Diffraction imaging and time-migration velocity analysis using oriented velocity continuation

Common-offset oriented velocity continuation

To adopt the general theory described above to the case of common-offset 2D oriented velocity continuation, we can substitute equation (2) into (7), arriving at the equation

$$\frac{\partial \widehat{I}}{\partial v} = \left(\frac{h^2}{v... ...artial x} + v p^3 \frac{\partial \widehat{I}}{\partial p}\;.$$ (8)

which describes image propagation in the time-space-slope coordinates rather than the usual time-space coordinates. After this kind of extrapolation, regular images can be reconstructed by stacking over offset and slope.

Slope gathers, analogous to dip-angle gathers, can be extracted before stacking over slope by analyzing $\{t,p\}$ panels for different image locations $x$ and velocities $v$. Measuring flatness of diffraction events in these gathers provides a means for estimating migration velocity (Landa et al., 2008; Reshef and Landa, 2009).

For practical implementation, the formulation of oriented velocity continuation can be simplified by employing a stretch from the regular time coordinate to squared time $\sigma=t^2$ (Fomel, 2003b). According to this transformation, the Hamilton-Jacobi equation (1) becomes

$$\frac{\partial \sigma}{\partial v} = \frac{v}{2} \left(\frac{\partial \sigma}{\partial x}\right)^2 + 2\frac{h^2}{v^3} ,$$ (9)

which leads to the simpler form of the oriented equation

$$\frac{\partial \widehat{I}}{\partial v} = \left(2\frac{h^2}{... ...al \sigma} - v q \frac{\partial \widehat{I}}{\partial x}\;.$$ (10)

where $q$ corresponds to $\frac{\partial \sigma}{\partial x}$, and the image is constructed in $\{\sigma,x,q,h\}$ coordinates instead of $\{t,x,p,h\}$ coordinates. Applying the Fourier transform, we can further transform equation (10) to

$$\frac{\partial \tilde{I}}{\partial v} = i \omega \left(\fr... ... 2 \frac{h^2}{v^3}\right)\tilde{I}- i v q k \tilde{I}\;,$$ (11)

where $\tilde{I}(\omega,k,q,v,h)$ is the double Fourier transform of $\widehat{I}(\sigma,x,q,v,h)$ in $\sigma$ and $x$. Equation (11) has the analytical solution:

$$\tilde{I}(\omega,k,q,v,h) = \tilde{I}(\omega,k,q,v_0,h) e^{... ...2) + i\omega h^2 (\frac{1}{v^2} - \frac{1}{v_0^2})}\;\new{,}$$ (12)

where $v_0$ is a constant non-zero initial migration velocity.

Stacking over offset provides a slope-decomposed formulation for oriented velocity continuation:

$$\tilde{I}(\omega,k,q,v) = \sum_{h}\tilde{I}(\omega,k,q,v_0,h... ...2-v_0^2) + i\omega h^2 (\frac{1}{v^2} - \frac{1}{v_0^2})}\;.$$ (13)

This derivation suggests the following algorithm for time-domain imaging using common-offset 2D oriented velocity continuation:

1. Start with initial time migration with a constant velocity $v_0$ to generate $I(t,x,v_0,h)$.
2. Apply vertical time stretch to transform from $t$ to $\sigma$.
3. Apply Fourier transform from $\sigma$ to $\omega$.
4. Perform slope decomposition (described in the next section) to generate $\hat{I}(\omega,x,q,v_0,h)$. Note that this operation is parallel in $\omega$ and $h$.
5. Apply Fourier transform from $x$ to $k$ to generate $\tilde{I}(\omega,k,q,v_0,h)$. Note that this operation is parallel in $q$ and $h$.
6. Apply the phase-shift filter from equation (12) to generate $\tilde{I}(\omega,k,q,v,h)$ for multiple values of $v$. Note that this operation is data-intensive but parallel in $q$, $k$, and $h$.
7. Stack over offset to generate $\tilde{I}(\omega,k,q,v)$.
8. Apply inverse double Fourier transform to generate $\hat{I}(\sigma,x,q,v)$.
9. Apply inverse time stretch from $\sigma$ to $t$.
10. Stack over $q$ and extract the slice at time-migration velocity $v_m(t,x)$ to generate the final time-migrated image $I(t,x,v_m(t,x))$.

In order to estimate the velocity $v_m(t,x)$, we apply the workflow described above to diffraction imaging and modify it as follows:

• Before Step 1, we separate reflections and diffractions in the common-offset data using local plane-wave destruction (Fomel, 2002; Decker et al., 2013; Fomel et al., 2007).
• After Step 9, we analyze slope gathers $\hat{I}(t,x,q,v)$ and automatically pick the velocity $v_m(t,x)$ that corresponds to the maximum flatness (semblance) over $q$ using the picking algorithm described by Fomel (2009b). This approach follows the principle of flatness of diffraction events in slope gathers (Landa et al., 2008; Reshef and Landa, 2009; Klokov and Fomel, 2012).

The computational cost associated with determining velocity using oriented velocity continuation is linear with the number of time samples, spatial samples, offsets, velocities, and slopes considered. It is parallel in spatial samples, offsets, velocity, and slope. The cost may then be considered as $O \left( \frac{N_t N_x N_h N_v N_p}{N_c} \right)$ where $N_c$ is the number of cores available.

Diffraction imaging and time-migration velocity analysis using oriented velocity continuation