Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 42

(,,)=

∬ Φ(,) ()

(27)

Equation (27) is a useful representation of the potential field and forms the basis for
derivation of many commonly encountered results.

6.3 A random interface
Potential fields that are caused by stochastic sources are of two types: a random interface
separating two homogeneous media (e.g. sedimentary rocks overlying a granitic basement)
and a horizontal layer of finite thickness within the density or magnetization varying
randomly. We can relate the stochastic character of the potential fields to this interface or
layer with the aim of determining some gross features of the source model (e.g. depth to
interface). Figure 2 shows a homogeneous random interface separating two media. Let f(x,
y) be a homogeneous (stationary) stochastic field. The spectral representation (or Cramer
representation) of a homogeneous random field is given by (Yaglom, 1962)

(,)=

(28)

∬ dF(,)exp[(+)]

Where

dF(u, v) = F(u + du, v + dv) - F(u, v), (du, dv)→0

and F(u, v) is the generalized Fourier transform of f(x, y). Some of the properties of dF(u, v)
include:
i.

(,)=[(,)]:==0

= 0: (u, v) ≠ 0
ii. (,). ∗ ( , )=0:(,)≠( , )
iii. When the two points overlap [i.e. (u, v) = (u', v')],
then

(,). ∗ (,)= (,)

Where S f (u, v) is the spectrum (power spectrum) of the stochastic field, f(x, y).
We can further obtain some basic properties of the random potential field in free space. If
(x, y, z) is a random potential field in free space, then similar to equation (26),

(,,)=

∬ dΦ(,)(,,)exp [ (+) ]

(29)

Where H(u, v, z) is selected so that (x, y, z) satisfies Laplace equation and d(u, v)
is a random function. When equation (29) is substituted in the Laplace equation, we obtain

( , , )
−( + )=0

whose solution is

( ,, ) =exp[±√ + ]

and so equation (29) becomes