Radio tomography of the ionosphere: Analysis of an underdetermined, ill-posed inverse problem, and regional application

Authors


Abstract

[1] After analysing the forward and inverse problems of radio tomography, a time varying three-dimensional imaging method of the ionosphere from GPS slant TEC data is described and applied at regional scale. Our approach is based on local basis parametrisation of electron density, and constrained by NeQuick ionosphere model and its spatial gradients. Our inversion scheme is fundamentally different from the data assimilation approach because it is not based on a physical ionosphere model. Preliminary results obtained with European GPS receiver data validate the stability of our method which is able to retrieve small-scale ionosphere features in properly resolved volumes. Many improvements are still possible on the algorithm, and a careful validation of inverted models by independent data is still necessary. However, our inversion algorithm is shown to improve the determination of the small-scale features of a coarse a priori ionosphere model.

1. Introduction

[2] The present development of Global Navigation Satellite Systems (GNSS) receivers, both on ground and onboard satellites, and the advent of new GNSS satellite constellations is opening a new era for ionosphere imaging by increasing greatly the amount of data available for retrieval of time varying 3D electron density models. Moreover, the development of Faraday rotation measurements on-board satellites opens the way to magnetospheric radio tomography [Zhai and Cummer, 2006]. The applications of these observational tools is wide. For example, they allow us to investigate the physical phenomena at work in ionosphere and plasmasphere dynamics [Schunk and Nagy, 2000]. In addition, these observations allow us to image the coupling of the ionosphere with its external forcing such as solar radiation [Basu et al., 2005; Stolle et al., 2006], atmospheric gravity waves [Hooke, 1968; Huang et al., 1998], oceanic gravity waves [Artru et al., 2005; Occhipinti et al., 2006] and infrasonic perturbations created by solid Earth vibrations [Ducic et al., 2003; Artru et al., 2004; Garcia et al., 2005; Lognonné et al., 2006].

[3] A short review of slant total electron content data extraction from GPS receiver measurements is first presented with an analysis of the forward radio tomography problem. The level of underdetermination of the inverse problem is discussed, and the different strategies for imposing a priori constraints are explored. The inversion method is then described in detail. Finally, a regional application of the four dimensional ionosphere imaging using the data of European permanent GPS receivers networks is presented. Preliminary results, limitations and future developments are also discussed.

2. GPS Data and Forward Problem

2.1. GPS Data

[4] The main data source for radio tomography of the ionosphere is actually coming from dual-frequency receivers of Global Positioning System (GPS) both on ground and onboard Low Earth Orbit (LEO) satellites. Among these systems, the data coming from the NAVSTAR constellation are largely dominating in terms of data amount. Consequently, we will restrict our analysis to these observations, usually referred to as GPS receiver data. However, the analysis of the forward and inverse problems will remain valid for any dual-frequency bistatic radio measurements with frequencies well above the maximum plasma frequency of the ionosphere (about 30 MHz), and able to provide information on the Slant Total Electron Content (STEC) of the ionosphere along the radio wave ray paths.

[5] The integrated ionospheric electron content is extracted from dual frequency GPS receivers through the Geometry Free (GF) combination of L1 and L2 phases:

equation image

where the average 〈〉 is performed over at least 100 epochs at 30 seconds sampling along the same satellite-receiver arc in order to remove the phase ambiguity of geometry free combination. The other parameters are defined by:

equation image
equation image
equation image

where λ1, λ2 are the wavelengths corresponding to frequencies f1 and f2 respectively, e is the charge of the electron, me is the mass of the electron and ∈0 is the vacuum permittivity. L1 and L2 are the phase signals, P1 and P2 are the code signals of GPS receivers respectively at frequencies f1 and f2. The coefficient K is derived from the second order approximation of the refractive index of the ionosphere [Bassiri and Hajj, 1993]. All nondispersive effects on pseudo-ranges and phase measurements are avoided by the geometry-free linear combinations (2) and (3).

[6] The elementary datum dit at time t is given by:

equation image

where ray(i) is the GPS ray from satellite k to receiver l, Ne(r,t) is the ionospheric electron density at position r and time step t, the integral of electron density is called Slant Total Electron Content (STEC) expressed in TEC units (1 TECU = 1016e/m2), and TGDk(t) and IFBl(t) are the inter-frequency biases of satellite k and of receiver l at time step t. The satellite and receiver biases in this equation system are defined relative to a constant which is fixed by imposing an additional condition on the biases. This condition can be to fix the bias of one receiver to an arbitrary value, or to impose that the sum of the biases is equal to zero. These biases are however varying slowly in time both for satellites and receivers [Sardon and Zarraoa, 1997; Mannucci et al., 1999].

[7] In addition to the problem of estimating the satellite and receiver biases, various noise sources are affecting the GF combination. The main noise source is multipath effect which preferentially occurred at low elevation angles. Because of the repetition of the GPS satellite orbits, the multipath noise can be reduced by various methods [Bock et al., 2000; Nikolaidis et al., 2001; Garcia et al., 2005], once the correct orbit period of GPS satellites is taken into account [Choi et al., 2004]. Other noise sources are affecting this combination at high frequencies, but their contribution is small compared to the STEC absolute value.

2.2. Forward Problem

[8] The forward problem summarized in equation (5) implies the knowledge of GPS ray paths for frequencies f1 and f2. Because the relative perturbation of light speed created by the electron content of the ionosphere is of the order of 10−6 in this frequency range, the ray paths can be approximated by straight lines, and the two frequencies have approximately the same ray paths with a maximum discrepancy of 500 m. This approximation allows a fast computation of the integral in equation (5) by the choice of an appropriate model parametrization [Garcia et al., 2005].

[9] Along their paths, the rays are sensitive to refractive index variations inside the Fresnel volume of the radio waves. At GPS frequencies f1 and f2, the size of the Fresnel zone perpendicular to the ray path is smaller than 0.25 km2 [Crespon, 2007]. Therefore, the GPS data are sensitive to the ionosphere small-scales.

3. Underdetermination and a Priori Information

3.1. An Underdetermined and Ill-Posed Inverse Problem

[10] Equation (5) sets the inverse problem in the class of integral inverse problems [Tenorio, 2001]. At this point, let us investigate its underdetermination level. Because the radius of the Fresnel volume is about 300 m, in order to have an over-determined inverse problem above Europe (100–700 km altitude range over 45°*45° latitude*longitude range) more than 200 million rays with different geometries are required. Therefore, the radio tomography of the ionosphere is necessarily a strongly underdetermined inverse problem.

[11] Moreover, the presence of satellite and receiver biases in the equation (5) makes the inverse problem ill-posed. Effectively, the satellites biases are correlated to the plasmasphere electron content, at altitudes above 1000 km, because all the receivers on ground or onboard Low Earth Orbit (LEO) satellites look at one GPS satellite along approximately the same direction. Consequently, the GPS satellite bias and the plasmasphere electron content along this direction cannot be completely separated from receiver data only.

[12] Strategies using single and double differences of GPS receiver geometry free combinations can remove completely the biases of both GPS receivers and satellites. However, these differential data create a new indetermination because they are defined relative to a constant. Moreover, a proper treatment of this inverse problem requires to take into account the complex data correlations through the a priori covariance matrix, which complicates the resolution of the inverse problem. The problem of biases estimates will not be further considered, and we will solve the homogeneous inverse problem corresponding to equation (5) corrected for biases.

3.2. A Priori Information: Assimilation Versus Constrained Inversion

[13] Because of the strong underdetermination of the inverse problem, good a priori information is necessary to solve it. The bottom line to constrain under-determined inverse problem is usually “the more physics you put in the problem, the better constrained it is.”

[14] With such an assumption, the best way to constrain the inverse problem is to use a physical model of the ionosphere and to invert the data to recover the fundamental parameters governing the spatial and temporal evolution of the electron density model. In that case, data assimilation is performed through 4D variational or Kalman filter methods [Chunming et al., 2004; Scherliess et al., 2004]. The 4D variational method is a simultaneous nonlinear inversion of data at different time steps, and Kalman filter method consists of an inversion at each time step and a time evolution of the a priori model. The main advantages of using a physics-based ionosphere model are to reduce the number of inverted parameters to a few parameters justified by the physical model, and to allow a prediction of the ionosphere state evolution. However, such models present disadvantages. First, the nonlinearity of the forward problem precludes a proper estimate of error and resolution of physics-based model parameters. In addition, the differences between the physical model and the true ionosphere dynamics are creating nonlinear modeling errors which are difficult to estimate. In particular, the small scales of the ionosphere dynamics cannot be reproduced through coarse physics-based model parametrization. And finally, the uneven coverage of the ionosphere is not taken into account into the model retrieval.

[15] Another approach consists in an inversion of the ionosphere electron density model parameters Ne(r,t) at each time step. The a priori model of electron density Ne0(r,t) can be an empirical model such as IRI [Bilitza, 2001] or NeQuick models [Radicella and Leitinger, 2001; Coïsson et al., 2006], a model describing the dynamics of the ionosphere through its physics like SAMI2 [Huba et al., 2000], or an electron density model obtained after inversion or assimilation of radio tomography data [Chunming et al., 2004]. The last type of model should be avoided due to its possible correlation with the data used in the inversion. The advantages of such a priori model and parametrization are to estimate error and resolution of the electron density model parameters, and to recover the small scales of the ionosphere dynamics in regions where the data sampling is high. The main drawback of such approach is that the ionosphere state evolution cannot be predicted. However, because the a priori constraints and model parametrizations are different, we believe that physics-based assimilation models and constrained inversion models are complementary, in particular in regions of dense data sampling such as North America, Europe and Japan. Physics-based models of the ionosphere will not be considered in the following discussion. However, the points discussed hereafter are relevant to the “optimization” of data assimilation algorithms in physics-based models.

4. Solving the Inverse Problem

4.1. A Nonlinear Inverse Problem

[16] The parameter to invert is the electron density in the ionosphere Ne(r,t). This parameter is strictly positive, so the state of zero information is not corresponding to a constant probability density [Tarantola, 1987]. Because the least square inversion theory imposes a parameter with a Gaussian probability density, the parameter to invert is not Ne(r,t), but m(r,t) = log [Ne(r,t)]. By using such a parameter, the integral equation (5), corrected for biases, is now a nonlinear inverse problem of the form:

equation image

where the function gi is a nonlinear function of m(r,t). There are two main ways of resolving such a nonlinear inverse problem. The first one is to investigate the inverse problem through nonlinear inversion schemes by using an a priori probability density of the model parameters centered on m0(r,t) = logNe0(r,t), and investigate the posterior probability density function by model search methods [Tarantola, 1987]. However, owing to the large number of unknowns and the strong underdetermination of the inverse problem, the amount of computation time required precludes the use of such methods. The second method is to linearize the problem by solving the forward problem for the a priori model m0(r,t), and invert the data residuals to retrieve the model perturbations through least square estimates. By doing so, we implicitly make the assumption of small model perturbations relative to the a priori model.

[17] Such a linearization of the inverse problem is described by the following equations:

equation image
equation image

In this last equation, the model parameters m(r,t) have been projected onto a finite basis of functions of the model space, and the parameters mj are the coefficients of this projection.

[18] The linearization of the inverse problem is a strong assumption. It imposes the a priori model to be close enough to the true ionosphere state. If the a priori model does not correctly explain the data, the nonlinear inverse problem can be solved by successive iterations of linear inversions. However, in that case, the non-unicity of the solutions generates instabilities that are causing divergence of the model. Consequently, our linearized inversion method is limited to small perturbations (about 10–15%) relative to the a priori model, and so the reliability of the results is limited by the accuracy of the a priori model.

4.2. Choice of Model Space Basis Functions

[19] At this point, it is important to discuss the choice of the model space basis functions, because the inversion results and efficiency will depend strongly on this choice. In a previous study [Garcia et al., 2005], it was demonstrated that constant blocks in cubed sphere coordinates [Sadourny, 1972; Ronchi et al., 1996] allow an optimal computation of the forward ray tracing problem. However, the fast forward computation is not an argument for the choice of the basis functions, because the electron density projected on the cubed sphere basis can be projected into another basis before solving the inverse problem.

[20] There are three main types of basis functions: nonlocal, local and wavelet basis functions. The nonlocal basis functions can be for example spherical harmonics along horizontal coordinates, and radial functions (splines, Chapman profiles…). The main advantages of these basis functions are the possibility to introduce a priori information on the model parameters directly through the choice of the basis functions, and to produce smooth models with a low numbers of model parameters. For example, if the radial profile is parametrized by a Chapman profile, the shape of the profile is imposed, and only three parameters are needed (altitude, maximum value and width of electron density peak). Another recent example is using a basis of empirical orthogonal functions extracted from the principal component analysis of model simulations [Zhai and Cummer, 2006]. However, the a priori imposed by the choice of nonlocal basis functions is often too strong, and it can create artifacts in areas poorly sampled by data. In the case of Chapman radial parametrization, the upper part of the model can be biased due to the imposed shape. Moreover, additional a priori constraints on time and space derivatives (see next section) are difficult to incorporate into the inverse problem with nonlocal basis functions.

[21] For local basis functions, the model parameters are directly related to the model values in one region of space and time. For example, blocks of constant model parameters, or model values at the nodes of a grid interpolated between the grid points are typical local basis functions [Zhai and Cummer, 2005; Stolle et al., 2005; Garcia et al., 2005]. The main advantage of these functions is that the constraints on the time and spatial gradients of the model are easy to incorporate in the inversion. An additional advantage is that the shape of the model is not imposed in the limit of infinitely small blocks. The main disadvantage of these functions is the high number of model parameters required to cover the model space.

[22] Another kind of basis functions are the wavelet basis functions. Depending on the scale parameter of the wavelet, the support of the function will vary in size. Such kind of basis also requires a large number of parameters, but due to their compression properties, only a few nonzero parameters are needed in equation (8). This property has been used recently to reduce the size of inverse problems in seismology [Chevrot and Zhao, 2007] for which the observable is sensitive to a large number of model parameters (large Fresnel volume of low frequency seismic waves). However, the small Fresnel volume of radio waves is not requiring such a tool. Moreover, the constraints on time and spatial gradients are difficult to handle with wavelet basis functions.

[23] Whatever the choice of basis functions, a strong idea in the resolution of under-determined inverse problems is that the final model should depend only on the a priori constraints, and not on the number of model space basis functions, or on the size of grid blocks. The only way to achieve this goal is to properly constrain the time and spatial gradients of the model parameters. Thus, we will prefer purely local basis functions such as blocks of constant electronic density, for which these constraints can be easily implemented. With such a parametrization, equations (7) and (8) are written:

equation image
equation image

where Lij is the length of ray(i) in block j, mj0 and Δmj the values of a priori model parameters and inverted model perturbations in block j. This equation can be written in matrix form:

equation image

where Δd is the observation vector, G is the sensitivity matrix of model parameters, and Δm is the model perturbation vector of the linearized inverse problem. The time dependence is not explicit in this equation because different time steps may be inverted simultaneously, but the data, the model and the sensitivity are evolving with time.

4.3. Constraints on the Inversion

[24] The under-determination of the inverse problem is solved by imposing additional constraints on the inverse problem. The usual way to constrain the inverse problem, once it is linearized in (11), is through damping and smoothing of the model parameters. As an example, let us write the cost function to be minimized in order to solve the inverse problem at one time step (optimization part of a Kalman filter):

equation image

where ∇Δm is a vector corresponding to the space derivatives of the model perturbations, ∥·∥d, ∥·∥grad and ∥·∥m are norms defined in the data, model gradient and model spaces by respective covariance matrices, λ and α are parameters to adjust the relative weights of smoothing and damping constraints. Previous studies [Ory and Pratt, 1995] demonstrated that the damping constraint biases the model norm, whereas model smoothing is not. Therefore, ideal inversion of linear inverse problems should use α = 0. However, because the problem here is not linear, the linearization impose that equation image≪1. In order to respect this criterion, we have to choose α such as model perturbations have a variance relative to the a priori model smaller than about 10–15%. So, the linearization imposes to have a priori model reliability at the 10–15% level.

[25] The solution to the inverse problem is obtained through the following generalized inverse:

equation image

where Cd is the data covariance matrix, Cm is the a priori covariance matrix of the model, and R is the finite difference model operator normalized to the a priori model finite differences [Ory and Pratt, 1995]. In this matrix, each model gradient is normalized by the same gradient computed in the a priori model, and the sum of the column vectors is equal to zero. This last property ensures that the model estimate is not biased if smoothing only is applied (α = 0).

[26] The smoothing and damping constraints depend completely on the values of the couple (λ,α). Assuming that the value of α is properly justified by an a priori on model gradients in time, only the λ value has to be determined. Various methods for the determination of optimal regularisation parameters are presented in a review by Tenorio [2001]. The most usual one is the L-curve estimate. This method is not optimal, but it requires less computation time than cross-validation methods. The method consists in solving the inverse problem for different values of the parameter λ, and to investigate the trade-off curve (L-curve) between model gradient norm (∥∇Δmgrad) and data residuals (∥ΔdGΔmd). The best value of λ is obtained at the maximum curvature of the L-curve where the trade-off is optimal. An example is shown in Figure 1.

Figure 1.

Example of L curve analysis. (a) L curve: data residual (∥Δd-GΔmd) as a function of model perturbation gradient norm (∥∇Δmgrad) in arbitrary units. (b) Curvature as a function of smoothing parameter in logarithmic scale. The filled circles stand for different inversion runs, and the filled star indicates the optimal inversion run corresponding to the highest curvature along the L-curve.

[27] Once the parameter λ is determined, a posteriori covariance of the model perturbation (CmP) and resolution (RmP) matrices are given by:

equation image
equation image

Because the model space sampling by one GPS ray is sparse, the matrix G is sparse. The inverse problems involving such large linear systems with sparse matrices have been extensively studied, and various algorithms are available to resolve them [Saad, 2003].

4.4. Time Evolution

[28] The ionosphere state at different time steps can be inverted simultaneously. However, in that case, the a priori model is not improved by the inversion results at previous time step, which are critical to remain in the linear approximation assumption, and the computer time increases significantly due to the inversion of larger matrices.

[29] So, we have chosen to use a Kalman like time evolution of the a priori model. Once a model perturbation value Δmest(t) is obtained, the inverted model mest(t) = m0(t) + Δmest(t) is computed. In order to compute the a priori model for the next time step, simple physical constraints are imposed: the model mest(t) is rotated around the Earth's spin axis in order to maintain a ionosphere model constant in local time reference frame, and the model space areas not sampled by data are set to the average between mest(t) and the a priori empirical model. Because the ionosphere is evolving with time, both a priori model and a priori model gradients are changing with time. In our approach, a priori model values Ne0(r,t) extracted from an empirical ionosphere model are used only for the first inversion step, and for the model space areas not sampled by the data. However, the matrix R is scaled by a priori model gradients at each time step.

[30] Because the a priori ionosphere state is based on our estimate at previous time step, the linearization of the inverse problem is justified if the ionosphere electron content is changing by less than 15% during 30 s. The statistical analysis of NeQuick ionosphere model [Coïsson et al., 2006] indicates that this criterion is satisfied. However, ionosphere state evolution can be much faster, in particular during geomagnetically active periods. For such conditions, GPS data at 1 second sampling should be used.

[31] Owing to the movement of GPS satellites, the data sampling is changing with time, and so the level of underdetermination of the inverse problem. As a consequence, the smoothing parameter λ is estimated at each time step from L-curve analysis to ensure that this parameter is properly adapted to the actual inverse problem. In addition, the computed model error and resolution estimates (CmP and RmP) will change as a function of the data sampling evolution. This is the reason why the computation of the model resolution and error are necessary in order to estimate the time evolution of the reliability of the model.

[32] Because the inversion procedure described above is smoothing only the model perturbation Δmest(t), and not the final model mest(t), the long term stability of the ionosphere model is not a priori ensured. In particular, the low vertical resolution inherent to ground based GPS data can create inverted models with spurious features, such as successive layers of high and low electron density, which are able to properly fit the data. However, the proper data selection, the use of the a priori empirical model in unsampled areas and for model gradient scaling, and the model perturbation smoothing avoid such effects. A flow chart of the whole inversion procedure is presented in Figure 2.

Figure 2.

Flow chart of the inversion procedure.

5. Regional Application

5.1. Some Details on Inversion Algorithm

[33] Our ionosphere tomography approach is applied to European GPS data preprocessed by the “Service for Electron Content and Troposphere Refractive index above Europe from GPS data” (SPECTRE) [Lognonné et al., 2006]. This service provide the STEC data corrected from satellite and receiver biases which are extracted from a 2D TEC map inversion. Data from about 300 European GPS receivers sampled at 30 s are available through this service for scientific applications since April 2004 at www.noveltis.fr/spectre.

[34] The a priori empirical ionosphere model is the latest version (v2) of NeQuick ionosphere model, because from our limited experience, a better data fit than for the IRI2001 model is usually obtained, due to the part of the model above 700 km height [Coïsson et al., 2006]. The only input parameter used in NeQuick model is f10.7 solar flux daily value. The components of the finite difference model operator R are normalized to the a priori model finite differences. By assuming independent data and model parameters, matrices Cm = σm2Id and Cd = σd2Id are multiples of the identity matrix Id with σm = 0.05 and σd = 1 TECU. As a result, the computation of their inverse is trivial. By imposing σm = 5% and α = 1, the model perturbations are limited to 5% root mean square variations.

[35] The model is parametrized by a 32 × 32 × 32 cubed sphere grid [Garcia et al., 2005] centered on Europe with an average cell size of 1.4°× 2° × 18 km in (latitude; longitude; altitude) coordinates (see Figure 4). The altitude coverage is from 100 to 700 km. The STEC data are corrected for ionosphere electron density contributions in between 700 km height and the GPS satellite position by a computation in NeQuick a priori ionosphere model. The average number of model blocks sampled by all the rays is about 12 000 (about one third of the model space).

[36] Because a priori model computations through NeQuick and ray tracing in the cubed sphere grid are fast, the most time consuming part of the algorithm is the resolution of equations (13), (14) and (15). The resolution of equation (13), including factorization, takes about 3.5 minutes on a 64 bits Sun WS2100 with bi-processor Opteron 250 with sparse sequential SuperLU routines [Demmel et al., 1999]. However, the computation of the L-curve requires repeating this resolution for each point of the curve. Moreover, the computation of model error and resolution through equations (14) and (15) in the actual state of the code is requiring a factor 100 more time than solving equation (13). However, the configuration of the GPS constellation is changing slowly, so the computation time can be slightly reduced by computing λ parameter, model error and resolution only every 10 or 20 epochs.

5.2. Preliminary Results

[37] The ionosphere state has been inverted over Europe for the last day of year 2005. On that day, f10.7 value is 84.5 solar flux units. The inversion is started at 6 h UT from NeQuick a priori model. Figure 3 is plotting the time evolution of variance reduction and data number. NeQuick model explains properly the GPS STEC data during day time (>90% variance reduction), but it faces some discrepancies at sunrise, and during the geomagnetically active period starting at 16 h UT (DST index < −15 nT). NeQuick model is of course not able to retrieve high frequency ionosphere variations because the model uses the daily average of f10.7 value as input. The strong decreases of variance reductions on short time scales are related to the presence of some wrong data because this phenomenon is seen both for NeQuick and inverted ionosphere models. After initialisation, the variance reduction of the inverted ionosphere model is rapidly increasing to reach values higher than 95%. Then it maintains at values higher than 87% during the geomagnetically active period, producing a good fit to GPS STEC data even during periods of high space and time variations of electron density. The number of STEC data is evolving during the day with the GPS constellation configuration relative to European GPS receivers. The data selection process excludes about one third of STEC data because a part of their ray path in the 100–700 km altitude range falls outside the inverted volume.

Figure 3.

(top) DST index time evolution from hourly values. (bottom) Time evolution of some inversion parameters: variance reduction (in %) after inversion (thin plain line), variance reduction of NeQuick model (dashed line), initial number of STEC data (dotted line) and number of STEC data after data selection (thick plain line).

[38] Figure 4 presents the diagonal values of the resolution matrix RmP of model perturbations Δmest(t), at different time steps. Low values are obtained due to the damping constraint. Despite the smoothing constraint, the best resolved regions are restricted to volumes along ray paths around the peak of electron density, and mainly above the center of the European GPS receiver network. Moreover, the properly resolved regions evolve rapidly in time with GPS satellites movements. However, small scale features can be mapped inside volumes with the highest resolution.

Figure 4.

Isosurfaces of resolution at two different times: 12 h UT (left) and 13h UT (right) seen from the top (top), west (middle) and south (bottom). Vertical scale is multiplied by a factor of three to enhance readability. Red dots indicate inverted grid points (ranging from 100 km to 700 km altitude).

[39] A comparison between the inverted ionosphere state and NeQuick model is presented on Figure 5 along a North-South cut at two different times. At 12 h UT, the inverted ionosphere model is very similar to NeQuick model because the a priori variance reduction is high, and so the inverted model perturbations are small. At 21 h UT, during the active geomagnetic period, the properly resolved areas in the middle of the model space present a high electron content to explain the STEC GPS data. In contrast, in the poorly resolved areas, the model is similar to the NeQuick a priori model. In the best resolved volumes, small-scale features are mapped on top of the smooth a priori model and the electron density profile is modified. Variance reduction and resolution analysis validate these features.

Figure 5.

Comparison between the inverted ionosphere model (left) and NeQuick model (right) at two different times (12 h and 21 h UT). (two top panels) Isosurfaces along a north–south cut in the three-dimensional model passing by the approximate center of the grid at 15° longitude. (bottom) Isosurfaces of the three-dimensional model at 21 h UT. The logarithm of electron density is plotted as colored isosurfaces and background color along the cut. Color bar is given on the right side. Vertical scale is multiplied by a factor of three to enhance readability.

5.3. Estimate of Linear Approximation Error

[40] The linear approximation error is estimated by performing a Gauss-Newton non linear inversion at some specific time steps with successive linearized inversions. The only objective measure of the data fit quality available is the χ2 estimator. χ2 is defined has the ratio between residual data variance and the square of a priori data error. A value close to the number (N) of inverted data demonstrates that data fit is within the data error bar. So, equation image is a criterion ensuring convergence of the Gauss-Newton nonlinear inversion. The a priori slant TEC data error is usually estimated to be in the 1-2 TEC units range.

[41] Figure 6a describes the evolution of equation image parameter as a function of the number of successive linear inversions performed at the same time step. These curves present a decrease of equation image toward an asymptotic value close to 2, if we assume a 1 TEC unit a priori data error. The asymptotic pattern of these curves demonstrate that, in the limit of smooth model perturbations, the model is converging to a final model estimate and that the error on the data is close to equation image TEC units. The asymptotic “final” models present more small scale features in order to better fit the data. The convergence speed is dependent of the initial data fit (or equation image value). Models with a poor initial data fit are converging more slowly than models with a starting point closer to the asymptotic solution.

Figure 6.

(top) (a) Evolution of equation image as a function of nonlinear iteration number at 12 h (circles), 13 h (squares) and 21 h (diamonds). Curves are shown for different a priori data errors of 1 TEC unit (plain lines and filled symbols) and 2 TEC units (dashed lines and open symbols). (bottom) Histograms of model parameter differences (in %) between the first and the 400th iteration, as a proxy of linearisation error, at (b) 12 h, (c) 13 h and (d) 21 h.

[42] Figures 6b–6d present histograms of model parameter differences (in %) between the first and the 400th iteration, as a proxy of linearisation error, at 12 h (b), 13 h (c) and 21 h (d). Most of the model parameters present a linear approximation error less than 10%.

[43] The linear approximation errors can be improved by either performing a Gauss-Newton nonlinear inversion at each time step, at the expense of computation time, and/or decreasing the value of the damping parameter α, at the expense of inversion stability.

5.4. Limitations and Future Evolutions

[44] Our ionosphere imaging method is limited by the linearization of the inverse problem. This assumption imposes to have an a priori ionosphere model reliability of about 10–20%, and a rate of change of electron density less than 10–20% at 30 seconds sampling. However, better a priori models and high rate GPS data will overcome these limitations.

[45] These preliminary results validate the inversion approach, the a priori NeQuick model, and the stability of the model time evolution. Because our results are strongly tied to the quality of the a priori ionosphere model, and because the data assimilation in physical ionosphere models can be improved by a better knowledge of the small-scale features of the ionosphere, we believe that our inversion algorithm can improve kalman like assimilation algorithms The algorithm can be improved by a better determination of the damping parameter α through a statistical study of the rate of change of electron density inside the a priori model as a function of solar flux conditions. Computation time can also be improved by a new procedure for computation of matrices RmP and CmP. Additional data such as ionospheric sounders, GPS occultation [Garcia-Fernandez et al., 2005; Stolle et al., 2005], DORIS and altimeter satellite data can be easily included in the inversion process. Finally, long term run and statistical validation by independent data are absolutely necessary to validate completely these very preliminary results.

6. Conclusion and Perspectives

[46] After analysing the forward and inverse problems of radio tomography, a time varying three-dimensional imaging method of the ionosphere through GPS slant TEC data has been described, justified and applied at regional scale. The preliminary results obtained with European GPS receiver data validate the stability of the method, even if a lot of improvements are still possible on the algorithm and a careful validation by independent data is necessary. Because additional slant TEC estimates from different data sources can be easily inserted in the processing, the model resolution and reliability can still be improved. The approach presented here is different of GPS data assimilation in physics-based ionosphere model. However, because of its capability to image small-scale variations in properly resolved areas where assimilation models are limited by their grid cell size, we believe that our approach is complementary to the assimilation approach.

[47] The ionosphere imaging methods discussed here will be greatly improved in the next years due to the advent of new GNSS systems and the improvement of actual ones. These new capabilities will improve our knowledge of ionosphere dynamics and its couplings to external forcing (sun, atmosphere, ocean, solid earth …).

Acknowledgments

[48] We acknowledge the three anonymous reviewers for their comments improving greatly the manuscript in different directions. We thank Pierdavide Coïsson and NeQuick team for providing the latest version of NeQuick model and Sébastien Chevrot for helpful discussions and debates on the “inversion philosophy.” The following permanent GPS receiver networks or corresponding institutions are acknowledged for providing freely their data at 30 seconds sampling: EUREF, GREF, RENAG, REGAL, IGN, IGEX, ASI, ICC. We also thank the following international institutions: the International GNSS Service for providing satellite orbits and some GPS data, and the World Data Center for Geomagnetism for providing DST index. Movies of the above figures are available at http://w3.dtp.obs-mip.fr/∼garcia/iono.html. We thank the SPECTRE team for the data preprocessing. This study was funded by the French ministry of research, CNES and ESA respectively through the RTE SPECTRE project, a space research scientific project and the SWENET SPECTRE project.

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