COMPARISON TO FIRST-ORDER THEORY

Black and Brzostowski 1993 derive simple expressions for the positioning error of time migration. We have used some of their intermediate results in equations(A-4)-(A-6) to calculate the theoretical plume response. In this section we review Black and Brzostowski's first-order theory of plumes and compare it to our previous results.

The geometry used for the derivation of the theoretical result is illustrated in Figure 10. Starting at the point diffractor location (x_m,z_m), a normal ray is traced to the surface where it emerges at x₀. Time migration then translates us to the point x_k. Black and Brzostowski go on from x_k along the image ray to the point Q. The difference between Q and (x_m,z_m) is the error after the image ray correction. Since we are concerned with the response of time migration to lateral velocity variation, we are interested in the time migrated position (x_k,t_k).

 

fig2
Figure 10 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.

The velocity gradient which gives rise to the plumes is due to the velocity contrast across the dipping layer and the dip angle $\theta_1$.To simplify the analysis velocity contrast is defined as:

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

The analysis is then carried out to first order in $\gamma$ and $\sin\theta_1$.This yields the surface location x₀ and travel time t₀:  

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

By varying $\theta$, equations(4) and (5) define a diffraction curve like that given in Figure 2. The first term of the equations yields the constant velocity hyperbola, the second term contains the effects of vertical velocity gradients, the third and fourth terms contain the effects of lateral velocity changes.

Time migration is carried out by performing the common-tangent construction. Matching the slope of the hyperbola in equation (1) to the time dip D of the event leads to equations(2) and (3). Expanding these to first order in $\gamma$ and $\sin\theta_1$ gives the spatial and temporal position of the event after time migration:Black and Brzostowski (1993).  

$$x_k \approx x_m + A(x_m,z_m)(1+3\tan^2\theta)$$ (7)

 

$$\frac{v_2t_k}{2} \approx \frac{v_2t_m}{2} - 2A(x_m,z_m)\tan^3\theta$$ (8)

We can vary $\theta$ in equations (7) and (8) to get a first order analytic representation for the plume. This is done for propagation angles of $\pm 60^{\circ}$ in Figure 11.

The theoretical result is well positioned relative to the synthetic result. The synthetic plume is skewed upwards whereas the theoretical plume is symmetric about a horizontal axis drawn through its apex. The lower limb of the theoretical plume overlies the synthetic result very closely, however the upper limbs do not match. Both the theoretical and synthetic plume open in the same direction and display the same type of limb curvature: convex upward on the upper limb and convex downward on the lower limb.

Although the first order theory contains some of the features of the plume, it is incomplete. It is clear that a higher-order theory is needed to explain all the features of time migration plumes.

 

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Figure 11 Comparison of the analytic expression for the plume to the Kirchhoff migration result. The migration result is identical to Figure 7, but here we overlay the theoretical result derived by Black and Brzostowski.