Fast 3D velocity updates using the pre-stack exploding reflector model

PERM using a common-azimuth migrated image

In the way PERM is formulated, there is no restriction on the dimensionality of the pre-stack image used as the initial condition for the modeling, which means that if the original data have sufficient cross-line offsets as in the acquisition geometries with wide range of azimuths (Kapoor et al., 2007; Regone, 2007), the initial conditions are a five-dimensional hypercube on $\bf x$ , $h_x$ and $h_y$ . To synthesize PERM data starting with the five-dimensional initial conditions such that no crosstalk is generated during migration, the minimum number of modeling experiments is $2n_{h_{x}}n_{h_{y}}$ , where $n_{h_{x}}$ and $n_{h_{y}}$ are the number of subsurface offsets in the $x$ and $y$ directions. Considering usual parameters, the number of modeling experiments may be as low as several hundreds. This data reduction is very substantial if we compare, for instance, with data reduction achieved by 3D-plane-wave decomposition. Using plane waves, to obtain artifact-free SODCIGs due to the lack of illumination from some propagation directions we need to migrate few thousands of plane waves. This means that 3D-PERM data size can be one order of magnitude smaller than 3D-plane wave data.

Despite the recent good migration results obtained in geologically complex areas using wide-azimuth data, narrow-azimuth acquisition is still the industry standard. Narrow-azimuth data can be efficiently imaged by common-azimuth wave-equation migration (CAM) (Biondi and Palacharla, 1996). By assuming zero cross-line offset in contrast with the full-azimuth migration, instead of a five-dimensional hypercube, CAM images are four-dimensional hypercubes in $\bf x$ and $h_x$ . Because of the lower dimensionality of CAM images, when using them as the initial conditions to synthesize PERM data, the SODCIGs in the cross-line direction can be sampled continuously, as depicted in Figure 1b. Contrast this case with the five-dimensional initial conditions for the full azimuth case of Figure 1a. The continuous sampling of SODCIGs in the cross-line direction yields one more order of magnitude of data reduction. Therefore, under the common-azimuth assumption, 3D-PERM data size can be two orders of magnitude smaller than 3D-plane wave data.

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Figure 1. The initial conditions for synthesizing PERM data from CAM images can be specified as in b) because no pre-stack information exists in the cross-line direction, in contrast with the full azimuth situation in a).

To illustrate the validity of the above assumptions, a split-spread data with maximum offset of 1587.5 m was computed using 3D-Born modeling on a 30$^{\circ}$ dipping reflector with 45$^{\circ}$ azimuth with respect to the acquisition direction. There are 96 in-lines and cross-lines spaced 25 m apart. The offset interval is 25 m. The velocity used in the modeling is the 1D function $v(z) = (1500 + 0.5z)$ m/s. The Born data are input to CAM with a 5$\%$ slower velocity. Migration results can be seen in Figures 2a and 2b for SODCIGs positioned at $(x = 750$ m, $y = 600$ m$)$ and $(x = 750$ m, $y = 1000$ m$)$ , respectively. The panel on the left is the SODCIG, which contains 21 subsurface offsets ranging from $-250$ m to $250$ m. The panel in the middle is the in-line at zero subsurface offset, with $y = 600$ m (Figure 2a) and $y = 1000$ m (Figure 2b). The panel on the right is the cross-line at zero subsurface offset, with $x = 750$ m.

In the common-azimuth regime, the computation of the dip-independent initial conditions is performed by simply rotating the SODCIGs in the in-line direction, since no cross-line offset is computed in migration.

PERM source and receiver wavefields were modeled using as the initial conditions combined SODCIGs from the CAM image (Figure 2) with continuous sampling along the cross-line direction and sampling period of 48 in the in-line direction. This period is sufficient to avoid crosstalk during the areal-shot migration, given that the number of subsurface-offsets of the pre-stack image is 21. One synthesized 3D receiver wavefield is shown in Figure 3. The left panel is the in-line at $y = 1200$ m, the right panel is the cross-line at $x = 1400$ m, and the top panel is the time-slice at $t = 0.5$ s.

The 3D migration of the 48 areal shots with the velocity underestimated by 5$\%$ is shown in Figures 4a and 4b for SODCIGs positioned at $(x = 750$ m, $y = 600$ m$)$ and $(x = 750$ m, $y = 1000$ m$)$ , respectively. The kinematics of the SODCIGs computed with PERM wavefields matches those of the SODCIGs computed with CAM. This enables the use of 3D PERM wavefields computed from CAM images in optimization of migration velocity, as will be shown next for a 3D survey from the North Sea.

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Figure 2. CAM of the 3D-Born data. Middle: in-line at zero-subsurface offset, and $y = 600$ m (Figure 2a) and $y = 1000$ m (Figure 2b). Right: cross-line at zero-subsurface offset, and $x = 750$ m. Left: SODCIGs.

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Figure 3. 3D-PERM receiver wavefield. The left panel is the in-line at $y = 1200$ m, the right panel is the cross-line at $x = 1400$ m, and the top panel is the time-slice at $t = 0.5$ s.

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Figure 4. 3D-areal-shot migration of PERM data. Middle: in-line at zero-subsurface offset, and $y = 600$ m (Figure 4a) and $y = 1000$ m (Figure 4b). Right: cross-line at zero-subsurface offset, and $x = 750$ m. Left: SODCIGs.

Fast 3D velocity updates using the pre-stack exploding reflector model