We describe an adaptive, Monte Carlo importance sampling algorithm (AMI) to efficiently select models from a large model space and we apply this algorithm to estimate a-posteriori, marginal probability density functions (p.d.f.s) for physical quantities given some data and a physical theory. For an example problem, we examine Rayleigh group dispersion for a given, layered-earth model (the "true" model). We assign likelihoods to each sampled model based on a comparison of the dispersion data predicted by theory for the sample model and that predicted for the "true" model. Instead of showing the results in the parameter space used for the search, we transform the results to easily understood, velocity-depth plots showing marginal p.d.f.s of velocity at each depth for 1) the sampling density, 2) the conjunction of the sampling density and the information given by the theory and data, and 3) the a-posteriori estimate of S velocity. We also show the "acceptable" and "best" models obtained from the search.

With this presentation of the results we examine the behaviour and efficiency of the AMI method relative to a "crude" Monte Carlo search (CMC), and we discuss the resulting velocity-depth images with regards to the parameterization of the problem, the "true" model and the physics of the forward problem.

We find that, because it uses importance sampling, the AMI algorithm is about 4 to 8 times more efficient than the CMC in identifying "acceptable" solutions for our example problem without significant loss in thoroughness of the search of the model space. The results show that when a large model space is searched the p.d.f.s give a good indication of uncertainty, resolution and correlations in the problem, that there are many "acceptable" solutions, and that no single solution may be very similar to the "true" model.

We also apply the genetic algorithm directed search method to our example problem. Though the genetic algorithm rapidly identifies a large number of "acceptable" solutions, we show that it uses a sparse and poorly distributed sampling of the model space, and consequently produces strongly clustered sets of solutions and unstable and incomplete estimates of the p.d.f.'s.