FOURIER METHODS 121

4.4. FOURIER METHODS 121

Figure 4.56: The f -k spectrum of the data of Figure 4.57: The (k x ,k z ) spectrum of the data Figure 4.31.

of Figure 4.54.

28 end

29 if( floor(j/kpflag)*kpflag == j)

30 disp([’finished wavenumber ’ int2str(j)]);

31 end

32 end

33 %inverse transform

34 [seismig,tmig,xmig]=ifktran(fkspec,f,kx);

End Code

As a last example, consider the computation of the discrete f -k spectra of one of the preceding examples before and after migration. This should produce a graphical confirmation of the mapping of Figure 4.49. This is very simply done using fktran . Specifically, the case of the synthetic of Figure 4.31 and its migration in Figure 4.54 is shown. If seis is the unmigrated seismic matrix, then the command [fkspec,f,kx]=fktran(seis,t,x) computes the complex-valued f -k spectrum and plotimage(abs(fkspec),f,kx) produces the result shown in Figure 4.56. In a similar manner, the (k x ,k z ) spectrum after migration can be computed and is shown in Figure 4.57. Comparison of these figures shows that the spectral mapping has been performed as described.

4.4.3 f -k wavefield extrapolation

Stolt (1978) provided an approximate technique to adapt f -k migration to v(z). This method used

a pre-migration step called the Stolt stretch that was followed by an f -k migration. The idea was to perform a one-dimensional time-to-depth conversion with v(z) and then convert back to a pseudo time with a constant reference velocity. The f -k migration is then performed with this reference velocity. (Stolt actually recommended the time-to-depth conversion be done with a special velocity function derived from v(z) called the Stolt velocity.) This method is now known to progressively lose accuracy with increasing dip and has lost favor.

A technique that can handle all scattering angles in v(z) is the phase-shift method of Gazdag (1978). Unlike the direct f -k migration, phase shift is a recursive algorithm that treats v(z) as

a system of constant velocity layers. In the limit as the layer thickness shrinks to infinitesimal, any v(z) variation can be modelled. The method can be derived starting from equation (4.49).

122 CHAPTER 4. ELEMENTARY MIGRATION METHODS Considering the first velocity layer only, this result is valid for any depth within that layer provided

that ˆ v is replaced by the first layer velocity, ˆ v 1 . If the thickness of the first layer is ∆z 1 , then the ERM wavefield just above the interface between layers 1 and 2 can be written as

ψ(x, z = ∆z , t) = φ (k , f )e 1 2πi(k 0 x x x−k z1 ∆z 1 −ft) dk x df. (4.56)

Equation (4.56) is an expression for downward continuation or extrapolation of the ERM wavefield to the depth ∆z 1 . The extrapolated wavefield is distinguished from the surface recorded wavefield by the presence of the term e 2πik z1 ∆z 1 under the integral sign that is a specific form of the Fourier extrapolation operator, e 2πik z ∆z . Any extrapolated wavefield is a temporal seismogram more akin to a surface recording than to a migrated depth section. In the phase-shift method, as with any recursive technique, the migrated depth section is built little-by-little from each extrapolated seismogram. The extrapolated wavefield is a simulation of what would have been recorded had the receivers actually been at z = ∆z rather than z = 0 . Since any extrapolated section intersects the depth section at (z = ∆z, t = 0) each extrapolation can contribute one depth sample to the migration (see Figure 4.24). This process of evaluating the extrapolated section at t=0 was discussed in section

4.2.6 as the post-stack imaging condition. For the wavelike portion of (k x , f ) space, the extrapolation operator has unit amplitude and a phase of 2πk z ∆z. For evanescent spectral components, it is a real exponential e ±2π|k z |∆z . Forward wavefield propagation must obey physical law and propagate evanescent spectral components using the minus sign in the exponent, e.g. e −2π|k z |∆z . Therefore, inverse wavefield extrapolation, as is done for migration, should use e +2π|k z |∆z . However, this inversion of evanescent spectral components is a practical impossibility because they have decayed far below the noise level in forward propagation. The practical approach is to use e −2π|k z |∆z for the evanescent spectral components for both forward and inverse extrapolation. Even more practical is to simply zero the evanescent spectral components on inverse extrapolation. In all that follows, it is assumed that e 2πik z ∆z has one of these two practical extensions implemented for evanescent spectral components.

The wavefield extrapolation expression (equation (4.56)) is more simply written in the Fourier domain to suppress the integration that performs the inverse Fourier transform:

(4.58) Now consider a further extrapolation to estimate the wavefield at the bottom of layer 2 (z =

φ(k , z = ∆z ,f)=φ (k , f )e 2πik z1 ∆z 1 x 1 0 x .

∆z 1 + ∆z 2 ). This can be written as φ(k x , z = ∆z 1 + ∆z 2 , f ) = φ(k x , z = ∆z 1 , f )T (k x , f )e 2πik z2 ∆z 2 (4.59) where T (k x , f ) is a correction factor for the transmission loss endured by the upgoing wave as it

crossed from layer 2 into layer 1. If transmission loss correction is to be incorporated, then the data must not have had such amplitude corrections applied already. This is extremely unlikely because seismic data is generally treated with a statistical amplitude balance that will compensate for transmission losses. Also, correcting for transmission losses at each extrapolation step can be numerically unstable because the correction factors are generally greater than unity. Consequently, it is customary to set t(k x , f ) = 1. With this and incorporating equation (4.58), equation (4.59)