Practical applications of uniqueness theorems in gravimetry: Part II—Pragmatic incorporation of concrete geologic information

Abstract

We illustrate the importance of establishing solution uniqueness through mathematical restrictions reflecting a source attribute. We also illustrate the validity and utility of a guideline derived in an accompanying paper for constructing sound gravity inversion methods for the class of sources presenting either homogeneous or depth‐independent density distributions. The two‐part guideline is (1) to introduce a priori information favoring uniqueness, either by assuming that a nonnull density distribution depending only on x and y is confined to the interior of a horizontal slab with known position or by limiting the class of possible solutions to homogeneous, simply connected polygons (or polyhedra) with known density, displaying no fancy shapes and no curling apophyses at their borders, and (2) to introduce information favoring solution stability by estimating only the features of the source which may be resolved by the data. Following the guideline, we apply different methods to gravity data using interpretation models consisting of a grid of cells on the x‐y and x‐z planes. In both cases the estimates are very close to the true synthetic source. The data produced by the distribution varying with x and z are also inverted using the method, which minimizes the norm of the first‐order derivative of the density. This constraint does not reflect a true source attribute but is strong enough to stabilize the solution and to guarantee its uniqueness. Because of the strong bias imposed to the solution, the estimated distribution, although unique and stable, is far from the true source, concentrating most of the anomalous mass at the surface. Finally, we present an alternative method which redistributes the estimated anomalous mass downward. To be effective, this technique requires prior knowledge about the source depth to the top. In addition, the source should not be too small and deep. Although being able to produce good results, this alternative method requires a great dose of the interpreter's art.