E(xxt ) The normal condition in (4.5) gives us as the optimal linear estimate of x\+l based on x by Lx = E{x\xt )E{xx1) (4.8) 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Substitution of the model equation (4.7) into (4.8) results in Lx = i?((M jX* + rji)xT)E(xxT)x — [M.iE('xt ixT)E(xxT) + E(r]iXT)E(xxT)]x = Mj E ( x \ x t ) E ( x x t )x = M iKx (4.9) were we used the requirement that for the system noise EfarjJ) = 0 for % ± j and as the estimate x only depends on the previous system noises we can conclude E(r]iXT) = 0. The practical impact of this is that once we gained an optimal estimate for time U we can get an optimal estimate for time f j+i by using the matrix of the model equation (4.7). This is the formal justification for the computation of the estimate x{ for x\+l in the forecast step x { +1 = M jx “ (4.10) based on the apriori estimate x“ for x\. The error covariance of x{+1 is given by P/+ 1 = E{{x\ + 1 - x{+l){x\+l - x{+1)T) - M iP?M j + Qi (4.11) 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3.2 U p d a te Step Without any additional measurements the forecast estimate x^(tk) is the best estimate of the system state at time t& . Assume we obtain at time tj in addition observations in the form of y? = iZizjj. + e* (4-12) with E(ek) = 0 and E(e[ek) = Rk- If we would estimate x\ solely based on this new data y°, we would give up all the information collected in the past. What we want to do is improvement of the forecast estimate based on the additional information about x\. The need to update is caused by d(tk) = y0{tk) — H(tk)xf (ffc), the so called inno vation vector, which we can interpret as a indicator for how wrong we predicted the state. The updated estimator for time tk has the form xa(tk) — xJ(tk) + Kd(tk)- We again will use the projection theorem to find a K such that E(\\xa( t k ) - A t k)\\2) = E{\\Kd(tk) + xf{tk)-x\tk)\\2) (4.13) becomes minimized. With the normal equation in (4.5) we know that the optimal K is given by K = E {{x \tk )-x f{tk))d{tk)T)[E{d{tk)d(tk)r )]-1 (4.14) 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Expanding the innovation vector d(ffc) and using that x? (t^) is uncorrelated to e(tfc) we get: E(didJ) = E M - H i x l M - H i x l f ] = E[(Hi{x\ - x{) + ei)(Hi(x\ - x{) + ei)T] = H i E [ { x \ - x{)(xl - x{)t ]H? + E(eief) = H iP /H f + Ri (4.15) E Kxi - xi)dJ] = E Kxi ~ xi)(Hi{xi ~ xi) + eif] = E[(x\ - x{) (xl - x{)T}Hj = P f H l (4.16) Further substitution of (4.15) and (4.16) into (4.14) results in K = P /H ? (H i P / H .f + R{ r 1 (4.17) the so called Kalman gain. This constitutes the so called update step which gives us a linear minimum variance estimate x\ for the system state x\ at time U based on the a priori estimate x{ and the measurements obtained in (4.12) xi = x { + K H i ( x * i - x { ) = x{ + P f H j (.HiPlHj + R{T'Hiixl - x {) (4.18) 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Whereby the error covariance of this estimate is ^[(*i -*<)(*“ = Pf-KiHiPf (4.19) 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Com putational R esults 5.1 Structure of the Simulation Process In order to test the correctness of the Kalman filter implementation we use our sim plified model of earth’s ionosphere, which we derived in chapter 2, and use the filter in the context of an Observation System Simulation Experiment (OSSE). An OSSE is an controlled simulated experiment where data synthesized based on the ‘true’ state of the medium being studied, and then assimilated after adding the proper kind of noise. In our OSSE scenario (Figure 5.1) we can see the five steps building a single OSSE- run: 1. Grid Generation-. A grid for the ionosphere is created according to the space discretization described in Section 2.2.2. A graphical representation is given in Figure 2.2. In our simulation the ionosphere was separated into 876 regions. 2. Create Initial State: To create a realistic initial state of the ionosphere we start a ion density of zero in all volume elements. Then we use the mathematical model for earth’s ionosphere introduced in chapter 2 to simulate the change of state for the next 48 hours. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C re a te grid Truth Run K alm an filter K alm an Run C re a te initial s ta te G en eratio n of noisy d a ta Figure 5.1: Flow chart for th e O bservation System S im ulation E xperim ent (OSSE). 3. Truth Run: The ‘true’ state of the ionosphere is generated by the same iono spheric model, but in time steps of 6 minutes we disturb the state. We add a normal distributed noise with zero mean and standard deviation asys. This disturbed state is stored on hard disk and used as input state to simulate the next 6 minutes. For the simulations in this work, we generated ‘true’ states for a time of 2 hours i.e. 21 files containing the states after 0, 6, ... , 120 minutes. Therefore each file contains data of the form given by (4.1). 4. Generation of Data: For the purpose of our study we assumed 28 GPS trans mitters and 47 GPS receivers. In the simulated two hours more than 6000 data links were created. Figure 5.2 shows data links for 4 of the GPS transmitters and 5 of the ground receivers over this two hours. Given the receiver-transmitter geometry, synthetic TEC values are generated based on the ‘underlying contin uum of the true’ electron density. In order to simulate the observational error 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. •3 -3 -2 - 1 0 1 2 3 x 1 G 7 Figure 5.2: D a ta links created by 5 ground receiver an d 4 G PS tra n sm itte rs in 2 hours. a normal distributed noise with zero mean and standard derivation a0t,s is add. The data containing all measurements for the two hours is saved on disk. This data is of the form given by (4.12). 5. Kalman filter/Kalman Run Once the ‘true’ states and observation data is gen erated we can start the Kalman filter. As input we need • initial state and standard deviation asys the system noise of the model. • the observation data and the standard deviation aQ bs of the observation noise. In order to show that our Kalman filter implementation works correct we have to show that our optimality criterion E[(x* — xa)T(xt — xa)\ is reached as predicted. By repeating the OSSE described above for 100 times we want to give an empirical proof that is actually the case. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Theoretical Results for Comparison We can get the theoretical value for the norm of the error e“ at time U through the error covariance matrix P “ of x\\ e? = £ ( ||4 - x \ ||2) = E ( ( x \ - 4 > f ( 4 - i f ) ) = trace[£((4 ~ 4 ) ( 4 - 4 ) T)] = trace(Pf) (5.1) We will use this value as comparison for the average error e in our simulation runs, e is computed as ^ = ll^i-^ll2 (5-2) r = l where the sum is over the N OSSE runs and i is the time index. In reality it is not possible to calculate the theoretical error as the true values of the model states are unknown. The only way we can express the ‘goodness’ of the estimates produced by the Kalman filter is the innovation vector dy. Therefore another goal of the OSSE could be to investigate if it is possible to infer from the innovation vector the quality of the estimates gained by the Kalman filter. If we have at any time the error covariance matrix Q for the system noise and R for the observation noise the expected norm of the innovation vector dyi at time i is given by E(\\dyi\\2) = E(dyfdyi) = trace[E(HPf H T + R { )], (5.3) 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where we use the covariance matrix of dyi derived in (4.15). 5.3 Evaluation of the Simulation Runs For a system noise with asys = 101 0 and an observation noise with a0 b s = 101 1 we run 100 OSSEs. The comparison between the norm of the estimation error is shown in Figure 5.3. The thick dashed line shows the theoretical value, the solid line the mean error norm after the 100 runs. experimental error theoretical error lime (In steps of 6 minutes) Figure 5.3: The theoretical error compared to the experimental error. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the same runs the norm of the innovation vector is shown in Figure 5.4. The thick dashed line shows again the theoretical norm, the solid line the observed norm. 12 TEC 10 8 6 4 2 experimental value theoretical value 0 20 5 25 10 15 0 time (in steps of 6 minutes) Figure 5.4: The theoretical norm of the innovation vector compared to the experim ental value. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 A ppendix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] C. Wang, ’ ’Notes on Ionosphere Modeling and Simulation Assimilation Using GPS Measurements”, Department of Mathematics at USC, Los Angeles, 1999. [2] Michael C. Kelley, ’ ’The Earth’s Ionosphere”, Academic Press, New York, 1989 [3 ] David G. Luenberg, ’ ’Optimization by Vector Space Methods”, John Wiley & Sons, Inc., New York, 1969 [4 ] Peter S. Maybeck, ’ ’Stochastic Models, Estimation and Control”, Academic Press, New York, 1979 [5] G.A. Hajj, L.C.Lee, X.Pi, L.J. Romans, W.S.Schreiner, P.R. Strauss, C. Wang, ’ ’COSMIC GPS Ionospheric Sensing and Space Weather”, Department of Math ematics at USC, Los Angeles, 1999. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Lehn, Michael Christian (author)

Core Title A Kalman filter approach for ionospheric data analysis

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Degree Master of Science

Degree Program Applied Mathematics

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

Tag Mathematics,OAI-PMH Harvest,Physics, Atmospheric Science

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Advisor Wang, Chunming (committee chair), Hajj, George (committee member), Rosen, Gary (committee member)

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