Reversible jump MCMC inversion of the ultra-deep azimuthal resistivity LWD tools using Cauchy proposal function

Since the ultra-deep azimuthal resistivity logging while drilling (LWD) service was first introduced a few years ago, the new service has been widely used in the well placement, the reservoir mapping, the geo-stopping, etc. The tools use multi-components and multi-spacings in a range from several meters to tens of meters with different frequencies rang from several hundred hertz to a few hertz. The wider detection range and multi-components measurements not only provide much richer information compared with the conventional directional propagation resistivity tools but also bring great troubles for operators due to the complexity of the tool response characteristics and data processing or inversion. In inversion, people can't confirm the number of layers in priori which leads to kinds of results. Operators have to face the huge uncertainty about the formation in front of bit. On the other hand, the abundant measurements also supply conditions to process the logging data without bias. That is, the low-resolution seismic data and logging data with shallow detection are no longer necessary. To quantify the uncertainty of the inversion results and fully mine the data, in this paper, we establish a completely data driven inversion method based on the transdimensional or reversible jump Markov chain Monte Carlo (MCMC) algorithm with a Cauchy distribution as the proposal function. A study is conducted to show the Cauchy distribution still has the ‘parsimony’ as the Gaussian distribution generally used in previous works. A synthetic example demonstrates that the uncertainty quantification inversion method can be crucial to assess the ultra-deep azimuthal propagation resistivity technology and helpful to promote the understanding of the investigation characteristics.