Many inverse problems are most naturally formulated as implicit problems where data cannot be expressed in closed form as a function of the unknown model parameters. Notable examples are cases where the data satisfy a partial differential equation with model parameters as constants. When the problem can be recast as an ordinary explicit inverse problem the price to be paid is often that the functional relationship between data and model cannot be formulated in closed form. The consequence is typically that a computer time intensive numerical algorithm is needed to perform the forward calculation, thereby making an iterative solution of the problem unreasonably time consuming or impossible. In this paper we present a probabilistic procedure to solve rather general, implicit inverse problems through Monte Carlo sampling of feasible solutions. Our starting point is a probabilistic formulation of inverse problems, and our goal is to produce near-independent samples from the posterior distribution in model space. From these samples important information on the model and its resolution can be obtained. The proposed algorithm is applied to an implicit real-data problem involving analysis of the fluid motions at the Earth's core-mantle boundary from geomagnetic data.

Algorithms used in geomagnetic main-field modelling have for the most part treated the noise in the field measurements as if it were white. A major component of the noise consists of the field due to magnetization in the crust and it has been realized for some time that such signals are highly correlated at satellite altitude. Hence approximation by white noise, while of undoubted utility, is of unknown validity. Langel, Estes & Sabaka (1989) were the first to evaluate the influence of correlations in the crustal magnetic field on main-field models. In this paper we study two plausible statistical models for the crustal magnetization described by Jackson (1994), in which the magnetization is a realization of a stationary, isotropic, random process. At a typical satellite altitude the associated fields exhibit significant correlation over ranges as great as 15" or more, which introduces off-diagonal elements into the covariance matrix, elements that have usually been neglected in modelling procedures. Dealing with a full covariance matrix for a large data set would present a formidable computational challenge, but fortunately most of the entries in the covariance matrix are so small that they can be replaced by zeros. The resultant matrix comprises only about 3 per cent non-zero entries and thus we can take advantage of efficient sparse matrix techniques to solve the numerical system.We construct several main-field models based on vertical-component data from a selected 5" by 5" data set derived from the Magsat mission. Models with and without off-diagonal terms are compared. For one of the two Jackson crustal models, k3, we find significant changes in the main-field coefficients, with maximum discrepancies near degree 11 of about 27 per cent. The second crustal spectrum gives rise to much smaller effects for the data set used here, because the correlation lengths are typically shorter than the data spacing. k, also significantly underpredicts the observed magnetic spectrum around degree 15. We conclude that there is no difficulty in computing mainfield models that include off-diagonal terms in the covariance matrix when sparse matrix techniques are employed; we find that there may be important effects in the computed models, particularly if we wish to make full use of dense data sets. Until a definitive crustal field spectrum has been determined, the precise size of the effect remains uncertain. Obtaining such a statistical model should be a high priority in preparation for the analysis of future low-noise satellite data.

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