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:  

$${t_0^2} = {t_k^2} + 4{\left[{x_0 - x_k \over V_{mig}}\right]}^2$$ (3)

Defining the velocity contrast across the dipping reflector in Figure 6 as

$$\gamma = \frac{v_2-v_1}{v_2},$$

and the dip angle as $\theta_1$, Black and Brzostowski 1993 use the geometry of Figure 6 to express the true diffraction curve to first order in $\gamma$ and $\sin\theta_1$ as:  

$$x_0-x_m \approx -z_m\tan\theta + \gamma d_1 \frac{\sin\theta... ...m \frac{1}{\cos^4\theta} + A\frac{1+\sin^2\theta}{\cos^4\theta}$$ (4)

 

$$\frac{v_2t_0}{2} \approx \frac{z_m}{\cos\theta} + \gamma d_... ...sin\theta}{\cos^4\theta} - 2A\frac{\sin^3\theta}{\cos^4\theta},$$ (5)

where the quantity A is defined as

$$A(x_m,z_m)=d_2\gamma \sin\theta_1.$$ (6)

 

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Figure 6 Geometry and raypaths for the dipping layer model. The diffractor is positioned at (x_m,z_m). The spatial and temporal coordinates of the diffraction curve and time migrated image are given by (x₀,z₀) and (x_k,z_k), 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 (x_k,t_k):  

$$x_k = x_0 - {\left[{V_{mig}D \over 2}\right] \left[{V_{mig}t_0 \over 2}\right]} ,$$ (7)

 

$$t_k = t_0 \sqrt {1 - {\left[{V_{mig}D \over 2}\right]}^2},$$ (8)

where the time dip D can be expressed to first order in $\gamma$ and $\sin\theta_1$ as:  

$$\frac{v_2D}{2} \approx -\sin\theta -\gamma\frac{\sin\theta_1}{\cos\theta}.$$ (9)

Black and Brzostowski 1993 define the RMS migration velocity V(x_m,t_m) at the correct migrated position (x_m,t_m) to first-order in $\gamma$ and $\sin\theta_1$ as:  

$$\left[\frac{V(x_m,t_m)}{v_2}\right]^2 \approx 1-2\gamma\frac{d_1}{z_m}.$$ (10)

Since we are interested in the variation of the plume operator with migration velocity, we define  

$$V_{mig}= V(x_m,t_m) + \triangle V_{mig} = V(x_m,t_m)\left(1+\frac{\triangle V_{mig}} {V(x_m,t_m)}\right).$$ (11)

Considering small values of $\triangle V_{mig}$ so that

$$\frac{\triangle V_{mig}}{V(x_m,t_m)} \leq \gamma,$$

we can expand V_mig² retaining only first-order terms as:  

$$V_{mig}^2 \approx V(x_m,t_m)^2\left(1+2\frac{\triangle V_{mig}}{V(x_m,t_m)}\right).$$ (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 $\gamma$ and $\sin\theta_1$allows us to express the first order theory for a perturbation in migration velocity by the following equations:  

$$x_k = x_m + A( 1 + 3\tan^2\theta ) + {{2\triangle V_{mig}}\over{V_{mig}}} z_m \tan\theta$$ (13)

 

$${{v_2 t_k}\over{2}} = {{v_2 t_m}\over{2}} - 2A\tan^3\theta - {{\triangle V_{mig}}\over{V_{mig}}} z_m \tan^2\theta$$ (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 $\triangle V_{mig} = 0$ equations  (13) and  (14) reduce to equations  (1) and  (2). The center frame of Figure 7 ($\triangle V_{mig} = 0$) is generated using the RMS well velocity. It is the same as the first-order curve overlaid on the migration result in Figure 2.

 

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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 ($\triangle V_{mig} = 0$) 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.