Practical applications of uniqueness theorems in gravimetry: Part I—Constructing sound interpretation methods

Abstract

To obtain a unique and stable solution to the gravity inverse problem, a priori information reflecting geological attributes of the gravity source must be used. Mathematical conditions to obtain stable solutions are established in Tikhonov's regularization method, where the a priori information is introduced via a stabilizing functional, which may be suitably designed to incorporate some relevant geological information. However, there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem. Rather, there are many theorems, usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution. In Part I of these companion papers, we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inversion methods. As an example, we analyze three uniqueness theorems in gravimetry restricted to the class of homogeneous bodies with known density and show that the uniqueness conditions established by them are more probably met if the solution is constrained to be a compact body without curled protrusions at their borders. These conditions, together with stabilizing conditions (assuming a simple shape for the source), form a guideline to construct sound gravity inversion methods. A historical review of the gravity interpretation methods shows that several methods implicitly follow this guideline. In Part II we use synthetic examples to illustrate the theoretical results derived in Part I. We also illustrate that the presented guideline is not the only way to design sound inversion methods for the class of homogeneous bodies. We present an alternative approach which produces good results but whose design requires a good dose of the interpreter's art.