Deriving Sensitivity Kernels of Coda-Wave Travel Times to Velocity Changes Based on the Three-Dimensional Single Isotropic Scattering Model

Abstract

Recently, coda-wave interferometry has been used to monitor temporal changes in subsurface structures. Seismic velocity changes have been detected by coda-wave interferometry in association with large earthquakes and volcanic eruptions. To constrain the spatial extent of the velocity changes, spatial homogeneity is often assumed. However, it is important to locate the region of the velocity changes correctly to understand physical mechanisms causing them. In this paper, we are concerned with the sensitivity kernels relating travel times of coda waves to velocity changes. In previous studies, sensitivity kernels have been formulated for two-dimensional single scattering and multiple scattering, three-dimensional multiple scattering, and diffusion. In this paper, we formulate and derive analytical expressions of the sensitivity kernels for three-dimensional single-scattering case. These sensitivity kernels show two peaks at both source and receiver locations, which is similar to the previous studies using different scattering models. The two peaks are more pronounced for later lapse time. We validate our formulation by comparing it with finite-difference simulations of acoustic wave propagation. Our formulation enables us to evaluate the sensitivity kernels analytically, which is particularly useful for the analysis of body waves from deeper earthquakes.

Appendix

We show how to evaluate the infinitesimal volume element in the prolate spheroidal coordinate shown in Fig. 9. We extend 2D configuration of Pacheco and Snieder (2006) to 3D configuration. The infinitesimal areal element dA at x (a point of velocity change) is expressed as:

$$ {\text{d}}A({\mathbf{x}}) = {\text{d}}l{\text{d}}e\frac{s}{{r_{\text{s}} }}\sin \phi . $$

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Fig. 9

An elliptical coordinate on which a source, a receiver, a point of velocity change, and a scatterer are located for calculating the infinitesimal areal element dA

Here, de is defined along a spheroidal scattering shell and calculated as:

$$ \begin{aligned} {\text{d}}e &= \sqrt {\left( {\frac{{\partial x_{\text{bs}} }}{\partial \theta }} \right)^{2} + \left( {\frac{{\partial y_{\text{bs}} }}{\partial \theta }} \right)^{2} + \left( {\frac{{\partial z_{\text{bs}} }}{\partial \theta }} \right)^{2} } {\text{d}}\theta \hfill \\ &= h\sqrt {\varepsilon^{2} \sin^{2} \theta + (\varepsilon^{2} - 1)\cos^{2} \theta } {\text{d}}\theta \hfill \\ & = h\sqrt {\varepsilon^{2} - \cos^{2} \theta } {\text{d}}\theta = h\sqrt {\left( {\frac{{v_{0} t}}{2h}} \right)^{2} - \cos^{2} \theta } {\text{d}}\theta . \hfill \\ \end{aligned} $$

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Note that ${\mathbf{x}}_{{{\mathbf{bs}}}} = (x_{\text{bs}} ,\,y_{\text{bs}} ,\,z_{\text{bs}} )$ is the location of a scatterer on the scattering shell for the path ls. In the derivation, the following relations are used:

$$ \begin{aligned} &\frac{{{\partial}x_{\text{bs}} }}{{{\partial}\theta }} = \frac{\partial}{{{\partial}\theta }}\left( {h\varepsilon \cos \theta } \right) = - h\varepsilon \sin \theta , \hfill \\ &\frac{{{\partial}y_{\text{bs}} }}{{{\partial}\theta }} = \frac{\partial}{{{\partial}\theta }}\left( {h\sqrt {\varepsilon^{2} - 1} \sin \theta \cos \psi } \right) = h\sqrt {\varepsilon^{2} - 1} \cos \theta \cos \psi \hfill \\ &\frac{{{\partial}z_{\text{bs}} }}{{{\partial}\theta }} = \frac{\partial}{{{\partial}\theta }}\left( {h\sqrt {\varepsilon^{2} - 1} \sin \theta \sin \psi } \right) = h\sqrt {\varepsilon^{2} - 1} \cos \theta \sin \psi . \hfill \\ \end{aligned} , $$

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The unit vector n directing away from the source along the path ls is:

$$ {\mathbf{n}} = \frac{1}{{\sqrt {(h + x_{\text{bs}} )^{2} + y_{\text{bs}}^{2} + z_{\text{bs}}^{2} } }}\left( {\begin{array}{*{20}c} {h + x_{\text{bs}} } \\ {y_{\text{bs}} } \\ {z_{\text{bs}} } \\ \end{array} } \right) = \frac{1}{\varepsilon + \cos \theta }\left( {\begin{array}{*{20}c} {\varepsilon \cos \theta + 1} \\ {\sqrt {\varepsilon^{2} - 1} \sin \theta \cos \psi } \\ {\sqrt {\varepsilon^{2} - 1} \sin \theta \sin \psi } \\ \end{array} } \right) . $$

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The unit vector e directing along the edge of the ellipse is:

$$ \,{\mathbf{e}} = \frac{1}{{\sqrt {\left( {\frac{{\partial x_{\text{bs}} }}{\partial \theta }} \right)^{2} + \left( {\frac{{\partial y_{\text{bs}} }}{\partial \theta }} \right)^{2} + \left( {\frac{{\partial z_{\text{bs}} }}{\partial \theta }} \right)^{2} } }}\left( {\begin{array}{*{20}c} {\frac{{\partial x_{\text{bs}} }}{\partial \theta }} \\ {\frac{{\partial y_{\text{bs}} }}{\partial \theta }} \\ {\frac{{\partial z_{\text{bs}} }}{\partial \theta }} \\ \end{array} } \right) = \frac{1}{{\sqrt {\varepsilon^{2} - \cos^{2} \theta } }}\left( {\begin{array}{*{20}c} { - \varepsilon \sin \theta } \\ {\sqrt {\varepsilon^{2} - 1} \cos \theta \cos \psi } \\ {\sqrt {\varepsilon^{2} - 1} \cos \theta \sin \psi } \\ \end{array} } \right) . $$

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The term of $\,\sin \phi$ can be calculated using the vector product between n and e as:

$$ \sin \phi = |{\mathbf{n}} \times {\mathbf{e}}| = \frac{{\sqrt {\varepsilon^{2} - 1} }}{{\sqrt {\varepsilon^{2} - \cos^{2} \theta } }} = \frac{{\sqrt {\left( {\frac{{v_{0} t}}{2h}} \right)^{2} - 1} }}{{\sqrt {\left( {\frac{{v_{0} t}}{2h}} \right)^{2} - \cos^{2} \theta } }} . $$

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Equations (26), (27), (31), (7) and (10) give the volume element dV at x as:

$$ {\text{d}}V({\mathbf{x}}) = h^{2} \left\{ {\left( {\frac{{v_{0} t}}{2h}} \right)^{2} - 1} \right\}\frac{s}{{r_{\text{s}} }}\sin \theta \;{\text{d}}\theta {\text{d}}l{\text{d}}\psi . $$

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