Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 40

6. Spectrum of random fields
6.1 Random functions
A random variable is a real-valued function defined on the events of probability system. A
random variable, f(t) emanates from a random or stochastic process: a process developing in
time and controlled by probabilistic laws. A random (or stochastic) process is an ensemble
or set of functions of some parameter (usually time, t) together with a probability measure
by which we can determine that any member, or a group of members has certain statistical
properties. Like any other functions, random processes can either be discrete or continuous.
At any point in a medium, a unit mass or a unit charge or a unit magnetic pole experiences a
certain force. This force will be a force of attraction in the case of gravitational field. It will be
a force of attraction or repulsion when two unit charges or two magnetic monopoles of
opposite or same polarity are brought close to each other. Every mass in space is associated
with a gravitational force of attraction. This force has both magnitude and direction. For
gravitational field, the force of attraction will be between two masses along a line joining the
bodies. For electrostatic, magnetostatic and direct current flow fields, the direction of the
field will be tangential to any point of observation. These forces produce force fields. These
fields, either global or man-made local fields are used to quantitatively estimate some
physical properties at every point in a medium.
Most geophysical potential fields, in particular gravity and magnetic fields are caused by an
ensemble of sources distributed in some complex manner, which may be best described in a
stochastic or random framework. We shall examine some characteristics of these fields.

6.2 Geophysical potential fields
The potential field, ϕ (x, y, z) in free space (i.e. without sources) satisfies the Laplace equation

+ + =0

(21)

When sources are present, the potential fields satisfy the so-called Poisson equation

+ + =−(,,)

(22)

Where ρ(x,y,z) stands for the density, magnetization or conductivity depending opun
whether stands for gravity, magnetic or electric potential respectively. It is important to
know that both global or local fields are subject to inverse square law attenuation of the
signal strengths. It is at its simple peak in gravity work where the field due to a point mass
is inversely proportional to the square of the distance from the mass, and the constant of
proportionality (the gravitational constant, G) is invariant. Magnetic fields, though complex,
also obey the inverse square law. The fact that their strength is, in principle, modified by the
permeability of the medium, is irrelevant in most geophysical work, where measurements
are made in either air or water. Magnetic sources are, however, essentially bipolar and the
modifications to the simple inverse-square law due to this fact are important. A dipole here
consists of equal-strength positive and negative point sources: a very small distance apart.
Field strength here decreases as the inverse cube of distance and both strength and direction
change with “latitude” (inclination) of the Earth's magnetic field. The intensity of the field at
a point on a dipole axis is double the intensity at a point, the same distance away on the
dipole “equator”, and in the opposite direction.