The Kirchhoff diffraction integral

4.5.2 The Kirchhoff diffraction integral

Let ψ be a solution to the scalar wave equation

1 ∂ 2 2 ψ(#x, t)

∇ ψ(#x, t) =

v 2 ∂t 2

where the velocity v may depend upon position or it may be constant. To eliminate the time dependence in equation (4.85), let ψ be given by a single Fourier component ψ(#x, t) = ˆ ψ(#x)e −2πift . Then equation (4.85) becomes the Helmholtz equation

2 ˆ ∇ 2 ψ(#x) = −k ψ(#x) ˆ

where k 2 = 4π 2 f 2 /v 2 . Now, let g(#x; #x 0 ) be the solution to ∇ 2 g(#x; #x

0 )−k 0 g(#x; #x 0 ) = δ(#x − #x 0 )

(4.87) where k 2 2 2 0 2 = 4π f /v 0 with v 0 constant over all space and δ(#x − #x 0 ) represents a source at #x = #x 0 .

The analytic solution to equation (4.87) is well known (e.g. Morse and Feshbach (1953) page 810) and can be build from a linear combination of the two functions

e ±ik 0 r

g ± (#x; #x 0 )=

where r = |#x − #x 0 | and three spatial dimensions are assumed. In two dimensions, the solution must

be expressed with Hankel functions that have the asymptotic form

g ± (#x; #x 0 )∼

e ±ikr+iπ/4

, r → ∞.

kr

4.5. KIRCHHOFF METHODS 133 Since g(#x; #x 0 )e −2πift is the time-dependent Green’s function, it is apparent that g + =r −1 e ik 0 r

corresponds to a wavefield traveling out from r = 0 while g − =r −1 e −ik 0 r is a wavefield traveling inward towards r = 0. In modelling, g + is commonly used and is called the causal Green’s function while, in migration, it turns out that g − is appropriate and it is called the anticausal Green’s function.

Now, apply Greens theorem (equation 4.84) using ˆ ψ and g − to get

g − (#x; #x

∂ˆ ψ(#x)

∂g − (#x; #x 0 )

0 )∇ ψ(#x) − ˆ ψ(#x)∇ g (#x; #x 0 ) dvol =

g (#x; #x 0 )

Substituting equations (4.86) and (4.87) into the left hand side of this expression leads to

∂g (#x; #x 0 ) −k 0 (#x; #x 0 )ˆ ψ(#x) dvol+

∂ˆ ψ(#x)

ψ(#x)δ(#x−#x 0 ) dvol =

g (#x; #x 0 )

−ˆ ψ(#x) dsurf

Assuming that the point #x 0 is interior to the volume V , the delta function collapses the second integral on the left and this expression can be rewritten as

∂ˆ ψ(#x)

∂g − (#x; #x 0 )

ψ(#x 0 ) = Λ(#x 0 )+

g (#x; #x 0 )

Λ(#x 0 2 )≡ 2 −k 0 − (#x; #x 0 )ˆ ψ(#x) dvol.

Equation (4.92) estimates the wavefield ˆ ψ at the point #x 0 interior to V as a volume integral plus

a surface integral over ∂V . The surface integral is what is desired since we can hope to know ˆ ψ over the boundary of V . However, the volume integral involves the unknown ˆ ψ and is essentially not computable. The function Λ(#x 0 ) expresses this volume integral and can be seen to vanish if the reference medium v 0 is equivalent to the actual medium v over the entire volume. Since g has been chosen as a constant velocity Green’s function, Λ can only vanish precisely for constant velocity. However, in the variable velocity case, approximate, ray theoretic Green’s functions can be used to help minimize Λ (Docherty, 1991). To the extent that the reference medium does not equal the true medium, then Λ expresses the error in a ˆ ψ that is computed without Λ. In any case, the next step

is to drop Λ and substitute in g − =e −ik 0 r /r into equation (4.92) with the result

e ∂ˆ ψ(#x)

∂n −ˆ ψ(#x)

The normal derivative of g can now be resolved into two terms $ "

−ik # 0 r

e ∂ˆ ψ(#x)

ik 0 ψ(#x)e ˆ −ik 0 r ∂r

Multiplying both sides of this result by e −2πift and recalling that ψ(#x 0 , t) = ˆ ψ(#x 0 )e −2πift gives

$ " e −ik 0 r ∂ˆ ψ(#x)

ik 0 ψ(#x)e ˆ −ik 0 r

∂r # e −ik 0 r ∂r

ψ(#x 0 , t) = e −2πift

+ˆ ψ(#x)

dsurf (4.96)

∂V

∂n

∂n

r 2 ∂n

134 CHAPTER 4. ELEMENTARY MIGRATION METHODS or, using k 0 = 2πf /v 0 ,

e # −2πif(t+r/v 0 ∂ˆ ψ(#x) i2πf ˆ ψe −2πif(t+r/v 0 ) ∂r e −2πif(t+r/v 0 ) ∂r ψ(#x 0 , t) =

r 2 ∂n (4.97) Now, ˆ ψ(#x)e −2πif(t+r/v 0 ) = ψ(#x, t + r/v 0 ) is the wavefield ψ at the point #x but at the advanced time t + r/v 0 . It is customary to denote this quantity as [ψ] t+r/v 0 with the result

∂n

1 # 1 ∂r 1 ∂r ψ(#x 0 , t) =

t+r/v 0 dsurf (4.98)

∂V

r ∂n

t+r/v

v 0 r ∂n ∂t

t+r/v 0 ∂n

where the time derivative in the second term results from ∂ t ψ = −2πifψ. This is a famous result and is known as Kirchhoff’s diffraction integral. (In most textbooks this integral is derived for forward

modelling with the result that all of the terms are evaluated at the retarded time t − r/v 0 instead of the advanced time.) It expresses the wavefield at the observation point #x 0 at time t in terms of the wavefield on the boundary ∂V at the advanced time t + r/v 0 . As with Fourier theory, it appears that knowledge of both ψ and ∂ n ψ are necessary to reconstruct the wavefield at an internal point.