# Orthorhombic anisotropy

Orthorhombic anisotropy has three mutually orthogonal planes of symmetry. In the sedimentary crust, one of these is usually parallel to bedding. The other two are usually caused by one set (or two orthogonal sets) of fractures oriented perpendicular to this plane. Such orthogonal fractures are caused by the orthogonality of the stress tensor, in places where the tectonic history is simple. More complicated tectonic history may result in monoclinic anisotropy. Orthorhombic symmetry is the symmetry of a brick; crystallographers call it "orthotropic".

In seismics, the corresponding elastic stiffness matrix (symmetric) has nine independent components:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{c_{\alpha \beta}\}=\begin{pmatrix} c_{11}&c_{12}&c_{13}&0&0&0\\ c_{12}&c_{22}&c_{23}&0&0&0\\ c_{13}&c_{23}&c_{33}&0&0&0\\ 0&0&0&c_{44}&0&0\\ 0&0&0&0&c_{55}&0\\ 0&0&0&0&0&c_{66}\\ \end{pmatrix}}**

The resulting expressions for the seismic velocities at any azimuthal and incidence angle may be found with conventional algebraic techniques, as with polar anisotropy; they are quite complicated. However,for propagation within any of the three symmetry planes, they reduce without approximation to the well-known velocities of polar anisotropy (a different effective polar-anisotropic medium for each plane). These may be evaluated by sequentially parsing a wide-azimuth dataset. The assumption of weak polar anisotropy, in each case, makes this program feasible^{[1]}.

Of course, the indices in the matrix above refer to directions in the *natural coordinate system* of the medium. If the medium is flat-lying, the 3-direction is usually taken to be the same as in the *survey coordinate system*, i.e. vertical. The azimuthal orientation of the two vertical symmetry planes may be ascertained by examining residuals of arrivals, following correction for moveout, azimuthally invariant. The azimuthal orientations of these horizontal coordinate axes is normally different from those of the survey, initially defined.

However, if the medium is dipping, and the survey coordinate system is not, further analysis is required. This sort of **Tilted Orthorhombic Symmetry** is at the limit of feasibility, in 2020.

## External Links

- ↑ Tsvankin, I., 1997. Anisotropic parameters and P-wave velocity for orthorhombic media: Geoph., 62, 1292-1309