Johannes Norberg

Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

We present a novel ionospheric tomography reconstruction method. The method is based on Bayesian inference with the use of Gaussian Markov random field priors. We construct the priors as a system of stochastic partial differential equations. Numerical approximations of these equations can be represented with linear systems with sparse matrices, therefore providing computational efficiency. The method enables an interpretable scheme to build the prior distribution based on physical and empirical information on the structure of the ionosphere. We show through synthetic test cases in a two-dimensional setup of latitude-altitude slices how this method can be applied to satellite-based ionospheric tomography and how information about the structure of the ionosphere can be implemented in the prior. The technique is capable of being easily extended to multifrequency tomographic analysis or used for the inclusion of other data sets of ionospheric electron density, such as ground-based observations by radars or ionosondes.

Beacon satellite receiver for ionospheric tomography

We introduce a new coherent dual-channel beacon satellite receiver intended for ionospheric tomography. The measurement equation includes neutral atmosphere and ionosphere propagation effects, relative errors in satellite and receiver clocks, and residual Doppler shifts caused by errors in the satellite ephemeris. We also investigate the distribution of errors for phase curve measurements and the use of phase curve measurements for limited angle tomography using the framework of statistical linear inverse problems. We describe the design of our beacon satellite receiver software and present one possible hardware configuration. Finally, we present results obtained using a network of four newly developed receivers and compare the results with those of an existing ionospheric tomography network at Sodankyla Geophysical Observatory.

Multi-instrument ionospheric tomography in Scandinavia with Bayesian statistical inversion and correlation priors

We present a novel algorithm for ionospheric tomography and the latest results obtained with it. Ionospheric tomography is mathematically a sparse limited-angle tomography problem and therefore severely ill-posed. This means that to obtain reasonable reconstructions the problem needs a strong regularisation. In ionospheric tomography the regularisation is often implemented with a limited set of base functions, Tikhonov regularisation, initial profiles for iterative methods, or with a different combinations of these schemes. These methods have been shown to produce satisfactory reconstructions, but the role of the regularisation and how much the chosen method actually constraints the possible outcomes is not always clear. Our tomography scheme is implemented in the Bayesian statistical framework (Markkanen et al. Ann. Geophys., vol. 13, pp. 1277-1287, 1995). In Bayesian inference the regularisation is given as an a priori distribution. The a priori distribution contains the information about the unknown parameters before the measurements. The second step is to build a likelihood function for the unknown parameters, given the observed measurements. By combining the likelihood with the prior density function, we obtain the a posteriori distribution, from where we can obtain the estimate with the highest probability, based on the information in our disposal. Here we give the a priori distribution with the novel correlation priors (Roininen et al. Inverse Probl. Imag., vol. 5, issue 2, pp. 611-647, 2011). Essentially the correlation priors are Gaussian Markov random fields, wherein we can state the physical assumptions of the ionosphere in an interpretable manner. With this implementation we get a very thorough understanding of what is the role of the regularising a priori distribution. On addition to that, the statistical framework also gives a very natural way to combine different ionospheric measurements. Therefore we can use measurements from Low Earth Orbit (LEO) and Global Positioning System (GPS) satellites as well as ground based measurements in the same model. Starting from 2011, as a part of TomoScand project, Finnish Meteorological Institute has been installing a new receiver network for LEO beacon satellites. At the moment, together with the receiver network of Sodankyla Geophysical Observatory, we collect data from 11 receivers stations. The northern most receiver located in Svalbard and southern most in Estonia. We also collect data from 86 GPS receiver stations provided by Finnish company Geotrim. We present the latest results with the new network and algorithm in a 2-dimensional case, however, the goal in the near future is to extend the model to a 3-dimensional case to cover over Scandinavia.