New Approach for Solving Inverse Problems Encountered in Well-Logging and Geophysical Applications

This paper proposes a new method for solving inverse problems for which a large calibration database exists consisting of pairs of inputs and corresponding outputs. Distinct pairs of input and output data in the database correspond to different states of the underlying physical system whose properties are the subject of the inversion. Either empirical measurements or numerical computations, e.g., using a forward model, can be used to construct the database. The method uses radial basis function (RBF) interpolation, a method for approximating smooth and continuous multivariate functions of many variables. RBF interpolation is used to derive a non-linear mapping function from which the properties of physical systems (e.g., crude oils, sandstones, etc.) are predicted from input measurements that are not in the database. An advantage of this method over the traditional approach of fitting data to a forward model is that it can be used to solve well-logging and geophysical inverse problems associated with unknown forward models. In other cases where accurate forward models are known, the method can be used to solve inverse problems in real time, where these might otherwise be computationally too expensive or, as in many practical cases, lead to ill-behaved non-linear minimization. Construction of a robust database that spans the physical range of input and output data encountered in practice is essential for accurate predictions. Database construction is the most challenging part of applying the method. The method is intuitive and does not require iterative training. It is easier to implement than inversion methods based on artificial neural networks, which use non-intuitive multilayered networks and require lengthy iterative training. Testing and benchmarking of the inversion method is performed on three challenging example problems of current interest in well logging. Example 1 shows that viscosities of dead crude oils can be predicted from nuclear magnetic resonance (NMR) relaxation time measurements. The predicted viscosities are in good agreement with those measured by a Couette-type viscometer and are more accurate than viscosities predicted using published empirical correlations. The first example also shows that molecular compositions of dead crude oils can be predicted from NMR relaxation time measurements. The predicted compositions are in good overall agreement with those obtained from gas chromatography. The method used in this paper can also be applied to databases constructed from measurements made on live crude oils. Example 2 shows that borehole-corrected signals can be predicted from raw measurements made by a 3D induction tool. The predicted borehole-corrected signals are in excellent overall agreement with the target tool responses (i.e., for homogeneous media). Example 3 shows that mass density and molecular composition of dead crude oils can be predicted from near-infrared (NIR) spectra. These examples demonstrate that the methodology proposed in this paper is applicable to a large variety of problems encountered in well logging and geophysical applications.