Bernstein Copula-Based Spatial Stochastic Simulation of Petrophysical Properties Using Seismic Attributes as Secondary Variable

Abstract

A novel Bernstein copula-based spatial stochastic co-simulation (BCSCS) method for petrophysical properties using seismic attributes as a secondary variable is presented. The method is fully nonparametric, and it has the advantages of not requiring linear dependence between variables. The methodology is illustrated in a case study from a marine reservoir in the Gulf of Mexico, and the results are compared with sequential Gaussian co-simulation (SGCS) method.

Appendix A: Copula-Based Approach for Dependence Modeling

A theorem by Sklar (1959) proved that there exists a functional relationship between the joint probability distribution function of a random vector and its univariate marginal distribution functions. In the bivariate case, for example, if (X, Y) is a random vector with joint probability distribution ${F}_{XY}\left( x, y\right)= P\left( X\le x, Y\le y\right)$, then the marginal distribution functions of X and Y are ${F}_X(x)= P\left( X\le x\right)={F}_{X Y}\left( x,\infty \right)$ and ${F}_Y(y)= P\left( Y\le y\right)={F}_{XY}\left(\infty, \mathrm{y}\right)$, respectively, but in the marginalization of F _XY , some information is lost since the only knowledge of the marginal distributions F _X and F _Y is not generally possible to specify F _XY because the marginals only explain the probabilistic individual behavior of the random variables they represent. Sklar’s theorem proves that there exists a function ${C}_{XY}:{\left[0,1\right]}^2\to \left[0,1\right]$ such that

$$ {F}_{X Y}\left( x, y\right)={C}_{X Y}\left({F}_X(x),{F}_Y(y)\right) $$

C _XY is called copula function associated to (X, Y) and contains information about the dependence relationship between X and Y, independently from their marginal probabilistic behavior. C _XY is uniquely determined on $\mathrm{Ran}\ {F}_X\times \mathrm{Ran}\ {F}_Y$, and therefore, if F _X and F _Y are continuous, then C _XY is unique on [0, 1]². Among several properties of copula functions, see Nelsen (2006), we have the following:

• $C\left( u,0\right)=0= C\left(0, v\right)$

• $C\left( u,1\right)= u, C\left(1, v\right)= v$

• $C\left({u}_2,{v}_2\right)- C\left({u}_2,{v}_1\right)- C\left({u}_1,{v}_2\right)+ C\left({u}_1,{v}_1\right)\ge 0$ if ${u}_1\le {u}_2,{v}_1\le {v}_2$

• C is uniformly continuous on its domain [0, 1]².

• The horizontal, vertical, and diagonal sections of a copula C are all nondecreasing and uniformly continuous on [0, 1].

• $W\left( u, v\right)\le C\left( u, v\right)\le M\left( u, v\right)$ where $W\left( u, v\right)= \max \left( u+ v-1,\ 0\right)$ and $M\left( u, v\right)= \min \left( u, v\right)$ are also copulas known as the lower and upper Fréchet-Hoeffding bounds.

• A convex linear combination of copula functions is also a copula function.

• If X and Y are continuous random variables with copula C _XY , and if α and β are strictly increasing functions on Ran X and Ran Y, respectively, then ${C}_{\alpha (X)\beta (Y)}={C}_{XY}$. Thus, C _XY is invariant under strictly increasing transformations of X and Y.

Copula functions are a useful tool to build joint probability models in a more flexible way since we may choose separately the univariate models for the random variables of interest and the copula function that better represents the dependence among them, in each case in a parametric or nonparametric way. In the case of a multivariate normal model, for example, all the marginal distributions have to be normally distributed, with no tail dependence at all and with finite second moments for the correlations to be well defined. In fact, the multivariate normal model is a particular case when the underlying copula is Gaussian and all the univariate marginals are normally distributed.

In case F _X and F _Y are continuous, by elementary probability we know that $U={F}_X(X)$ and $V={F}_Y(Y)$ are continuous Uniform(0, 1) random variables and the underlying copula C for the random vector (U, V) is the same copula corresponding to (X, Y), and by Sklar’s theorem we have that the joint probability distribution function for (U, V) is equal to ${F}_{U V}\left( u, v\right)= C\left({F}_U(u),{F}_V(v)\right)= C\left( u, v\right)$. Therefore, in case F _X and F _Y are known and F _XY is unknown, if {(x ₁, y ₁), …, (x _n , y _n )} is an observed random sample from (X, Y), the set $\left\{\left({u}_k,{v}_k\right)=\left({F}_X\left({x}_k\right),{F}_Y\left({y}_k\right)\right): k=1,\dots, n\right\}$ would be an observed random sample from (U, V) with the same underlying copula C as (X, Y), and since $C={F}_{UV}$ we may use the (u _k , v _k ) values (called copula observations) to estimate C as a joint empirical distribution:

$$ \widehat{C}\left( u, v\right)=\frac{1}{n}\ {\displaystyle \sum_{k=1}^n}{1}_{\left\{{u}_k\ \le\ u, \kern0.5em {v}_k\ \le\ v\right\}} $$

Strictly speaking, the estimation Ĉ is not a copula since it is discontinuous and copulas are always continuous. If F _X , F _Y , and F _XY are all unknown (the usual case), we estimate F _X and F _Y by univariate empirical distribution functions:

$$ {\hat{F}}_X( x)=\frac{1}{n}{\displaystyle \sum_{k=1}^n}{1}_{\left\{{x}_k\le x\right\}}\kern0.5em {\hat{F}}_Y( y)=\frac{1}{n}{\displaystyle \sum_{k=1}^n}{1}_{\left\{{y}_k\le y\right\}} $$

Now the set of pairs $\left\{\left({u}_k,{v}_k\right)=\left({\hat{F}}_X\left({x}_k\right),{\hat{F}}_Y\left({y}_k\right)\right): k=1,\dots, n\right\}$ is referred to as copula pseudo-observations. It is straightforward to verify that ${\hat{F}}_X\left({x}_k\right)=\frac{1}{n}\mathrm{rank}\left({x}_k\right)$ and ${\hat{F}}_Y\left({y}_k\right)=\frac{1}{n}\mathrm{rank}\left({y}_k\right)$. In this case the concept of empirical copula, see Nelsen (2006), is defined as the following function ${C}_n:{I}_n^2\to \left[0,1\right]$, where ${I}_n = \left\{\frac{i}{n}: i=0,\dots, n\right\}$, given by:

$$ {C}_n\left(\frac{i}{n},\frac{j}{n}\right)=\frac{1}{n}\ {\displaystyle \sum_{k=1}^n}{1}_{\left\{\mathrm{rank}\left({x}_k\right)\le i,\ \mathrm{rank}\left({y}_k\right)\le j\right\}} $$

Again, C _n is not a copula, but it is an estimation of the underlying copula C on the grid I ² _n that may be extended to a copula on [0, 1]² by means of, for example, Bernstein polynomials, as proposed and studied in Sancetta and Satchell (2004), which leads to what is known as a Bernstein copula nonparametric estimation $\tilde{C}:{\left[0,1\right]}^2\to \left[0,1\right]$ given by:

$$ \tilde{C}\left( u, v\right)={\displaystyle \sum_{i=0}^n}{\displaystyle \sum_{j=0}^n}{C}_n\left(\frac{i}{n},,,\frac{j}{n}\right)\left(\begin{array}{c} n\\ {} i\end{array}\right){u}^i{\left(1- u\right)}^{n- i}\left(\begin{array}{c} n\\ {} j\end{array}\right){v}^j{\left(1- v\right)}^{n- j} $$

As summarized in Erdely and Diaz-Viera (2010) in order to simulate replications from the random vector (X, Y) with the dependence structure inferred from the observed data {(x ₁, y ₁), …, (x _n , y _n )}, we have the following:

Algorithm 1

• 1. Generate two independent and continuous Uniform(0, 1) random variates u and t.

• 2. Set $v={c}_u^{-1}(t)$ where ${c}_u(v)=\frac{\partial \tilde{C}\left( u, v\right)}{\partial u}$.

• 3. The desired pair is $\left( x, y\right)=\left({\tilde{Q}}_n(u),{\tilde{R}}_n(v)\right)$ where ${\tilde{Q}}_n$ and ${\tilde{R}}_n$ are empirical quantile functions for X and Y, respectively.

For a value x in the range of the random variable X and a given $0<\alpha <1$, let $y={\varphi}_{\alpha}(x)$ denote the solution to the equation $P\left( Y\le y\left| X\right.= x\right)=\upalpha$. Then the graph of $y={\varphi}_{\upalpha}(x)$ is the α-quantile regression curve of Y conditional on X = x. In Nelsen (2006), it is proven that:$P\left( Y\le y\left| X\right.= x\right)={c}_u(v)\Big|{}_{u={F}_X(x), v={F}_Y(y)}$

This result leads to the following algorithm to obtain the α-quantile regression curve of Y conditional on X = x:

Algorithm 2

• 1. Set ${c}_u(v)=\alpha$.

• 2. Solve for v the regression curve, say $v={g}_{\upalpha}(u)$.

• 3. Replace u by ${\tilde{Q}}_n^{-1}( x)$ and v by ${\tilde{R}}_n^{-1}( y)$.

• 4. Solve for y the regression curve, say $y={\varphi}_{\alpha}(x)$.