Wavefield extrapolation in pseudodepth domain

Impulse responses

The $\tau$ domain wave equations in the previous section are derived for 3D models. For simplicity, the examples in this paper is for D models. Since the change in the vertical axis acts on a single vertical velocity profile at a time, the conclusions in this section can be extended to 3D case. To test the accuracy of the $\tau$ domain wavefield extrapolation, we look at impulse responses of the $\tau$ domain migration operators and compare them with those obtained from the Cartesian domain extrapolations. A synthetic zero-offset section with three spikes are illustrated in Figure 4. Migration images obtained from this section are superpositions of the Green's functions due to a point source located at the center on the surface.

spike Figure 4. Zero-offset section with three impulse events equally spaced in time. The source has peak frequency of $20$ Hz. The lag between impulses is $0.5\mathrm {sec}$ .

Our first example is a lens velocity model, shown in Figure 5(a). The background velocity is $v = 2100 + .025z + .14x \; \mathrm{m/sec}$ and contains a negative anomaly of $-750 \; \mathrm{m/sec}$ . We use the same velocity to obtain vertical time $\tau$ , i.e. $v_m=v$ in Equation 2. The $\tau$ mesh is overlaid on the velocity model, a prominent ``pull-down'' near the bottom of the model is due to the slow velocity lens. By the same analogy, a ``push-up'' will appear in the $\tau$ domain beneath a positive velocity anomaly, for example a salt body. By applying the change of variable in Equation 2, the velocity is interpolated to the $\tau$ domain and plotted in Figure 5(b). The $\tau$ axis is discretized by $401$ samples to speedup extrapolation and honor the aliasing condition. The zero-offset section in Figure 4 is migrated using Equation 10 by extrapolating backward in time and applying zero-time imaging condition. The resulting image is shown in Figure 5(c). In the $\tau$ domain, the extrapolation is done by solving Equation 15. Following the same imaging condition, the migrated image is shown in Figure 5(d). This image is then interpolated back to the Cartesian domain using equation 4, and compared with Figure 5(c). The error shown in Figure 5(f) is due to the linear interpolation between Cartesian and $\tau$ meshes, and it is relatively small.

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Figure 5. A lens velocity model in (a) Cartesian and (b) $\tau$ domains overlaid with $\tau$ mesh. Migration images are shown for (c) Cartesian and (d) $\tau$ domains. (e) is image (d) interpolated to the Cartesian domain. (f) is the relative error between (c) and (e). The model is discretized with $501$ samples in depth and $401$ samples in vertical time.

The next examples demonstrate extrapolation in complex models. In Figure 6(a), we show a section of the isotropic Marmousi velocity model. Since the vertical time derived from this velocity has very ragged curvatures that may pose stability difficulties for the $\tau$ domain extrapolation, in Equation 2 we use instead a smoothed version of the Marmousi velocity as $v_m$ to compute $\tau$ . Interpolation of the Marmousi velocity on to the $\tau$ mesh is shown in Figure 6(b). Impulse responses are obtained by solving Equations 10 (Figure 6(c)) and 15 (Figure 6(d)). The error is plotted in Figure 6(f) and it is relatively small.

margC,margT,mariC,mariT,mariB,mariE
Figure 6. A portion of the Marmousi velocity in (a) Cartesian and (b) $\tau$ domains overlaid with the $\tau$ mesh. Migration images are shown for (c) Cartesian and (d) $\tau$ domains. (e) is image (d) interpolated to the Cartesian domain. (f) is the relative error between (c) and (e). The model is discretized with $751$ samples in depth and $501$ samples vertical time. The mapping velocity $v_m$ is smoothed from true velocity by using $8$ -point triangle filter.

Extrapolation of anisotropic wavefields in the $\tau$ coordinates system is also feasible. The propagation of quasi-acoustic P waves is characterized by three parameters: vertical $P$ -wave velocity $v_v$ , NMO velocity $v$ and the anellipticity $\eta$ . For example, the anisotropic parameters of the SEG/Hess model are shown in Figures 7(a) to 7(c).

hesm1,hesm2,hesm3
Figure 7. A portion of the SEG/Hess anisotropic velocity model. (a) Vertical velocity (b) NMO velocity (c) $\eta$ .

Similar to isotropic complex models, to alleviate the distortion of the $\tau$ mesh due to the strong velocity variation of the salt body, the $\tau$ mesh is constructed using a smooth background velocity $v_m$ . The resulting $\tau$ coordinate system is overlaid on vertical velocities in the Cartesian and $\tau$ domains, shown in Figure 8(a) and 8(b). The Cartesian domain impulse response is computed from Equation 20, Figure 8(c) shows the horizontal stress field $p_H$ . In the $\tau$ domain, the impulse response is obtained from equation 21, and the resulting $p_H$ field is shown in Figure 8(d). As expected, the wavefield beneath the salt is ``pushed up'' due to positive velocity anomaly at salt dome. The error in the migration image obtained in $\tau$ domain is plotted in Figure 8(f).

hesgC,hesgT,hesiC,hesiT,hesiB,hesiE
Figure 8. The vertical velocity in (a) the Cartesian and (b) the $\tau$ domains overlaid with $\tau$ mesh. Migration images are shown for (c) Cartesian and (d) $\tau$ domains. (e) is image (d) interpolated to the Cartesian domain. (f) is the relative error between (c) and (e). The model is discretized with $601$ samples in depth and $501$ samples vertical time. The mapping velocity $v_m$ is smoothed from true velocity by using $8$ -point triangle filter.

Table 1 summarizes the numerical cost of wavefield extrapolation in Cartesian and $\tau$ domains. For isotropic extrapolations, elapsed time is shorter in $\tau$ domain than in Cartesian domain, the percentage cut is close to the reduction of vertical sampling of the wavefield. For anisotropic extrapolations, the efficiency improvement is less significant, due to the increased number of derivatives on the right-hand side of equations 21 as compared to Cartesian extrapolator 20.

Model	$n_c$	$n_\tau$	reduction	$t_c$	$t_\tau$	speedup
Lens (Figure 5(a))	401	251	37.5%	18	12	33.3%
Marmousi (Figure 6(a))	751	501	33.3%	277	193	30.3%
SEG/Hess (Figure 8(a))	601	501	16.6%	102	98	3.9%

Table 1. Cost of wavefield extrapolation in Cartesian (c) and $\tau$ ($\tau$ ) domain, showing elapsed time $t$ in seconds and number of vertical samples $n$ . Reduction of samples are computed by $(n_c - n_\tau ) / n_c$ . Speedup is estimated by $(t_c - t_\tau ) / tc$ .

In addition to the reduced computational cost, $\tau$ anisotropic extrapolation also features attenuated shear wave artifacts. Figure 9 shows the impulse responses using a homogeneous anisotropic velocity model. The shear wave artifact is significant in the Cartesian domain, shown on the left. In the $\tau$ domain, the artifact is attenuated with the coarser vertical sampling, as shown on the right. Coarser sampling enhance numerical dispersion of the shear wave, which tends to spread the energy across the domain, thus attenuates the shear wave artifact.

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Figure 9. Anisotropic impulse response obtained in Cartesian (Left) and $\tau$ (Middle and Right) domains, showing attenuation of the shear wave artefact. In the Middle figure, the $\tau$ axis is sampled at $\Delta\tau = 2.5 \mathrm{msec}$ with $401$ samples. In the Right figure, the $\tau$ axis is sampled at $\Delta\tau = 3.0 \mathrm{msec}$ with $301$ samples. The anisotropic model has vertical velocity $2000 \mathrm{m/sec}$ , NMO velocity $1643 \mathrm{m/sec}$ and anellipticity $\eta =0.1$ .

Wavefield extrapolation in pseudodepth domain