We propose a method, namely OptAVO to build enhanced linear AVO approximations. The basis functions of the approximation are orthogonal and their coefficients represent a new set of AVO attributes. These attributes can directly be used for AVO classification or to obtain better estimates of the usual coefficients (e.g., intercept, gradient). The method will be illustrated for class I reflectors using large reflection angles. A real data example shows the applicability of the proposed approach.
Amplitudes of reflected and transmitted plane waves at a planar boundary of two elastic media are completely determined for all incidence angles by the Zoeppritz equations (Zoeppritz, 1919), and can be computed if the elastic parameters are known. A number of lineralized approximations to the Zoeppritz equations have been developed that give more insight into the factors that control amplitude changes with offset/angle and simplify computations. Typically, trigonometric functions of the reflection angle are building the basis for the approximations, assuming small elastic parameter changes across the interface (Aki and Richards, 1980) or small incident angles (Ursin and Dahl, 1992). Because of these assumptions the well-known linear AVO approximations can show some inaccuracies, e.g. close to the critical angle. The resulting approximation errors can affect the quality of the AVO analysis, e.g. by causing systematic errors in estimates of the seismic parameter contrasts. The OptAVO approach still uses a linear approximation, but the trigonometric functions are replaced by more general ones to obtain an optimal approximation. Available information on the seismic parameters is used to derive optimal basis functions for the particular application. The functions are orthogonal and derived using singular value decomposition.
In each of these models we used the Zoeppritz equations and calculated the curves for a discrete number of angle values. Each reference curve can therefore be represented by a vector. Organizing all vectors as columns in a matrix and applying singular value decomposition (SVD) we can write:
=
The matrices
and
are orthogonal and
is diagonal. All matrices are totally defined by the SVD. The new basis fi(q) functions are given by the columns of matrix
for the discrete reflection angle values used during modeling. Figure 2 shows the basis functions obtained by OptAVO for this particular case. The physical meaning of equation (2) is that, using as many terms as modeled reference curves, each of the reference functions can be exactly described by equation (1), with the coefficients Ci given by the columns of the weighting matrix
. The size of the weights in the matrix is related to the size of the singular values in the diagonal matrix
. As the singular values are sorted in decreasing order the largest weights are associated with the basis functions f1 and f2. As the singular values decrease usually very rapidly, we only need to consider two or three terms in practice. This will introduce only a small error. The coefficients Ci are representing the new OptAVO attribute space.
