Hybrid optimization for a binary inverse problem Richard A. Krahenbuhl* and Yaoguo Li Gravity and Magnetics Research Consortium, Department of Geophysics, Colorado School of Mines Summary We have developed a hybrid optimization algorithm for inversion of gravity data using a binary formulation. The new algorithm utilizes the Genetic Algorithm (GA) as a global search tool, while implementing Quenched Simulated Annealing (QSA) intermittently for local search. The hybrid has significantly decreased computational cost over GA or Simulated Annealing (SA) alone and has allowed for successful inversion of more realistic gravity problems. We illustrate the algorithm using a large 2½D model derived from the SEG/EAGE 3D salt model, which has a complex background density profile and a pronounced nil zone. Introduction When a salt body of uniform density is located at a depth where the sedimentary density is equal to the salt density within a depth interval, a region of salt referred to as a nil zone exists. Because of the zero contribution to surface gravity data from the nil zone, gravity inversion algorithms typically produce poor result near a nil zone. Furthermore, the cancellation of gravity effects due to salt above and below the nil zone renders portions of a salt body invisible to the surface gravity data and difficult to image without incorporating specific information. To overcome difficulties associated with nil zones, we use a binary formulation (Krahenbuhl and Li, 2002) that enables one to incorporate the density contrast values, a strength of non-linear interface inversion, while retaining the flexibility and linearity of density (cell based) inversion. Because the variable can only take on discrete values, 0 or 1 for sediment or salt respectively, derivative- based minimization techniques are no longer applicable. There are two obvious techniques for carrying out such discrete-variable minimization problems, namely, genetic algorithm (GA) and simulated annealing (SA). Both are ideally suited for working with binary values. The GA has the advantage of easily incorporating background density information and any previous inversion results simultaneously. It can also easily locate the general vicinity of the solution in the model space. The drawback is that it has a slow convergence and requires a large number of iterations to obtain the final solution. The SA, on the other hand, can zero in on a final solution rapidly but requires a great deal of work to reach this neighborhood. When combining the two methods, by modifying the GA for course global search and QSA for local search, the resulting hybrid method retains the advantages of both algorithms and has a much higher computational efficiency: it is faster by an order of magnitude than either GA or SA alone. The GA allows for rapid build-up of the larger model features, while QSA rapidly modifies small- scale structure within the model. As a result, the new algorithm enables us to solve realistic problems with a large number of parameters and a complex background density profile. In the following, we will first review the methodology of the binary formulation for gravity inversion and discuss its solution by genetic algorithm and simulated annealing alone. We will develop a hybrid formulation and illustrate it using a 2½D gravity problem with a large number of unknown parameters, a variable background density profile and a nil zone at depth. Binary formulation The difficulty caused by the presence of a nil zone, as discussed in the Introduction, can only be overcome by incorporating prior information to restrict the class of admissible models. We impose the condition that the density contrast must be the discrete values appropriate for the geologic problem. In the simplest form, density contrast is restricted to being either zero or a known value at a given depth. Similar binary approach has been used before. For example, Litman et al. (1998) invert for the shape of a scatterer by assuming a constant electrical conductivity value for the background and the scatter, respectively. We adopt the Tikhonov regularization approach and formulate the inversion for the general case of salt imaging at the presence of density reversal by working with discrete density contrasts. The problem then becomes one of minimizing a model objective function subject to allowing model parameters to attain only one of two values at each depth. The objective function, therefore, consists of the weighted sum of a model objective function m φ and data misfit d φ : { } . ) ( , 0 subject to , ) ( ) ( min. m d z ρ ρ ρ λφ ρ φ φ ∆ ∈ + = (1) where d φ is formulated as the 2 χ measure of our data