Subspace method for solving large-scale equivalent layer and density mapping problems

Abstract

The equivalent layer technique explores an intrinsic property of potential fields that any observed field at the ground surface can be reconstituted from a fictitious continuous distribution at an arbitrary flat surface that is indistinguishable from the field of the true sources. This continuous distribution can be represented by a set of discrete cells of known position and size but unknown physical property by solving a linear system, with size proportional to the number of data points measured. The density distribution at the equivalent layer carries information about the true sources because it is a scaled and downward-continued version of the field generated by the true sources at the level that the geophysical survey was undertaken. The computation of this downward-continued field is unstable, and an equivalent source evaluation is constrained by the intense computational demand required to solve the associated large linear system. A new formulation is developed to directly solve large-scale gravity equivalent layer problems using a subspace representation for the unknown density distribution. This subspace basis is constructed by applying the singular value decomposition to the matrix containing the gridded data set. A procedure to diminish (by two orders of magnitude) the number of forward model evaluations is introduced by exploring the symmetry of the gravity kernel and its evaluation on a regular mesh. The density distribution at the equivalent layer is used to outline the spatial distribution of contrasting underlying sources, to discriminate regions with predominant positive or negative density contrast, and to estimate the mass excess or deficiency for sources with positive or negative density contrast. This technique is applied to determine density models for the Carajás airborne gravity survey and to analyze density distributions associated with banded iron formations and structures of the Carajás Mineral Province, Brazil, providing mass estimates for specific geologic unities.