Monday, August 30, 2010

Isotropic and anisotropic earth media in exploration geophysics

When we are doing exploration seismology, we take seismic wave equations as our theoretical base, and often we mention acoustic wave equation, elastic wave equations of isotropic media and anisotropic media. Real earth media should be considered as anisotropic media. VTI (transversely isotropic with vertical axis) media or TTI (transversely isotropic with tilted axis) media are the best approximation for complex geologic areas, while in relatively simple geological areas, all the seismic data processing steps can be taken under the assumption that the media are acoustic. This simplification has its reason that acoustic wave equation techniques are much mature than elastic (anisotropic, even isotropic) wave equation techniques. If we consider elastic wave equation, then we must deal with converted waves such like PP, PS, SP, SS, and so on, waves, which will add much difficulties for practical seismic data processing.

When we talk about elastic media, we want to differentiate different kinds of elastic media by their definite physical characteristics. Elasticity is one significant (almost most significant) character of elastic media. It is characterized by second rank elasticity tensor. According to elasticity theory, a second rank elasticity tensor has limited symmetries, and different symmetries will result in different types of earth models. In exploration seismology, elastic tensor is used in the form of its corresponding 6 by 6 elastic matrix because of its symmetries. There are the following kinds of anisotropic media, and at last the isotropic media can be considered as a special type of anisotropic media.
1. Generally anisotropic continuum has an elastic matrix that is symmetric and has 21 independent entries.
2. Monoclinic continuum is a continuum whose symmetry group contains a reflection about a plane through the origin. The elasticity matrix is also a symmetry matrix with 12 independent entries.
3. Orthotropic continuum is a continuum that possesses three orthogonal symmetry planes. The elasticity matrix has 9 independent entries.
4. Tetragonal continuum is a continuum whose symmetry group contains a four-fold rotation and a reflection through the plane that contains the axis of rotation. The elasticity matrix of tetragonal continuum has 6 independent entries.
5. Transversely isotropic continuum is a continuum that is invariant with respect to a single rotation. The elasticity matrix of transversely isotropic continuum has 5 independent parameters. Transversely isotropic media is a kind of very important media in exploration seismology and reservoir geophysics, since either VTI or TTI media can be considered as the approximation of real sedimentary geology, where the sedimentary layers lay parallel layer by layer.
6. Isotropic continuum is a continuum whose symmetry group contains all orthogonal transformations. Only two independent parameters are needed to describe the elasticity matrix of isotropic continuum. And sometimes people will conveniently use Lame parameters, which can be expressed as the linear combination of these two independent parameters, to solve problems.

Researchers are now using more and more anisotropic techniques to implement seismic data processing, since practical cases indicate that in some complex areas, anisotropy is necessary. However, anisotropic methods still have a lot of problems and is still under construction.

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