Prestack time migration/inversion

In areas with weak lateral velocity variation, it has long been known that time migration followed by time-to-depth conversion is a good approximation for full prestack depth migration in a kinematic imaging sense, and is at least an order of magnitude more efficient. I claim the same is true for migration/inversion in a dynamic sense, where time migration approximations are used for the amplitude and ray dynamic variables as well as the standard kinematic approximations. Not everyone agrees (e.g., Berkhout, 1992, EAEG discussion period following the presentation by Lumley and Biondi, 1992).

In this study, I show the results of efficient time migration/inversion in a velocity model of weak lateral variation. I factor the WKBJ amplitudes A into components of source/receiver directivity and strength, geometric divergence, transmission loss, and intrinsic Q attenuation. In this example I implement WKBJ amplitude compensation for only source/receiver directivity and geometric divergence, assuming attenuation due to transmission and Q is negligible here. I represent the air-gun source with a $\cos^2\theta_s$ directivity to account for the free-surface and array effect. I represent the receiver directivity as $\cos^2\theta_r$ to account for the free-surface and hydrophone group effect. These directivity patterns are input parameters to be freely chosen by the user depending on source and receiver type, and whether marine or land acquisition.

I estimate the ray angles as follows. In a constant velocity earth, the straight raypath angle is given simply as:

 

$$\cos\theta_0 = \frac{\tau}{t} \;,$$ (10)

where $\tau$ is vertical migration pseudodepth, and t is migration hyperbolic traveltime. For the case of an rms time migration velocity field $v(x,y,\tau)$, I make the approximation that the take-off angle at the source or receiver surface is given by:

 

$$\cos\theta_{s,r} \approx \left[ \cos\theta_0 \right]^{+q} \;,$$ (11)

and the incident angle at the subsurface image point is:

 

$$\cos\theta_i \approx \left[ \cos\theta_0\right]^{-q} \;,$$ (12)

where the exponent q is defined as:

 

$$q = \frac{v(x,y,\tau=0)}{v(x,y,\tau)} \;.$$ (13)

For the divergence compensation, I use the standard geometric spreading approximation in an rms velocity field $v(x,y,\tau)$:

 

$$A_{div} \approx \frac{1}{v^2 t} \;.$$ (14)

These approximations are then used in the prestack migration/inversion to compute the gradient and Hessian terms of equations (4) and (5), resulting in final prestack time estimates of $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ and $\Theta({\bf x};{\bf x}_h)$. These are then easily converted to depth by a simple vertical stretch algorithm in weakly lateral variable media, or by reverse-time extrapolation through an interval velocity model with strong time migration lateral positioning errors. Of course, in strong v(x) media, time migration may not be appropriate at all even in a kinematic sense. However, many reservoir and exploration situations satisfy the time migration approximations well.