Robust ionospheric tomography using sparse regularization

T. Panicciari*1, N.D. Smith2, F. Da Dalt3 and C.N. Mitchell4

1Univeristy of Bath, Claverton Down, Bath, BA2 7AY, UK

t.panicciari@bath.ac.uk, +44 01225 386061

2n.smith@bath.ac.uk, 3f.da.dalt@bath.ac.uk, 4c.n.mitchell@bath.ac.uk

Abstract The ionosphere is a dynamic medium which can interact with electromagnetic waves and cause delay and

refraction/diffraction effects. Ionospheric structures occur with different scale in different locations. Therefore, the correct imaging of the location and scale of those structures is of particular importance. Computerized ionospheric tomography uses observations from ground-based receivers to reconstruct the state of the ionosphere and is denoted as an inverse problem. Globally, ground-based receivers tend to be unevenly and sparsely distributed and produce limited-angle observations. Therefore the definition of the problem in a regularized form is required. Furthermore, there is a need to accommodate inconsistencies in observations as the data rate increases, e.g. due to representativity error or residual dispersive offsets. This is the main focus of the paper. A new sparse regularization form, for ionospheric tomography, was introduced. It promotes sparsity in the reconstruction and it is tailored for wavelet representation due to their localization properties. An experiment using simulated data is proposed comparing sparse regularization with a standard approach for 3D ionospheric tomography. Results indicate sparse regularization as a promising technique showing a higher reliability of reconstructed ionospheric structures and robustness to observational inconsistencies compared to a more standard approach.

1. Introduction

The ionosphere is an ionized medium whose interaction with the electromagnetic wave causes delay in the

propagation speed and episodes of refraction/diffraction. The ionosphere is the major cause of errors in positioning using single frequency Global Navigation Satellite System (GNNS) signals. These effects of the ionosphere on GNSS signals take place due to structures that occur on multiple scale sizes. Computerized Ionospheric Tomography (CIT) is a technique that allows imaging the ionosphere. The technique uses slant observations measurements from ground based receivers in the form of Total Electron Content (TEC). Those slant TEC observations are then vertically projected to vertical TEC by means of the operation of inversion.

The potential of tomography for 2D imaging the ionosphere was formerly demonstrated by [1]. They used a

medical-based technique to reconstruct the state of the ionosphere using polar-orbit observations collected from ground-based receivers. Most of the algorithms were imported from medical applications (see for example [2, 3]) and based on the Radon transformation. The ionospheric tomography problem, unlike medical tomography, has geometric limitations such as uneven and sparse distribution of ground-based receivers and limited-angle observations [4, 5]. Those make the problem difficult to solve. To overcome the geometric limitations of the problem, regularization techniques were introduced based on Least Square Minimization (LSM) algorithms, and in particular the Tikhonov regularization [6]. Many research groups have successfully developed their CIT algorithms, and are reviewed in [7]. Another regularization is introduced by [8] which can, for example, exploits the sparse nature of the wavelet decomposition. Its application to CIT is new. The different scale sizes of ionospheric structures together with the uneven and sparse distribution of the receivers result in a requirement to image the ionosphere on multiple scale sizes. This makes sparse regularization particularly promising in ionospheric tomography.

The paper contribution concerns limitations in representing horizontal variations of electron density due to sparse

and uneven distribution of ground receivers, and the ability to accommodate observational inconsistencies. 3D ionospheric maps are reconstructed from simulated data at low and high resolution. High resolution maps require dense data coverage and high data sampling but can produce more inconsistencies in the observations. This causes higher variability of the measurements. Results show advantages and higher robustness and reliability of sparse regularization over a standard method. In particular, Section 2 describes method and observations and Section 3 the results. Conclusions are in Section 4.

2. Method

CIT uses slant TEC observations collected from ground receivers. The slant TEC is defined as the integrated

electron content between the satellite and the receivers along the ray path

(1)

978-1-4673-5225-3/14/$31.00 ©2014 IEEE

The observations are actually ointo slant TEC through a simplified appapproximation together with other effecEq. (1), and cause to be biased. Thereterms. Further details can be found in [1contribution of the offsets . The slant T

where is the geometric matrix that congrid and defines the forward-problem. Tof basis functions as

where are the basis function coefficiresolution maps require higher data ratinconsistencies include residual dispersiand what the geometric matrix can repro

The electron densities are obt

were considered, and both can guarantee• Tikhonov regularization. Spheric

function coefficients are estimate• Sparse regularization. The meth

compactly represent the energy Thresholding Algorithm (FISTA)

A 3-Dimensional grid of dimens

included part of North America and Eu22km in altitude. The data gap in the Adistribution of receivers. Fig. 1a showsground receiver distribution (black dotsModel (IRI) 2012. Some structures of di

Figure 1. a) Number of rays and GPSstructures.

Observations were obtained from

minutes and a sample rate of 30 secondsThe effect of a further Gaussian noise temodel the observational inconsistencies

The vertical profile of the elec(EOFs), while the horizontal variation i(discrete Meyer, DM) basis functions. Gaussian noise term. The low resolutionhigh resolution used the whole set of berror, which is calculated on the vertical

Figure 2 shows low resolution reonly where data coverage is defined. In coverage. Both the methods successfully(Fig 2a) tends to underestimate TEC vaTEC value and produces the better recoerror. With the Gaussian noise term the 2d) but DM produces a better image icoefficients used for the reconstruction.

obtained by measuring the delay between the receiver proximation of the Appleton-Hartree equation [9]. Thcts like multipath and non-dispersive contributions arefore, observations cannot be considered in absolute 10]. For this reason the contribution of the electron deEC measurements are then related with the electron

ntains the discretized lengths Δ of Eq. (1). Eq. (2) is

The horizontal and vertical variation of the electron con

ients, and basis functions are contained in the columte sampling, which can result in inconsistencies in ive offsets and representativity error (this is the differoduce).

tained by solving the inverse problem of Eq. (3). Twe a unique solution: cal harmonics are used to represent the horizontal vad by minimizing a regularized residual[10].

hod is particularly tailored to wavelet basis functioof the coefficients. The solution is implemented w

) [11].

3. Results

sion 64x64x22 voxels in longitude, latitude and altituurope defining a voxel of dimension 2x1 degrees in Atlantic Ocean describes well the limitations causeds the number of available rays summed along altituds). The state of the ionosphere was simulated using fferent scale (Fig. 1b) were then added to the backgrou

S ground stations (black); b) simulated ionosphere

m the simulated ionosphere using Eq. (2) and collecteds. The offsets were included by adding a constant offseerm with a variance of 2 TEC units was considered. Thdescribed in Section 2.

ctron density was constrained by means of Empiricn electron density was described by using spherical hThe results were compared at two different resolutn reconstruction was obtained by selecting a subset obasis functions. The comparison is done in terms of TEC where there is only data coverage.

econstructions using SH (Fig. 2 a-b) and DM (Fig. 2 cthis way, the issue of robustness is better examined r

y reconstruct a smooth ionosphere without the Gaussialue in Europe and America. On the contrary, DM (Fonstruction. This can be seen in Table 1 through the reconstruction is similar to the previous case for both in term of RMS error. This is shown in Table 1 to

and satellite and converted he errors introduced by the re not taken into account in terms but rather in relative

ensity is separated by the content and offsets by

(2) defined in a 3-Dimensional ntent is described by means

(3) mns of the matrix . High observations. Examples of

rence between observations

wo different regularizations

ariation of . All the basis

ons due to their ability to with the Fast Iterative Soft-

ude was selected. The grid longitude and latitude and

d by the uneven and sparse de at same location and the the International Reference und.

with IRI2012 with added

d within a time window of 8 et for each satellite-receiver. his is used to approximately

cal Orthonormal Functions harmonic (SH) and wavelet tions, with and without the of basis functions while the Root Mean Square (RMS)

c-d). TEC values are shown rather than the issue of data ian noise term although SH

Fig. 2c) better estimates the Root Mean Square (RMS) SH (Fig. 2b) and DM (Fig.

ogether with the number of

Figure 2. Low resolution reconstructions without Gaussian noise term for: a) spherical harmonics; b) discrete Meyer; and with Gaussian noise term for: a) spherical harmonics; b) discrete Meyer. Vertical TEC values where there is no ray coverage are masked out.

At high resolution (Fig. 3) the number of coefficients increases and the regularization is stronger. In the case

without the Gaussian noise term, SH (Fig. 3a) resolves better the small structures in America but the reconstruction seems noisy and strongly underestimates TEC values, mainly in Europe. DM (Fig. 3c) resolves well the small scale structures and still produces the best reconstruction in terms of RMS error (Table 1).

Figure 3. High resolution reconstructions without Gaussian noise term for: a) spherical harmonics; b) discrete Meyer; and with Gaussian noise term for: a) spherical harmonics; b) discrete Meyer. Vertical TEC values where there is no ray coverage are masked out.

With the Gaussian noise term, SH (Fig. 3b) needs a stronger regularization and the reconstruction becomes noisy.

This seems due to the fact that SH basis functions tend to replicate the rapid fluctuations introduced by the noise. DM (Fig. 3d) shows, instead, a more robust algorithm. Some of the information content is reduced but the reconstruction is still good. In terms of RMS error DM does the best reconstruction (see Table 1).

Table 1 shows that DM produces the best reconstruction in all the cases in terms of RMS error. This is due to the

sparse regularization. By minimizing the number of coefficients only the most significant are used in the reconstruction. This is illustrated in Table 1 by a smaller number of coefficients. The choice of which coefficients are used depends on the scale, position and intensity of structures detected from observations. The sparse regularization sets a threshold to the coefficients and filters out the ones with a contribution comparable to the oscillations introduced by the Gaussian noise term[12]. We used a simple Gaussian additive noise for which this Soft-Thresholding (FISTA) method is sensible; it

would be useful to test the method with correlated or non-Gaussian noise. For information regarding how the method reconstructs across data gaps, see [10]. Table 1. RMS error of reconstructed ionosphere for spherical harmonics and discrete Meyer with and without the Gaussian noise term. The percentage of basis functions with non-zero coefficients is shown and, within brackets, the number in absolute value.

Low Resolution High Resolution Discrete Meyer Spherical Harmonics Discrete Meyer Spherical Harmonics

Noise RMS error

Basis number

RMS error

Basis number

RMS error

Basis number

RMS error Basis number

No 6.6 36% (92) 10.9 100% (1089) 8.4 9% (355) 19.3 100% (16641)

Yes 7.7 30.8% (79) 12.5 100% (1089) 9.5 7.4% (304) 28.7 100% (16641)

4. Conclusions We have shown the advantages of using sparse regularization over the standard Tikhonov regularization. Those

benefits include robustness and reliability of the reconstruction for high resolution maps. Therefore, sparsity has been shown to be beneficial for Computerized Ionospheric Tomography (CIT) where observations are unevenly and sparsely distributed. Sparse regularization is new in CIT and results shown a strong improvement in terms of Root Mean Square (RMS) error. In some sense sparsity is naturally related with wavelet representation, but other basis constructions could produce further improvements. The previous knowledge of the scale structures we may expect at different locations might also help to drive the choice of basis functions for the reconstruction. This can improve the reconstruction particularly in zones with non-uniform or small number of observations.

Acknowledgments

We are grateful to the International Reference Ionosphere (IRI) project (http://iri.gsfc.nasa.gov/) for its

contribution to this paper. We also thank IGV and UNAVCO repositories for providing satellite and ground receiver data. This research activity was supported by a Marie Curie initial training network (TRANSMIT) within the 7th European Community Framework Programme under Marie Curie Actions.

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