Parameterization of A Priori Geological Knowledge in Seismic Inversion

Abstract—An approach to parameterization of prior geological knowledge concerning the changes in depositional environment in space and geological time for their quantitative use in the workflow of seismic inversion is presented. The idea is to describe the observed or expected facies diversity in terms of a few statistically independent factors (generalized geological variables). The topology and metrics of the model are determined by the set of basic depositional environments and the statistics of facies transitions. The introduced parameters make it possible to estimate the occurrence probability of different facies at each model point. The proposed technique can be applied for regions with various degree of detail of the existing geological knowledge and amount of available well logging data.

CALCULATION OF “GENERALIZED COORDINATES” OF FACIES FROM THE DIVERGENCE MATRIX

The “generalized geological variables” are calculated from quantities (17) with the use of the Torgerson algorithm (Torgerson, 1952) underlying the classical multidimensional scaling technique (Davison, 1988; Borg and Groenen, 2005). The method is based on the idea of reconstructing the coordinates of the objects from the given distances between them. The procedure includes calculation of a centered covariance matrix and its subsequent factorization.

Let there be given a divergence matrix D whose elements are the squared distances between N objects of the analyzed set (in the context of this work, reference environments):

$$D = \left( {\begin{array}{*{20}{c}} 0&{{{d}_{{12}}}}&{..}&{{{d}_{{1N}}}} \\ {{{d}_{{12}}}}&0&{..}&{{{d}_{{2N}}}} \\ :&:&:&: \\ {{{d}_{{1N}}}}&{{{d}_{{2N}}}}&{..}&0 \end{array}} \right).$$

((A.1))

The transition to the centered covariance matrix is conducted as follows:

$$R = - \frac{1}{2}JDJ.$$

((A.2))

Here, matrix J is calculated according to the formula

$$J = E - \frac{1}{N}{{l}^{T}}l,$$

where E is identity matrix, N is dimension of matrix D (the number of the objects under comparison), and l = {1, 1, …1} is a row vector composed of N unities.

In scalar notations, formula (A.2) is recorded in the following form (Davison, 1988):

$${{r}_{{ij}}} = - \frac{1}{2}\left( {{{d}_{{ij}}} - {{d}_{{i*}}} - {{d}_{{*j}}} + {{d}_{{**}}}} \right).$$

((A.3))

Here, d_ij is the element of matrix D (the squared distance between the ith and jth objects), d_i* is the mean value of the elements of the ith row, d_*j is the mean value of the elements of the jth column, and d_** is the mean value of the elements of matrix D.

Let also V be the matrix of column eigenvectors of matrix R (of a unit length) and Λ be the matrix whose diagonal is composed of the corresponding eigenvalues. The values of the objects’ coordinates in the new factorized orthogonal system are then determined by the row vectors of matrix X:

$$X = V{{\Lambda }^{{1/2}}}.$$

((A.4))

The covariance matrix is invariant to this coordinate transformation (since RV = VΛ and V^TV = E):

$$X{{X}^{T}} = V{{\Lambda }^{{1/2}}}{{\Lambda }^{{1/2}}}{{V}^{T}} = V\Lambda {{V}^{T}} = R.$$

On the other hand, as V^TV = E, matrix X^TX = Λ is diagonal. That is, the factors corresponding to the axes of the new coordinate system are statistically independent (the sample correlation coefficients of the values of coordinates are zero), whereas the sample variances of the values of the coordinates are equal to eigenvalues of matrix R. Therefore, the dimensionality of the coordinate space can be reduced without a significant loss of accuracy by discarding the terms that correspond to small eigenvalues. In this case, matrices Λ and V in (A.4) are replaced by matrices Λ₁ and V₁ corresponding to the largest eigenvalues, and instead of matrix X, a rectangular matrix of coordinates X₁ with relatively few (K < N) columns is obtained:

$${{X}_{1}} = {{V}_{1}}\Lambda _{1}^{{1/2}}.$$

((A.5))

As we know matrix W of the weights of reference environments (7), we can find the facies coordinates from coordinates X₁ of reference environments calculated according to formula (A.5):

$$Y = W{{X}_{1}}.$$

((A.6))

Here, the ith row of matrix Y contains K < N “generalized coordinates ” of the ith facies. Dimensionality K of the model is determined by the number of factors that are significant from the standpoint of accuracy of reconstruction of the initial divergence matrix (A.1).

The uncertainty of the obtained parameterization is determined by the sum of the rejected eigenvalues. This quantity can be used for estimating the probability distributions P(F_i|x) participating in the calculation of P(F_i) in (46) and (47) as well as for specifying the models of spatial correlation (variograms) during the interpolation into the interwell space.