Measurements of translation, rotation and strain: new approaches to seismic processing and inversion

Abstract

We propose a novel approach to seismic tomography based on the joint processing of translation, strain and rotation measurements. Our concept is based on the apparent S and P velocities, defined as the ratios of displacement velocity and rotation amplitude, and displacement velocity and divergence amplitude, respectively. To assess the capability of these new observables to constrain various aspects of 3D Earth structure, we study their corresponding finite-frequency kernels, computed with a combination of spectral-element simulations and adjoint techniques. The principal conclusion is that both the apparent S and P velocities are generally sensitive only to small-scale near-receiver structure, irrespective of the type of seismic wave considered. It follows that knowledge of deeper Earth structure would not be required in tomographic inversions for local structure based on the new observables. In a synthetic finite-perturbation test, we confirm the ability of the apparent S and P velocities to directly detect both the location and the sign of shallow lateral velocity variations.

Appendix: Details for the computation of sensitivity kernels

The forward wavefield is excited by a bandpass filtered Heaviside function between 20 and 200 s. The moment tensor components given in Nm are

$$ \begin{array}{rll} M_{\theta\theta}&=&0.710 \cdot 10^{19}\\[-4pt] M_{\phi\phi}&=&-0.356 \cdot 10^{19}\\[-4pt] M_{zz}&=&-0.355 \cdot 10^{19}\\[-4pt] M_{\theta\phi}&=&0.800 \cdot 10^{19}\\[-4pt] M_{\theta z}&=&0.315 \cdot 10^{19}\\[-4pt] M_{\phi z}&=&-1.150 \cdot 10^{19}. \end{array} $$

The sensitivity kernel K _m (v) is computed via the adjoint source time function

$$ f^v_k(\mathbf{x})=\frac{1}{\int\mathbf{v}^2(\mathbf{x}^r,t)\,dt} \ddot{u}_k(\mathbf{x}^r) \delta(\mathbf{x}-\mathbf{x}^r). $$

The adjoint sources for the sensitivity kernels K _m (w) and K _m (s) are dipolar sources described by the moment tensor M. The explicit moment tensor components for the computation of K _m (w) are

$$ \begin{array}{rll} M_{\theta\theta}&=&0\\ M_{\theta\phi}&=&\frac{1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_z(\mathbf{x}^r,t)\\ M_{\theta z}&=&\frac{-1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_{\phi}(\mathbf{x}^r,t)\\ M_{\phi\theta}&=&\frac{-1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_z(\mathbf{x}^r,t)\\ M_{\phi\phi}&=&0\\ M_{\phi z}&=&\frac{1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_{\theta}(\mathbf{x}^r,t)\\ M_{z \theta}&=&\frac{1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_{\phi}(\mathbf{x}^r,t)\\ M_{z \phi}&=&\frac{-1}{2\int \boldsymbol{\omega}^2(\mathbf{x}^r,t)\,dt}\omega_{\theta}(\mathbf{x}^r,t)\\ M_{zz}&=&0. \end{array} $$

The moment tensor components corresponding to K _m (s) are given by

$$ \begin{array}{rll} M_{\theta\theta}&=&\frac{-1}{\int [\text{tr}\,\mathbf{e}(\mathbf{x}^r,t)]^2\,dt}\text{tr}\,\mathbf{e}(\mathbf{x}^r,t)\\ M_{\theta\phi}&=&0\\ M_{\theta z}&=&0\\ M_{\phi\theta}&=&0\\ M_{\phi\phi}&=&\frac{-1}{\int [\text{tr}\,\mathbf{e}(\mathbf{x}^r,t)]^2\,dt}\text{tr}\,\mathbf{e}(\mathbf{x}^r,t)\\ M_{\phi z}&=&0\\ M_{z \theta}&=&0\\ M_{z \phi}&=&0\\ M_{zz}&=&\frac{-1}{\int [\text{tr}\,\mathbf{e}(\mathbf{x}^r,t)]^2\,dt}\text{tr}\,\mathbf{e}(\mathbf{x}^r,t). \end{array} $$

For a detailed derivation of the adjoint source time functions, we refer to Fichtner and Igel (2009).