The Inverse Problem behind the 3D Ionosphere Reconstruction

Kurzfassung

The ionosphere is the ionized part of the upper Earth's atmosphere lying between about 50 km and 1000 km above the Earth surface. The solar radiation and particle precipitation control the temporal and spatial variation of the ionization level in this near-Earth space strongly depending on day of year, day time and geographic location. From the applications perspective the electron density, Ne, is certainly one of the most important parameter of the ionosphere because it induce several effects on the radio signal propagation. Especially the critical frequency, foF2, which is related to the F2 layer peak electron density, NmF2, and builds the lower limit for the maximum usable frequency MUF, is of particular interest with regard to the HF radio communication applications [Davies, 1990]. The Global Navigation Satellite System (GNSS) has become the premier tool for measuring, monitoring and reconstructing the ionosphere. During the propagation through the ionosphere, L-band signals of GNSS satellites are delayed. The delay results in range errors of up to 100 meters. In a first order approximation the range error is proportional to the integral of the electron density along the ray path, i.e. TEC (total electron content) [Jakowski et al., 2011]. Thus, the information about the total electron content along the receiver-to-satellite ray path s can be obtained from the dual frequency measurements permanently transmitted by GNSS satellites. This measured TEC along the receiver-to-satellite ray path s is the integral of the electron density Ne along s. Obviously, the individual TEC measurements contain no information about the spatial variation of the electron concentration. Therefore, the task of finding the spatial distribution of the electron density requires a tomographic solution. To use the discrete inverse theory for the tomographic inversion, we have first to choose the basis functions. There are basically two categories. First, there are non-voxel based methods (c.f. [Kee et al., 2003]). In this category separable basis function, mostly spherical harmonics in the latitude and longitude dimensions and empirical orthogonal functions in the radial dimension are used to discretize the electron density. Voxel based functions form the second group. In our reconstruction approach, we choose this way. We set up a grid of n voxels, each bounded in latitude, longitude and altitude. By spatial discretization of the electron density Ne using the indicator functions of the voxels we transform the original integral equations to a linear system of equations. For each receiver-to-satellite ray path s we get an approximation of TEC along s by a sum of the electron density in the i-th voxel multipl. by the length of the propagation path of the ray path s through the i-th voxel. The resulting system of equations is underdetermined, inconsistent and ill-conditioned because of measurement geometry and measurement errors. Many approaches exist to compensate the missing information, to regularize and to solve this inverse problem. There are row action methods, such as algebraic reconstruction technique (ART) and multiplicative algebraic reconstruction technique (MART) on the one hand (c.f. [Heise, 2002]), and not iterative methods, such as singular value decomposition (c.f. [Erturk et al 2009]), Gauss-Markov Kalman filter approach etc. (c.f. [Bust et al., 2008]) on the other hand. As data base for our reconstruction approach we use the measured slant TEC along the receiver-to-satellite ray path from ground base stations as well as vertical TEC calculated from this measured slant TEC by means of a common single layer mapping function [Jakowski et al., 2011]. Additionally, in situ foF2 measurements of the ionosonde stations provide further significant information. The basic concept of our approach is the following one: We calculate an estimation of a solution of the linear system of eq. by means of an adapted row action method NART (Neustrelitz Algebraic Reconstruction Technique). To circumvent the fact that the problem is ill posed, the iteration procedure starts with a background ionosphere. To determine the initial guess for the iteration procedure we assimilate the vertical TEC as well as the ionosonde station measurements into the DLR empirical 3D ionosphere model by means of a successive corrections method. The iteration algorithm stops, when the predefined accuracy of the solution has been reached. To make the iteration result smoother and to increase the dependence of the result on the measurements, we realize the influence of the iteration changed voxels on the neighbored voxels by means of some weighting functions. At the conference our model assisted iterative assimilation approach will be presented in more details.