Nonlinear one-dimensional seismic waveform inversion using simulated annealing

Abstract

The seismic inverse problem involves finding a model m that either minimizes the error energy between the data and theoretical seismograms or maximizes the cross-correlation between the synthetics and the observations. We are, however, faced with two problems: (1) the model space is very large, typically of the order of 50 50 ; and, (2) the error energy function is multimodal. Existing calculus-based methods are local in scope and easily get trapped in local minima of the energy function. Other methods such as 'simulated annealing' and 'genetic algorithms' can be applied to such global optimization problems and they do not depend on the starting model. Both of these methods bear analogy to natural systems and are robust in nature. For example, simulated annealing is the analog to a physical process in which a solid in a 'heat bath' is heated by increasing the temperature, followed by slow cooling until it reaches the global minimum energy state where it forms a crystal. To use simulated annealing efficiently for 1-D seismic waveform inversion, we require a modeling method that rapidly performs the forward modeling calculation and a cooling schedule that will enable us to find the global minimum of the energy function rapidly. With the advent of vector computers, the reflectivity method has proved successful and the time of the calculation can be reduced substantially if only plane-wave seismograms are required. Thus, the principal problem with simulated annealing is to find the critical temperature, i.e., the temperature at which crystallization occurs. By initiating the simulated annealing process with different starting temperatures for a fixed number of iterations with a very slow cooling, we noticed that by starting very near but just above the critical temperature, we reach very close to the global minimum energy state very rapidly. We have applied this technique successfully to band-limited synthetic data in the presence of random noise. In most cases we find that we are able to obtain very good solutions using only a few plane wave seismograms.