Finite Difference Implicit Structural Modeling of Geological Structures

Abstract

We introduce a new method for implicit structural modeling. The main developments in this paper are the new regularization operators we propose by extending inherent properties of the classic one-dimensional discrete second derivative operator to higher dimensions. The proposed regularization operators discretize naturally on the Cartesian grid using finite differences, owing to the highly symmetric nature of the Cartesian grid. Furthermore, the proposed regularization operators do not require any special treatment on boundary nodes, and their generalization to higher dimensions is straightforward. As a result, the proposed method has the advantage of being simple to implement. Numerical examples show that the proposed method is robust and numerically efficient.

Appendix

The analytic expressions of terms appearing in the formulas for ${\mathcal {R}}(\phi )$, as proposed in Sect. 5, are given below

$$\begin{aligned} \begin{array}{rcl} \partial ^2_{[1\;0\;0]}\phi &{}=&{}\partial _x^2\phi ,\\ \partial ^2_{[0\;1\;0]}\phi &{}=&{}\partial _y^2\phi ,\\ \partial ^2_{[0\;0\;1]}\phi &{}=&{}\partial _z^2\phi ,\\ \sqrt{2}\partial _{[1\;0\;0]}\big (\partial _{[0\;1\;0]}\phi \big )&{}=&{}\sqrt{2}\partial _x\big (\partial _y\phi \big ),\\ \sqrt{2}\partial _{[1\;0\;0]}\big (\partial _{[0\;0\;1]}\phi \big )&{}=&{}\sqrt{2}\partial _x\big (\partial _z\phi \big ),\\ \sqrt{2}\partial _{[0\;1\;0]}\big (\partial _{[0\;0\;1]}\phi \big )&{}=&{}\sqrt{2}\partial _y\big (\partial _z\phi \big ),\\ \sqrt{2}\partial _{[1\;0\;0]}\big (\partial _{[0\;1\;1]}\phi \big )&{}=&{}\partial _x\big (\partial _y\phi +\partial _z\phi \big ),\\ \sqrt{2}\partial _{[1\;0\;0]}\big (\partial _{[0\;1\;-1]}\phi \big )&{}=&{}\partial _x\big (\partial _y\phi -\partial _z\phi \big ),\\ \sqrt{2}\partial _{[0\;1\;0]}\big (\partial _{[1\;0\;1]}\phi \big )&{}=&{}\partial _y\big (\partial _x\phi +\partial _z\phi \big ),\\ \sqrt{2}\partial _{[0\;1\;0]}\big (\partial _{[1\;0\;-1]}\phi \big )&{}=&{}\partial _y\big (\partial _x\phi -\partial _z\phi \big ),\\ \sqrt{2}\partial _{[0\;0\;1]}\big (\partial _{[1\;1\;0]}\phi \big )&{}=&{}\partial _z\big (\partial _x\phi +\partial _y\phi \big ),\\ \sqrt{2}\partial _{[0\;0\;1]}\big (\partial _{[1\;-1\;0]}\phi \big )&{}=&{}\partial _z\big (\partial _x\phi -\partial _y\phi \big ),\\ \partial ^2_{[0\;1\;-1]}\phi &{}=&{}\frac{1}{2}\big (\partial _y^2+\partial _z^2-2\partial _y\partial _z\big )\phi ,\\ \partial ^2_{[0\;1\;1]}\phi &{}=&{}\frac{1}{2}\big (\partial _y^2+\partial _z^2+2\partial _y\partial _z\big )\phi ,\\ \partial ^2_{[1\;0\;-1]}\phi &{}=&{}\frac{1}{2}\big (\partial _x^2+\partial _z^2-2\partial _x\partial _z\big )\phi ,\\ \partial ^2_{[1\;0\;1]}\phi &{}=&{}\frac{1}{2}\big (\partial _x^2+\partial _z^2+2\partial _x\partial _z\big )\phi ,\\ \partial ^2_{[1\;-1\;0]}\phi &{}=&{}\frac{1}{2}\big (\partial _x^2+\partial _y^2-2\partial _x\partial _y\big )\phi ,\\ \partial ^2_{[1\;1\;0]}\phi &{}=&{}\frac{1}{2}\big (\partial _x^2+\partial _y^2+2\partial _x\partial _y\big )\phi ,\\ \partial ^2_{[1\;-1\;-1]}\phi &{}=&{}\frac{1}{3}\big (\partial _x^2+\partial _y^2+\partial _z^2-2\partial _x\partial _y-2\partial _x\partial _z+2\partial _y\partial _z\big )\phi ,\\ \partial ^2_{[1\;-1\;1]}\phi &{}=&{}\frac{1}{3}\big (\partial _x^2+\partial _y^2+\partial _z^2-2\partial _x\partial _y+2\partial _x\partial _z-2\partial _y\partial _z\big )\phi ,\\ \partial ^2_{[1\;1\;-1]}\phi &{}=&{}\frac{1}{3}\big (\partial _x^2+\partial _y^2+\partial _z^2+2\partial _x\partial _y-2\partial _x\partial _z-2\partial _y\partial _z\big )\phi ,\\ \partial ^2_{[1\;1\;1]}\phi &{}=&{}\frac{1}{3}\big (\partial _x^2+\partial _y^2+\partial _z^2+2\partial _x\partial _y+2\partial _x\partial _z+2\partial _y\partial _z\big )\phi . \end{array} \end{aligned}$$