Gravity inversion using a binary formulation Richard A. Krahenbuhl* and Yaoguo Li Gravity and Magnetics Research Consortium, Department of Geophysics, Colorado School of Mines, Golden, CO Summary When a nil zone is present in the subsurface salt structure, it effectively creates an annihilator of density contrast that gives rise to zero gravity response on the surface. As a result, part of the salt structure is invisible to the surface data and inversion algorithms often have difficulties in recovering the salt structure correctly. We develop a binary inversion technique in which the density contrast is restricted to being one of the two possibilities: either zero or the value expected at a given depth. The binary condition places a strong restriction on the admissible models so that the non-uniqueness caused by nil zones can be resolved. In this presentation, we will outline the formulation, discuss the solution strategy, and illustrate it with numerical examples. Introduction Inversion methods for imaging salt structure using gravity data can be divided into two general categories. The first are interface inversions. These methods assume a simple topology for the salt body and known density contrast and construct the base of the salt (e.g., Jorgensen and Kisabeth 2000). Methods in the second category are generalized density inversions. These methods construct a density contrast distribution as a function of spatial position and image the base of salt by the transition in recovered density (e.g., Li 2001). The interface inversion has the advantage that it directly inputs the known density contrast at each depth and provides a direct image of base of salt. However, difficulties may arise from the nonlinear relationship between observations and salt boundaries. In addition, the assumed simple topology of salt creates difficulties when either regional field or small-scale residuals due to shallow sources are not completely removed. The density inversion has the flexibility of handling multiple anomalies and the solution is easier to obtain because the relationship between observations and the density contrast is linear. However, these methods, as they are currently formulated, are not well suited for the cases where nil zones are present. The reason is the following. These methods start by discretizing a 3D model into a large number of cuboidal cells, with each cell assuming a constant density value between a lower and upper bound. The solution is obtained by minimizing a model objective function measuring the structural complexity of the density model subject to fitting the data to an appropriate degree. This approach works well when the density contrast is single-signed, i.e., it is either entirely positive or negative. Difficulty arises, however, when the salt is located at a depth such that the sedimentary density is equal to the constant salt density within a depth interval inside the salt. When this occurs, the density contrast reverses sign as the depth increases and the gravity anomalies due to top and bottom portions of the salt cancel out. Consequently, a portion of the salt body is invisible to the surface gravity data. Inversion allowing continuous density values will in general produce a model that has little resemblance to the true structure. The data are satisfied by intermediate density values. To overcome difficulties associated with both methods, we propose a binary formulation that enables one to incorporate the density contrast values into the inversion while retaining the flexibility and linearity of the density inversion. Theory The difficulty of an annihilator can only be overcome by incorporating prior information to restrict the class of admissible models. We propose to impose the condition that the density contrast must be the discrete values appropriate for the geologic problem. In the simplest form, we restrict the density contrast to being either zero or a known value at a given depth. Staying within this constraint, the problem becomes one of minimizing an objective function subject to allowing model parameters to attain only one of two values at each depth: { } . ) z ( , subject to , ) ( ) ( min. m d ρ ρ ρ λφ ρ φ φ Δ ∈ + = 0 (1) where d φ is the data misfit function, m φ is the model objective function, and ) ( z ρ Δ is the expected density contrast at depth z . For convenience, we introduce a new binary model parameter τ( r )∈{0,1} such that ) ( ) ( z r ρ τ ρ Δ = . (2)