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Computerized ionospheric tomography (CIT) is a technique that allowsreconstructing the state of the ionosphere in terms of electron content froma set of slant total electron content (STEC) measurements. It is usuallydenoted as an inverse problem. In this experiment, the measurements areconsidered coming from the phase of the GPS signal and, therefore, affectedby bias. For this reason the STEC cannot be considered in absolute terms butrather in relative terms. Measurements are collected from receivers notevenly distributed in space and together with limitations such as angle anddensity of the observations, they are the cause of instability in theoperation of inversion. Furthermore, the ionosphere is a dynamic medium whoseprocesses are continuously changing in time and space. This can affect CIT bylimiting the accuracy in resolving structures and the processes that describethe ionosphere. Some inversion techniques are based on ℓ2minimization algorithms (i.e. Tikhonov regularization) and a standardapproach is implemented here using spherical harmonics as a reference tocompare the new method. A new approach is proposed for CIT that aims topermit sparsity in the reconstruction coefficients by using wavelet basisfunctions. It is based on the ℓ1 minimization technique and waveletbasis functions due to their properties of compact representation. Theℓ1 minimization is selected because it can optimize the result withan uneven distribution of observations by exploiting the localizationproperty of wavelets. Also illustrated is how the inter-frequency biases onthe STEC are calibrated within the operation of inversion, and this is usedas a way for evaluating the accuracy of the method. The technique isdemonstrated using a simulation, showing the advantage of ℓ1minimization to estimate the coefficients over the ℓ2 minimization.This is in particular true for an uneven observation geometry and especiallyfor multi-resolution CIT

Using sparse regularization for multi-resolution tomography of the ionosphere