Three-Dimensional Modelling of Geological Surfaces Using Generalized Interpolation with Radial Basis Functions

Abstract

A generalized interpolation framework using radial basis functions (RBF) is presented that implicitly models three-dimensional continuous geological surfaces from scattered multivariate structural data. Generalized interpolants can use multiple types of independent geological constraints by deriving for each, linearly independent functionals. This framework does not suffer from the limitations of previous RBF approaches developed for geological surface modelling that requires additional offset points to ensure uniqueness of the interpolant. A particularly useful application of generalized interpolants is that they allow augmenting on-contact constraints with gradient constraints as defined by strike-dip data with assigned polarity. This interpolation problem yields a linear system that is analogous in form to the previously developed potential field implicit interpolation method based on co-kriging of contact increments using parametric isotropic covariance functions. The general form of the mathematical framework presented herein allows us to further expand on solutions by: (1) including stratigraphic data from above and below the target surface as inequality constraints (2) modelling anisotropy by data-driven eigen analysis of gradient constraints and (3) incorporating additional constraints by adding linear functionals to the system, such as fold axis constraints. Case studies are presented that demonstrate the advantages and general performance of the surface modelling method in sparse data environments where the contacts that constrain geological surfaces are rarely exposed but structural and off-contact stratigraphic data can be plentiful.

Appendix: The action of linear functionals on radial functions

We shall expand the generalized RBF interpolant (Eq. (5)) using linear functionals (Eq. (7)) for data $\left\{ {\varvec{x}_i ,f( {\varvec{x}_i })} \right\} _{i=1}^\mu$,$\left\{ {\varvec{x}_i ,\nabla f( {\varvec{x}_i })} \right\} _{i=\mu +1}^{\mu +\sigma }$, and $\left\{ {\varvec{x}_i ,\left\langle {\nabla f( {\varvec{x}_i }),\varvec{t}_i } \right\rangle } \right\} _{i=\mu +\sigma +1}^{\mu +\sigma +\tau }$ and demonstrate the action of linear functionals on radial functions.

$$\begin{aligned} s(\varvec{x})&= \sum \limits _{j=1}^N {w_j } \lambda _j^{{\varvec{x}}^{'}} \Phi ( {\varvec{x,x}^{'}})+p({\varvec{x}}) =\sum \limits _{j=1}^\mu {a_j \delta _{{\varvec{x}}_j }^{{\varvec{x}}^{'}} } \Phi ( {\varvec{x,x}^{'}})\nonumber \\&+\sum \limits _{k=1}^\sigma {\left( \!{b_k \delta _{{\varvec{x}}_{\mu +k} }^{{\varvec{x}}^{'}} \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial { x}^{'}}+c_k \delta _{{\varvec{x}}_{\mu +k} }^{{\varvec{x}}^{'}} \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial y^{'}}+d_k \delta _{{\varvec{x}}_{\mu +k} }^{{\varvec{x}}^{'}} \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial z^{'}}}\!\!\right) } \nonumber \\&+\sum \limits _{l=1}^\tau e_l \left\{ t_{\mu +\sigma +l,{\varvec{x}}} \delta _{{\varvec{x}}_{\mu +\sigma +l} }^{{\varvec{x}}^{'}} \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial { x}^{'}}+t_{\mu +\sigma +l,y} \delta _{{\varvec{x}}_{\mu +\sigma +l} }^{{\varvec{x}}^{'}} \right. \nonumber \\&\left. \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial y^{'}}+t_{\mu +\sigma +l,z} \delta _{\varvec{x}_{\mu +\sigma +l} }^{{\varvec{x}}^{'}} \circ \frac{\partial \Phi ( {{\varvec{x,x}}^{'}})}{\partial z^{'}} \right\} \nonumber \\&+\sum \limits _{s=1}^Q {g_s } p_s (\varvec{x}) =\sum \limits _{j=1}^\mu {a_j } \Phi ( {{\varvec{x,x}}_j })+\sum \limits _{k=1}^\sigma \nonumber \\&\times \left( b_k \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +k} })}{\partial x^{'}}+c_k \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +k} })}{\partial y^{'}}+d_k \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +k} })}{\partial z^{'}}\right) \nonumber \\&+\sum \limits _{l=1}^\tau e_l \left\{ t_{\mu +\sigma +l,\mathbf{x}} \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +\sigma +l} })}{\partial { x}^{'}}+t_{\mu +\sigma +l,y} \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +\sigma +l} })}{\partial y^{'}}\right. \nonumber \\&\left. +t_{\mu +\sigma +l,z} \frac{\partial \Phi ( {{\varvec{x,x}}_{\mu +\sigma +l} })}{\partial z^{'}} \right\} +\sum \limits _{s=1}^Q {g_s } p_s (\varvec{x}) =\sum \limits _{j=1}^\mu {a_j } \Phi ( {{\varvec{x,x}}_j })+\sum \limits _{k=1}^\sigma \nonumber \\&\times {\left\langle {\mathbf{b}_k ,\nabla ^{'}\Phi ( {{\varvec{x,x}}_{\mu +k} })} \right\rangle } +\sum \limits _{l=1}^\tau {e_l \left\langle {\varvec{t}_{\mu +\sigma +l} ,\nabla ^{'}\Phi ( {{\varvec{x,x}}_{\mu +\sigma +l} })} \right\rangle } +\sum \limits _{s=1}^Q {g_s } p_s (\varvec{x}). \, \nonumber \\ \end{aligned}$$

(30)

Note that the point evaluation functional $\delta _{{\varvec{x}}_j }^{{\varvec{x}}^{'}}$acts on the ${\varvec{x}}^{'}$variable of the RBF $\Phi ( {\varvec{x,x}^{'}})$ and replaces that variable with the data point $\varvec{x}_j$.

To build the interpolation matrix (Eq. (12)), linear functionals for each constraint type are applied to the interpolant Eq. (9). For on-contact constraints $\left\{ {\varvec{x}_i ,f( {\varvec{x}_i })} \right\} _{i=1}^\mu$

$$\begin{aligned} \lambda _i^{\varvec{x}} s(\varvec{x})&= \delta _{{\varvec{x}}_i }^{\varvec{x}} s(\varvec{x})=\sum \limits _{j=1}^\mu {a_j } \overbrace{\Phi ( {\varvec{x}_i ,\varvec{x}_j })}^A+\sum \limits _{k=1}^\sigma \nonumber \\&\times {\left( {b_k \overbrace{\frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x^{'}}}^B+c_k \overbrace{\frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial y^{'}}}^C+d_k \overbrace{\frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial z^{'}}}^D}\right) }\nonumber \\&+\sum \limits _{l=1}^\tau {e_l \overbrace{\left\langle {\nabla \Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +\sigma +l} }),t_{\mu +\sigma +l} } \right\rangle }^E} +\sum \limits _{s=1}^Q {g_s } \overbrace{p_s ( {\varvec{x}_i })}^G. \end{aligned}$$

(31)

For gradient constraints $\left\{ {\varvec{x}_i ,\nabla f( {\varvec{x}_i })} \right\} _{i=\mu +1}^{\mu +\sigma }$ (x-component in this case)

$$\begin{aligned}&\lambda _i^{\varvec{x}} s(\varvec{x}) =\delta _{{\varvec{x}}_i }^{{\varvec{x}}} \circ \frac{\partial }{\partial x}s(\varvec{x}) =\sum \limits _{j=1}^\mu {a_j } \overbrace{\frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_j })}{\partial x}}^{B^T}+\sum \limits _{k=1}^\sigma \Bigg ( b_k \overbrace{\frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x\partial x^{'}}}^{H^{xx^{'}}} \nonumber \\&\quad +c_k \overbrace{\frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x\partial y^{'}}}^{H^{xy^{'}}}+d_k \overbrace{\frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x\partial z^{'}}}^{H^{xz^{'}}}\Bigg ) \nonumber \\&\quad +\sum \limits _{l=1}^\tau {e_l \overbrace{\left( {\varvec{t}_{\mu +\sigma +l,x} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial x^{'}}+\varvec{t}_{\mu +\sigma +l,y} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial y^{'}}+\varvec{t}_{\mu +\sigma +l,z} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial z^{'}}}\right) }^{J^x}} \nonumber \\&\quad +\sum \limits _{s=1}^Q {g_s } \overbrace{\frac{\partial p_s ( {\varvec{x}_i })}{\partial x}}^{G^x}. \end{aligned}$$

(32)

For tangent constraints $\left\{ {\varvec{x}_i ,\left\langle {\nabla f( {\varvec{x}_i }),\varvec{t}_i } \right\rangle } \right\} _{i=\mu +\sigma +1}^{\mu +\sigma +\tau }$

$$\begin{aligned} \lambda _i^{\varvec{x}} s(\varvec{x})&= \Big ( {t_{ix} \delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial { x}}+t_{iy} \delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial y}+t_{iz} \delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial z}}\Big )s( \varvec{x})\nonumber \\&= \sum \limits _{j=1}^\mu {a_j } \overbrace{\left( {t_{ix} \frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_j })}{\partial x}+t_{iy} \frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_j })}{\partial y}+t_{iz} \frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_j })}{\partial z}}\right) }^{E^T} \nonumber \\&+\sum \limits _{k=1}^\sigma {\left\{ {\begin{array}{l} b_k \overbrace{\left( {t_{ix} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x\partial x^{'}}+t_{iy} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial y\partial x^{'}}+t_{iz} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial z\partial x^{'}}}\right) }^{J^{_x^T }} \\ +c_k \overbrace{\left( {t_{ix} \frac{\partial ^2\Phi ( {x_i ,x_{\mu +k} })}{\partial x\partial y^{'}}+t_{iy} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial y\partial y^{'}}+t_{iz} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial z\partial y^{'}}}\right) }^{J^{_y^T }} \\ +d_k \overbrace{\left( {t_{ix} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial x\partial z^{'}}+t_{iy} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial y\partial z^{'}}+t_{iz} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{\mu +k} })}{\partial z\partial z^{'}}}\right) }^{J^{_z^T }} \\ \end{array}} \right\} } \nonumber \\&+\sum \limits _{l=1}^\tau {e_l \overbrace{\left\{ {\begin{array}{l} t_{ix} t_{\mu +\sigma +l,x} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial x^{'}}+t_{ix} t_{\mu +\sigma +l,y} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial y^{'}}+t_{ix} t_{\mu +\sigma +l,z} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial x\partial z^{'}} \\ +t_{iy} t_{\mu +\sigma +l,x} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial y\partial x^{'}}+t_{iy} t_{\mu +\sigma +l,y} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial y\partial y^{'}}+t_{iy} t_{\mu +\sigma +l,z} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial y\partial z^{'}} \\ +t_{iz} t_{\mu +\sigma +l,x} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial z\partial x^{'}}+t_{iz} t_{\mu +\sigma +l,y} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial z\partial y^{'}}+t_{iz} t_{\mu +\sigma +l,z} \frac{\partial ^2\Phi ( {\varvec{x}_i ,\varvec{x}_{{\mu +\sigma +l} } })}{\partial z\partial z^{'}} \\ \end{array}} \right\} }^L} \nonumber \\&+\sum \limits _{s=1}^Q {g_s } \overbrace{\left\{ {t_{ix} \frac{\partial p_s ( {\varvec{x}_i })}{\partial x}+t_{iy} \frac{\partial p_s ( {\varvec{x}_i })}{\partial y}+t_{iz} \frac{\partial p_s ( {\varvec{x}_i })}{\partial z}} \right\} }^M. \end{aligned}$$

(33)

Collecting the terms from previous equations, our interpolation matrix becomes

$$\begin{aligned} \left[ {{\begin{array}{*{20}c} A &{}\quad F &{}\quad E &{}\quad G \\ {F^\mathrm{T}} &{}\quad H &{}\quad J &{}\quad K \\ {E^\mathrm{T}} &{}\quad {J^\mathrm{T}} &{}\quad L &{}\quad M \\ {G^\mathrm{T}} &{}\quad {K^\mathrm{T}} &{}\quad {M^\mathrm{T}} &{}\quad 0 \\ \end{array} }} \right] \left[ {{\begin{array}{*{20}c} \varvec{a} \\ \varvec{f} \\ \varvec{e} \\ \varvec{g} \\ \end{array} }} \right] =\left[ {{\begin{array}{*{20}c} 0 \\ {\varvec{n}_i } \\ 0 \\ 0 \\ \end{array} }} \right] \!, \end{aligned}$$

(34)

where

$$\begin{aligned} \begin{array}{c} A_{ij} : 1\le i,j\le \mu ,\,F=\left[ {{\begin{array}{*{20}c} {B_{ij} } &{} {C_{ij} } &{} {D_{ij} } \\ \end{array} }} \right] : 1\le i\le \mu ,\,1\le j\le \sigma , \\ E_{ij} :1\le i\le \mu , \text{1 }\le j\le \tau ,\,G_{ij} :1\le i\le \mu , \text{1 }\le j\le Q,\, \\ H=\left[ {{\begin{array}{*{20}c} {H_{ij}^{xx^{'}} } &{} {H_{ij}^{xy^{'}} } &{} {H_{ij}^{xz^{'}} }\\ {H_{ij}^{yx^{'}} } &{} {H_{ij}^{yy^{'}} } &{} {H_{ij}^{yz^{'}} }\\ {H_{ij}^{zx^{'}} } &{} {H_{ij}^{zy^{'}} } &{} {H_{ij}^{zz^{'}} }\\ \end{array} }} \right] :1\le i,j\le \sigma ,J=\left[ {{\begin{array}{*{20}c} {J_{ij}^x } \\ {J_{ij}^y } \\ {J_{ij}^z } \\ \end{array} }} \right] :1\le i\le \sigma ,\,1\le j\le \tau \\ K=\left[ {{\begin{array}{*{20}c} {G_{ij}^x } \\ {G_{ij}^y } \\ {G_{ij}^z } \\ \end{array} }} \right] :1\le i\le \sigma ,\,1\le j\le Q,L_{ij} :1\le i,j\le \tau \\ M_{ij} :1\le i\le \tau ,\,1\le j\le Q,f=\left[ {{\begin{array}{*{20}c} \varvec{b} \\ \varvec{c} \\ \varvec{d} \\ \end{array} }} \right] ,\varvec{n}_i =\left[ {{\begin{array}{*{20}c} {n_{ix} } \\ {n_{iy} } \\ {n_{iz} } \\ \end{array} }} \right] :1\le i\le \sigma . \, \\ \end{array}. \end{aligned}$$

(35)

Most matrix elements are defined in Eqs. (31), (32), and (33). Remaining elements are defined by applying the linear functionals $\delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial y}$ and $\delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial z}$ to the interpolant $s(\varvec{x})$.

Some examples of partial derivatives of the cubic RBF

$$\begin{aligned} \Phi ( {\varvec{x,x}^{\prime }})=r^3=( {( {x-x^{\prime }})^2+( {y-y^{\prime }})^2+( {z-z^{\prime }})^2})^{3/2}, \end{aligned}$$

(36)

with input data

$$\begin{aligned} \Phi ( {\varvec{x}_i ,\varvec{x}_j })=r_{ij}^3 =( {( {x_i -x_j })^2+( {y_i -y_j })^2+( {z_i -z_j })^2})^{3/2}. \end{aligned}$$

(37)

Examples of first derivative are

$$\begin{aligned} \begin{aligned} ( {\delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial x}})( {\delta _{\varvec{x}_j }^{\varvec{x}^{\prime }} })\Phi ( {\varvec{x,x}^{\prime }})&= \frac{\partial \Phi ( {\varvec{x}_i ,\varvec{x}_j })}{\partial x}=3( {x_i -x_j })r_{ij}\\ ( {\delta _{{\varvec{x}}_i }^{\varvec{x}} })( {\delta _{\varvec{x}_j }^{\varvec{x}^{\prime }} \circ \frac{\partial }{\partial x^{'}}})\Phi ( {\varvec{x,x}^{\prime }})&= \frac{\partial \Phi ( {x_i ,x_j })}{\partial x^{\prime }}=-3( {x_i -x_j })r_{ij}. \end{aligned} \end{aligned}$$

(38)

Examples of second derivatives are

$$\begin{aligned} \begin{aligned} ( {\delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial x}})( {\delta _{\varvec{x}_j }^{\varvec{x}^{'}} \circ \frac{\partial }{\partial x^{'}}})\Phi ( {\varvec{x,x}^{'}})&= \frac{\partial \Phi ( {x_i ,x_j })}{\partial x\partial x^{'}}=-3( {r_{ij} +\frac{( {x_i -x_j })^2}{r_{ij} }}) \qquad \\ ( {\delta _{{\varvec{x}}_i }^{\varvec{x}} \circ \frac{\partial }{\partial x}})( {\delta _{\varvec{x}_j }^{\varvec{x}^{'}} \circ \frac{\partial }{\partial y^{'}}})\Phi ( {\varvec{x,x}^{'}})&= \frac{\partial \Phi ( {x_i ,x_j })}{\partial x\partial y^{'}}=-3\frac{( {x_i -x_j })( {y_i -y_j })}{r_{ij} }.\qquad \end{aligned}\qquad \end{aligned}$$

(39)