Bayesian Inversion of Logging-While-Drilling Extra-Deep Directional Resistivity Measurements Using Parallel Tempering Markov Chain Monte Carlo Sampling

Abstract

We present a Bayesian inversion scheme to extract multiple bed boundaries from extra-deep directional logging-while-drilling (LWD) resistivity measurements (EDDRM). The inversion uses a dimensionality reduction approach to simplify the three-dimensional inversion problem into a stitch of 1-D ones. The Bayesian framework associated with a parallel tempering (PT) Markov Chain Monte Carlo (MCMC) sampling algorithm is invoked to derive the multiple boundaries from the 1-D inverse model. The kernel of PT algorithm is that multiple chains with different temperatures execute parallelly and the states can be swapped between the chains. Compared with conventional single-chain MCMC, the PT strategy accelerates the convergence and has a better global search capability. In addition, a new 1-D inverse model is proposed. By letting all beds share the same anisotropy coefficient, the new model incorporates fewer parameters of interests. Therefore, the MCMC samples can be reduced significantly. Numerical experiments performed over synthetic examples are presented to verify the feasibility of the new model, to test the inversion performance and to obtain the best practice of the inversion. The uncertainty of inverted results is also assessed from the probability distributions of resistivity profile and histograms of relative dipping and formation anisotropy.