Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 41

Electric current flowing from an isolated point electrode embedded in a continuous
homogeneous ground provides a physical illustration of the significance of the inverse
square law. All the current leaving the electrode must cross any closed surface that
surrounds it. Usually the surface is spherical, concentric with the electrode and the same
function of the total current will cross each unit area on the surface of the sphere. The
current per unit area will be inversely proportional to the total surface (half-space) of 2πr 2 .
Current flow in the earth, however, is modified drastically by conductivity variation.
The potential fields of either gravity, magnetic or electrical fields are the ones given by
either the Laplace or Poisson equations. Some of the useful properties of (x, y, z) are (i)
given this potential field (scalar) over any plane, we can compute the primary or force field
(vector) at almost all points in the space by analytic continuation and (ii) the points where
the force field cannot be computed are the so-called singular points. A closed surface enclosing
all such singular points also encloses the sources which give rise to the potential field. Thus the
singularities of the potential field are confined to the region filled with sources.
All these properties are best described and accentuated in the Fourier domain. We shall
therefore express the Fourier transformation of the potential field in two or three
dimensions (see Section 4.5). In two dimensions, the Fourier transform pair,  (u, v) and its
inverse (x, y) are given by

Φ(,)=∬ (,)exp[−(+)]

(23)

and

(,)=

∬ Φ(,)exp[(+)]

(24)

Where here, u and v are coordinates of the Fourier plane. Equation (24) is also known as the
Fourier integral representation of (x, y). Equation (23) exists only if and only if

∬ |(,)|<∞ :

a condition generally not satisfied in most geophysical situations except for an isolated
anomaly (Roy, 2008). However, the Fourier transform of a real function in two dimensions
possesses the following symmetry:

Φ(,)=Φ ∗ (−,−),Φ(−,)=Φ ∗ (,−)

Φ(0,0)=∬ (,)

(25)

If (x, y, z) is the potential field on a plane z, satisfying the Laplace equation (equation (21)),
its Fourier integral representation is given by (Naidu 1987)

(,,)=

∬ Φ(,)(,,)exp[−(+)]

(26)

Where H(u, v, z) is to be determined by requiring that it also satisfies the Laplace equation
and is only true if it satisfies the differential equation

−( + )=0 ,
whose solution is H(u, v, z) = e  s z for all values of z, where =√ + . For z ≥ 0 equation
(26) becomes