FOURIER METHODS 125

4.4. FOURIER METHODS 125

seconds

meters

Figure 4.60: A diffraction chart showing the response of five point diffractors as recorded on the surface z = 0.

and, in summary,

(4.67) The focusing phase shift is angle-dependent and vanishes for k x = 0. That the static phase shift

total phase shift = µ = µ s +µ f .

accomplishes a bulk time delay can be seen as a consequence of the phase shift theorem of digital signal theory (e.g. Karl (1989) page 87). This well-known result from time-series analysis says that

a time shift is accomplished in the Fourier domain by a phase shift where the phase is a linear function of frequency. The slope of the linear phase function determines the magnitude of the time shift. Rather than merely quote this result, it is instructive to actually demonstrate it. The static phase shift can be applied to the ERM seismogram by

s (x, t) =

φ 0 (k x , f )e −µ s

+2πi(k

x x−ft) dk x df. (4.68)

Since the static phase shift is independent of k x , the k x integral can be done directly to give

s (x, t) =

v)

ψ ˆ 0 (x, f )e −2πi(ft+f∆z/ˆ df (4.69)

where ˆ ψ 0 (x, f ) is the temporal Fourier transform of the ERM seismogram. Letting τ = ∆z/ˆ v, this becomes

ψ s (x, t) =

ψ ˆ 0 (x, f )e −2πif(t+τ )

The last step follows because the integral is an inverse Fourier transform for the time coordinate t + τ . This result is simply a proof of the phase shift theorem in the present context.

Wavefield extrapolation in the space-time domain Since the extrapolation operator is applied with a multiplication in the Fourier domain, it must be

a convolution in the space-time domain. In section 4.2.4 constant-velocity migration was shown to

be a nonstationary convolution. Constant-velocity extrapolation, which is a much simpler process

126 CHAPTER 4. ELEMENTARY MIGRATION METHODS

seconds

seconds

meters

meters

Figure 4.61: The first hyperbola of diffraction chart of Figure 4.60 is shown convolved with the its time reverse. (A) The focusing term only is applied. (B) Both the focusing and the thin-lens term are applied

than migration, is a stationary convolution. Figure 4.60 shows the response of five different point diffractors, at depths of 200 m, 400 m, 600 m, 800 m, and 1000 m, as recorded at z = 0. This diffraction chart was constructed with an exploding reflector velocity ˆ v = 2000 m/s. The chart represents an idealized ERM seismogram for a geology that consists on only five point diffractors. It is desired to determine the space-time shape of the extrapolation operator that will extrapolate this seismogram to 200 m. Since this is the depth of the first diffractor, the first diffraction curve should focus to a point and shift to time zero. The other diffraction curves should all focus somewhat and shift to earlier times by the same amount. In fact, the second hyperbola should be transformed into the first, the third into the second, and so on.

In section 4.2.4 it was seen that replacing each point in the ERM seismogram by a wavefront circle (the operator ) will focus all hyperbolae at once. With extrapolation, only the first hyperbola should focus and the operator (i.e. the replacement curve) should be the same for all points. Clearly, the operator needs to be concave (⌣) to focus the convex (⌢) diffraction curves. One process that will focus the first diffraction curve is to cross correlate the seismogram with the first diffraction curve itself. To visualize this, imagine tracing the first diffraction curve on a clear piece of film (being careful to mark the coordinate origin on the film) and then placing the apex of this curve at some point on the diffraction chart. The value of the cross correlation is computed by taking the sample-by-sample product of the ERM seismogram and the section represented by the film, and then summing all of the resulting products. Assuming that all amplitudes are either zero (white) or one (black) for simplicity, this reduces to summing the samples in the ERM seismogram that are coincident in space-time with the diffraction curve on the film. This cross-correlation value is then assigned to the location on the ERM seismogram of the coordinate origin on the film. Clearly, when the film is positioned such that its diffraction curve exactly overlies the first diffraction curve, a large value for the cross correlation will result. In fact, since this is the only exact match between the ERMseismogram and the curve on the film, this will be the largest value for the cross correlation. This exact match will occur when the coordinate origin on the film coincides with that on the ERM seismogram.