Modified radial basis function interpolation

In order to interpolate the rock properties predicted by several well logs, Karimi and Fomel (2015) used a weighted summation of the painting predictions. The weights were calculated by radial basis function (RBF) $\phi(d) = \phi(\vert x_k-x_r\vert)$, whose value decays along the increasing distance (Micchelli, 1984). An inverse quadratic form of the RBF is:

  $$\phi(d) = \frac{1}{1 + \left(d/d_0\right)^2},\;\;\mathrm{where}\;d_0>0.$$ (2)

Then the interpolation can be calculated as below:

  $$S(\mathbf{x}) = \frac{\sum_{r=1}^N \phi(\vert x-x_r\vert)S_r(\mathbf{x})} {\sum_{r=1}^N \phi(\vert x-x_r\vert)},$$ (3)

where $S_r$ is the result of spreading well log at well location $x_r$ into the seismic data and $N$ represents the total number of used wells.

We propose to extend the definition of $d$ from simply the horizontal distance $\vert x_k-x_r\vert$ to the geologic distance that is measured along the seismic horizon:

  $$\tilde d(\mathbf{x}_k,\mathbf{x}_r) = \int_{\mathbf{x}_r}^{\math... ...mathbf{x}) + \lambda f(\mathbf{x}) \right)\;\mathrm{d}\mathbf{h}(\mathbf{x}),$$ (4)

in which $\mathbf{x}_r$, $\mathbf{x}_k$ are points along a seismic horizon $\mathbf{h}(\mathbf{x})$; $\Delta l(\mathbf{x})$ is the curve length of the horizon, it can be calculated by $\Delta l(\mathbf{x}) = \Delta x\sqrt{1+p^2}$ given local slope estimation $p$; $f(\mathbf{x})$ is fault attribute and $\lambda$ represents the distance penalty parameter, this term exaggerates the distance across fault. The geologic distance indicates the decay of information confidence from the reference trace. We perform this integration efficiently by accumulative predictive painting.