Analyses of Geo-statistical Modelling, in the Wavenumber Domain, for Multi-dimensional Models

The probabilistic seismic inversion program, Promise, has been equipped with a module that is able to account for lateral continuity. In this module, the prior probability density function (pdf) is generated using two point statistics. Bayes rule is used to account for the observations; the result is a posterior pdf. The observations consist of seismic traces and arrival times of interpreted horizons. An iterative algorithm is deployed to sample the posterior pdf. This visits all the locations in its model, numerous times in succession and effectively creates a Metropolis algorithm. Unexpected results are sometimes generated by this. Consequently, a closer examination of the algorithm and its theoretical background has been partaken. The models that are used in Promise are multi-dimensional; a one to one mapping between the model parameters and the observations does not always exist. This makes the analyses more complicated. The uncertainty in the observations is represented by the seismic noise. For convenience, it is often assumed that the seismic noise at different locations is statistically independent. This is often an erroneous assertion and causes the modeling results to be adversely affected. To make the analyses comprehensible, they are carried out in the wavenumber domain, for models with a linear relation between the model parameters and the observations. Some of the learnings will be discussed for models that are based on a 1D grid.