Introduction

Seismic velocity estimation or travel time inversion of the earth's vertical cross-section structure is one of the main objectives of seismic data processing. Velocities and changes in velocities provide seismologists valuable clues about subsurface structure. The analyzed seismic data sets are composed of the travel times of the first arriving seismic waves from sources (which can be earthquakes or explosions) to receivers on the surface (Figure 1). To generalize the problem, the sources can be considered inputs to a black box which denotes an unknown velocity model, and the seismic reflection data, outputs generated from the black box. Based on these inputs and outputs, seismic velocity estimation seeks to reconstruct the unknown velocity model (Figure 2). This kind of problem is usually multi-dimensional, multi-modal, and highly non-linear.

  

Figure 1: Seismic wave propagates from a source to a receiver through an unknown structure.

  

Figure 2: Seismic inversion problem.

The velocity estimation problem can be considered a search problem. In velocity model space, search algorithms try to find a model which makes the calculated travel time fit the observed travel times the best. There are many methods in seismology to estimate velocities using seismic field data with first-arrival times [Pullammanappallil, 1994]. Simulated annealing (SA) and genetic algorithms (GAs) are two approaches that make few assumptions about the search space.

Pullammanappallil [Pullammanappallil, 1994] used Simulated annealing to approach the search problem. The SA algorithm is motivated by an analogy to the statistical mechanics of annealing of solids. If we consider a physical system at a high temperature, the large number of atoms in the system is in a highly disordered state. To get the atoms into a more orderly state, we need to reduce the energy of the system by lowering the temperature. The system will be in thermal equilibrium when the probability of a certain state is governed by a Boltzmann distribution given below:

$$Pr(\triangle E) \sim exp((-\triangle E)/kT)$$

We generate a candidate configuration by randomly perturbing the current configuration and calculate its objective function value. If the objective function value is lower than the current value, then we accept the displacement. Otherwise this new displacement is accepted with a probability given by the Boltzmann distribution in the above Equation. Thus there is always a nonzero probability of accepting worse solutions. This gives the algorithm a probability of escaping a local minimum and leads to a global optimum if annealing proceeds slowly enough. Currently simulated annealing defines the state of the art in seismic inversion and is being commercially applied. We therefore experimentally compare simulated annealing with genetic algorithms in this paper.

Genetic algorithms (GAs) were also designed to work on such non-linear, multi-modal and poorly understood problems [Holland, 1975,Goldberg, 1989]. They have been applied to seismic inversion for one-dimensional (1D) and two-dimensional (2D) velocity inversions [Sen and Stoffa, 1992,Ozalaybey et al., 1995,Li et al., 1995] with geological and geophysical knowledge used to increase performance. In this paper, we will use minimal domain knowledge to search for better velocity models, try several different domain independent crossover operators to increase performance and compare results with those obtained from simulated annealing.

The rest of the paper is organized as follows. In section 2, we describe the genetic algorithm used in this paper. Section 4 presents and analyzes our results and compares them with results from simulated annealing. The last section provides conclusions.