Sensitivity Kernel for the Weighted Norm of the Frequency-Dependent Phase Correlation

Abstract

Full-3D waveform tomography (F3DT) is often formulated as an optimization problem, in which an objective function defined in terms of the misfit between observed and model-predicted (i.e., synthetic) waveforms is minimized by varying the earth structure model from which the synthetic waveforms are calculated. Because of the large dimension of the model space and the computational cost for solving the 3D seismic wave equation, it is often mandatory to use Newton-type local optimization algorithms; in which case, spurious local optima in the objective function can prevent the global convergence of the descent algorithm if the initial estimate of the structure model is not close enough to the global optimum. By appropriate design of the objective function, it is possible to enlarge the attraction domain of the global optimum so that Newton-type local optimization algorithms can achieve global convergence. In this article, an objective function based on a weighted L ₂ norm of the frequency-dependent phase correlation between observed and synthetic waveforms is proposed and studied, and its full-3D Fréchet kernel is constructed using the adjoint state method. The relation between the proposed objective function and the conventional frequency-dependent group-delay is analyzed and illustrated using numerical examples. The methodology has been successfully applied on a set of ambient-noise Green’s function observations collected in northern California to derive a full-3D crustal structure model.

Appendix A

The perturbation of the phase correlation, as defined in Eq. (1), can be expressed in terms of the perturbation of the synthetic waveform, as in Eq. (7). In this appendix, we provide a more detailed derivation of Eq. (7).

From Eq. (1), the perturbation of the phase correlation can be expressed as

$$ \begin{aligned} \delta \hat{C}(\omega ) & = \delta \left( {\frac{{\hat{\bar{u}}^{ * } (\omega )\hat{u}(\omega )}}{{|\hat{\bar{u}}^{*} (\omega )\hat{u}(\omega )|}}} \right) \\ & = \frac{{\hat{\bar{u}}^{ * } (\omega )}}{{|\hat{\bar{u}}^{*} (\omega )\hat{u}(\omega )|}}\delta \hat{u}(\omega ) + \hat{\bar{u}}^{ * } (\omega )\hat{u}(\omega )\delta \left\{ {\left[ {\hat{\bar{u}}^{*} (\omega )\hat{\bar{u}}(\omega )\hat{u}^{*} (\omega )\hat{u}(\omega )} \right]^{ - 1/2} } \right\}. \\ \end{aligned} $$

(36)

The second term on the right-hand-side can be expressed as,

$$ \begin{aligned} & \delta \left\{ {\left[ {\hat{\bar{u}}^{*} (\omega )\hat{\bar{u}}(\omega )\hat{u}^{*} (\omega )\hat{u}(\omega )} \right]^{ - 1/2} } \right\} \\ & \quad = - \frac{1}{2}\left[ {\hat{\bar{u}}^{*} (\omega )\hat{\bar{u}}(\omega )\hat{u}^{*} (\omega )\hat{u}(\omega )} \right]^{ - 3/2} \left[ {\hat{\bar{u}}^{*} (\omega )\hat{\bar{u}}(\omega )} \right]\delta \left[ {\hat{u}^{*} (\omega )\hat{u}(\omega )} \right] \\ & \quad = - \frac{1}{2}\left[ {\hat{\bar{u}}^{*} (\omega )\hat{\bar{u}}(\omega )} \right]^{ - 1/2} \left[ {\hat{u}^{*} (\omega )\hat{u}(\omega )} \right]^{ - 3/2} \left[ {\hat{u}^{*} (\omega )\delta \hat{u}(\omega ) + \hat{u}(\omega )\delta \hat{u}^{*} (\omega )} \right] \\ \end{aligned} . $$

(37)

Bring (37) into (36) and collect the terms containing $\delta \hat{u}(\omega )$ and the terms containing$\delta \hat{u}^{*} (\omega )$, respectively, we obtain Eq. (7).