KIRCHHOFF METHODS 135

4.5. KIRCHHOFF METHODS 135

source x s

x receiver θ

scatterpoint S z

image source

Figure 4.65: The geometry for Kirchhoff migration is shown. The integration surface is S 0 +S z +S ∞ and it is argued that only S 0 contributes meaningfully to the estimation of

the backscattered field at vx 0 .

S ∞ , though it may contribute, can never be realized due to finite aperture limitations and its neglect may introduce unavoidable artifacts. With these considerations, equation (4.98) becomes

−1 # 1 ∂r

ψ(#x 0 , t) =

dsurf (4.99)

∂z t+r/v 0 v 0 r ∂z ∂t t+r/v 0

where the signs on the terms arise because #n is the outward normal and z is increasing downward so that ∂ n = −∂ z .

Now, ∂ z ψ must be evaluated. Figure 4.65 shows the source wavefield being scattered from the reflector at #x 0 which is called the scatterpoint. A simple model for ψ is that it is approximately the wavefield from a point source, placed at the image source location, that passes through the scatterpoint to the receiver. This can be expressed as

1 [A] t−r/v

ψ(#x, t) ∼ A(t − r/v) =

where A(t) is the source waveform at the scatterpoint. Using the chain rule gives

∂ψ # ∂r ∂ψ ∂r −1 [A]

− t−r/v .

∂z vr ∂t t−r/v

If the second term is neglected (a near-field term), this becomes

∂z vr ∂t t−r/v

∂z v ∂t

When this is substituted into equation 4.99, the two terms in square brackets become similar both involving the time derivative of the advanced wavefield. These will combine if v 0 is now taken to be the same as v. Thus

2 ∂r

ψ(#x 0 , t) =

dsurf

S 0 vr ∂z ∂t t+r/v

136 CHAPTER 4. ELEMENTARY MIGRATION METHODS Finally, consider ∂ z r. Since r = (x − x 0 ) 2 + (y − y 0 ) 2 + (z − z 0 ) 2 this can be written

∂r

(x − x 0 ) 2 + (y − y 0 ) 2 + (z − z 0 ) 2 = z = cos θ (4.104)

∂z

∂z

where θ is the vertical angle between the receiver location and ray to the scatterpoint. With this, the final formula for the scattered wavefield just above the reflector is

2 cos θ

ψ(#x 0 , t) =

∂t t+r/v

Equation (4.105) is not yet a migration equation. As mentioned, it provides an estimate of the scattered wavefield just above the scatterpoint. Thus it is a form of wavefield extrapolation though it is direct, not recursive. A migration equation must purport to estimate reflectivity, not just the scattered wavefield and for this purpose a model relating the wavefield to the reflectivity is required. The simplest such model is the exploding reflector model (section 4.2.6) which asserts that the reflectivity is identical to the downward continued scattered wavefield at t = 0 provided that the downward continuation is done with ˆ v = v/2. Thus, an ERM migration equation follows immediately from equation (4.105) as

∂t 2r/v

This result, derived by many authors including Schneider (1978) and Scales (1995), expresses mi- gration by summation along hyperbolic travelpaths through the input data space. The hyperbolic summation is somewhat hidden by the notation but is indicated by [∂ t ψ] 2r/v . Recall that this no- tation means that the expression in square brackets is to be evaluated at the time indicated by the subscript. That is, as ∂ t ψ(#x, t) is integrated over the z = 0 plane, only those specific traveltimes values are selected that obey

which is the equation of a zero-offset diffraction hyperbola. When squared, this result is a three dimensional version of equation (4.36).

In addition to diffraction summation, equation (4.106) requires that the data be scaled by

4 cos θ/(vr) and that the time derivative be taken before summation. These additional details were not indicated by the simple geometric theory of section 4.2.4 and are major benefits of Kirch- hoff theory. It is these sort of corrections that are necessary to move towards the goal of creating bandlimited reflectivity. The same correction procedures are contained implicitly in f -k migration.

The considerations taken in deriving equation (4.106) suggest why ERM migration does not achieve correct amplitudes. As mentioned following equation (4.105), a model linking the backscat- tered wavefield to reflectivity was required. A more physical model will illustrate the shortcomings of the exploding reflector model. Such a model has been advanced by Berkhout (1985) and others. They consider that the source wavefield propagates directly to the reflector, undergoing transmission losses and geometrical spreading but without multiples and converted modes. At the reflector it is scattered upward with an angle-dependent reflection coefficient and then propagates upward to the receivers, with further geometrical spreading and transmission losses but again without multiples and mode conversions. This allows an interpretation of equation (4.105) as equivalent to the wave-

4.6. FINITE DIFFERENCE METHODS 137 field propagated from the source to the scatterpoint and scaled by the reflection coefficient. Thus, a

better way to estimate the reflectivity is to produce a model of the source wavefield as propagated down to the scatterpoint and divide the result of equation (4.105) by this modeled wavefield. This is

a very general imaging condition called the deconvolution imaging condition that works for pre-stack and zero offset data. The ERM imaging condition is kinematically correct but does not achieve the same amplitudes as the deconvolution imaging condition.

Kirchhoff migration is one of the most adaptable migration schemes available. It can be eas- ily modified to account for such difficulties as topography, irregular recording geometry, pre-stack migration, and converted wave imaging. When formulated as a depth migration, it tends to be a slow method because great care must be taken in the raytracing (Gray, 1986). When formulated as

a time migration, straight-ray calculations using RMS velocities can be done to greatly speed the process. Another advantage of Kirchhoff methods is the ability to perform ”target-oriented” migra- tions. That is, equation (4.106) need only be evaluated for those points in (x,z) which comprise the target. Since the cost of computation is directly proportional to the number of output points, this can greatly reduce the run times and makes migration parameter testing very feasible.