Finite difference extrapolation by Taylor series

4.6.1 Finite difference extrapolation by Taylor series

Finite difference techniques are perhaps the most direct, but often the least intuitive, of migration methods. As the name implies, they involve the approximation of analytic derivatives of the wave equation with finite difference expressions. To see how this might be done, consider the definition of the derivative of an arbitrary function ψ(z)

dψ(z)

ψ(z + ∆z) − ψ(z)

A simple finite difference approximation for this derivative simply involves omitting the limit

dψ(z)

ψ(z + ∆z) − ψ(z)

z ψ(z) ≡

δ 1+ z is the first-order forward-difference operator. Here ∆z is assumed small with respect to length scales of interest (i.e. wavelengths) but is still finite.

Finite difference operators can be used to predict or extrapolate a function. Suppose that values for ψ(z) and its first derivative are known at z, then equation 4.109 can be rearranged to predict ψ(z + ∆z)

dψ(z)

ψ(z + ∆z) ≈ ψ(z) +

This can be regarded as a truncated Taylor series. Taylor’s theorem provides a method for ex- trapolation of a function provided that the function and all of its derivatives are known at a single point

d 1 ψ(z)

1 d 2 ψ(z)

1 d 2 3 ψ(z)

ψ(z + ∆z) = ψ(z) +

If all derivative exist, up to infinite order at z, then there is no limit to the extrapolation distance. Essentially, if the function and all its derivatives are known at one location, then it is determined everywhere.

138 CHAPTER 4. ELEMENTARY MIGRATION METHODS If finite difference extrapolation is equivalent to using a truncated Taylor series, then what

relation does wavefield extrapolation by phase shift have with Taylor series? Recall that if φ(z) is a solution to the scalar wave equation in the Fourier domain, then it can be extrapolated by

(4.112) Here, k z is the vertical wavenumber whose value may be determined from the scalar wave dispersion

φ(z + ∆z) = φ(z)e 2πik z ∆z .

relation, k z = f 2 /v 2 −k 2 x . Furthermore, if φ 0 (k x , f ) is the (k x , f ) spectrum of data measured at z = 0, then, for constant velocity, φ(z) is given by

(4.113) As an illustration of the connection between Taylor series and the phase-shift extrapolator, consider

φ(z) = φ 0 e 2πik z z .

using Taylor series applied to equation (4.113) to derive something equivalent to equation (4.112). From equation (4.113) the various derivatives can be estimated with the result that the n th derivative is

d n φ(z) dz n

(4.114) Then, using this result, a Taylor series expression for the extrapolation of φ(z) to φ(z + ∆z) is

= [2πik z ] n φ(z).

φ(z + ∆z) = φ(z) + [2πik z ]φ(z)∆z + [2πik z ] 2 φ(z)∆z 2 + [2πik φ(z)∆z 3 z ] 3 +... (4.115)

or

φ(z + ∆z) = φ(z)

1 + 2πik z ∆z + [2πik

z ∆z] + [2πik z ∆z] 3 +... . (4.116)

At this point, recall the expression for the series expansion of an exponential is e x =1+x+x 2 /2 + x 3 /6 + . . . which affords the conclusion that the infinite series in square brackets of equation (4.116) may be summed to obtain

6 . Thus, if infinitely many terms are retained in the series, equation (4.116) is equivalent to equation (4.112).

z ∆z + 1 2 [2πik z ∆z] 2 + 1 [2πik z ∆z] 3 +...

2πik z ∆z

It may be concluded that wavefield extrapolation by phase shift is equivalent to extrapolation with an infinite-order Taylor series and there is no upper limit on the allowed size of ∆z (in constant velocity). Alternatively, wavefield extrapolation by finite difference approximations is equivalent to extrapolation with a truncated Taylor series and the ∆z step size will have a definite upper limit of validity.