Next: CONCLUSION Up: Black et. al.: Plume Previous: AGREEMENT WITH GEOMETRICAL CONSTRUCTION

# FIRST-ORDER THEORY OF PLUME VELOCITY DEPENDENCE

The first-order theory of plume velocity dependence is developed here following steps similar to Black and Brzostowski 1993 and Paper 1.

We begin with the zero-offset time migration curve:
 (3)
Defining the velocity contrast across the dipping reflector in Figure 6 as

and the dip angle as , Black and Brzostowski 1993 use the geometry of Figure 6 to express the true diffraction curve to first order in and as:
 (4)

 (5)
where the quantity A is defined as
 (6)

fig2
Figure 6
Geometry and raypaths for the dipping layer model. The diffractor is positioned at (xm,zm). The spatial and temporal coordinates of the diffraction curve and time migrated image are given by (x0,z0) and (xk,zk), respectively.

Matching the slope of the time migration curve with the time dip D of the true diffraction curve leads to the following formulas for the position of the time migrated point (xk,tk):
 (7)

 (8)
where the time dip D can be expressed to first order in and as:
 (9)

Black and Brzostowski 1993 define the RMS migration velocity V(xm,tm) at the correct migrated position (xm,tm) to first-order in and as:
 (10)

Since we are interested in the variation of the plume operator with migration velocity, we define
 (11)
Considering small values of so that

we can expand Vmig2 retaining only first-order terms as:
 (12)
We are finally ready to derive the first-order theory of plume velocity dependence.

Time migration is carried out by inserting equations (4), (5), (10), and (12) into equations (7) and (8). Retaining only the terms which are first order in and allows us to express the first order theory for a perturbation in migration velocity by the following equations:
 (13)

 (14)

The theoretical curves given by equations (13) and  (14) are plotted in Figure 7 for the same values of migration velocity that are used in Figures 4 and  5. Notice that if equations  (13) and  (14) reduce to equations  (1) and  (2). The center frame of Figure 7 () is generated using the RMS well velocity. It is the same as the first-order curve overlaid on the migration result in Figure 2.

theory
Figure 7
First-order theory of plume velocity dependence. The same migration velocities used in Figures 4 and 5 are used here. The center frame Delta Vmig = 0.0 is generated using the RMS well velocity of 8.84 kft/s. It is the same curve that is overlaid on Figure 2.

The first-order theory does not match the tangent construction (Figure 5) and Kirchhoff migration (Figure 4) results exactly; however, we can qualitatively follow the plume formation in Figure 7 just as we did in the other two cases. The curve progresses from an undermigrated frown to an overmigrated smile as the migration velocity increases. It is evident how the left limb of the frown curls around to form the upper limb of the plume () and later, as velocity increases even more, into the right side of the smile. Similarly, the right limb of the frown becomes the lower limb of the plume, and as velocity increases, it becomes the left side of the smile. We do not observe the slight clockwise rotation in the three middle frames with this simple first-order theory.

Next: CONCLUSION Up: Black et. al.: Plume Previous: AGREEMENT WITH GEOMETRICAL CONSTRUCTION
Stanford Exploration Project
11/17/1997