Computerized ionospheric tomography (CIT) is a technique that allows
reconstructing the state of the ionosphere in terms of electron content from
a set of slant total electron content (STEC) measurements. It is usually
denoted as an inverse problem. In this experiment, the measurements are
considered coming from the phase of the GPS signal and, therefore, affected
by bias. For this reason the STEC cannot be considered in absolute terms but
rather in relative terms. Measurements are collected from receivers not
evenly distributed in space and together with limitations such as angle and
density of the observations, they are the cause of instability in the
operation of inversion. Furthermore, the ionosphere is a dynamic medium whose
processes are continuously changing in time and space. This can affect CIT by
limiting the accuracy in resolving structures and the processes that describe
the ionosphere. Some inversion techniques are based on

Tomographic imaging is an important tool for understanding the ionosphere, its behaviour and its effects on radio propagation. Ionospheric disturbances can persist for days in particular conditions. The electron density is the main measure that can tell us about the state of the ionosphere. Enhancements or depletions in the electron density produce irregularities or structures that are present in the ionosphere with different scales and vary with geographical location, time and sun activity. The correct localization of the irregularities can therefore play an important role. The spatial and temporal variability of ionospheric structures justifies the wavelet approach that we will describe in this paper.

CIT (computerized ionospheric tomography) is mainly an underdetermined inverse problem. The goal is to find the spatial and temporal distribution of electron density from a series of relative observations that are collected in the form of slant total electron content (STEC) measured from ground receivers along the signal path. The term relative is introduced because the observations are uncalibrated, hence the calibration becomes part of the inverse problem. Receivers are generally unevenly distributed on the Earth and together with the limited angle geometry of the rays the problem is difficult to solve (Yeh and Raymund, 1991; Na and Lee, 1992) and potentially unstable. Even with increasing observations from new constellations of satellites there is always the problem of the movement of the ionospheric medium in the time taken for the Global Navigation Satellite System (GNSS) satellites to cross the sky.

In this type of mathematical problem a functional cost is minimized (Geophysical Inverse Theory and Regularization Problems). A conventional approach is based on Tikhonov regularization (Tikhonov and Arsenin, 1977) and aims to balance the solution for good data agreement and to compensate (regularize) where no data are available. In general a proper regularization is needed to ensure stability, and to reduce artefacts and therefore noise in the reconstruction due to lack of data.

Another recent approach uses the

The advantage of having a compact representation is not only in terms of data. It also allows the removal of noise terms (Tsaig and Donoho, 2006), and in the case of ionospheric tomography it can also potentially better handle the uneven data distribution (Schmidt, 2007).

Sparse regularization techniques which minimize the

This paper describes an alternative method based on the

Section 2 gives the definition and the mathematical notations of the problem
including biases and basis functions. It also gives an overview of the

As in most geophysical applications of tomography, CIT is an undetermined problem. The receiver-satellite geometry and the uneven distribution of the receivers make the inversion a difficult operation. While the vertical sensitivity can be partially improved by means of Empirical Orthonormal Functions (EOFs) (Fremouw et al., 1992; Sutton and Na, 1994), the estimation of horizontal structures can be limited by the presence of artefacts especially when the number of coefficients to estimate increases considerably (e.g. for global or high-resolution maps).

In this section we will firstly define the observations and biases that are involved in the forward problem notation. Then we will describe the inverse problem in terms of basis functions and the regularization techniques.

In CIT observations are collected from ground-based receivers. The
measurement

Observations of differential phase can be generally considered noise free from the point of view of the instruments which inherently smooth over noise, but the measurement arc between a single receiver and satellite is uncalibrated or biased. In this section we will discuss the nature of the biases and the mathematical notation we will use to include them in the inversion algorithm.

Equation (1) relates the observation (STEC) with the electron density

Following the Mannucci (Mannucci et al., 1999) notation the recorded
pseudorange

It is possible to use a more accurate estimation of the STEC from the carrier
phase but unfortunately with the disadvantage that calibration is required.
To explain, and contrasting with the pseudorange in Eq. (2), the carrier
phase of the signal

Measurements for each receiver-satellite pair are contained in the

The problem of Eq. (7) is defined on a 3-D grid spacing in altitude, latitude
and longitude and it is known as a forward-problem where

The quantity we want to estimate is represented in Eq. (7) in terms of
electron content

The functional

The solution

In this section the mathematical notation used to decompose the ionosphere through basis functions is provided. Basis functions are used to extract the information and to emphasize some properties in the reconstructed ionosphere, in this case wave number for spherical harmonics and spatial localization and scale for wavelets. In particular, the vertical profile of electron density is described in terms of basis functions (EOFs) while the horizontal distribution with spherical harmonics and wavelet basis functions. EOFs are obtained from Chapman profiles (Chapman, 1931) and are used to constrain the vertical profile (Hargreaves, 1995). These are taken directly from the standard MIDAS approach as published in 2003.

The inverse problem of Eq. (11) is now expressed, in terms of associated
functional, as

The choice of

Figure 1 shows a one-dimensional example of the basis functions (normalized to one) that will be used in the experiment. Figure 1a and b illustrate two wavelets at the same scale and position for discrete Meyer (DM) and Daubechies 4 (DB4). They have a spatial compact support that makes them particularly useful to resolve localized structures. Figure 1c shows a single harmonic (normalized to one) that has to be multiplied with the Lagrange polynomial (along latitude) to produce a spherical harmonic (SH). They have a longer spatial support and work well to describe periodicities in the ionosphere.

Different regularizations exist to stabilize Eq. (15) and make the solution
unique and physically meaningful. In this section the two regularizations
based on the

The main goal of regularization is finding the best representation of the ionosphere that matches the observations and at the same time obviates the lack of data we usually face (e.g. in the oceans between continents).

Regularization techniques exploit the fact that Eq. (15) can be convex, i.e. that by minimizing it a global minimum is guaranteed, but it does not mean that different regularizations may have the same minima. The minimizer becomes the best representation we can have, and its properties will strongly depend on the chosen regularization term.

The

The regularization term of Eq. (15) can be expressed in different ways. The
classical approach is by using an

A different measure comes from the sparsity which involves the number of
nonzero coefficients in

A more feasible solution is obtained with the

The minimization of Eq. (15) with Eq. (20) is implemented with the Fast
Iterative Shrinkage-Thresholding Algorithm (FISTA, see Beck and Teboulle,
2009). It applies a non-linear thresholding (or soft-thresholding) (Donoho
and Johnstone, 1994) to the estimated coefficients

A similar stabilization can be introduced by selecting a subset of

We selected a grid that spans from North America to Europe. This is a good
example to show the limitation imposed, in this case by the ocean, on the
density of the receivers. We selected a grid of dimension

Data were simulated with the international reference ionosphere (IRI) model.
Some structures were then added in order to test the efficiency of the
algorithm to resolve them. We considered uncalibrated observations that were
obtained by adding a constant bias to each receiver-satellite pair and we
collected observations within a time window of 8

Figure 2 shows the Vertical Total Electron Content (VTEC) map that was used as truth while Fig. 3 illustrates the number of rays that was used in the reconstruction (black dots are the ground stations). The number of rays is obtained by summing the intersections along the altitude within voxels of the grid. The VTEC is calculated by integrating the electron content in a certain latitude and longitude location along the altitude. The ray coverage strictly depends on the density of ground stations, data (STEC) sampling rate and, in our case, the time window within which we run the reconstruction. The selection of the grid is also important as a finer grid will increase the number of voxels that are not intercepted by a ray and the number of coefficients to estimate.

Some structures were located where data coverage is particularly low. In those locations the reconstruction will struggle to recover the actual value independently from the regularization that has been used. The behaviour of the algorithm in those zones will strongly depend on the regularization term.

Simulated ionosphere with structures added to IRI2012. Values are in
TECU (

We used EOFs obtained from Chapman profiles (Hargreaves, 1995), and wavelets (DB4 and DM) and Spherical Harmonics (SH) to represent the horizontal distribution of structures in the ionosphere. By selecting a subset of larger horizontal basis functions we also limited the resolution in the reconstruction, i.e. the smallest scale structures that can be resolved.

For the aim of this paper it will be considered a standard implementation of
Eqs. (18) and (20), i.e. the matrix

The reconstructions are shown in Fig. 4 for low resolution and Fig. 5 for high resolution. Each figure shows the behaviour of the algorithm using different basis functions: SH (top), DM (middle) and DB4 (bottom). In order to highlight the regularization effects where only data coverage was present, we applied a mask (left) to the reconstruction (right). In fact, each regularization technique will handle the absence of data in different ways but we want to compare their ability to resolve structures where data are available.

Number of rays with ground stations (black dots).

Reconstructions obtained at low resolution with masked out VTEC
values where there is no ray coverage for

Reconstructions obtained at high resolution with masked out VTEC
values where there is no ray coverage for

At low resolution the reconstruction looks reasonable for both methods. The structures appear smoothed and with little detail (Fig. 4a, b, c). SH seems to produce some oscillations outside the data coverage (Fig. 4d), mainly in the Atlantic Ocean. This is due to the sinusoidal nature of SH that makes it problematic to represent localized structures. Wavelets do not produce oscillations and the reconstruction looks reasonably smoothed for this resolution, but there are some edge effects, especially for DM, between Canada and Greenland. Furthermore, DB4 unlike DM tends to fill the data gap in the ocean (Fig. 4f).

As the resolution increases (and therefore the number of coefficients to
estimate) the inversion needs in general a stronger regularization. This is
shown in Fig. 5. With SH the regularization damps many coefficients down but
it seems to resolve some of the structures well (north UK and US) where good
data coverage is present (Fig. 5a, b, c). However the reconstruction presents
the ring oscillation phenomenon that is an indication of the limitation of
the method when a high number of basis functions are used (Fig. 5d). The
stronger regularization has reduced most of the coefficients, and the VTEC is
in general underestimated. In fact, we are expecting a VTEC of 40 in central
Europe but the reconstruction shows a VTEC less than 30. Where data are not
available the regularization forces the VTEC to go rapidly toward zero. With
wavelets the regularization aims to minimize the number of non-zero
coefficients. Therefore the smallest basis functions are contributing with
the largest (smoother) ones to add detail to the reconstruction only where
good data coverage is available (this concept is regarded as
multi-resolution, which will be explained later). Where data are not enough to
resolve a small structure the solution will be approximated with a bigger and
smoothed one. By looking at the VTEC values, wavelets are perfectly
recovering the value of 40 VTEC units in Europe (Fig. 5b, c). This is mainly
due to the fact that the regularization term, by exploiting the localization
properties of wavelets, is adding the smallest basis only if they detect a
significant enhancement over the threshold

RMS error (values are in TECU) of the VTEC map obtained with spherical harmonics and wavelets at two different resolutions. Only the VTEC coefficients where there is ray coverage were considered. The percentage of basis functions with non-zero coefficients is also shown and, within brackets, the number in absolute value.

For each reconstruction the root mean square (RMS) error of the VTEC between the true and the reconstructed ionosphere was calculated. The RMS error is taking into account only the VTEC values where there is ray coverage. Values where there is no ray are, in fact, less meaningful for this statistic.

Table 1 shows the RMS error and the number of basis functions for each reconstruction at the two different resolutions. The number of basis functions is shown in percentage and in absolute values within the brackets. The increasing of RMS error with resolution is caused by the attempt of the basis functions to describe the small variations in STEC due to non-uniform data coverage (especially in north Norway). Wavelets need less than 50 % of basis functions at low resolution and even less at high resolution. The small number of basis functions help to stabilize the inversion as only fewer coefficients have to be estimated.

The offsets, obtained from Eq. (14) and averaged for each receiver, are also
very well recovered at low resolution by SH (Fig. 6a) and DM (Fig. 6b).
Figure 7a and b show the scatter plot of the original offsets (

As introduced earlier, another concept that can be exploited with wavelets is multi-resolution analysis. A similar concept was already used in (Schmidt, 2007). Wavelets allow the detection of structures according to their scale and position. Small-scale basis functions are therefore selected to represent small variations, otherwise only the basis functions with bigger scales are used. The ability of the algorithm to recognize small variations depends on the data availability and, therefore, the resolution (here intended as the smallest scale we can resolve in a certain position in the map) will depend on data.

Scatter plot of the estimated offsets (

Scatter plot of the estimated offsets (

Multi-Resolution (MR) Map for the high-resolution case with discrete Meyer basis functions. Each box represents the scale of the basis function and its position.

Figure 8 aims to explain multi-resolution with DM basis functions. Each square box indicates where the wavelet is centered in the map and the size that the wavelet is contributing with (i.e. the scale of the wavelet, which we selected the same level for each box). This is valid only in principle as a wavelet can be defined in a longer domain than the one defined by the square. The algorithm selects smaller scale basis functions where data coverage is good, trying, as a consequence, to match better the observations. In regions where data are not available or not enough, only the biggest scale wavelets are selected and therefore the solution will look smoother. This is not possible to obtain with SH as they are longer functions and are defined over the whole globe. It is interesting to notice how small-scale wavelets are not used if there is not a comparable (to the scale of the wavelet) enhancement from the data. This is the case in east and south Europe where, even if good data coverage is provided, only large-scale wavelets are used.

We stated at the beginning that wavelets allow the better removal of noisy terms in the reconstruction. Actually ground stations produce observations that can be considered generally noiseless. The noise term that we intended comes from the fact that the ionosphere is a dynamic medium, where different scale structures evolve with time according to complicated physics laws in a complex environment.

In order to test the effect of variability in the observations, we decided to add a zero mean Gaussian noise to each observation with a standard deviation of 1 TEC unit. A similar approach was used by Chartier et al. (2012, 2014).

Figure 9a and b show the reconstruction obtained with SH and DM. SH reconstruction (Fig. 9a) is quite sensitive to the noise, which causes additional oscillations and artefacts. DM reconstruction (Fig. 9b) shows a better robustness to noise, and the reconstruction is similar to the ones in Fig. 5e–f. This is mainly due to the sparse regularization which aims to minimize the number of nonzero coefficients. When the soft-thresholding of Eq. (20) is applied with FISTA, a subset of the most significant coefficients is selected. Those coefficients will contain the most important part of the energy (or information) (Donoho and Johnstone, 1994). In general, it would not be possible to make the same considerations if the energy was evenly distributed among all the coefficients, like in the case of SH.

The RMS error obtained from Fig. 9a and b is shown also in this case in
Table 2 together with the percentage of number of basis functions with
non-zero coefficients. The number of basis functions used with DM is slightly
decreased compared to the case without noise. This is due to the higher
threshold

We implemented a model-aided inversion by imaging the residual after removing
from the observations a background model of the ionosphere. This is called
three-dimensional variational (3DVar) data assimilation and assumes the
knowledge of a priori information about the state of the ionosphere. This is
generally obtained with an empirical model (like IRI2012) or a first
principle physics model. For the sake of this paper we wanted to test the
algorithms with Eqs. (18) and (20) under these conditions. Therefore, we
considered there was almost perfect knowledge of the ionosphere, i.e. we set
the background model

RMS error (values are in TECU) of the VTEC map obtained with spherical harmonics and discrete Meyer with a noise term added to the observations. Only the VTEC coefficients where there is ray coverage were considered. The percentage of basis functions with non-zero coefficients is also shown and, within brackets, the number in absolute value.

As we expected both methods work well. The only remarkable difference is that SH basis functions are picking up some noisy coefficients which result in a noisier reconstruction than with DM. Table 3 summarizes the RMS error obtained for these reconstructions.

Reconstruction with

Model-aided reconstruction obtained with

By perfectly removing the background the algorithm needs to resolve only few relatively smooth structures at a different scale. This scenario can be considered as the best case, where we had background knowledge of the ionosphere, in comparison with the worst case of the previous subsection where such knowledge was lacking. Actually, we will never have a perfect knowledge of the ionosphere and, therefore, a background model cannot aid the reconstruction as in the above example. This mismatching with the truth means that the algorithm with an approximated background model will have performances between the worst and best case.

RMS error (values are in TECU) of the VTEC map obtained with a 3DVar scheme using spherical harmonics and discrete Meyer with a noise term added to the observations. Only the VTEC coefficients where there is ray coverage were considered. The percentage of basis functions with non-zero coefficients is also shown and within brackets the number in absolute value.

Sparse regularization has been shown to be a valid alternative to standard method based on Tikhonov regularization and is particularly suitable with wavelets.

The method has been tested to estimate the offsets of the observations and,
even though it was applied to the specific case of the computerized
ionospheric tomography (CIT), it can be used for general inverse problems
where unknown offsets must be estimated. The method gave good performances in
recovering the offsets, but a useful remark is that there is a tendency to
overestimate them as

Sparsity allows a better noise removal and a more stable regularization when the number of coefficients to estimate increases considerably. We have shown tomographic reconstructions obtained with Spherical Harmonics (SH) and two different wavelets, Daubechies 4 (DB4) and Discrete Meyer (DM) in a worst and best case. The best case was obtained by selecting a background model which exactly represented the smoothed ionosphere, whilst the worst case was without any background model. In both cases wavelets were shown to produce the best reconstruction in terms of the root mean square (RMS) error and oscillations (artefacts). An important characteristic in this new approach is the ability of wavelets to handle the uneven distribution of the observations. We have explained this ability through the multi-resolution map showing how the resolution is adapted to the data coverage and the ionospheric structures observed by the measurements.

It is noted that CIT is actually a time-dependent inversion problem and in this paper it has been simplified in the simulation to a case where the ionosphere does not change in time. The work in this paper has shown the potential of the method when the ionosphere does not considerably change within a short time window, e.g. under quiet geomagnetic conditions. For more active conditions a full 4-D imaging would be required. This factor will be studied in further research.

In conclusion sparse regularization techniques can produce significant improvements to CIT and to inverse problems in general. They demonstrate properties of noise robustness and adaptability to data coverage. The choice of wavelet basis functions is not critical, but we believe that other wavelet constructions could lead to further improvements.

We thank the International Reference Ionosphere (IRI) project
(