Logging-while-drilling (LWD) Resistivity Measurements in high-angle and horizontal wells cannot be used for quantitative calculation directly, since they are easily influenced by borehole/formation geometry, surrounding beds and other factors. Although Least-Squares (LS) inversion method is widely used to reconstruct the actual reserve resistivity, it assumes that the measurement data are corrupted with pure Gaussian noise. This assumption makes it cannot work when the measured data are contaminated by non-Gaussian noise. Furthermore, in highly deviated wells, LWD apparent resistivity measurements always show "horns" near the bed boundaries where the resistivity contrastsare high. These "horns" can also decrease the inversion accuracy.
In this paper, we propose a new robust nonlinear inversion algorithm that uses Huber criterion as a solution for handling the measured data mentioned above. Compared with Least-Squares inversion, this method requires one additional parameter, namely, the threshold of Huber criteria, δ. This parameter is very important and must be chosen carefully. By varying δ, Huber inversion method can be divided into two parts. If the absolute error of simulated response (compared to the measured response) is greater than δ, l1 norm inversion is used. Otherwise, l2 norm inversion method is used. This method combines advantages of both l1 and l2 inversion and works best if the resistivity data contains non-Gaussian noises as well as "horns". Meanwhile, during the inversion process, we introduce a new approximate method for computing the Jacobian matrix and desired step, which could improve the calculation results. Besides, since currentmulti-resolution LWD resistivity tools could providemultiple compensated resistivity measurements, a linear optimization combination method of iterative stepsis introduced for multi-resolution resistivity curves. The weights can be adjustedaccording to the LWD resistivity sensitivity for borehole deviation, resistivity contrast at bed boundaries, and the contaminated extent by noise. This optimal procedure could further improve the computation accuracy.
A series of numerical simulations for different conditions are analyzed and discussed, the comparison of LS and Huber inversion shows that Huber algorithm is more robust and stable when the measurements contain both data of "horns" and non-Gaussian noise. Therefore, this method is more suitable for routine petrophysical interpretation and quantitative formation evaluation.