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

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

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

CC BY-SA 4.0
2019-07-12T05:22:42.620Z
2019-07-12T05:22:42.620Z
Erik
<p>Does the <a href="http://sepwww.stanford.edu/sep/prof/bei/sg/paper_html/node7.html" rel="nofollow noreferrer">reciprocity principle</a> hold true in an <a href="https://wiki.seg.org/wiki/Seismic_anisotropy" rel="nofollow noreferrer">anisotropic</a> subsurface environment?</p>

<seismic>
Seismic wavefield theory: velocity anisotropy
nate
# Seismic wavefield theory: velocity anisotropy

(Asked by: username_0 on 2019-07-11)

Does the [reciprocity principle](http://sepwww.stanford.edu/sep/prof/bei/sg/paper_html/node7.html) hold true in an [anisotropic](https://wiki.seg.org/wiki/Seismic_anisotropy) subsurface environment?

**Tags:** seismic


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## 1 answers

### Answer by username_1 on 2019-07-12. Score: 2 | Accepted 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.


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