Single frequency 2D acoustic full waveform inversion

Appendix

Our objective is to invert the frequency-dependent velocity function $c(\omega,\mathbf{x})$ from observed solutions to our postulated surface wave propagation equation 1 in the frequency domain, where $F(\omega)$ is an arbitrary excitation source waveform. As has been noted above, 2D waveform inversion can be directly applied to this problem, with the only exception that inversion results for individual frequencies are treated as separate values of the frequency-dependent velocity $c(\omega,\mathbf{x})$ and not applied in a multi-scale velocity refinement of the standard for the waveform inversion process. While the applicability of 2D the waveform inversion is conceptually obvious, technical challenges arise in adapting specific implementations of the waveform inversion to equation 2. Our time domain CUDA implementation of the 2D acoustic equation solves equation 2 with an excitation source of the form

$$A\ \mathrm{sin}(\omega_o t + \phi )\delta(\mathbf{x}-\mathbf{x}_s).$$ (15)

The choice of the source 15 is dictated by the considerations of our CUDA implementation. We will now show how the time-domain kernel working with sources 15 can be adapted to solve the waveform inversion problem in the frequency domain for an arbitrary waveform source $F(\omega)$ . First, Fourier-transforming 2, we can see that recovering the velocity from a solution to 2 with the source 15 is equivalent to recovering the velocity from a solution to

$$\Delta u + \frac{\omega^2 u}{c^2}=\frac{A}{2 i}\left[ e^{i \phi}\delta(\omega-\omega_0)-e^{-i\phi}\delta(\omega+\omega_0)\right].$$ (16)

By choosing $\phi=\pi/2$ we get

$$\frac{A}{2}\left[ \delta(\omega-\omega_0)+\delta(\omega+\omega_0) \right]$$

in the right-hand side of 16. Now assuming that the velocity is an even function of the frequency, and that we attempt to reconstruct the velocity from 2 with $F(\omega)=F(-\omega)$ , we can see from equation 16 that we can equivalently run 2D waveform inversion 2 for frequencies $\omega \in [0,Nyq]$ with $A=2F(\omega),\phi=\pi/2$ in 15. For an odd waveform (but still even velocity) for each $\omega$ we use $A=2F(\omega),\phi=0$ . An arbitrary waveform can be represented as the sum of an even and odd components, and equation 2 will have a linear combination of two source terms 15 in the right-hand side.

Single frequency 2D acoustic full waveform inversion