Abstract
Abstract. 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 ℓ2 minimization algorithms (i.e. Tikhonov regularization) and a standard approach is implemented here using spherical harmonics as a reference to compare the new method. A new approach is proposed for CIT that aims to permit sparsity in the reconstruction coefficients by using wavelet basis functions. It is based on the ℓ1 minimization technique and wavelet basis functions due to their properties of compact representation. The ℓ1 minimization is selected because it can optimize the result with an uneven distribution of observations by exploiting the localization property of wavelets. Also illustrated is how the inter-frequency biases on the STEC are calibrated within the operation of inversion, and this is used as a way for evaluating the accuracy of the method. The technique is demonstrated using a simulation, showing the advantage of ℓ1 minimization to estimate the coefficients over the ℓ2 minimization. This is in particular true for an uneven observation geometry and especially for multi-resolution CIT.
Highlights
Tomographic imaging is an important tool for understanding the ionosphere, its behaviour and its effects on radio propagation
This paper describes an alternative method based on the 1 norm, using wavelet basis functions, in relation to the 2 norm, using spherical harmonics, for Computerized ionospheric tomography (CIT)
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
Summary
Tomographic imaging is an important tool for understanding the ionosphere, its behaviour and its effects on radio propagation. In general a proper regularization is needed to ensure stability, and to reduce artefacts and noise in the reconstruction due to lack of data Another recent approach uses the 1 norm as the metric to regularize the solution. An implementation of this is given by the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) (Beck and Teboulle, 2009; Daubechies et al, 2004). This algorithm is tailored with wavelets and, under certain conditions, aims to minimize the number of basis functions that can be used to represent the structures in the ionosphere.
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