Interpret Model Complexity: Trans-Dimensional MCMC Method

Abstract

In the previous chapters, we witness the power of statistical inverse methods that used to sample from the posterior distribution of earth model parameters given the observed azimuthal resistivity measurements. The statistical inversion resolves the local minimum problem in the deterministic methods and tells the uncertainty of model parameters via the statistical distribution. However, the effect of using traditional MCMC methods is challenged when handling the ultra-deep azimuthal resistivity data. Besides, we have to answer the second question: how many parameters do we need to describe an earth model when solving model-based inversion? Stressing on the problems illustrated, a new method, trans-dimensional Markov chain Monte Carlo (tMCMC), is studied in this chapter. The method relaxes the problem dimensionality as an unknown parameter to be inferred. A novel algorithm named Birth-death algorithm will be introduced as one realization to implement tMCMC on an inverse problem. In this chapter, we will include one field case as example to demonstrate the miracle of tMCMC on the inference of model complexity.