Introduction

Most geophysical inverse problems require, explicitly or implicitly, that the final model is within a range of acceptable values. For instance, densities should always be positives, P-waves velocity should be greater than S-waves velocity, etc. These bounds, or constraints, on the final model are called simple bounds. Many different techniques have been developed by the optimization community to solve inverse problems with simple bounds. What makes these methods attractive is their cost and ease of use.

Among the variety of bound constrained methods, the L-BFGS-B technique Zhu et al. (1997) is very appealing. The L-BFGS-B method is an extension of the well-known, quasi-Newton, limited-memory BFGS technique Broyden (1969); Fletcher (1970); Goldfarb (1970); Liu and Nocedal (1989); Nocedal (1980); Shanno (1970) that can impose simple bounds on the model parameters. L-BFGS-B (Limited-memory BFGS with Bounds) incorporates a trust region method with update of the Hessian (or second derivative) and a line search. There are three main reasons why L-BFGS-B is chosen for this task. First, the L-BFGS method has been already successfully applied on different geophysical optimization problems such as minimization of the Huber norm Guitton and Symes (2003) or multiple attenuation with sparse radon transforms Sava and Guitton (2003). Second, the memory requirements and the computation costs for L-BFGS-B are limited. Finally, the user interface is very simple, making its implementation within existing code very simple.

In this paper, the L-BFGS-B method is presented with a few algorithmic details. These details include the gradient projection technique and the trust region method. Both elements are blended inside the L-BFGS-B algorithm to insure that the estimated models are within a desired range of values. As an illustration, L-BFGS-B is tested on the dip estimation problem as formulated by Fomel (2002). L-BFGS-B is an improvement over the conventional conjugate gradient approach because it allows the incorporation of simple bound constraints on the dips. Several examples illustrate the dip estimation results.