a block iterative algorithm for tomographic reconstruction of ionospheric electron density

A Block Iterative Algorithm for Tomographic Reconstruction of Ionospheric Electron Density

Eric Sutton and Helen Na

Department of Electrical and Computer Engineering, 4000 Engineering Building, University of Iowa, Iowa City, Iowa 52242

ABSTRACT

Ionospheric tomography is a technique whereby a vertical cross section through ionospheric electron density can be imaged. The vertical resolution of ionospheric tomography systems is inherently poor, but can be improved by using a priori information in the tomographic reconstruction algorithm. Care must be exercised in using a priori information, since if too much of it is used, the reconstruction algorithm may discard some of the information contained in the tomographic data in favor of satisfying some of the a priori assumptions. Orthogonal decomposition (OD) is an existing technique that uses a priori information to constrain t h e reconstruction to lie in a space of reasonable images without weighting the reconstruction toward any particular a priori image. In this way a priori information can be used in a manner that does not overwhelm information contained in the data. Gauss-Seidel (GS) is an iterative algorithm that is used to calculate solutions for large systems of linear equations. In this article, a block version of the GS algorithm will be used to calculate the solution of the least-squares problem that is created using OD. The complete algorithm presented here will be called the residual correction method (RCM), since it involves calculation of successively better approximations based on the residual error. RCM is a fast and numerically stable algorithm that extracts as much information from the data as possible. A numerical example demonstrating the properties of RCM will also be presented. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION The distribution of electron density in the ionosphere exhibits a complicated structure and dynamic behavior that has been studied for many years. The ionosphere affects the reliability of satellite communications and the accuracy of radio astronomy. Information about the ionosphere can also be used to locate the source of radio transmissions that have been reflected from the ionosphere. The ionosphere is also a naturally occurring laboratory for plasma physics.

There are several techniques for measuring ionospheric electron density. The ionosonde is essentially a one-dimensional radar looking straight up. Electron density can be measured with an ionosonde only up to the altitude of the maximum electron density. Ionospheric electron density can also be measured with an incoherent scatter radar. Incoherent scatter radar measures electron density in three dimensions, but there are only eight or nine of these radars in the world, so coverage is limited.

Received 2 February 1996; revised manuscript received 5 April 1996

Ionospheric tomography is another technique that can be used to obtain information about ionospheric electron density [ 1-61, An ionospheric tomography system consists of a navigational satellite orbiting at about 1000 km altitude and several ground stations in a line directly underneath the satellite’s path. A typical configuration for an ionospheric tomography system is shown in Figure 1 . The data measured at the ground stations are total electron content (TEC). The TEC data can be modeled as

TEC = 1 N ( i ) dl , (1) P

where N(;) is the electron density, and p is the path between the receiver and the satellite. Thus, the data take the same form as those used for computed tomography. Ionospheric tomography systems are much less expensive than incoherent scatter radar systems and offer the possibility of worldwide coverage. The practical feasibility of ionospheric tomography has been demonstrated in several recent studies [6-131.

Notice in Figure 1 that all of the paths are between the two concentric circles formed by the surface of the earth and the satellite path; none of the paths lie across the ionosphere. This is a fundamental limitation associated with ionospheric tomography systems, and it is for this reason that ionospheric tomography systems have very poor vertical resolution. However, the problem with poor vertical resolution can be minimized through careful design of the tomographic reconstruction algorithm [ 14-20].

For many traditional applications of tomography, the data are

International Journal of Imaging Systems and Technology, Vol. 7, 238-245 ( I 996) 0 1996 John Wiley & Sons, Inc.

\

Earth Surface 7 - / Earth Center

Figure 1. Geometry of ionospheric tomography system.

CCC 0899-9457/96/030238-08

gathered in a set of parallel projections evenly spaced over a range of 180". If the data from an ionospheric tomography system are sorted into parallel projections using the geometry shown in Figure 2, then the limitations inherent in ionospheric tomography systems can be stated as follows:

The range of view angles is limited; that is, the angle that each data ray makes with the vertical is limited, and none of the rays pass horizontally through the ionosphere. The number of data points in each projection can be no greater than the number of ground stations. The data points within each projection are unevenly distribut- ed.

Therefore, compared to a traditional tomography system, the ionospheric tomography system suffers from missing or sparse data. The traditional algorithms used for tomography applications where complete data are available do not work well for ionospheric tomography without significant modification. Practical ionospheric tomography algorithms must use a priori information to minimize the effects of incomplete data and enhance the vertical resolution of the system.

Many different techniques have been used for ionospheric tomography reconstruction. Reconstruction techniques fall into two broad classes: 1 ) techniques that use pixels as the basic element of reconstruction, and 2 ) techniques that use a set of' functions other than pixels as the basic element of reconstruction. In addition, the way in which a priori information is used for reconstruction varies significantly from one algorithm to another.

The most commonly used algorithms based upon pixels are algebraic reconstruction technique (ART), simultaneous iterative reconstruction technique (SIRT), and multiplicative algebraic reconstruction technique (MART) [1,2,6,8,21]. ART, SIRT, and MART use a priori information in the form of an initial guess at the solution. An algorithm that uses the a priori information contained in a set of model ionospheres to supplement the information contained in the data has been proposed by Raymund et al. [22]. Another algorithm that uses smoothness constraints instead of model ionospheres has recently been proposed by Fehmers [23,24]. A technique called phase-difference tomography

/ /

has been used to remove calibration offsets associated with the receivers [5,7,25,26].

Reconstruction can also be performed using functions other than pixels as the basic element of reconstruction. The image is converted to pixels only as a final step before display. There are two separate rationales for using nonpixel functions: 1 ) Functions can be chosen so that fewer functions are needed to adequately approximate the ionospheric electron density, and 2) functions can be chosen so that the reconstruction algorithm is simplified. Fremouw and Secan proposed an algorithm based on stochastic inverse theory [4,9], and Raymund et al. proposed expansion in model ionospheres [19,21,27]. These algorithms use a priori information to determine what functions to use as the basic element of reconstruction. In addition, the traditional algorithms filtered back projection (FBJ) and direct Fourier method (DFM) have been adapted for use in ionospheric tomography [28,29,31].

None of the existing algorithms for ionospheric tomography has been shown to be clearly superior to the others; each algorithm has its strengths and weaknesses. All ionospheric tomography algorithms use a priori information to improve vertical resolution; however, some algorithms use a priori information in a very limited way, while others use a priori information in such a way that some of the information contained in the data could potentially be neglected. Also, some of the existing algorithms are very computationally intensive.

This article will present a new algorithm for ionospheric tomography called the residual correction method (RCM). The new algorithm is based on the theoretical framework provided by orthogonal decomposition (OD) [3,20,29,32], and on the numerical technique known as the Gauss-Seidel (GS) algorithm [33].

Orthogonal decomposition is used to create a model equation for data collection that must be inverted to calculate the solution. However, since the model equation does not have a unique solution, it is not possible to invert i t without additional a priori assumptions. Since OD already provides a priori information, it is desirable that any additional a priori information be no stronger than necessary to calculate a unique solution.

The GS algorithm is an iterative algorithm used to solve large systems of linear equations. This article presents a block version of the GS algorithm for ionospheric tomography. The formulation of the GS algorithm presented here is designed to take advantage of the special structure of the model equation. A priori information is used to determine the composition of the blocks and the order in which they are processed.

II. BACKGROUND A. Orthogonal Decomposition. Orthogonal decomposition provides a theoretical framework upon which RCM is based [3,20,29,32]. For OD, the source image g(?) is expressed as the weighted sum of a set of orthonormal basis images:

Center of Rotation

J

R ( i ) = c (2) ,=I

Usually, the basis functions $(?) are orthonormal as defined by

( 1 , i f i = j ; (3) 0 , otherwise, +,(f')d$?) dA =

where R is the region of reconstruction. The objective is to calculate estimates of the parameters xj so that the source image can be reconstructed using Equation (2). To relate the projection

\

Figure 2. Geometry for sorting data into projections.

Vol. 7, 238-245 (1996) 239

data to the weights xJ, Equation (2) is integrated along each ray corresponding to an element of the projection data:

(4)

where p, is the ray corresponding to the ith element of the projection data. Then Equation (4) represents a set of I simultaneous linear equations that must be solved for the parameters x , . . . x , . Equation (4) can be expressed in matrix notation by making the following definitions:

b = [ b , b, . . . b, ] ' , ( 5 )

where

b, = I,, s ( 3 dl .

Thus, b is the vector of projection data.

x = [ x , x , . . . x,]' (7)

Thus, x is the vector of the unknown parameters x i .

A = [a , , ] > (8)

where

The matrix A is called the system matrix. The system matrix depends upon the set of paths p, and the basis functions In other words, A is determined by the imaging system geometry and the basis functions chosen for the source domain.

Equation (4) can then be written in matrix form as

A x = b . (10)

Equation (10) can be interpreted as a model equation for data collection; given an image of ionospheric electron density contained in x , the data contained in b can easily be calculated. The tomography problem then consists of inverting Equation ( 1 0) and solving for x . Since there are normally many more equations than unknowns, Equation (10) must usually be solved in the least- squares sense. For ionospheric tomography, the problem is usually mixed under and over determined.

Orthogonal decomposition provides a way of using a priori information in the reconstruction algorithm. A priori information is entered into the OD algorithm by choosing the basis functions so that the solution is constrained to lie in the space spanned by a set of model ionospheres. The model ionospheres should be chosen to span the space of all reasonable solutions to the reconstruction problem.

B. Separable Basis Functions. The basis functions used for OD are separable in polar coordinates. Each basis function can be expressed as

orthogonal is the section of the annulus defined by 13, < 0 < 0, and r I < r < r , .

The set {4,J(x, y ) } is orthonormal if each of the sets {,f,(O)} and {qJ(r)} are orthonormal. The set {[,(O)} must satisfy

i f m = n ; 0 , otherwise,

and the set {v,(r)} must satisfy

Note that there is a radial weighting factor that must be included in the construction of {v,(r)} to take into account the curvature of the imaging region.

For ionospheric tomography, since horizontal resolution is generally good, a priori information is needed primarily for the vertical direction. Therefore, basis functions that span the entire space in the horizontal direction may be used. Pixel basis functions, Fourier basis functions, and Legendre polynomials are all possible choices. The horizontal resolution of the reconstruction depends, in part, on the number of basis functions used.

If the vertical basis functions span the entire space in the vertical direction, then the vertical resolution will be very poor, since no a priori information has been used. Instead, the vertical resolution can be improved by choosing vertical basis functions based on a set of model ionospheres. In this way, the space in which the reconstruction will lie is limited based upon a priori information.

Vertical basis functions are calculated from a set of sample vertical profiles obtained from an ionospheric model [20,32]. Let the samples of a set of model ionosphere vertical profiles be entered as the columns of the matrix F :

F = [.f, f, f, ' ' .I 3 (14)

where ,l, are vectors containing the samples of a vertical profile of the ionosphere obtained from an ionospheric model. Let R be a diagonal matrix with the same number of rows as F , where each diagonal entry is proportional to the radial coordinate of the sample in the corresponding row of F :

where r,,? is the distance from the center of the Earth of the sample contained in the mth element of J;,. The product R"*F may be decomposed using the singular value decomposition as follows:

R"'F = uzv' , (16)

where the columns of U are on an orthonormal basis for the column space of R' / 'F , the columns of V are an orthonormal basis for the row space of R ' / * F , and 2 is a diagonal matrix of singular values. Define the matrix U, as

U, = R - ' / ' U .

Then F can be written in the form

F = U,zV'. (18)

The samples of the vertical basis function r],(r) are obtained from where 0 is latitude and r is distance from the center of the Earth. The region R in Equation (3) over which the set {+,,(I3, r ) } must be

240 Vol. 7, 238-245 (1996)

thej th column of UR, and the set of vertical basis functions {q,(r)} will be orthogonal with a radial weighting factor as in Equation (13), since

(19)

The vertical resolution of the reconstruction is enhanced, because the solution is restricted in the vertical direction to a space of reasonable solutions. On the other hand, the solution is free to lie anywhere in the space of reasonable solutions without being weighted toward one particular most probable solution.

UZRU, = (R'"UR)T(R1/2UR) = UrU = I .

I l l . RESIDUAL CORRECTION METHOD The GS algorithm is an iterative method that is used to solve large systems of linear equations [33]. A block version of the GS algorithm is used as part of the RCM. The central concept behind RCM is that if a priori information is used to construct the basis functions, then additional a priori assumptions should not be necessary to perform the reconstruction. The block version of the GS algorithm allows the reconstruction to be performed without additional a priori assumptions in the form of regularization.

For RCM the system matrix is partitioned into submatrices which are then inverted separately. RCM iterates through the submatrices, using the residual vector from the previous iteration as the new data vector. In this way, an approximate solution is first obtained then improved as the iteration continues. In addition, the submatrices can be chosen so that the rate of convergence is accelerated during the first few iterations.

A. The Algorithm. Suppose there are P vertical basis functions, Q horizontal basis functions, and T data points. Then the system matrix A is T X PQ and contains a row for each data point and a column for each basis function. Let the columns of A be ordered so that all columns corresponding to basis functions containing the first vertical basis function come first, followed by all columns corresponding to basis functions containing the second vertical basis function, etc. The matrix A and vector x can then be partitioned as

where each A,> is T X Q and each x,, is Q X 1. Since each of the submatrices of A contain the columns corresponding to only one vertical basis function, when the submatrices are considered separately the problem reduces from one two-dimensional reconstruction to a set of P one-dimensional reconstructions. Assuming that each of submatrix represents an overdetermined problem, define the generalized inverses of the submatrices as

A," = (AKAp)- 'A;

Then the RCM algorithm is initialized using

x ( o ) = 0 , (22)

6"" = b , (23)

The vector x ( ' ~ ) is the initial solution vector, and the vector b"" is the initial residual vector. The algorithm proceeds by iterating through

(24)

x;') + &, (25)

- A g & , (26)

& = A;?.bb"-'),

b " ' = b " - l ~

where the subscripts i on A t and x, are interpreted modulo P. Equation (24) calculates a correction to the solution based on one submatrix of A. Equation (25) adds the correction to the solution vector, and Equation (26) adjusts the residual vector to reflect the new solution. Each iteration of Equations (24), (25), and (26) will be called a subiteration; and a set of P iterations, where P is the number of vertical basis functions, will be called an iteration. The magnitude of the residual vector b"' is reduced at each subiteration, and the iteration is halted when the magnitude of b"' is decreasing at a sufficiently slow rate.

To facilitate analysis of convergence, it is convenient to write the algorithm in a more compact form. Equations (24) and (25) can be combined and written in terms of the vector x as follows:

Then define

B, 1 [ 0 Y X T

Also, the residual vector can be written as

(29)

Combining Equations (27), (28), and (29), the complete iteration can be written as

b " - l ' = b - A x " - l '

+ B,(b - A x " - " ) , (30) x ( I = x ( I - I

where the subscript i on B, is interpreted modulo P. The algorithm is initialized using Equations (22) and (23) in the same way as before.

Since electron density is a nonnegative quantity, nonnegativity is a constraint commonly applied to ionospheric tomography problems. The solution calculated by the RCM algorithm tends toward a nonnegative solution, although a nonnegative solution is not guaranteed. If the solution produced by the RCM algorithm contains negative electron densities, then it is likely that the tomographic data are inconsistent with the a priori assumptions used in creating the model equation.

B. Example. A simple example will illustrate how RCM works. The original image to be reconstructed is shown in Figure 3. Data were simulated for 15 receivers evenly spaced from -28" to 28" latitude. Satellite positions were in 2" increments from -39" to 39" latitude. The minimum elevation for any data ray was 18", not unlike that of Figure 1.

Three vertical basis functions will be used for the reconstruction. The vertical basis functions were calculated from the original image using the methods discussed in Section IIB. The vertical basis functions are shown in Figure 4. Notice that all of the vertical

Vol. 7, 238-245 (1996) 241

0.3 0.4d 0.2

al 0.1

al C O

800 - h

U 3

n, Q

c Y .- U

a 400 = -0.1

v.9- -0.2 0.

200 -0.3

0 -30 -20 -10 0 10 20 30

Latitude (degrees)

Figure 3. Original image,

basis functions decay to zero at extreme low and high altitude. This is because ionospheric electron density decays to zero at extreme low and high altitude.

Twenty-one horizontal basis functions will be used for the reconstruction. Legendre polynomials are used for the horizontal basis functions. The first eight horizontal basis functions are shown in Figure 5. Since there is much more horizontal information than vertical information contained in the data, there are many more horizontal basis functions than vertical basis functions. The horizontal basis functions are chosen to span the entire space in the horizontal direction.

Since there are three vertical basis functions, the system matrix A and the unknown vector x are subdivided as follows:

[ A , A , A, ] xz = b (31) [::I The submatrices are ordered so that the first submatrix contains

all of the columns corresponding to the first vertical basis function. Because of the properties of the SVD, the first vertical basis

Altitude (km)

Figure 4. Vertical basis functions (VBFs).

I -40 -20 0 20 40

Latitude (degrees)

-0.4'

Figure 5. First eight horizontal basis functions.

function is an average ionospheric electron density profile. This accelerates the convergence and contributes some a priori information that helps make the solution nonnegative.

The algorithm is initialized using Equations (22) and (23), and the first subiteration is performed as given by Equations (24), ( 2 5 ) , and (26). The original image is reproduced for convenience in Figure 6a, and the solution after the first subiteration is shown in Figure 6b. The reconstruction at this stage shows the horizontal structure of the original but not the vertical structure. The reconstruction consists of the first vertical basis function multiplied by a weight in the horizontal direction. The reconstruction after the first subiteration serves as an initial guess at the final solution.

The solution after the second subiteration is shown in Figure 6c. The solution now shows the effect of the inclusion of the second

I I Latitude

(b)

I 1 Latitude

(4 Figure 6. Comparison of original image and solution after first three subiterations. (a) Original image. (b) Solution after first subiteration. (c) Solution after second subiteration. (d) Solution after third subiteration.

242 Vol. 7, 238-245 (1996)

vertical basis function. The vertical structure of the original image is evident in the reconstruction.

The solution after the third subiteration is shown in Figure 6d. Even though the magnitude of the residual vector decreases from the second subiteration to the third subiteration, there is no obvious difference in the solution. This is because the vertical basis functions were chosen so that most of the energy of the original image is accounted for by the first two vertical basis functions.

The final solution after 300 iterations is shown in Figure 7. In the regions below -30" latitude and above 30" latitude the data are too sparse for the solution to be meaningful.

C. Convergence. There are two important issues relating to iterative algorithms: 1 ) Does it converge to a solution? and 2) To what solution does it converge? The second issue will be discussed first.

The algorithm cannot cycle through different solutions in the subiterations and arrive back at the same point at the end of an iteration, because different elements of the unknown vector x are changed in each subiteration. Therefore, convergence will be considered in terms of iterations rather than subiterations. Equation (30) shows that if the algorithm converges, then it converges to a point x where

B,(b - A x ( ' - I)) = 0 V1 5 i 5 P . (32)

Equation (32) can be written as

[ B1](b - Ax) = 0 , (33) BP

and after omitting rows of zeros from the matrix on the left, Equation (32) can then be put into the form

Ar(b - A x ) = 0 . (34)

Since the block diagonal matrix on the left in Equation (34) is obviously invertible, the algorithm converges to a point where

Ar(b - A x ) = 0 . ( 3 5 )

J -30 -20 -10 0 10 20 30

Latitude (degrees)

Figure 7. Reconstruction

In other words, the algorithm converges to a least-squares solution, though not necessarily the minimum norm least-squares solution. This is as expected, since RCM incorporates a block version of the GS algorithm.

It remains to be shown that RCM actually does converge. The solution to which the algorithm converges satisfies

x = x + Bi(b - Ax) . (36)

If Equation (36) is subtracted from Equation (30), then

x ( ~ ) - = x ( ~ - I J - x +B,(b - A x " - ' ) ) - B , ( b - A X ) (37)

Define the error at the ith subiteration as

and define the error propagation matrix as

P

M = n ( I - B,A) . ,,=I

Then the error propagates according to the equation

The spectral radius of a matrix is defined as the maximum of the magnitudes of the eigenvalues of the matrix. If the error propagation matrix M satisfies the condition

where r, ,( .) denotes the spectral radius, then the algorithm converges. This condition can be checked before the iteration begins.

The rate of convergence depends upon the properties of the matrix M , but is also significantly affected by the order in which the submatrices appear in the algorithm. The submatrices are ordered in the same sequence that the corresponding vertical basis functions appear in the SVD matrix UR in Equation (17). There- fore, if the image is reasonably close to the a priori data set, then it is expected that

11x1 II >> llxzll>> . . . >> IlXPll . (44)

If condition (44) is satisfied, then the first few iterations take the solution very close to the final solution.

For the example problem of Section IIIB, the spectral radius of M is given by

rJM) = 0.99999999982897 ;

however, the algorithm actually converges much more quickly than the spectral radius of M would indicate. The spectral radius of M indicates a worst case rate of convergence that is never en- countered in practice.

Figure 8 shows the convergence curve for the example problem of Section IIIB. For this example, three subiterations are equivalent to one iteration. The algorithm converges very quickly during the first few iterations.

Vol. 7, 238-245 ( 1996) 243

types of noise is to reduce the vertical resolution of the reconstruction.

N 0.2

5 1 200 400 600 800 1 3 Sub-iteration number Sub-iteration number

(4 (b) Figure 8. Error curves in data domain: lib- Ax‘”llnli~bl12. (a) Error curve for i = O to 6. (b) Error curve for i = 7 to 900 with expanded vertical scale.

IV. CONCLUSIONS A new algorithm for ionospheric tomography has been presented. The new algorithm is built on a foundation of OD with separable basis functions. The vertical basis functions are based on an a priori set of ionospheric electron density profiles. In this way the reconstruction is limited to lie in a space consisting of reasonable solutions, thus improving the vertical resolution of the reconstruction. O D by itself does not provide a solution, because even with basis functions based on a priori information the solution is not unique.

The residual correction method is a fast, numerically stable, iterative algorithm for solving for the weights associated with the basis functions. R C M takes advantage of the structure of the system matrix to accelerate convergence. The first vertical basis function serves as an initial guess at the vertical structure of the electron density. Additional vertical basis functions are used to reconstruct variations in the vertical structure of the electron density. Convergence of R C M depends upon the spectral radius of the error propagation matrix. The stability and convergence of RCM have been demonstrated with a numerical example.

For ionospheric tomography systems noise comes from several sources: 1 ) random calibration offsets associated with each receiver, 2) measurement noise associated with individual data elements, and 3) discretization noise within the reconstruction algorithm. The R C M algorithm can b e easily modified to calculate the receiver calibration offsets simultaneously with reconstruction [30]. The noise associated with the differential Doppler phase measurements is very low. T h e typical signal to noise ratio is around 4 0 d B ; therefore, this is not a significant source of error. Furthermore, the convergence of the R C M algorithm depends on the properties of the error propagation matrix, and the error propagation matrix is not a function of the measurements; thus, the convergence of the algorithm is not affected by measurement noise. In the R C M algorithm the discretization noise is reduced, though not entirely eliminated, by using smooth basis functions instead of pixels for the reconstruction. The most significant effect of all three

ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant ATM-9419552, and the Office of Naval Research under Grant N00014-95-1-0850.

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a block iterative algorithm for tomographic reconstruction of ionospheric electron density

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