We introduce non-stationary Matérn field priors with stochastic partial differential equations, and construct correlation length-scaling with hyperpriors. We model both the hyperprior and the Matérn prior as continuous-parameter random fields. As hypermodels, we use Cauchy and Gaussian random fields, which we map suitably to a desired correlation length-scaling range. For computations, we discretise the models with finite difference methods. We consider the convergence of the discretised prior and posterior to the discretisation limit. We apply the developed methodology to certain interpolation and numerical differentiation problems, and show numerically that we can make Bayesian inversion which promotes competing constraints of smoothness and edge-preservation. For computing the conditional mean estimator of the posterior distribution, we use a combination of Gibbs and Metropolis-within-Gibbs sampling algorithms.

We study Cauchy-distributed difference priors for edge-preserving Bayesian statistical inverse problems. On the contrary to the well-known total variation priors, one-dimensional Cauchy priors are non-Gaussian priors also in the discretization limit. Cauchy priors have independent and identically distributed increments. One-dimensional Cauchy and Gaussian random walks are special cases of Lévy α-stable random walks with α = 1 and α = 2, respectively. Both random walks can be written in closed-form, and as priors, they provide smoothing and edge-preserving properties. We briefly discuss also continuous and discrete Lévy α-stable random walks, and generalize the methodology to two-dimensional priors.We apply the developed algorithm to one-dimensional deconvolution and two-dimensional X-ray tomography problems. We compute conditional mean estimates with single-component Metropolis-Hastings and maximum a posteriori estimates with Gauss-Newton-type optimization method. We compare the proposed tomography reconstruction method to filtered back-projection estimate and conditional mean estimates with Gaussian and total variation priors.

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Abstract. The stochastic inversion method in ionospheric radiotomography is reviewed with a special emphasis on regularization used in the inversion process. Regularization is used both for preventing vigorous point-to-point oscillations and for controlling the peak altitude and thickness of the inversion result. The latter usually means importing a priori information on the layer height and thickness to the solver. In this paper it is pointed out some information on the ionospheric altitude and the profile shape even in the case of a purely horizontally stratified layer. If this information could be used in choosing an appropriate regularization, no additional information would be needed. Simulation tests are presented which indicate that the altitude of a horizontally stratified layer can be determined with a reasonable accuracy without any a priori information. An attempt is also made to use the data for determining the shape of a proper regularization profile. Although some success is achieved in this effort, it is concluded that available a priori information, for example, ionosonde or incoherent scatter measurements, should be used in choosing the regularization profile. The ideas are tested with true data obtained from difference Doppler measurements carried out in Scandinavia, and the results are compared with simultaneous observations made by the European incoherent scatter radar. The comparison shows a reasonable agreement, although clear discrepancies also occur, for instance, in the shape of the bottomside profile.

Abstract. Incoherent scatter radar autocorrelation function estimates produced by alternating codes are found to contain significant correlations between different lags and different heights. While no correlations are caused by the background thermal noise, the special structure of the alternating code sequences causes clutter or the self-noise contribution of the codes themselves to introduce maximal positive correlations between the different lag estimates. Because of this, postintegration of adjacent heights does not improve the results as much as could be expected. For a high signal-to-noise ratio, alternating code results are worse than estimates produced by similar random codes. In this paper we show that we can correct the situation by randomizing the alternating codes. This is accomplished by multiplying each alternating code sequence in the alternating code cycle by a fixed random-looking sequence. This multiplying sequence can be changed and randomly chosen at the end of each alternating code cycle. The multiplying sequence can also be kept constant with almost equally good results. If this is done, the special structure of the alternating code sequences actually causes the correlations to become less than the correlations produced by random codes. The improvement in accuracy is discussed in various situations. The reasons for the appearance of maximal positive correlations with alternating codes are explained in a mathematical appendix.