Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 28

accomplished without compromising the quality of the original data. Even where the
picture of the time-domain data 'looks good”, we can perform further analyses on the signal
for correlation, improvement and enhancement purposes. There are many time-domain and
frequency-domain tools for these purposes. This is the reason for this chapter.
We shall be exploring the uses of some tools for the analyses and interpretations of
geophysical potential fields under the banner of Spectral Analysis of Geophysical Data.
The first section of the chapter covers the treatment and analysis of periodic and aperiodic
functions by means of Fourier methods, the second section develops the concept of spectra
and possible applications and the third section covers spectrum of random fields; ending
with an application to synthetic and real (field) data.

2. Periodic and aperiodic functions
A periodic function of time, t can be defined as f(t) = f(t+T), where T is the smallest
constant, called the period which satisfies the relation. In general f(t) = f(t+NT), where N
is an integer other than zero. As an example, we can find the period of a function such as
f(t) = cos t/3 + cos t/4. If this function is periodic, with a period, T, then f(t) = f(t+T).
Using the relation cosθ = cos(θ + 2πm), m = 0, ±1, ±2, ..., we can compute the period of this
function to be T = 24π.
Aperiodic function, on the other hand, is a function that is not periodic in the finite sense of
time. We can say that an aperiodic function can be taken to be periodic at some infinite time,
where the time period, T = ∞ .
A function f(t) is even if and only if f(t) = f(-t) and odd if f(t) = -f(-t).

3. Sampling theorem

A function, f which is defined in some neighbourhood of c is said to be continuous at c
provided (1) the function has a definite finite value f(c) at c and (2) as t approaches c, f(t)
approaches f(c) as limit, i.e. lim → ( ) =() . If a function is continuous at all points of an
interval a≤t≤b (or a<t<b, etc), then it is said to be continuous on or in that interval. A
function f(t) = t 2 is a continuous function that satisfies the two conditions above.
A graph of a function that is continuous on an interval a≤t≤b is an unbroken curve over that
interval. In practical sense, it is possible to sketch a continuous curve by constructing a table
of values (t, f(t)), plotting relatively few points from this table and then sketching a
continuous (unbroken) curve through these points.
A function that is not continuous at a point t = c is said to be discontinuous or to have a
discontinuity at t = c. The function, f(t) = 1/t is discontinuous at t = 0 as the two
requirements for continuity above are violated at t = 0. This discontinuity cannot be
removed. On the other hand, a function, f(t) =

has a removable discontinuity at t = 0,
even though the formula is not valid at t = 0. We can therefore extend the domain of f(t) to
include the origin, as f(0) = 1.
A continuous function, f(t) can be sampled at regular intervals such that the value of the
function at each digitized point is f n , with n = 0, ±1, ±2, ±3, ... (Fig. 1).
It will appear as if the time domain function f(t) versus t is transformed into a digitized form
domain (f n versus n).