All Faculty | Duke Electrical and Computer …people.ee.duke.edu/~cummer/reprints/043_Zhai05_RadSci_To...A flexible and robust direct reconstruction method for magnetospheric radio

A flexible and robust direct reconstruction method

for magnetospheric radio tomography

Yuhu Zhai and Steven A. Cummer

Electrical and Computer Engineering Department, Duke University, Durham, North Carolina, USA

Received 31 May 2004; revised 15 October 2004; accepted 4 January 2005; published 12 May 2005.

[1] We present a flexible and robust direct reconstruction method for magnetosphericradio tomography. We show that for a combined reconstruction of plasma density Ne andmagnetic field B, the direct reconstruction method performs as well as popular iterativemethods such as algebraic reconstruction technique (ART) and multiplicative algebraicreconstruction technique (MART) for large number of satellites, but it performssignificantly better when the number of satellites is small. The main advantages of thismethod are that extra information, such as in situ measurements, can be easily and flexiblyincorporated into the reconstruction; it is relatively robust in the presence of noise; and it isless sensitive than other methods to numerical reconstruction parameters like assumed gridsize. We demonstrate the good performance of this method in reconstructing electrondensity and magnetic field using constellations of relatively few satellites (11 and fewer)in a single orbit in a variety of magnetospheric regions. Although this method is relativelyrobust to noisy measurements, local measurements can significantly increase thereconstruction accuracy.

Citation: Zhai, Y., and S. A. Cummer (2005), A flexible and robust direct reconstruction method for magnetospheric radio

tomography, Radio Sci., 40, RS3004, doi:10.1029/2004RS003100.

1. Introduction

[2] Radio tomography was introduced by Austen et al.[1988] and has been successfully used over the pastdecades for the characterization of ionospheric structureby integrated line-of-sight measurements [Andreeva etal., 1990; Kunitsyn and Tereshchenko, 1992; Fougere,1995; Walker et al., 1996; Sutton and Na, 1996; Kunitsynet al., 1997; Kamalabadi et al., 1999; Andreeva et al.,2001]. Various reconstruction techniques [Fougere,1995; Nygren et al., 1996; Sutton and Na, 1996;Fehmers et al., 1998] can generate ionospheric electrondensity images from a series of line integrals, the totalelectron content (TEC). Experimental implementationsof these reconstruction techniques have been carried outsince the first experimental tomography image waspublished [Andreeva et al., 1990; Pryse and Kersley,1992; Pryse, 2003]. Literature reviews on the develop-ment of tomographic methods for reconstructing thestructure of ionosphere electron density, as well as theirexperimental implementations are well summarized inthe work of Pryse [2003], Leitinger [1999], and Kunitsynand Tereshchenko [2003]. Radio tomography has been

recently shown to be a promising new technique for alarge scale remote sensing of Earth’s magnetosphere[Kunitsyn et al., 1997; Ergun et al., 2000; Ganguly etal., 2000]. The ability to accurately image both electrondensity and magnetic field on a large scale from thesimultaneous measurements of group delay (or phasedifference) and Faraday rotation of two coherent radiosignals would be a major advance for magnetosphericscience to address many currently unanswered funda-mental questions [Ergun et al., 2000; Cummer et al.,2001].[3] Unlike traditional computer tomographic problems

widely used in medical applications, where transform-based reconstruction techniques are popular [Kak andSlaney, 2001], most ionosphere and magnetospheretomographic methods are based on ray approximation.The main reason is that in ionospheric or magnetospherictomography, we usually do not have the freedom tochoose measurement ray paths crossing the region ofinterest. Ionospheric and magnetospheric tomographytypically solves the ill-conditioned inverse problem bya direct or iterative inversion of the ray projectionmatrices. In addition to the incomplete ray geometry,background noise and measurement noise are inevitablyattached to the observed phase difference and Faradayrotation. Therefore radio tomography is more ill-posed

RADIO SCIENCE, VOL. 40, RS3004, doi:10.1029/2004RS003100, 2005

Copyright 2005 by the American Geophysical Union.

0048-6604/05/2004RS003100$11.00

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compared to the widely known computer tomography inmedical applications.[4] Two distinct forms have been proposed for

imaging the magnetosphere with radio tomography[Ergun et al., 2000; Ganguly et al., 2000]. One is toimage the electron density Ne from the TEC measure-ments from either the phase difference or the groupdelay of two radio signals [Ergun et al., 2000]. It wasshown that 16 satellites placed in 2 polar orbits cangive a good reconstruction of Ne. Fewer satellites,however, may result in an incorrect reconstruction.The other is a combined reconstruction of the electrondensity Ne and the magnetic field B from TEC andFaraday rotation measurements [Ganguly et al., 2000]. Adetailed reconstruction algorithm based on ART-type(Algebraic Reconstruction Technique) iterative methodsis introduced [Ganguly et al., 2000] and results fromcombined TEC and Faraday rotation measurementsusing 18 satellites in 2 orbits are shown to producegood reconstructions of magnetospheric parameterswhen tested in model problems with good spatialcoverage of the probed region.[5] Although simulations have demonstrated the

applicability of radio tomography in the magnetosphere,difficulties may occur if fewer satellites are used (whichmay be a practical constraint on the technique). Impor-tant research goals include finding image reconstructiontechniques that work well and robustly with fewersatellites and that can incorporate additional information(from other measurements or from models) flexibly intothe reconstruction. In conventional ART-type iterativemethods, other measurements may be enforced as con-straints in every iteration; the methods are, however,either not able to incorporate in situ measurements or cando so only in a too local or too stiff fashion.[6] In this paper, our primary goal is to develop and

demonstrate a robust reconstruction algorithm that per-forms well over a wide range of the number of path-integrated measurements and that is sufficiently flexibleto incorporate into the reconstruction many classes ofadditional information. Under realistic circumstances weanticipate a reconstruction method that trades off thepath-integrated TEC and Faraday rotation measure-ments, any additional measurements (for example, insitu measurements), and the smoothness of the recon-structed images such that they degrade gracefully whenthe number of satellites becomes smaller. Our approachis to formulate the ill-conditioned inverse problem into aconstrained and weighted least squares minimizationproblem and directly solve the discretized inverse prob-lem. The weighting scheme trades off all valuablefactors to improve robustness of the reconstructionalgorithm. Numerical examples show that the regular-ized direct method performs as good as popular iterativemethods with large number of satellites but it performs

significantly better with few satellites. We also demon-strate that extra information such as in situ measure-ments can be easily incorporated into the reconstructionprocess and the direct method is robust within somereasonable range of mesh grid size and measurementnoise.

2. Magnetospheric Radio Tomography

[7] Tomographic imaging requires reconstruction of animage in the probed region from an ensemble of path-integrated projections measured at different angles oralong different ray paths. Mathematically, reconstructingimages from measurements within or around the probingregion belongs to the class of so-called inverse problems,in which the information of interest, such as the distri-bution of electron density is not directly available.Instead, what is available is certain measurements of atransformation or a projection of this information. Inpractice, however, these measurements are both anincomplete sampling of information and corrupted bynoise. This makes most inverse problems ill-posed and adirect inversion not possible.[8] Two major classes of tomographic reconstruction

techniques widely used in medical or geophysical appli-cations are the transform-based methods [Kak andSlaney, 2001], and the algebraic methods formulatedeither statistically [Andrews and Hunt, 1977; Nygren etal., 1996] or deterministically [Frey et al., 1998; Kak andSlaney, 2001]. The transform-based methods are widelyused for medical applications due to their accuracy andspeed of implementation. However, these methodsrequire the measurement of a large number of projec-tions, which is not feasible for radio tomographicimaging in space physics, where only few satellites areavailable for the measurement of TEC and Faradayrotation. TEC is a line integration of the electron densityalong each measurement ray path, and Faraday rotationangle is the integral of the product of electron densityand the magnetic field projected onto the measurementray paths. The statistical formulations of the algebraicmethods such as the Baeysian methods lead to theMaximum A Posteriori (MAP) solutions. Although thepopular Maximum Likelihood (ML) method has beenwidely used for iteratively reconstructing the originaldistribution in emission tomography [Shepp and Vardi,1982; Sheng and Liu, 2004], simulations indicate that itdoes not perform well for magnetospheric imaging [Freyet al., 1998]. The deterministic formulations of thealgebraic methods solve for an array of unknowns eitherdirectly or iteratively by setting up algebraic equationsfor the unknowns in terms of the measured path-integrated projection data. They are by far the mostcommonly used method for radio tomography and arethe focus of this paper.

RS3004 ZHAI AND CUMMER: MAGNETOSPHERIC RADIO TOMOGRAPHY

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[9] An integral v of the unknown distribution u alonglines l between observation positions within a probingregion is given as

v ¼Z s¼l

s¼0

u sð Þds; ð1Þ

with s the distance along the integration path. Computertomography is then used to determine the distributionof u from a set of line integral measurements v. Inpractice, for a 2-D image reconstruction, we discretizethe probing region into small pixels, with the value ineach pixel representing the mean of the unknowndistribution u. A line integral (1) projects the unknowndistribution x, which is a discrete version of u, onto asubspace spanned by the columns of a projection matrixA, to obtain the physical quantity y that can bemeasured. A discrete version of the projection can thusbe written as

y ¼ Ax: ð2Þ

The vector x is a finite representation of the unknowndistribution at certain discrete points in the probingregion. Each component of x is the level of activity ordensity in a 2-D pixel for a given discretization of theregion. Each component of y represents the number ofmeasurements obtained at a given location of theimaging device. The matrix A, which is a discreteversion of the line integral in equation (1), has adimension of m by n, where m is the total number ofmeasurements and n is the number of discrete pixels. Ina hypothetical configuration for magnetospheric radiotomography, we generally have m � n. An inversionalgorithm for magnetospheric radio tomography mustproperly transform underdetermined, unevenly distrib-uted TEC measurements and Faraday rotation anglesinto images of the magnetospheric structures.

2.1. Direct Method With Regularization

[10] In general, A is very ill-conditioned and non-invertible, which implies that equation (2) can only besolved in the least square sense. For a unique solution ofthe underdetermined inverse problem, classical recon-struction methods such as the minimum norm leastsquares solution or the maximum entropy choosesolution vector x with minimum norm or with maximumentropy [Andrews and Hunt, 1977]. The minimum normleast squares solution amounts to solving the optimiza-tion problem under the constraint

minimize : r xð Þ k ATy ¼ ATAx� �

; ð3Þ

where r(x) is the norm of x or any other criteria ofconstraint leading to a unique solution. Such solutionsthrough the principle component analysis, however,

could be unstable in practice [Kamalabadi et al.,1999]. A slight modification of equation (3) by

minimize : k Ax� y k2 þlr xð Þ� �

ð4Þ

will provide simultaneously uniqueness and stability.The formulation in equation (3) is a constrained leastsquares minimization, whereas the formulation inequation (4) is the minimization of a weighted sumcontaining a least squares term and a side constraint. Themaximum entropy solution is therefore obtained by

minimize : k Ax� y k2 þ lXni¼1

xi ln xið Þ( )

: ð5Þ

[11] For a discrete line integral with specificallydesigned geometry for measurement, matrix ATA inequation (3) is degenerate, with a nontrivial null space.However, if we add any multiple l times a nondegene-rate quadratic form k Hx k2, which is a finite differencediscrete version of the regularizer r(x) in equation (4),then minimization of

& xð Þ ¼k Ax� y k2 þ l k Hx k2

leads to a unique and stable solution for x. Otherdiscretization schemes such as finite elements, waveletsand Fourier series may also be used. Careful selection ofthe regularizer or stabilizer r(x) in equation (4) is thechallenge for a solution that meets all physical andmathematical requirements.

2.2. Robust Reconstruction WithOther Observations

[12] We now formulate the ill-conditioned inverseproblem into a constrained and weighted least squaresminimization problem by incorporating any auxiliaryobservations expressed as z = Px, where the projectionmatrix P can be easily calculated from known raygeometry. For instance, based on specific local measure-ments, P is simply a sparse, rectangular permutationmatrix full of mostly zeros, but with a few ones of itselements for the case of in situ measurements. Let Sy andSz be the noise covariance matrices for the line-of-sightmeasurements y and auxiliary measurements z respec-tively, or simply set Sy and Sz the identity matrix if thenoise covariance information is unknown. We then solvethe inverse problem

minimize : kAx�yk2S�1yþ l1 kPx� z k2S�1

zþl2 kHxk2

ð6Þ


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by taking the derivative of equation (6) with respect tounknown x, the solution is obtained as

x ¼ ATS�1y Aþ l1P

TS�1z P þ l2H

TH� ��1

� ATS�1y yþ l1P

TS�1z z

� �: ð7Þ

[13] Each term in equation (6) represents respectivelycontributions to the reconstruction from the path-integrated measurements, the in situ measurements,the smoothing interpolation and regularization of theunknowns. For instance, increasing or decreasing l1 orl2 in equation (6) automatically adjusts the relativeweight of in situ measurements and global smoothnessrespectively, such that a good solution that trades off thepath-integrated measurements, the auxiliary observations,and the global smoothness is easily achieved. Each termcontributes to the reconstruction in such a flexible waythat by minimizing the functional we establish efficiencyand accuracy of the direct reconstruction.[14] A key advantage of this approach is that a wide

variety of information can be included and easily con-trolled. The end result is a reconstruction that agreesoptimally with a weighted combination of all the mea-surements and is maximally smooth in some sense.The noise covariance terms automatically decrease theweight of known noisy measurements since in reality themeasurements are likely uncorrelated, especially formagnetospheric tomography with widely separated sat-ellites. For instance, if noise in one of the path-integratedmeasurements is relatively big, the weight of that noisemeasurement is automatically suppressed by an increaseof the corresponding diagonal term in the noise covari-ance matrix without affecting other information in thereconstruction. Therefore the individual path-integratedmeasurements can be weighted differently, as would beappropriate in the case of a spatially varying noiseenvironment expected in the magnetosphere. Similarly,the influence of the path-integrated measurements rela-tive to any in situ measurement can be controlled toreflect the quality and certainty of these measurements.[15] For any prior known statistical models of the

probed magnetospheric region, a statistical regularizationterm l3 k x k2

S�1mod

can be easily added to equation (6), with

Smod the covariance matrix of the statistical model. Thisstatistical regularization produces fields that are statisti-cally similar to model fields. This is a very effective wayof including existing magnetospheric knowledge withoutforcing the solution toward a specific magnetosphericfeature, which would occur if a model magnetospherewere used as an initial guess in ART-type iterativemethods. The incorporation of a statistical regularizationto the direct reconstruction will be investigated in futurework.

[16] A similar model-independent direct method hasbeen introduced for ionospheric tomography based onprior knowledge of ionosphere electron density distribu-tion [Fehmers et al., 1998]. A three quadratic termregularization is used by Fehmers et al. [1998] thatincludes terms that specifically model statistical informa-tion of the electron density distribution. For the more ill-conditioned magnetosphere tomographic problem, theregularization in our paper, however, is proposed tosimultaneously reconstruct the electron density and themagnetic field of Earth’s magnetosphere. Therefore afinite difference approximation of the more generalLaplacian operator is employed to impose globalsmoothness of the solution, along with the regularizationterm that incorporates in situ measurements, which areshown in Section 3 to significantly improve quality ofthe direct reconstruction.[17] With noisy data, the selection of l1 and l2 is

crucial to a successful reconstruction. In general l2should be large enough to enforce stronger smoothnessto the solution compared to the noiseless case. However, atoo largel2 suppresses sharpness of the solution. A simplefirst trial for l1 and l2 is suggested by balancing the orderof magnitude of various terms in equation (6) and used inthis paper as follows [Press et al., 1992, p. 802]

l1 ¼trace ATS�1

y A� �

trace PTS�1z P

; l2 ¼trace ATS�1

y A� �

trace HTH : ð8Þ

[18] Proper selection of the deterministic regularizer Hin equation (6) is crucial to accurate tomographic imagereconstructions. Although there are many different formsof regularization, an optimal regularizer that agrees wellwith the prior knowledge of electron density andmagnetic field distribution is problem dependent. Thel2 norm quadratic energy regularizer generally enforcessmoothness but smears out sharpness on the solution.The l1 norm edge-preserving regularizer enforces moresharpness on the solution [Kamalabadi et al., 2002]. Theprincipal solution from a singular value decomposition isa zeroth-order regularization, in which a minimum normof the solution vector is selected.[19] A popular two-dimensional five-point finite dif-

ference approximation of the second order Laplacianoperator [Press et al., 1992; Kaup, 1999], which givesthe regularization matrix

H ¼

. . . . . . . . . . . . . . .

. . . . . . �1 . . . . . .

. . . �1 4 �1 . . .

. . . . . . �1 . . . . . .

. . . . . . . . . . . . . . .

266664377775;

is used throughout this work since we find that theLaplacian regularizer performs well in a variety of


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conditions. The H matrix is modified appropriately nearthe boundaries.

2.3. Algorithm Validation

[20] A simple test problem is used to validate theproposed method and to compare with iterative methodsART and MART that have been applied to magneto-spheric radio tomography and widely used to ionosphericradio tomography. The original image and its line-of-sight paths are shown in Figures 1 and 2. The ART isimplemented with the exact formulation by Kak andSlaney [2001, p. 283] and the MART is implementedwith the exact formulation by Ganguly et al. [2000]. Thenoise covariance matrix Sy is set to be the identity matrixin the direct method.[21] Figure 3 shows clearly the mean square errors

committed by the direct method with regularization aremuch smaller than that committed by the iterativemethods such as ART and MART, especially for a smallnumber of satellites. Unlike the iterative methods, thedirect method with regularization is less sensitive to thenumber of line integral measurements. The better per-formance mainly attributes to the smoothing term in

equation (6), which accordingly fills the gaps amongthe line integral measurements of the unknowns such thatan optimal solution with global smoothness and goodagreement with the line integral measurements, as well asthe in situ measurements is achieved. The good perfor-mance is also partly due to the background image beingsmooth. Figure 3 shows theoretically the convergence ofvarious reconstruction methods. In practice, however, wedo not have the freedom to see the convergence of anyreconstruction methods due to the constraint of availablesatellites.[22] Recent studies in ionospheric radio tomography

with ART-type iterative methods have shown that adiscretization of the line integral of electron density withpiecewise-planar approximation performs better thanthe maybe too-stiff piecewise-constant approximation[Andreeva et al., 2001]. In our proposed directreconstruction method, smoothness is imposed in thedeterministic regularization term with a finite differenceapproximation of the second-order Laplacian operator inthe reconstruction. The piecewise-constant approxima-tion is, therefore, combined with the finite differenceapproximation of the second-order Laplacian operator

Figure 1. Image of a Gaussian function used in the test problem. See color version of this figurein the HTML.


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for magnetospheric tomography. In the regularizeddirect reconstruction, the discretized Laplacian operatorenforcing global smoothness of the solution significantlyimproves the stiff behavior of the piecewise-constantapproximation. This is one primary advantage of ourdirect reconstruction method over the ART-type iterativemethods, which either not able to incorporate globalsmoothness or can do so only in a too local or too stifffashion. Moreover, the piecewise-constant approxima-tion is simpler and much easier to implement. It providesan easier way to enforce a prior knowledge on thepositiveness of the solution, while higher order approx-imations could be troublesome.[23] Themesh grid size used for reconstruction can have

a significant impact on the reconstruction error [Frey etal., 1998]. Figure 4 shows clearly a significant increaseof the reconstruction error committed by the iterativemethods when mesh grid size becomes small. This is dueto the fact that for a fixed number of satellites, when thediscrete mesh becomes finer, there are fewer measure-ment rays pass across each grid cell. Thanks to thesmoothing interpolation term in equation (6) and thetrade-off strategy, the direct method with regularization,

is relatively insensitive to mesh grid size and thus morerobust compared to the iterative methods.

3. Magnetospheric Image Reconstruction

[24] Magnetospheric plasma density Ne and themagnetic field B are two primary physical parametersof Earth’s magnetosphere. In situ observations of thesetwo parameters have been the primary means in the pastseveral decades for studying basic magnetosphericstructures and dynamics. However, valuable informationgathered in local regions does not reflect large-scalenature of magnetospheric processes, how small-scaleprocesses couple with each other over large distances, orsequence of events in large-scale processes. Thereforemany open questions on the global nature of magneto-spheric processes cannot be answered with current data[Ergun et al., 2000]. A global-scale observation thatprovides simultaneous measurements of plasma param-eters Ne and B over an extended region of magneto-sphere would answer many fundamental scientificquestions.

Figure 2. Line-integration paths used in the test problem.


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[25] Recent experiments [Cummer et al., 2001] success-fully measured the Faraday rotation on a magnetospherictransmission, experimentally validating some of the basicmagnetospheric radio tomography concepts. In thissection, we demonstrate performance of the proposedtomographic method for realistic magnetospheric radiotomography with relatively few satellites using resultsfrom magnetohydrodynamic (MHD) simulations.

3.1. Phase Difference and Faraday Rotation

[26] To measure path integrated Ne, each satellitetransmits coherently phased pairs of discrete radiofrequency signals to be received by all other satellites.Since the refractive index of the medium with a plasmafrequency wp is

n wð Þ ¼ c

up¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

w2p

w2

s; ð9Þ

where w is the wave circular frequency and wp2 =

Nee2

�0me

,

the phase velocity of an electromagnetic wave in a

plasma with w wp is thus approximated by truncatingthe higher order term in the Taylor expansion

up ¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�w2p

w2

s c 1þw2p

2w2

!: ð10Þ

The expected phase difference (Df) at a fixed timebetween two signals with frequencies f1 and f2 withrespect to f1 is given as

Df ¼ f1

f2f2 � f1 ¼

2pf1c

� �Z 1

0

n f2ð Þ � n f1ð Þð Þds;

ð11Þ

where e, me are electron charge and mass, �0 is free spacepermittivity, c is the speed of light, and Ne is the plasmadensity in probed region. It is showed that the groupdelay and differential phase measurements between twosignals yield an accurate reconstruction of Ne with amoderate number of satellites placed in two orbits[Ergun et al., 2000].[27] Faraday rotation is the rotation of polarization of

a linearly polarized wave as it travels through an aniso-

Figure 3. Comparison of reconstruction errors against total number of satellites.


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tropic medium like magnetospheric plasma. In cold-plasma approximation, the change in Faraday rotationangle F along the radio wave propagation path per unitlength s can be approximated via the Quasi-Longitudinal(QL) VHF approximation, where the signal frequency fis higher than both the electron gyrofrequency and theplasma frequency and the propagation is not too close to90�, when the propagation direction is perpendicular tothe magnetic field B. Under these conditions,

dF

ds¼ � k

f 2

� �NeB cos q; ð12Þ

and thus the Faraday rotation angle of radio signals withtwo different frequencies can be written as

DF ¼ �k1

f 21� 1

f 22

� �Z s¼l

s¼0

NeB cos qds; ð13Þ

where k =e3

8p2�0cm2e

= 2.36� 104 m2 T�1 s�2 is a constant

factor, and B is the magnitude of the magnetic field B.[28] The phase difference and group delay provide

integrated measurements of electron density only, whilethe Faraday rotation provides the product of magnetic

field B and electron density Ne and is thus sensitive tochanges in both quantities. Therefore a combination ofFaraday rotation and group delay or phase differencemeasurements enables independent measurement of themagnetic field in the plane of propagation paths and theplasma density. Recent work [Ganguly et al., 2000]showed how interspacecraft Faraday rotation measure-ments combined with phase different measurementsenable concurrent tomographic reconstructions of non-local magnetospheric plasma density Ne and in-planemagnetic field B through iterative methods MART andIART under relatively ideal situations where 18 satellitesare placed in 2 orbits.[29] If the region of interest is divided into n discrete

cells, then the phase difference measured through ray i,shown in equation (11) directly proportional to the TEC,can be expressed in a discrete form as follows

Ti ¼ tf11

f 21� 1

f 22

� �Xnj¼1

LijNj; ð14Þ

for i = 1, 2, .., m, where Lij is the length of ray i in cell j,

and t =e2

4p�0cme

= 4.211 � 10�7 C2 s F�1 kg�1 is a

constant factor. The Faraday rotation angle for ray i in

Figure 4. Comparison of reconstruction errors against mesh grid with 16 satellites.


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equation (13) can be estimated with a discrete sum asfollows

DFi ¼ �k1

f 21� 1

f 22

� �Xnj¼1

Lij ui � Bj

Nj; ð15Þ

for i = 1, 2, .., m, where ui is the radio wave propagationunit vector for ray i, Bj and Nj are the magnetic field andelectron density in cell j respectively, and m is the totalnumber of rays or measurements. In matrix form,equation (14) can be written as

Tm�1 ¼ tLm�nNn�1; ð16Þ

where t = tf11

f 21� 1

f 22

� �and equation (15) can be written

as

�Fi ¼� k1

f 21� 1

f 22

� �Li;1 Li;2 . . . Li;n� �

�

N1 ui;1B1;1 þ ui;2B1;2

N2 ui;1B2;1 þ ui;2B2;2

. . .

Nn ui;1Bn;1 þ ui;2Bn;2

8>>><>>>:

9>>>=>>>; ð17Þ

To further express

�F ¼

�F1

�F2

. . .�Fm

8>>>><>>>>:

9>>>>=>>>>;into a matrix form that facilities a vector fieldreconstruction, we simply augment the matrix size onthe right-hand side of equation (17) to obtain a newprojection matrix for all vector components, DF is thuswritten as

�F ¼� k1

f 21� 1

f 22

� �

�

L1;1u1;1 L1;1u1;2 . . . L1;nu1;1 L1;nu1;2

L2;1u2;1 L2;1u2;2 . . . L2;nu2;1 L2;nu2;2

L3;1u3;1 L3;1u3;2 . . . L3;nu3;1 L3;nu3;2

. . . . . . . . . . . . . . .

Lm;1um;1 Lm;1um;2 . . . Lm;num;1 Lm;num;2

26666664

37777775

�

N1B1;1

N1B1;2

N2B2;1

N2B2;2

. . .

NnBn;1

NnBn;2

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;: ð18Þ

Let k = �k1

f 21� 1

f 22

� �, which depends on frequency

only, we obtain

DFm�1 ¼ keLm�2n Q2n�1; ð19Þ

where vector Q is the product of electron density and themagnetic field in each discrete cell. The discrete electrondensity N is solved inversely from T in equation (16) andQ is solved from DF in equation (19). Components ofthe magnetic field are then decomposed from Q and Nreconstructed. Equations (16) and (19) are in the formof equation (2) discussed in Section 2. Thereforeequation (7) is applied for the reconstruction of N and Q.[30] In general, tomographic imaging of a vector field

in equation (19) is more ill-posed compared to a scalarfield imaging in equation (16), as the column size of theill-conditioned projection matrix eL is twice as large asthat in L for the same amount of measurements. Thechallenge is to reconstruct two components of a vectorfield when only the measured ‘total’ Faraday rotationangle, which is a sum of the rotation along the x and yaxes in a 2-D plane, is given.[31] Proper scaling of relevant physical parameters for

the combined tomographic imaging of the magnetic fieldand electron density is crucial to an accurate reconstruction.For example, in the reconstruction of B from the Faradayrotation measurement, renormalizing the deterministicregularization term by electron density Ne already foundfrom the phase difference measurement improves thesolution significantly by enforcing smoothness on Bitself rather than the measured product Q = NeB.

3.2. Algorithm Implementation andNumerical Examples

[32] With examples involving 2-D images of Ne and Bfrom the MHD simulations for the magnetotail and bowshock of Earth’s magnetosphere, we demonstrate perfor-mance of the direct reconstruction algorithm and itscomparison with ART-type iterative algorithms that havebeen applied for magnetospheric radio tomography inprevious work. The magnetospheric electron density andmagnetic field configurations were computed from anMHD simulation [DeZeeuw et al., 2000] availablethrough the Community Coordinated Modeling Center(CCMC) at NASA Goddard Space Flight Center. Theelectron density is computed from the MHD ion densityassuming quasi-neutrality. The probing frequency isselected based on the characteristic physical parameterssuch as plasma frequency and electron gyrofrequency ofeach region of interest in the following examples. Thegrid cell size in the detailed MHD simulation is 0.25 RE.The grid cell size in modeling the example tomographicproblems is 0.25–0.5 RE.3.2.1. Performance of the Proposed Algorithm[33] An 11-satellite tomographic imaging of Earth’s

magnetotail of the X-Yplane plasma sheet from an MHD


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Figure 5. MHD simulated electron density Ne (logarithm) and magnetic field B in an 11-satelliteprobed region with frequencies 1.9 MHz and 2 MHz. See color version of this figure in theHTML.


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Figure 6. Reconstructed electron density Ne (logarithm) and magnetic field B in an 11-satelliteprobed region with frequencies 1.9 MHz and 2 MHz. See color version of this figure in theHTML.


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Figure 7. Reconstructed Ne (logarithm) and B under 10% WGN in the measured phasedifference and Faraday rotation without other observations. See color version of this figurein the HTML.


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Figure 8. Reconstructed Ne (logarithm) and B under 10% WGN in the measured phasedifference and Faraday rotation with other observations. See color version of this figure inthe HTML.


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simulation is shown in Figures 5 and 6, where brightnessscale denotes the electron density and arrow the magneticfield vector. Figure 6 shows the reconstructed image forthe probed region without noise and in situ measure-ments. Reconstruction errors for the electron density Ne

and the dominant component of the magnetic field vectorBx are 7% and 14% respectively, which show that thedirect algorithm works well for cases without noise or insitu measurements.[34] To test robustness of the reconstruction method

with respect to noise, a 10% White Gaussian Noise(WGN) with zero mean is added to the ‘measured’phase difference for TEC and Faraday rotation angle.The WGN is constructed by scaling up a randomWGN by a factor of 10% of the ‘measured’ maxi-mum phase difference for TEC or Faraday rotationangle. Therefore the magnitude of the WGN in themeasurements is roughly 3 degrees of phase differ-ence or 0.2 degrees of Faraday rotation. Figures 7and 8 show the reconstructed image for the probedregion with 10% WGN in TEC and Faraday rotationangle respectively. The reconstruction error of themagnetic field B is directly related to the error of the

electron density Ne. With the rational selection ofregularization parameters l1 and l2 from equation (8),Figure 7 shows that a reasonable reconstruction can beachieved (roughly 12% error in Ne and 17% error in Bx).Therefore our direct reconstruction technique is robust inthe presence of modest noise.[35] The main advantage of the proposed reconstruc-

tion method is its adaptability and extensibility to incor-porate in situ measurements or other prior observationsto improve solution. In situ measurements can improvereconstruction significantly in the presence of noise.With regularization that incorporates local observationsat each satellite as in equation (7), Figure 8 shows abetter reconstruction with 9.3% error in Ne and 9.7%error in Bx, is obtained. In fact, the reconstruction isrobust and insensitive to the weighting parameters.Compared to the reconstruction without local measure-ments, the B reconstruction is improved more by the insitu measurements because the B reconstruction by itselfis more ill-conditioned.[36] Although the 11-satellite orbit configuration in

Figure 5 may not be optimal from a mission-designperspective, it is used to demonstrate good performance

Figure 9. MHD simulated electron density Ne and magnetic field B for an 7-satellite radiotomography mission with frequencies 200 kHz and 400 kHz. See color version of this figurein the HTML.


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Figure 10. Reconstructed electron density Ne and magnetic field B for an 7-satellite radiotomography mission with frequencies 200 kHz and 400 kHz. See color version of this figurein the HTML.


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Figure 11. MHD simulated electron density Ne and magnetic field B for an 11-satellite radiotomography mission with frequencies 200 kHz and 400 kHz. See color version of this figurein the HTML.


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of the proposed direct reconstruction method when con-stellation is in a favorable static configuration. In amission design for true tomographic reconstruction, inwhich continuous Faraday rotation or phase differencemeasurements are necessary to imaging the time-varyingelectron density and magnetic field, a more realistic orbitstrategy would be, for example, one at perigee, two atapogee, and four at the upward and downward leg of orbitwith larger distances toward perigee and shorter towardapogee. At any time in such a configuration, there wouldbe at least 10 satellites that could perform TEC or Faradayrotation measurements continuously. The one at perigeeblocked by the solid Earth for line-of-sight measurementcould be used for continuous in situ measurements.[37] The proposed reconstruction method in fact per-

forms well even with fewer satellites. Figure 9 shows theMHD simulated electron density and magnetic field ofEarth’s magnetotail in the X-Z plane and the location ofseven satellites in a single orbit for radio tomography.The mean square errors of the reconstruction shown inFigure 10 committed by the direct method with deter-ministic regularization are 10% for the electron densityNe and 13.3% for the Bx component, which is dominant

in the region. This demonstrates flexibility of the directreconstruction method under realistic situations withfewer satellites.[38] The regularized direct reconstruction method per-

forms well even in magnetospheric regions with sharpdensity and field gradients. The bow shock of theEarth’s magnetosphere is characterized by an increasein plasma density, and increased magnetic field turbu-lence. Figure 11 shows the MHD simulated electrondensity and magnetic field of the bow shock and thelocation of eleven satellites in a single orbit for radiotomography. The mean square errors of the reconstruc-tion shown in Figure 12 committed by the direct methodwith deterministic regularization are 3% for the electrondensity Ne and 18% for the By component, which isdominant in the region. The larger error in By is mainlydue to oversmoothing of the magnetic field on the leftside of the image. By incorporating in situ measurementsat each satellite, the reconstruction errors are signifi-cantly improved to 2.8% for the electron density Ne and7.3% for the By component. This demonstrates againflexibility and robustness of the direct reconstructionmethod.

Figure 12. Reconstructed electron density Ne and magnetic field B for an 11-satellite radiotomography mission with frequencies 200 kHz and 400 kHz. See color version of this figurein the HTML.


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3.2.2. Comparison With Iterative Algorithms[39] The ART-type iterative methods such as MART

and IART are shown in the work of Ganguly et al. [2000]and Ergun et al. [2000] to reconstruct Ne and Breasonably well under relatively ideal situations, wherea moderate number of satellites placed in at least twoorbits are employed for a good path coverage over theprobed region. To demonstrate performance of the directreconstruction algorithm, we have implemented exactlythe MART and ART that have been used by Ganguly etal. [2000] and Kak and Slaney [2001, p. 283] with thesame initialization and convergence criteria used byGanguly et al. [2000]. Tables 1 and 2 show the meansquare errors committed by various reconstructionmethods for the example in Figures 9 and 10, where asingle satellite orbit is used to probe an X-Z slice of theplasma sheet in the magnetotail. Horizontal measure-ments through satellites in a second orbit on the rightside of the probed region are crucial for a goodreconstruction in ART-type iterative methods. This isbecause in the satellite probed region, smooth Ne and Bx

(dominant B component) from MHD simulations varymainly along Z direction, meaning that the gradient of Ne

and Bx is much larger in the Z direction than that in the Xdirection, therefore horizontal line-of-sight measure-ments that detect Z direction gradient are more importantthan the vertical measurements. Even 49 satellites in asingle orbit are still not sufficient and a second orbit isnecessary for ART and MART to obtain accurate results.The reconstruction errors increase drastically for ARTand MART when satellites become sparse. The mainreason is that in ART-type iterative methods, reconstruc-tion in each discrete cell is carried out only if there isline-of-sight path across it. When satellites becomesparse, the number of cells that intersect with ray pathdecreases drastically, so is the quality of the reconstruc-tion. In other words, for a simultaneous reconstruction ofNe and B, at least 25–30 satellites placed in two carefullydesigned orbits are needed and the convergence may beslow. In Tables 1 and 2, we have selected about theoptimal discrete mesh-grid size for ART and MART. Thedirect method with regularization, on the other hand, isless sensitive to the mesh-grid size and the spatialcoverage of satellites for the studied example where theelectron density Ne from MHD simulations is smooth

and the magnetic field has a dominant Bx component.The regularization term fills out gaps smoothly in theseblind cells that are ignored in iterative methods. It isrobust even with fewer satellites, and generally speakingit is faster and still easy to implement.

3.3. Limitations and Optimal Reconstruction

[40] One limitation of our technique is its tendency tooversmooth sharp gradients, particularly in the magneticfield. In situations where the solution is not globallysmooth, an l2 norm quadratic energy regularizer mayoverly enforce smoothness to the solution. Therefore anl1 norm edge-preserving regularizer that enforces piece-wise smoothness of the solution may perform better[Kamalabadi et al., 2002]. The implementation of an l1norm edge-preserving regularizer, however, is notstraightforward for a direct reconstruction algorithm.An iterative method such as conjugate gradient methodmay be required.[41] In practice, the TEC and Faraday rotation mea-

surements from satellite transmitter and receiver areuseful in both directions. For instance, measurementsfrom satellite i to satellite j are slightly different frommeasurements from satellite j to satellite i due to thesmall time variation (the two measurements cannot bedone at the same time). Therefore in reality this could beviewed as additional useful information we have for fieldreconstruction and it may also reduce noise in themeasurements. This is particularly useful for vector fieldreconstruction.

4. Conclusions

[42] We introduce a flexible and robust direct recon-struction technique for radio tomographic imaging ofelectron density and vector magnetic field in the magne-tosphere. Numerical simulations based on MHD modelmagnetospheric parameters were presented and weshowed that this technique has a number of advantagescompared to other reconstruction techniques that havebeen applied to magnetospheric tomography. Mostimportantly, this technique is robust and reasonablyaccurate even when relatively few satellites (as fewas 7) are used and performs significantly better in thiscase than ART and MART techniques. The techniquecontains a very flexible approach for introducing addi-

Table 1. Mean Square Errors of Electron Density Ne

Committed by MART and Direct Method With Regularization

Reconstruction Methods

Number of Satellites

7 13 25 49

MART, % 21.9 17.8 9.68 7.95Direct, % 5.87 4.73 3.74 3.69

Table 2. Mean Square Errors of Magnetic Field B Committed

by ART and Direct Method With Regularization

Reconstruction Methods

Number of Satellites

7 13 25 49

ART, % 51.0 38.3 23.1 21.7Direct, % 10.9 8.11 5.94 4.92


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tional information, such as in situ measurements, into thereconstruction, which can improve the quality of theresulting image. In the framework of this approach it iseasy to control the trade-off between enforcing agree-ment with the path-integrated measurements, any addi-tional information, and overall smoothness of the image.Simulations show that this technique gives good resultsin dramatically different regions (i.e., the plasma sheetand the bow shock) of the magnetosphere without majormodification. Even with practical limitations imposed bythe realities of satellite missions, radio tomographyshould enable the large-scale measurements needed toresolve many open questions in magnetospheric physics.

[43] Acknowledgment. This research was supported byNASA Geospace Sciences grant NAG5-12072.

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��S. A. Cummer and Y. Zhai, Electrical and Computer

Engineering Department, Duke University, Box 90291,

Durham, NC 27708, USA. ([email protected])


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