Characterization of the null space of a generally anisotropic medium in linearized cross-well tomography

Details Characterization of the null space of a generally anisotropic medium in linearized cross-well tomography Characterization of the null space of a generally anisotropic medium in linearized cross-well tomography

Apr 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998geoji.133...65b&link_type=abstract

Geophysical Journal International, Volume 133, Issue 1, pp. 65-84.

Statistics

Computation

4

Scientific paper

Anisotropic traveltime tomography can potentially determine many useful rock properties, such as crack density or pore shape, that cannot be found from isotropic methods. Similar to isotropic cross-well transmission tomography, many features of an anisotropic model are poorly resolved owing to limited ray-path coverage; the lateral smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the cross-well transmission traveltimes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null-space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation.
As expected, the anisotropic null space is larger than the isotropic null space owing to the ambiguity amongst the elastic parameters. We identify three categories of model perturbations in the anisotropic null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself in the isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null-space elements: perturbations which are polynomials in the depth variable with coefficients which are functions of the horizontal variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions in the wells for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent anomalies that are completely contained in the interior of the model and yet are in the null space; this behaviour is in marked contrast to the isotropic problem. These categories are sufficient to describe the anisotropic null space completely. We demonstrate that every model perturbation in the null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives a rich family of examples of sums of null-space elements in the first and third categories, and thereby gives a sense of just how large the anisotropic null space is.
Moreover, we show that the traveltime perturbations determine only a small number of features of an elastic perturbation which distinguish between the several elastic parameters. We identify these features precisely: they are functions of depth representing horizontal averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the horizontal velocity appear more prominently in these features than those influencing the vertical velocity. All other features of the anisotropic model are ambiguous amongst the elastic parameters: except for these features, it is completely impossible from the traveltime perturbations alone to determine which elastic parameter (or which combination of the elastic parameters) gives rise to given traveltime perturbations.

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