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Geophysical inference is characterized by non-linear relationships between model and data parameters, large model spaces describing spatial distributions of media properties, and intensive computations related to the numerical resolution of the forward problem. Although sampling approaches are convenient to solve such inverse problems, sometimes the involved computations are demanding. We consider here sampling techniques directed to improve the efficiency of sampling procedures in large real geophysical applications. We propose a sampling algorithm incorporating classical importance sampling within a two-step (or multistep) Markov chain sampler set-up. The first step of the algorithm is a Metropolis sampler ergodic to an importance density function and the second step is a Metropolis sampler correcting from the bias introduced by the importance density function; the combined algorithm samples the posterior probability density function asymptotically. The importance density is a combination of the prior density and an importance likelihood function, obtained as an approximation of the data likelihood function. Hence, the importance density is an approximation of the posterior density. Although rejection rates of the combined algorithm are larger than the rejection rates of the simple step Metropolis algorithm, the computational sampling efficiency is improved when the calculation of the importance density is much easier than the calculation of the actual posterior density. The algorithm is characterized by high acceptance rates in its second step because the first step serves as a barrier rejecting unlikely samples. Data likelihood approximations can be obtained in several ways : using simplified geophysical simulation, using the likelihood with partial (smaller) sets of the observed data, or using information obtained by preliminary analysis of the data. The technique has been successfully used for the inversion of phase arrival times from an offset vertical seismic profile (OVSP) data.

We model and analyze the horizontal fluid flow near the core-mantle boundary from geomagnetic observations. The non-linear core motion problem is studied by means of a Bayesian Monte Carlo approach under the frozen flux assumption and subject to a relaxed geostrophic bound. In relaxing the geostrophic constraint we allow a nonzero Lorentz force to enter into the force balance of the core. We loosen the geostrophic constraint in steps and present a series of fluid flow models deviating from geostrophy by up to 29%. Models with up to a few percent ageostrophy have structures that are generally well resolved and are clearly under strong influence of the geostrophic constraint. Models of higher ageostrophy have structures that are generally poorly resolved as a result of the relaxation of the geostrophic bound. However, we find a certain well-determined structure of upwelling underneath the east Indian Ocean in our full series of models.

We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly.

Static Light Scattering.- Thermodynamic Geometry Determines Optimal Temperature Profile in Distillation Column.- Inversion of Seismic AVO Data.- Generalized Fourier Transforms and Inverse Scattering.- Multi-Step Samplers for Improving Efficiency in Probabilistic Geophysical Inference.- Full Waveform Inversion of Seismic Data for a Viscoelastic Medium.- Integrating Surface Reflection Data Inversion and Offest-VSP Modeling for Imaging of Converted S-Waves in Gulf Coast Hydrocarbon Reservoirs.- Inversion for the Moment Tensor and Source Location Using Love- and Rayleigh-Type Waves.- Resolution of Cole-Cole Parameters Based on Induced Polarization Data.- The Resolution Function in Linearized Born and Kirchoff Inversion.- SVD Analysis of a 3D Inverse Thermal Model.- Optimization Tools for Tikhonov Regularization of Nonlinear Equations Using the L-Curve and its Dual.- The PP-TSVD Algorithm for Image Restoration Problems.- Smooth versus Sharp Frechet Kernels in Time-Distance Helioseismology.- Covariance Modelling for Merging of Multi-Sensor Ocean Surface Data.- Inversion of 2D Geoelectrical Data by Iterated Multichannel Deconvolution.- Image Restoration as an Inverse Problem.- Resolution Studies of Fluid Flow Models Near the Core-Mantle Boundary Using Bayesian Inversion.- Inversion of GPS Occultation Data for Atomspheric Profiling.- A Limited Memory BFGS Method for an Inverse Problem in Atmospheric Imaging.