各向异性（地震）

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• 一般弹性张量(stiffness 或它的逆compliance) 是一个关于应力和应变的3X3X3X3张量。 在它的81个组件中，它包含多达21个独立常量，数量取决于对称性 (参见symmetry systems). 由于对称性，该张量可以写成6X6矩阵。 在一般情况下，对于平面波传播的每个方向，存在一个“准纵向”波和两个“准剪切波”（具有两个特征偏振角和两个不同速度，导致 "shear-wave splitting".[1]
• isotropic media（最简单的情况），矩阵的12个非零元素中只有两个独立的常量。 这些可以作为纵向模量M和剪切模量 ${\displaystyle \mu }$. 非零线性组合，${\displaystyle \lambda =M-(2*\mu )}$ 也出现在矩阵中。 其他弹性参数，如体积模量 ${\displaystyle K=M-(4/3)\mu }$ 也可以从这些计算出来，参见E-5. 在各向同性的情况下，纵波恰好是纵向的（极化矢量与波矢完全平行），并且两个剪切波退化为1（具有与极化方向无关的速度，其可能位于平面内的任何位置） 正好垂直于波矢，因此没有剪切波分裂）。
• Polar anisotropy (transverse isotropy) is the simplest geophysical case of anisotropy. It has elastic properties that are independent of the azimuth about a polar axis (hence the name) of symmetry, which is usually vertical. It is associated with, for example, unfractured shales or thinly-bedded sequences (see Physical Causes of Anisotropy, below). An example is given in Figure A-14. Polar anisotropy has five independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. However, the anisotropic seismic behaviour is governed directly by certain combinations of these five stiffnesses; for body waves they may be taken as the Thomsen anisotropic parameters.[2][3][4]
• Azimuthal anisotropy is a general term describing all lower symmetries, which all have azimuthal variation of elastic properties. In exploration geophysics, it is usually caused by aligned fractures (see Physical Causes of Anisotropy).
• The simplest plausible case of azimuthal anisotropy, in exploration geophysics, is that of Orthorhombic anisotropy (more properly: "orthotropic"). It has the symmetry of a brick, with nine independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. In exploration geophysics, it is usually caused by a single set of aligned fractures in an otherwise polar anisotropic medium, or perhaps two such sets, orthogonal to each other (see Physical Causes of Anisotropy). Despite the complexity, it is commonly feasible to analyze modern wide-azimuth (WAZ) datasets in terms of orthorhombic anisotropy [5].
• The most realistic case of anisotropy is that of monoclinic anisotropy. It has twelve independent constants among 18 nonzero elements of the stiffness (or compliance) matrix.

各向异性的物理机理

• At the scale of a single crystal (giga-Hertz frequencies), the small-scale structure is the arrangement of atomic cells.
• At the scale of a core (mega-Hertz frequencies), the small-scale structure is the micro-geometry of the arrangement of grains and pores. In some rocks, the crystals are randomly oriented, so that although each is intrinsically anisotropic, the random averaging means that the rock itself is isotropic (this might describe some sandstones). In other rocks (e.g. shales), there is a preferred orientation of some crystals (e.g. clays, which often have the shapes of platelets, and commonly lie horizontally), usually established by the direction of gravity during sedimentation and lithification. (The complementary pore space, occupied by fluid or kerogen, is then also flat-lying.) By itself, this usually leads to polar anisotropy.

Also at the core scale, there may be other physical causes in special circumstances. For example, if a salt body has flowed into place (rather than precipitated into place), the flow process can include re-crystallization, with preferential orientation of salt grains, related to the flow. (Tectonic flow in the upper mantle may include re-crystallization of olivine and other minerals, with preferred orientations related to that flow.)

At this core scale, there may also be micro-fractures, related to the state of stress, or a previous state of stress. It is sometimes thought that the stress itself (if the 3 principle stresses are not equal) makes for anisotropy . However, this direct stress-anisotropy is purely elastic, and so disappears once the stress is removed. By contrast, if the finite stress changes the micro-structure of the rock, e.g. by creating micro-fractures, this is called indirect stress-anisotropy, and may not be reversible, with the removal of the stress. (This could happen, for example, if the fractures were open for a long time, and minerals precipitated out of the fluid, on the crack-faces, so that the cracks do not close when the stress is removed.) In the lab, it is possible to distinguish these cases; normally one finds that the direct effect of stress is much smaller than the indirect effect.

If stress-aligned micro-fractures are present, they may be preferentially aligned with flat faces perpendicular to the direction of least compressive stress. There may be a second set, preferentially aligned with flat faces perpendicular to the direction of intermediate compressive stress. Rarely, there is a third set, preferentially aligned with flat faces perpendicular to the direction of most compressive stress. If (as is common in sedimentary basins), the maximum stress is oriented vertically, then the first two sets mentioned here are vertical, and orientated in two orthogonal horizontal directions.

• At the logging scale (kilo-Hertz frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the sonic wavelengths. By itself, this usually leads to polar anisotropy. The thin-layering need not not be periodic, although if the statistics of the layering are not stationary, then the sequence is effectively inhomogeneous, as well as anisotropic[6]. There may also be stress-induced anisotropy (both direct and indirect), caused by the stress concentrations near the borehole, which accompany the creation of the borehole wall. This will cause lower symmetry of anisotropy.
• At the reservoir scale, (seismic frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the seismic wavelength. By itself, this usually leads to polar anisotropy. (In the sub-crustal lithosphere, there may be azimuthal anisotropy at this scale, caused by sheeted-dike emplacement at spreading ridges.)

There may also be both direct and indirect (or crack-related) stress-induced anisotropy; in field data, it may not be possible to distinguish between these causes. Hence, the observed anisotropy orientations might indicate directions of either the current stress-state, or a previous stress-state, or both. If the fractures yield monoclinic anisotropy, this is an indication of a complex geologic history, which places its complex character on the current anisotropy.

If aligned cracks are present, their shapes may be limited, top and bottom, by fracture-resistant beds. Since there is no corresponding limitation in the horizontal direction, they may be "ribbon-shaped" joints, rather than "penny-shaped" micro-cracks. Such joints usually dominate the hydraulic anisotropy, although they may or may not dominate the seismic anisotropy.

参考文献

1. Crampin, S., 1981, A review of wave motion in anisotropic and cracked elastic media: Wave Motion, 3, 363–390.
2. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.
3. Alkhalifah, T. and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550–1566.
4. Thomsen, L.,2002. Understanding seismic anisotropy in exploration and exploitation: SEG-EAEG Distinguished Instructor Series #5: Soc. Expl. Geophys.(Second Edition 2014)
5. Tsvankin, I., 1997. Anisotropic parameters and P-wave velocity for orthorhombic media: Geoph., 62, 1292-1309.
6. Backus, 1962. Long-wave elastic anisotropy produced by horizontal layering: J. Geoph. Res., 67(11), 4427.