Abstract：
The radial basis function (RBF) interpolation approach proposed by Freedman is used to solve inverse problems encountered in well-logging and other petrophysical issues. The approach is to predict petrophysical properties in the laboratory on the basis of physical rock datasets, which include the formation factor, viscosity, permeability, and molecular composition. However, this approach does not consider the effect of spatial distribution of the calibration data on the interpolation result. This study proposes a new RBF interpolation approach based on the Freedman's RBF interpolation approach, by which the unit basis functions are uniformly populated in the space domain. The inverse results of the two approaches are comparatively analyzed by using our datasets. We determine that although the interpolation effects of the two approaches are equivalent, the new approach is more flexible and beneficial for reducing the number of basis functions when the database is large, resulting in simplification of the interpolation function expression. However, the predicted results of the central data are not sufficiently satisfied when the data clusters are far apart.
ZOU You-Long,HU Fa-Long,ZHOU Can-Can et al. Analysis of radial basis function interpolation approach[J]. APPLIED GEOPHYSICS, 2013, 10(4): 397-410.
[1]
Anand, V., and Freedman, R., 2009, New methods for predicting properties of live oils from NMR: SPWLA 50th Annual Logging Symposium, paper DD.
[2]
Franke, R., 1982, Scattered data interpolation: Tests of some methods: Mathematics of Computation, 38(157), 181 - 200.
[3]
Freedman, R., 2006, New approach for solving inverse problems encountered in well-logging and geophysical applications: Petrophysics, 47(2), 93 - 111.
[4]
Freedman, R., Lo, S., Flaum, M., Hirasaki, G. J., Matteson, A., and Sezginer, A., 2001, A new NMR method of fluid characterization in reservoir rocks: Experimental confirmation and simulation results: SPE Journal, 6(4), 452 - 464.
[5]
Gao, B., Wu, J., Chen, S., Kwak, H., and Funk, J., 2011, New method for predicting capillary curves from NMR data in carbonate rocks: SPWLA 52th Annual Logging Symposium.
Kansa, E. J., 1990, Multiquadrics- A scattered data approximation scheme with applications to computational fluid dynamics - II, Solutions to Parabolic, Hyperbolic and Elliptic partial differential equations: Computers and Mathematics with Applications, 19(8 - 9), 147 - 161.
[8]
Lukaszyk, S., 2004, A new concept of probability metric and its applications in approximation of scattered data sets: Computational Mechanics, 33(4), 299 - 304.
[9]
Micchelli, C. A., 1986, Interpolation of scattered data: Distance matrices and conditionally positive definite functions: Constructive Approximation, 2, 11 - 22.
[10]
Powell, M. J. D., 2001, Radial basis function methods for interpolation to functions of many variables: Presented at the Fifth Hellenic-European Computer Mathematics and its Applications, 1 - 23.
2013, Vol. 10 
