  In a broad and fundamental type of ''inverse problems'' in science, one
infers a spatially distributed physical attribute based on observations of
processes that are controlled by the spatial attribute in question. The
data-generating field processes, known as ''forward processes'', are usually
nonlinear with respect to the spatial attribute, and are often defined
non-analytically by a numerical model. The data often contain a large number of
elements with significant inter-correlation. We propose a general statistical
method to tackle this problem. The method is centered on a parameterization
device called ''anchors'' and an iterative algorithm for deriving the
distribution of anchors conditional on the observed data. The algorithm draws
upon techniques of importance sampling and multivariate kernel density
estimation with weighted samples. Anchors are selected automatically; the
selection evolves in iterations in a way that is tailored to important features
of the attribute field. The method and the algorithm are general with respect
to the scientific nature and technical details of the forward processes.
Conceptual and technical components render the method in contrast to standard
approaches that are based on regularization or optimization. Some important
features of the proposed method are demonstrated by examples from the earth
sciences, including groundwater flow, rainfall-runoff, and seismic tomography.
