Spectral Analysis of Geophysical Data - Advances in Data, Methods, Models and Their Applications in Geoscience - page 45

be one of the basic steps in data processing, but the FFT algorithm is most often used for the
computation when the data should have been windowed appropriately.
Spectral windowing, as we have earlier noted comes in because potential data are available
over a finite area (as against the infinite length demanded from the mathematics). The
finiteness of data is either because no measurements were made outside the area of
investigation or the data are found to be homogeneous only over a finite area. Such a
situation may be modeled as a product of a homogeneous random process and a window
function. Thus the model of the finite data is

f 0 (m, n) = f(m, n).w 0 (m, n)
Where f 0 (m, n) is the observed field over a finite area and w 0 (m, n) is a discrete window
function and f(m, n) is the random potential field function. Hence from equation (28), we have

(,)=

∬ dF(,)exp [ (+) ]

(36a)

(,)=

∬ W (,)exp[(+)]

(36b)

(,)=(,). (,)

=

∬ F (,)exp[(+)]

(36c)

Where

( , ) =

∬ ( , ) (− ,− )

.

The DFT coefficients of f 0 (m, n), 0 ≤ m ≤ M-1 and 0 ≤ n ≤ N-1 (area of investigation is a
rectangle of size M x N) and so for 0 ≤ k ≤ M-1 and 0 ≤ l ≤ N-1

(,)=

∬ ( , ) ( − , − )

(37)

And so the spectrum of a finite random potential field is (Kay 1989)

( , ) =

| (,) |

− ) ′′

=

∬ ( , ) ( − ,

(38)

Equation (38) gives a relationship between the determined spectrum from finite 2D data
which are observed and the spectrum of infinite 2D data which are not observed. The factor
| (,)| in the spectrum expression is the spectrum of the window function. The angular
variation of the spectrum of the potential field may be distorted unless the window
possesses the following properties:
a.

Window spectrum most be closer to a delta function. A time domain delta function,
δ(t) has that δ(t) ≥ 0 for some finite value of t, δ(t - a ) = 0 for t ≠ a and ()=1

∞

.

A window spectrum must be closer to the properties of δ(t).

b. The leakage of power is minimum which is possible by controlling the height of

sidelobes of window spectrum and

c. The spectrum of the window must be isotropic or close to being isotropic.