Computational analysis of extended full waveform inversion

Model-Space Extensions

Biondi and Almomin (2012) introduced an extension to full waveform inversion that can mitigate the cycle-skipping problem and allow for a larger error in the initial model. This is achieved by extending the velocity model along the subsurface offset and then solving the corresponding extended wave equation. The modeled data becomes:

$$d(\omega, \mathbf x_r, \mathbf x_s; v(\mathbf x, \mathbf x_h)) = ... ...hbf x, \mathbf x_s; v(\mathbf x, \mathbf x_h)) \delta(\mathbf x_r - \mathbf x),$$ (7)

and the extended Green's function satisfies:

$$\left( v^2(\mathbf x, \mathbf x_h) *^{-1} \omega^2 + \nabla^2 \ri... ...f x, \mathbf x_s; v(\mathbf x, \mathbf x_h)) = \delta(\mathbf x_s - \mathbf x),$$ (8)

where $\mathbf x_h$ is subsurface offset and $*^{-1}$ denotes a deconvolution operator over subsurface offset. This extended wave equation convolves each time slice by all subsurface offsets of velocity. The cost of extended forward modeling becomes:

$$C_{\rm EFWI-F} = N_x N_y N_z N_{\rm source} (N_{tp} C_{\rm FDTD} + N_{tp} N_{hx} N_{hy}),$$ (9)

where $N_{hx}$ and $N_{hy}$ are number of subsurface offsets along the $x$ and $y$ axes, respectively. By linearizing equation 7 over the velocity squared, we can compute the adjoint as:

$$\Delta v^2(\mathbf x, \mathbf x_h) = \sum_{\omega, \mathbf x_r, \mathbf x_s} \nabla^2 f(\omega, \mathbf x_s) G(\omega, \mathbf x - \mathbf h, \mathbf x_s; v(\mathbf x, \mathbf x_h))$$

$$G(\omega, \mathbf x + \mathbf h, \mathbf x_r; v(\mathbf x, \mathbf x_h)) \Delta d^*(\omega, \mathbf x_r, \mathbf x_s; v(\mathbf x, \mathbf x_h)).$$ (10)

Therefore, the cost of computing the adjoint of EFWI can be written as

$$C_{\rm EFWI-A} = N_x N_y N_z N_{\rm source} (2 \times N_{tp} C_{\rm FDTD} + 2 \times N_{tp} N_{hx} N_{hy} + N_{ti} N_{hx} N_{hy})$$ (11)

and the total cost of one iteration of EFWI becomes

$$C_{\rm EFWI} = N_x N_y N_z N_{\rm source} (6 \times N_{tp} C_{\rm FDTD} + 6 \times N_{tp} N_{hx} N_{hy} + N_{ti} N_{hx} N_{hy}).$$ (12)

We can see that the computational cost becomes extremely high when we include the subsurface offsets in the velocity model. One way to reduce the cost is presented in Biondi (2012) where the velocity model is extended over time instead of horizontal offset. In that case, the cost becomes a function of one time-lag parameter instead of two horizontal lags in 3D:

$$C_{\rm Time EFWI} = N_x N_y N_z N_{\rm source} (6 \times N_{tp} C_{\rm FDTD} + 6 \times N_{tp} N_{\tau} + N_{ti} N_{\tau}),$$ (13)

where $N_{\tau}$ is the number of time lags. The computational disadvantage is that several time slices need to be held in memory for each time instead of the conventional two slices.

Computational analysis of extended full waveform inversion