Geophysical inversion using approximate equality constraints

Current inversion methods estimating a physical property on a grid use two kinds of constraints: (1) absolute constraints requiring that the solution vector be close to a prespecified vector as, for example, proximity to the null vector in the ridge regression, and (2) relative constraints requiring that the elements of the solution vector be close to each other as, for example, smoothness imposed by methods requiring that spatial derivatives of the physical property be continuous. Using just absolute constraints may produce a biased solution conflicting with some true geological attributes of the source. On the other hand, using just relative constraints may be insufficient to stabilize the inversion. We present a stable inversion method incorporating absolute constraints only at points where the physical property is known as, for example, outcroppings and boreholes. Elsewhere, relative constraints are introduced according to a linear relationship for the spatial distribution of the physical property. The proposed method is analogous to an interpolation method applied to the physical property distribution: the interpolating function must satisfy some property, like continuity, and pass through the data. The proposed method is versatile in incorporating not only factual, but also indirect information such as continuity, symmetry, and trends. In addition, it may be applied both to linear and nonlinear problems. The relevant results obtained from applying the method to synthetic potential-field data were: (1) ability to produce reliable mappings of the basement relief of a sedimentary basin (from gravity data) and of an areal distribution of magnetization, using only smoothness constraints and restricted surface and borehole information; (2) operational simplicity because of the solution insensitivity to the damping parameter. The method is also applied to the Bouguer anomaly over San Jacinto Graben, California, producing results consistent with previous interpretations.