Transmission PS Migration Theory

Converted-wave migration migrates either reflected or transmitted mode-converted events to their place of conversion at the reflecting or transmitting boundary. Compared to PP reflection migration the advantage of migrating transmitted converted waves in crosswell and VSP surveys is that boundaries roughly parallel to the wells can be imaged, as demonstrated in Figure 2.1. To demonstrate this capability Figure 2.2 shows several migration isochrons, the curves along which PP reflections (Figure 2.2a) and converted PS-waves (Figure 2.2b) will be smeared by Kirchhoff migration. Each technique is suitable for imaging boundaries tangent to the migration isochrons. For a single trace and a homogeneous medium PP reflection migration smears along isochrons that form concentric ellipses, while, PS migration smears along isochrons that form oblate ellipsoids. The region near the tip of the oblate ellipsoids (dashed lines in Figure 2.2b) is where the P-waves convert to transmitted S-waves, which is the event this thesis will exploit in imaging vertical boundaries. Note, there are no vertical tangents between the wells for the PP reflection isochrons, meaning that such boundaries are invisible to PP reflection imaging. Contrast this with the many vertical or near-vertical tangents for the transmitted PS-wave isochrons.

Similar to PP-reflection migration, the migration algorithm for transmitted PS arrivals, in a single trace of crosswell or VSP data, consists of three steps: 1.) Calculate the traveltime $\it {\tau_{sr}^{P}}$ for P-waves to propagate from the source to an image point using the P-wave velocity, 2.) Calculate the traveltime $\it {\tau_{rg}^{S}}$ for S-waves to propagate from an image point to the receiver using the S-wave velocity, 3.) Smear the amplitude from the trace at time equal to $\it {\tau_{sr}^{P}}$ + $\it {\tau_{rg}^{S}}$ to the image point denoted by $\bf {r}$:

$$m(\it {\bf {r}}) = \int h(\bf {s,r,g}) \it {s}(z_{g}, \tau_{sr}^{P} + \tau_{rg}^{S})dz_{g} ,$$ (1)

where $m(\it {\bf {r}})$ is the migration image at $\bf {r}$, $s(z_{g},t)$ is the seismic trace recorded at the depth denoted by $z_g$ and

is the threshold filter that distinguishes transmitted rays from reflected rays. Migrating the data in this manner requires determining both the P- and S-wave velocity models. Figure 2.2b shows the isochrons for $\tau_{sr}^{P} + \tau_{rg}^{S}$, the imaging condition for both PS reflection (solid lines) and transmission PS-waves (dashed lines). If $\tau_{rg}^{S} \rightarrow \tau_{rg}^{P}$ then equation 2.1 is appropriate for PP reflection migration, with the associated isochrons shown in the Figure 2.2a.

The importance of Figure 2.2 is that the tangents to the transmission PS-wave isochrons are vertical or nearly vertical, while those for PS reflected waves or PP reflections have less dip between the wells. This means that PS transmission migration is appropriate for imaging steep dips compared to the limitation of imaging only shallower dips by PP and PS reflection migration.

Figure 2.1: Division of energy from an incident P-wave. The transmitted arrivals are extant in the receiver well, no reflections can be recorded.

Figure 2.2: Comparison of (a) PP reflection migration isochrons and (b) PS converted-wave isochrons in seconds. The PP isochrons form ellipses with the source and receiver as foci, while the PS converted-wave isochrons form oblate ellipsoid (egg shaped) isochrons. The tangents to the dashed portions of the isochrons denote the interfaces that can give rise to transmitted PS arrivals; solid lines indicate those for reflected waves. A transmission ray is shown in green and reflections in red.