Fourier methods

4.4 Fourier methods

The Fourier methods are developed from an exact solution to the wave equation using Fourier trans- forms. They provide a high-fidelity migration that illustrates precisely how the migrated spectrum is formed. There are two fundamental approaches and many variations of each. f -k migration (Stolt, 1978) is an exact solution to the migration problem for constant velocity. It is a direct method that is the fastest known migration technique. Phase-shift migration (Gazdag, 1978) is a recursive approach that uses a series of constant velocity extrapolations to build a v(z) migration.

4.4.1 f -k migration

Stolt (1978) showed that the migration problem can be solved by Fourier transform. Here, Stolt’s solution will be developed from a formal solution of the wave equation using Fourier transforms. It will be developed in 2D and the direct 3D extension will be stated at the end.

Let ψ(x, z, t) be a scalar wavefield that is a solution to

(4.40) where ˆ v is the constant ERM velocity. It is desired to calculate ψ(x, z, t = 0) given knowledge of

ψ(x, z = 0, t). The wavefield can be written as an inverse Fourier transform of its (k x , f ) spectrum as

ψ(x, z, t) =

φ(k x , z, f )e 2πi(k x x−ft) dk x df (4.41)

4.4. FOURIER METHODS 113 where cyclical wavenumbers and frequencies are used and the Fourier transform convention uses a +

sign in the complex exponential for spatial components and a − sign for temporal components. (The notation for the integration domain is explained in section 4.1.3.) If equation (4.41) is substituted into equation (4.40), the various partial derivatives can be immediately brought inside the integral where they can be readily evaluated. The result is

2 φ(z)

+ 4π 2 2πi(k x−ft)

−k x φ(z) e dk x

df = 0 (4.42)

∂z 2

where the (k x , f ) dependence in φ(z) has been suppressed for simplicity of notation. The derivation of equation (4.42) does not require that ˆ v be constant; however, the next step does. If ˆ v is constant 1 , then the left-hand-side of equation (4.42) is the inverse Fourier transform of the term in curly brackets. The uniqueness property of Fourier transforms (that there is a unique spectrum for a given function and vice-versa) guarantees that is a function vanishes everywhere in one domain, it must do so in another. Put another way, the zero function has a zero spectrum. Thus, it results that

∂ 2 φ(z)

2 + 4π k 2 z φ(z) = 0

∂z

where the wavenumber k z is defined by

v ˆ 2 −k x

Equation (4.44) is called the dispersion relation for scalar waves though the phrase is somewhat misleading since there is no dispersion in this case.

Equations (4.43) and (4.44) are a complete reformulation of the problem in the (k x , f ) domain. The boundary condition is now φ(k x , z = 0, f ) which is the Fourier transform, over (x, t), of ψ(x, z =

0, t). Equation (4.43) is a second-order ordinary differential equation for fixed (k x , f ). Either of the functions e ±2πik z z solve it exactly as is easily verified by substitution. Thus the unique, general solution can be written as

(4.45) where A(k x , f ) and B(k x , f ) are arbitrary functions of (k x , f ) to be determined from the boundary

φ(k x , z, f ) = A(k

x , f )e

2πik z

+ B(k x , f )e −2πik z

condition(s). The two terms on the right-hand-side of equation (4.45) have the interpretation of a downgoing wavefield, A(k x , f )e 2πik z z , and an upgoing wavefield, B(k x , f )e −2πik z z . This can be seen by substituting equation (4.45) into equation (4.41) and determining the direction of motion of the individual Fourier plane waves as described in section 2.9. It should be recalled that z increases downward.

Given only one boundary condition, φ(k x , z = 0, f ), it is now apparent that this problem cannot

be solved unambiguously. It is a fundamental result from the theory of partial differential equations that Cauchy boundary conditions (e.g. knowledge of both ψ and ∂ z ψ) are required on an open surface in order for the wave equation to have a unique solution. Since this is not the case here, the migration problem is said to be ill-posed. If both conditions were available, A and B could be found as the solutions to

φ(z = 0) ≡ φ 0 =A+B

1 Actually ˆ v (z) could be tolerated here. The necessary condition is that ˆ v must not depend upon x or t.

114 CHAPTER 4. ELEMENTARY MIGRATION METHODS and

(z = 0) ≡ φ z0 = 2πik z

A − 2πik z B. (4.47)

∂z

When faced with the need to proceed to a solution despite the fact that the stated problem does not have a unique solution, a common approach is to assume some limited model that removes the ambiguity. The customary assumption of one-waywaves achieves this end. That is, ψ(x, z, t) is considered to contain only upgoing waves. This allows the solution

A(k x ,f)=0 and B(k x ,f)=φ 0 (k x , f ) ≡ φ(k x , z = 0, f ). (4.48) Then, the ERM wavefield can be expressed as the inverse Fourier transform

ψ(x, z, t) =

φ 0 (k x , f )e 2πi(k x x−k z z−ft) dk x df. (4.49)

The migrated solution is

ψ(x, z, t = 0) =

, f )e 0 2πi(k (k x x x−k z z) dk x df. (4.50)

Equation (4.50) gives a migrated depth section as a double integration of φ 0 (k x , f ) over f and k x . Though formally complete, it has the disadvantage that only one of the integrations, that over k x , is a Fourier transform that can be done rapidly as a numerical FFT. The f integration is not

a Fourier transform because the Fourier kernel e −2πft was lost when the imaging condition (setting t = 0) was invoked. Inspection of equation (4.50) shows that another complex exponential e −2πik z z is available. Stolt (1978) suggested a change of variables from (k x , f ) to (k x ,k z ) to obtain a result in which both integrations are Fourier transforms. The change of variables is actually defined by equation (4.44) that can be solved for f to give

(4.51) Performing the change of variables from f to k z according to the rules of calculus transforms equation