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Neural spline flow multi-constraint NURBS method for three-dimensional automatic geological modeling with multiple constraints

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Abstract

The strike and the dip angle are vital for describing the geometry of the rock formations. However, in the interpolation and geological modeling, only the coordinates are considered; the strike and the dip angle are ignored. To this end, the neural spline flow (NSF) multi-constraint non-uniform rational B-splines (McNURBS) method is proposed in this study. Any complex high-dimensional joint distribution can be learned using the deep generative model, NSF; thus, the NSF model was used to perform the exact maximum likelihood estimation and joint sampling of three-dimensional (3D) geological point coordinates, strike, and dip angle, overcoming the shortcomings of conventional statistical models, which are difficult to extend to high-dimensional problems. In addition, the conventional single-constraint NURBS modeling method based on geological point coordinates was improved to obtain the McNURBS modeling method, which considers the geological point coordinates, strike, and dip angle during the modeling process. The practical application results show that by using the proposed method, a 3D geological model can be flexibly and automatically established considering both geological point coordinates and strike and dip angle constraints. Moreover, the fitting relative error (RE) of the proposed method was reduced by 52.4% compared to the conventional NURBS method. This study provides a convenient and efficient means to automatically build a reliable 3D geological model.

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Acknowledgements

This research was funded by the Yalong River Joint Funds of the National Natural Science Foundation of China (U1965207) and National Natural Science Foundation of China (Grant No. 51779169).

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  1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, 300072, Tianjin, China

    Mingming Lyu, Bingyu Ren, Xiaoling Wang, Jiajun Wang, Jia Yu & Shuyang Han

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Appendices

Appendix 1

Equation (8) can be expanded as Eq. (A1), where the unknown variables are n + 1 control points {Pi}; {Ni, p(t)} is calculated using the pseudo-code presented below. The value of __tk is calculated using Eq. (A2), and d is the total length of the polyline formed by {Qk} (Eq. (A3)). The knot vector U={t0, t1,…, tm } using for calculating {Ni, p(t)} can be obtained by Eq. (A4).

$$\left[ {\begin{array}{*{20}{c}} {\text{1}}&{}&{}&{}&{}&{}&{}&{}&{} \\ {{N_{0,3}}\left( {\overline {{{t_1}}} } \right)}&{{N_{1,3}}\left( {\overline {{{t_1}}} } \right)}&{{N_{2,3}}\left( {\overline {{{t_1}}} } \right)}&{{N_{{\text{3}},3}}\left( {\overline {{{t_1}}} } \right)}&{}&{}&{}&{}&{} \\ {{N_{0,3}}\left( {\overline {{{t_{\text{2}}}}} } \right)}&{{N_{1,3}}\left( {\overline {{{t_{\text{2}}}}} } \right)}&{{N_{2,3}}\left( {\overline {{{t_{\text{2}}}}} } \right)}&{{N_{3,3}}\left( {\overline {{{t_{\text{2}}}}} } \right)}&{}&{}&{}&{}&{} \\ {}&{{N_{1,3}}\left( {\overline {{{t_{\text{3}}}}} } \right)}&{{N_{2,3}}\left( {\overline {{{t_{\text{3}}}}} } \right)}&{{N_{3,3}}\left( {\overline {{{t_{\text{3}}}}} } \right)}&{{N_{{\text{4}},3}}\left( {\overline {{{t_{\text{3}}}}} } \right)}&{}&{}&{}&{} \\ {}&{}&{}&{}& \vdots &{}&{}&{}&{} \\ {}&{}&{}&{}&{{N_{n - {\text{4}},3}}\left( {\overline {{{t_{n - {\text{3}}}}}} } \right)}&{{N_{n - {\text{3}},3}}\left( {\overline {{{t_{n - {\text{3}}}}}} } \right)}&{{N_{n - {\text{2}},3}}\left( {\overline {{{t_{n - {\text{3}}}}}} } \right)}&{{N_{n - {\text{1}},3}}\left( {\overline {{{t_{n - {\text{3}}}}}} } \right)}&{} \\ {}&{}&{}&{}&{}&{{N_{n - {\text{3}},3}}\left( {\overline {{{t_{n - {\text{2}}}}}} } \right)}&{{N_{n - {\text{2}},3}}\left( {\overline {{{t_{n - {\text{2}}}}}} } \right)}&{{N_{n - {\text{1}},3}}\left( {\overline {{{t_{n - {\text{2}}}}}} } \right)}&{{N_{n,3}}\left( {\overline {{{t_{n - {\text{2}}}}}} } \right)} \\ {}&{}&{}&{}&{}&{{N_{n - {\text{3}},3}}\left( {\overline {{{t_{n - {\text{1}}}}}} } \right)}&{{N_{n - {\text{2}},3}}\left( {\overline {{{t_{n - {\text{1}}}}}} } \right)}&{{N_{n - {\text{1}},3}}\left( {\overline {{{t_{n - {\text{1}}}}}} } \right)}&{{N_{n,3}}\left( {\overline {{{t_{n - {\text{1}}}}}} } \right)} \\ {}&{}&{}&{}&{}&{}&{}&{}&{\text{1}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\mathbf{P}}_{\text{0}}}} \\ {{{\mathbf{P}}_{\text{1}}}} \\ {{{\mathbf{P}}_{\text{2}}}} \\ {{{\mathbf{P}}_{\text{3}}}} \\ \vdots \\ {{{\mathbf{P}}_{n - {\text{3}}}}} \\ {{{\mathbf{P}}_{n - {\text{2}}}}} \\ {{{\mathbf{P}}_{n - 1}}} \\ {{{\mathbf{P}}_n}} \end{array}} \right]{\text{=}}\left[ {\begin{array}{*{20}{c}} {{Q_0}} \\ {{Q_{\text{1}}}} \\ {{Q_{\text{2}}}} \\ {{Q_{\text{3}}}} \\ \vdots \\ {{Q_{n - {\text{3}}}}} \\ {{Q_{n - {\text{2}}}}} \\ {{Q_{n - 1}}} \\ {{Q_n}} \end{array}} \right]$$
(A1)
$$\left\{ \begin{gathered} \overline {{{t_{\text{0}}}}} =0 \hfill \\ \overline {{{t_k}}} =\overline {{{t_{k - 1}}}} +\frac{{\left| {{Q_i} - {Q_{i - 1}}} \right|}}{d},k=1,2, \cdots ,n - 1 \hfill \\ \overline {{{t_n}}} =1 \hfill \\ \end{gathered} \right.$$
(A2)
$$d=\sum\nolimits_{{i=1}}^{n} {\left| {{Q_i} - {Q_{i - 1}}} \right|}$$
(A3)
$$\left\{ \begin{gathered} {t_{\text{0}}}={t_{\text{1}}}={t_{\text{2}}}={t_{\text{3}}}=0 \hfill \\ {t_{j+3}}=\frac{1}{3}\sum\nolimits_{{i=j}}^{{j+2}} {\overline {{{t_i}}} } ,j=1,2, \cdots ,n - 3 \hfill \\ {t_{m - 3}}={t_{m - 2}}={t_{m - 1}}={t_m}=1 \hfill \\ \end{gathered} \right.$$
(A4)

The Pseudo-code for computing the B-spline basis function is cited from the literature [55].

Appendix 2

The (n + 1) equations representing Qk and Vk are shown in Eqs. (A6, A7).

$${Q_k}=C\left( {\overline {{{t_k}}} } \right)=\sum\nolimits_{{i=0}}^{{2n+{\text{1}}}} {{N_{i,p}}\left( {\overline {{{t_k}}} } \right){{\mathbf{P}}_i}}.$$
(A6)
$${V_k}=C^{\prime}\left( {\overline {{{t_k}}} } \right)=\sum\nolimits_{{i=0}}^{{2n+1}} {{{N^{\prime}}_{i,p}}\left( {\overline {{{t_k}}} } \right){{\mathbf{P}}_i}}.$$
(A7)

The solution methods for Ni, p(t) and N(k) i, p(t) are shown in Appendix 1.

Equation (A8) is a system of linear equations with a coefficient matrix 2(n + 1) × 2(n + 1) generated by combining Eqs. (A6) and (A7) in an alternating fashion. The construction of a parametric NURBS curve C(t) based on coordinate and derivative constraints is achieved by solving the following system of equations:

$$\left\{ {\begin{array}{*{20}{c}} {{{\mathbf{P}}_0}}&{}&{}&{}&{}&{}&{}&{}&{}&{}&=&{{Q_0}} \\ { - {{\mathbf{P}}_0}}&+&{{{\mathbf{P}}_1}}&{}&{}&{}&{}&{}&{}&{}&=&{\frac{{{t_4}}}{3}{V_0}} \\ {}&{}&{{N_{1,3}}\left( {\overline {{{t_1}}} } \right){{\mathbf{P}}_1}}&+& \cdots &+&{{N_{4,3}}\left( {\overline {{{t_1}}} } \right){{\mathbf{P}}_4}}&{}&{}&{}&=&{{Q_1}} \\ {}&{}&{{{N^{\prime}}_{1,3}}\left( {\overline {{{t_1}}} } \right){{\mathbf{P}}_1}}&+& \cdots &+&{{{N^{\prime}}_{4,3}}\left( {\overline {{{t_1}}} } \right){{\mathbf{P}}_4}}&{}&{}&{}&=&{{V_1}} \\ {}&{}&{}& \vdots &{}&{}& \vdots &{}&{}&{}& \vdots &{} \\ {}&{}&{}&{{N_{2n - 3,3}}\left( {\overline {{{t_{n - 1}}}} } \right){{\mathbf{P}}_{2n - 3}}}&+& \cdots &+&{{N_{2n,3}}\left( {\overline {{{t_{n - 1}}}} } \right){{\mathbf{P}}_{2n}}}&{}&{}&=&{{Q_{n - 1}}} \\ {}&{}&{}&{{{N^{\prime}}_{2n - 3,3}}\left( {\overline {{{t_{n - 1}}}} } \right){{\mathbf{P}}_{2n - 3}}}&+& \cdots &+&{{{N^{\prime}}_{2n,3}}\left( {\overline {{{t_{n - 1}}}} } \right){{\mathbf{P}}_{2n}}}&{}&{}&=&{{V_{n - 1}}} \\ {}&{}&{}&{}&{}&{}&{}&{ - {{\mathbf{P}}_{2n}}}&+&{{{\mathbf{P}}_{2n+1}}}&=&{\frac{{1 - {t_{m - 4}}}}{3}{V_n}} \\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{{{\mathbf{P}}_{2n+1}}}&=&{{Q_n}} \end{array}} \right..$$
(A8)

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Lyu, M., Ren, B., Wang, X. et al. Neural spline flow multi-constraint NURBS method for three-dimensional automatic geological modeling with multiple constraints. Comput Geosci 27, 407–424 (2023). https://doi.org/10.1007/s10596-023-10202-9

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