F O U R I E R T H E O RY
Content
INTRODUCTION
REPRESENTATION OF SEISMIC SIGNAL
INTRODUCTION
Seismic signals such as ground velocity and water pressure are not normally examined in an
analytical mathematical expressions. Usually, they are broken down into simpler constituent
components by Fourier decomposition or analysis. Using a Fourier representation of seismic signals
we can examine otherwise hidden characteristics using the frequency and phase information. A
seismic signal can always be reconstructed from the parts by Fourier reconstruction.
A seismic signal and actually ANY continuous representative time series (i.e., stationary) can be
analyzed using Fourier Theory. This portion of the signal we analyze should be able to predict the
general behavior that we are attempting to describe.
Theoretically, if the signal is too short then the theory breaks down or if there are gradual
changes within the portion of data being analyzed and which are too large to observe within the small
amount of data. Similarly sudden changes are hard to analyze and are an important limitation of the
technique.
Fourier transforms calculated on computers deal with sampled data at discrete intervals of time,
i.e., values at a specific interval of time, which is usually fixed, say 2 ms, or 1 ms.
There are three properties we assume implicitly when processing seismic data. Processing
seismic data means that we apply algorithms via computer software to enhance usable characteristics
of the most desirable seismic data, also know strictly as seismic signal. Unwanted data is known as
seismic noise.
We assume during seismic processing that it is just as easy to break the data down into smaller
independent parts treat them independently and then rebuild the signal.( superposition) and that if
somehow the signal were originally twice its size then all the parts would be twice the size as well
(linear superposition). In other words, that the total output signal is the sum of the individual parts.
This implies that energy is conserved throughout the process.
REPRESENTATION OF SEISMIC SIGNAL
In one dimension:
u  t   A0 cos   
2
T
(1)
We have seen in the past few weeks that this can also be represented as
u  t   A0
 i 2T t   

e 
,


Using the Euler formula or Euler identity, we have already seen.
(2)
where
2
is known as the angular frequency (  ) and  , the phase. Note that a positive phase
T
delays the signal and that a negative phase advances the signal in time, contrary to first hunch of
many. Phase has units of time. The use of angular frequency reminds us of the endless periodic
nature of waves, that is fundamental to Fourier theory. Angular frequency has units of seconds-1, as
do Hertz, or regular frequency.
We can now write (2) as:
u  t   A0 eit   