# Seismic wavefield theory: velocity anisotropy

Does the reciprocity principle hold true in an anisotropic subsurface environment?

## 1 Answer

Yes, or at least certainly in theory. If you look up any derivation of the elastodynamic representation theorem you're bound to find that it holds for the most general version of the elastic wave equation: \begin{align}\rho \frac{\partial^2 u_i(x,t)}{\partial t^2}& = \frac{\partial \sigma_{ij}}{\partial x_j} \\ \sigma_{ij}(x,t) & = c_{ijkl}(x)\varepsilon_{kl}(x,t)\end{align} with $$u$$ the displacement, $$\sigma$$ the stress, $$\rho$$ the density and $$c_{ijkl}$$ Hooke's tensor. You'll have run into this tensor whenever you researched anisotropy, because it is how one describes anisotropy.

The main reference is a book by Aki & Richards (1982): https://www.ldeo.columbia.edu/~richards/Aki_Richards.html , but it is also derived in these handouts, see the bottom of page 15: http://www.geo.uu.nl/~paulssen/Theoreticalseismology/handout.pdf .

The notation $$G_{nm}(\xi_2,\tau|\xi_1,t)= G_{mn}(\xi_1,\tau|\xi_2,t)$$ is the reciprocity relation. $$G_{nm}(\xi_2,\tau|\xi_1,t)$$ is the measured displacement due to a point source in direction $$m$$ at time $$t$$ at location $$\xi_1$$ as measured at time $$\tau$$ at $$\xi_2$$ in direction $$n$$. The reciprocity relation says that this recorded signal is identical to an experiment where a source in direction $$n$$ at time $$t$$ at location $$\xi_2$$ is measured at time $$\tau$$ at $$\xi_1$$ in direction $$m$$. Hence, you can interchange source and receiver when you interchange source and receiver dirctions too.