RMO velocity analysis

The computation of traveltime errors along the reflectors picked in the poststack image requires RMO velocity analysis. Measuring RMO implies relative movement of the reflection events in the common reflection point (CRP) gathers according to the different travel paths that each event follows.

Conventionally, RMO velocity analysis is performed in the common surface location (CSL) gather after prestack migration. The RMO velocity obtained from the CSL gather after prestack migration does not reflect the correct residual movement of CRP images because CRP images are not located in a CSL gather when a reflector has a dip. Therefore, we need to apply residual dip moveout (RDMO) before RMO velocity analysis of CSL gathers Etgen (1990) or else apply a dip-dependent RMO velocity analysis that is very difficult to implement or is not practical Zhang (1990).

Even though surface-oriented planewave synthesis imaging also shows residual moveout in CSL gathers, it is not easy to quantify in terms of residual velocity. This is because of the lack of the information about the implied ray trajectories. Since both the source and the received wave field used in planewave synthesis imaging are plane waves, we do not know which source and receiver locations correspond to an image. Therefore, we are forced to update the model qualitatively using the curvature of the RMO curves Whitmore and Garing (1993).

This drawback can be overcome by using reflector-oriented planewave synthesis imaging. With this method, the local incidence angle on top of the reflector is predefined. Thus, the image obtained can be interpreted as a wave field that follows the ray trajectory extended from the local incidence angle to the surface (Figure ).

 

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Figure 6 Ray trajectory extended using the local incidence angel on top of a reflection point.

For example, I applied reflector-dependent planewave synthesis imaging for each reflector picked in the previous section (Figure ). Figures  through  show some of the resulting CSL gathers. In contrast to the CSL gathers obtained from surface-oriented PWS imaging, each reflector image in the corresponding reflector-oriented PWS imaging shows a symmetric RMO pattern with respect to the normal incidence angle. The reason of the symmetric RMO pattern is that a ray that follows the shortest path from reflector to surface would have normal incidence angle to the reflector.

To quantify the residual moveout shown in the reflector-oriented PWS imaging, we can use the accurate dip-dependent RMO equation derived by Zhang 1990. However, the residual moveout equation is derived under the assumption that the medium above the reflector has constant velocity and the ray trajectories are straight lines, which is not true when we are dealing with a variable-velocity medium. Therefore, for the convenience of velocity analysis, I derived a simplified RMO equation for a dipping event that resembles the RMO equation for a flat reflector.

 

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Figure 7 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at the surface.

 

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Figure 8 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at reflector number 2.

 

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Figure 9 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at reflector number 3.

 

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Figure 10 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at reflector number 4.

 

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Figure 11 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at reflector number 5.

Let us consider a dipping reflector as shown in Figure . The depth to a reflection point is z, the average slowness of the medium to the reflector is $\bar w$,and t is the recorded traveltime. If a planewave source has the incidence angle $\alpha$ to the reflector, the ray path from and to the reflector will be a straight line, and the presummed half offset h of the ray path can be approximately calculated from the incidence angle $\alpha$ of the planewave, the reflector depth z, and the dip of the reflector $\theta$, as follows :

$$h = {l \tan{\alpha}},$$ (27)

where $l = z/\cos(\theta)$ and represents the normal distance from reflector to the surface.

If we assume n=m (see Figure ), the traveltime is given by  

$$t = 2\sqrt{h^2+l^2} \cdot \bar w.$$ (28)

After migration with the slowness of the medium, the image under a surface location will be at the same depth z. If we migrate with an average slowness $\bar w_m$instead of the slowness of the medium itself, the image under a surface location will be at depth z_m. In this case, the traveltime is given by  

$$t = 2\sqrt{h^2+l_m^2}\cdot \bar w_m,$$ (29)

where $l_m = z_m/\cos(\theta)$.

 

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Figure 12 For a reflector of dipping angle $\theta$, a plane wave satisfies Snell's law.

Note that the traveltime, t, is the same in equations () and () because it is an observed quantity. Eliminating t from equations () and (), we obtain  

$$z_m = \sqrt{\gamma^2 z^2+(\gamma^2 -1)h^2\cos^2(\theta)},$$ (30)

where

$$\gamma = {\bar w \over \bar w_m}.$$ (31)

Equation () gives a relation between the apparent depth, z_m, and the actual depth, z. They are linked through the parameter $\gamma$.Note that they are equal regardless of the offset or incidence angle of the planewave when the slowness used in migration is equal to the slowness of the medium ($\gamma = 1$). This is the essence of the velocity analysis principle in the CSL gather; the image in a CLS gather is aligned horizontally if the velocity model is correct. When $\gamma$ is not equal to 1, there is both a moveout as a function of offset and a shift at zero offset.

At each depth point along the reflector, RMO is defined by the parameter $\gamma$ in equation (). The data is then summed along this curved trajectory. The summation is done for a range of $\gamma$,and the sum is largest for the value of $\gamma$that matches the curvature. Because some signals may be weaker than others, the sum is normalized. This normalized summation is similar to the normalized summation along a hyperbola using the NMO equation, commonly known as semblance Taner and Koehler (1969). If the data in a CSL gather is $p(z_m,\alpha)$,then searching for curvature produces the semblance panel $g(z_m,\gamma)$, defined as

 

$$g(z_m,\gamma) = {[ \sum_\alpha p( z=\sqrt{z_m^2 + (\gamma^2-... ...qrt{z_m^2 + (\gamma^2-1)h(\alpha)^2 \cos^2\theta },\alpha)]^2}.$$ (32)

Figure  shows that the semblance velocity analysis panel for the reflectors picked with $\gamma$ values, which are the maximum semblance values for each CSL gather, are on top of semblance panel. As Figure  shows, some portions of the reflectors have much lower semblance values than other portions.

In order to avoid possible bias in inversion from this erroneous information, I excluded reflection locations that have semblance values below 0.4. The rest of the reflectors to be used in the inversion are shown in Figure .

 

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Figure 13 Semblance stack panels for the picked reflectors. The $\gamma$ values that correspond to the maximum semblance value are overlaid on top the semblance panel with lines.

 

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Figure 14 The portions of the picked reflectors to be used for determining traveltime error. The reflector locations that have below .4 in semblance value were excluded in traveltime error calculation.