Summary
The combination of neural network with a mixture Gaussian probability density model can jointly invert elastic parameters into the joint probability density function (PDF) of reservoir parameters quickly. In this paper, we used well-logging as input data for building a diagonal mixture density network (MDN) applied to invert seismic elastic parameters into porosity and shale content. The accuracy and resolution of the inversion results can be improved by increasing the number of Gaussian kernels. We applied this method not only to transform seismic elastic parameters inverted by prestack seismic AVO inversion into porosity and shale content, but also to give the uncertainty of the inversion results, which shows that the MDN is a good nonlinear method for Paleogene braided-river delta reservoir parameter inversion in Bohai oil field.
Introduction
Predicting reservoir parameters from elastic parameters has traditionally been done by deterministic inversion methods, in which a forward physical model such as Biot-Gassmann equation connects reservoir parameters with elastic parameters. However, these methods could not determine the uncertainty of the inversion results. In recent decades, statistical methods based on Bayesian theory were proposed to describe the inversion results as a posterior probability distribution (Tarantola, 1987). Neural network is one of statistical inversion methods, which completes the mapping relations between input data and output data by a transformation encoded in the network weights. This processing is completely data-driven method as it does not need a forward physical model (Schltz, 1994).
An MDN introduced by Bishop (1995) is a neural network to compute a mixture probability density function (PDF). McLachlan and Peel (2002) presented that a mixture of densities with Gaussian PDFs can approximate any arbitrary PDF to any desired accuracy. Williams (1996) proposed a multidimensional Gaussian PDF with a full covariance matrix. However, his work is inapplicable in practical problems for instable and expensive computing. Meier et al. (2007) used a mixture Gaussian PDF with isotropic covariance matrix to inverse global crustal thickness, while isotropic assumption is not available for the multidimensional models with a large variation of uncertainty. Shahraeeni and Curtis (2011) proposed a diagonal MDN, which used a mixture Gaussian PDF with a covariance matrix with unequal diagonal element to model any arbitrary PDF. The diagonal MDN can estimate parameters with different uncertainties quickly. In this paper, we applied the diagonal MDN to inverse reservoir parameters (porosity and shale content) from elastic parameters (density, P-wave and S-wave velocity) in Bohai oil field, China.
