A New Differentiable Parameterization Based on Principal Component Analysis for the Low-Dimensional Representation of Complex Geological Models

Abstract

A new approach based on principal component analysis (PCA) for the representation of complex geological models in terms of a small number of parameters is presented. The basis matrix required by the method is constructed from a set of prior geological realizations generated using a geostatistical algorithm. Unlike standard PCA-based methods, in which the high-dimensional model is constructed from a (small) set of parameters by simply performing a multiplication using the basis matrix, in this method the mapping is formulated as an optimization problem. This enables the inclusion of bound constraints and regularization, which are shown to be useful for capturing highly connected geological features and binary/bimodal (rather than Gaussian) property distributions. The approach, referred to as optimization-based PCA (O-PCA), is applied here mainly for binary-facies systems, in which case the requisite optimization problem is separable and convex. The analytical solution of the optimization problem, as well as the derivative of the model with respect to the parameters, is obtained analytically. It is shown that the O-PCA mapping can also be viewed as a post-processing of the standard PCA model. The O-PCA procedure is applied both to generate new (random) realizations and for gradient-based history matching. For the latter, two- and three-dimensional systems, involving channelized and deltaic-fan geological models, are considered. The O-PCA method is shown to perform very well for these history matching problems, and to provide models that capture the key sand–sand and sand–shale connectivities evident in the true model. Finally, the approach is extended to generate bimodal systems in which the properties of both facies are characterized by Gaussian distributions. MATLAB code with the O-PCA implementation, and examples demonstrating its use are provided online as Supplementary Materials.

Appendix: O-PCA Analytical Solution and Derivative for Binary Models

In this Appendix, we solve the minimization problem (10), defined in Sect. 2.3. We proceed by introducing the Lagrangian $L(x_i)$, given by Luenberger and Ye (2008)

$$\begin{aligned} L(x_i) \!=\! (1\!-\!\gamma )x^2_i \!-\! 2\left( a_i\!-\!\frac{\gamma }{2}\right) x_i\!-\! \mu _i x_i \!-\! \eta _i(1\!-\!x_i), \ \ \mu _i \ge 0, \ \eta _i \ge 0, \ {x_i} \in {[ 0,1]},\nonumber \\ \end{aligned}$$

(18)

where $\mu _i$ and $\eta _i \ \forall \ i \in \{1,\ldots ,N_\mathrm{c}\}$ are the Lagrange multipliers. The Karush–Kuhn–Tucker (KKT) optimality conditions for (18) are

$$\begin{aligned}&\displaystyle \frac{\partial L(x_i)}{\partial x_i}= 2(1-\gamma )x_i - 2\left( a_i-\frac{\gamma }{2}\right) -\mu _i +\eta _i =0, \end{aligned}$$

(19)

$$\begin{aligned}&\displaystyle \mu _i x_i = 0, \end{aligned}$$

(20)

$$\begin{aligned}&\displaystyle \eta _i (1-x_i) = 0, \end{aligned}$$

(21)

$$\begin{aligned}&\displaystyle \mu _i \ge 0,\ \eta _i \ge 0,\ 0 \le x_i \le 1. \end{aligned}$$

(22)

The stationary point for the quadratic convex objective function without bound constraints, $x^\mathrm{s}_i = (a_i-\gamma /2)/(1-\gamma )$, is first computed. Given the fact that the objective function is quadratic and convex with $0 \le \gamma <1$, the minimal point for the bound-constrained problem, designated $x^*_i$, is either $x^\mathrm{s}_i, 0$, or $1$. The following three cases must be considered:

• (i) Case 1: If $a_i \le \gamma /2$, then $x^\mathrm{s}_i \le 0$, and the minimum point is $x^*_i = 0$. From (21) we have $\eta _i = 0$. Substituting $x^*_i = 0$ and $\eta _i = 0$ in (19), we obtain $\mu _i = \gamma - 2a_i$. We can readily verify that, for $x^\mathrm{s}_i \le 0$ and $0 \le \gamma <1$, we have $\mu _i = \gamma - 2a_i \ge 0$, so (22) will be satisfied.

• (ii) Case 2: If $a_i\ge 1-\gamma /2$, then $x^\mathrm{s}_i \ge 1$, and the minimum point is $x^*_i = 1$. From (20) we have $\mu _i = 0$. Substituting $x^*_i = 1$ and $\mu _i = 0$ in (19), we obtain $\eta _i = 2a_i + \gamma - 2$. It is easy to verify that if $x^\mathrm{s}_i \ge 1$ and $0 \le \gamma <1$, we also have $\eta _i = 2a_i + \gamma - 2 \ge 0$. Thus (22) will be satisfied.

• (iii) Case 3: If $\gamma /2<a_i<1-\gamma /2$, we have $0<x^\mathrm{s}_i<1$, and the minimum point of the constrained problem is $x^*_i = x^\mathrm{s}_i$. We also have $\mu _i =\eta _i =0$ from (20) and (21), which satisfies (22).

Using the Lagrange multipliers computed above, we can apply the multivariate implicit function theorem (Moskowitz and Paliogiannis 2011) to determine the derivative ${\partial x_i}$/${\partial \xi _j}$, which will be required for gradient-based data assimilation. We first define, from (19), (20) and (21),

$$\begin{aligned} F_1(x_i, \mu _i, \eta _i, \varvec{\xi })&= 2(1-\gamma )x_i - 2\left( a_i(\varvec{\xi })-\frac{\gamma }{2}\right) -\mu _i +\eta _i , \nonumber \\ F_2(x_i, \mu _i, \eta _i, \varvec{\xi })&= \mu _i x_i , \nonumber \\ F_3(x_i, \mu _i, \eta _i, \varvec{\xi })&= \eta _i (1-x_i). \end{aligned}$$

(23)

Denoting partial derivatives as $F_{1,x_i}\!\! =\!\!{\partial F_1}/{\partial x_i}, F_{1,\mu _i}\!=\!\! {\partial F_1}/{\partial \mu _i}, F_{1,\eta _i} \!\!=\!\! {\partial F_1}/{\partial \eta _i}, F_{1, \xi _j}={\partial F_1}/{\partial \xi _j}$, etc., through application of the implicit function theorem for a system of equations along with Cramer’s rule, we have

$$\begin{aligned} \frac{\partial x_i}{\partial \xi _j} = -\frac{\mathrm{det}\begin{pmatrix} F_{1, \xi _j} &{} F_{1, \mu _i} &{} F_{1, \eta _i}\\ F_{2, \xi _j} &{} F_{2, \mu _i} &{} F_{2, \eta _i}\\ F_{3, \xi _j} &{} F_{3, \mu _i} &{} F_{3, \eta _i}\\ \end{pmatrix}}{\mathrm{det}\begin{pmatrix} F_{1, x_i} &{} F_{1, \mu _i} &{} F_{1, \eta _i}\\ F_{2, x_i} &{} F_{2, \mu _i} &{} F_{2, \eta _i}\\ F_{3, x_i} &{} F_{3, \mu _i} &{} F_{3, \eta _i}\\ \end{pmatrix}} \end{aligned}$$

(24)

By noting that $a_i=\bar{m}_i + \sum _{j=1}^{l} \Phi _{ij} \xi _j$, all of the partial derivatives in (23) can be readily determined. After evaluating the determinants of the two matrices in (24), we have

$$\begin{aligned} \frac{\partial x_i}{\partial \xi _j} = \frac{2 ({\partial a_i}/{\partial \xi _j}) x_i (1-x_i)}{\mu _i+2 x_i +\eta _i x_i -2 \gamma x_i - \mu _i x_i +2 \gamma x^2_i - 2 x^2_i}, \end{aligned}$$

(25)

where ${\partial a_i}/{\partial \xi _j}=\Phi _{ij}$.

The analytical solutions for $x_i$ and ${\partial x_i}/{\partial \xi _j}$, although derived from the O-PCA formulation introduced here, can be interpreted in terms of a post-processing (histogram transform or soft-thresholding) of the underlying PCA model (designated $a_i$). Namely, for this binary case, with the regularization term given by $\gamma x_i(1-x_i)$, O-PCA maps PCA solutions with $a_i \le \gamma /2$ to $x_i=0$, and solutions with $a_i\ge 1-\gamma /2$ to $x_i=1$. For PCA solutions with $\gamma /2 < a_i < 1-\gamma /2$, O-PCA applies the linear mapping $x_i = (a_i-\gamma /2)/(1-\gamma )$. Consistent with this, it can be seen from (25) that, if $x_i=1$ or $x_i=0, {\partial x_i}/{\partial \xi _j}=0$. Otherwise, ${\partial x_i}/{\partial \xi _j}=\Phi _{ij}/({1-\gamma })$.

This analytical description of the post-processing applied by O-PCA is possible in this case because of the simple form of the regularization term. As shown in Sect. 5, O-PCA can also be applied for channelized models described by bimodal log-permeability distributions. For such cases, the regularization term is significantly more complicated, and the analytical solution of the minimization problem cannot be achieved. As a result, the specific mapping introduced by O-PCA cannot be described analytically (though the derivative ${\partial x_i}/{\partial \xi _j}$ can be obtained semi-analytically).

In the development above, $\mathbf{x}$ is the optimization variable. As noted in Sect. 2.3, after the solution and its derivative are found, they are designated $\mathbf{m}$ and ${\mathrm{d}\mathbf m}/{\mathrm{d}\varvec{\xi }}$. To verify our derivation, the analytical solution for $\mathbf{m}$ was compared to solutions obtained by applying a numerical optimization to the minimization problem defined by (8). Agreement to within numerical tolerance was consistently achieved. In addition, the analytical expression (25) for ${\mathrm{d}\mathbf m}/{\mathrm{d}\varvec{\xi }}$ was compared with numerical derivatives computed using finite differences. The Frobenius norm of the difference between the two derivative matrices was less than $10^{-6}$. These results serve as verification of the analytical solutions derived here. All results presented in this paper for binary models were generated using these analytical solutions.