Frequency domain seismic forward modelling: A tool for waveform ...

for k = 0; 1;:::;N ,1 and
h r =
N,1
X
k=0
H k e i2kr=N ; (1.8)
for r =0;1;:::;N,1, where r is a time sample index, k is a frequency domain sample
index, H k is the k-th Fourier transform coecient, h r is the time series. Provided
each representation is complete (the time series or the frequency components), each
series can be uniquely recovered from the other, using these formulas.
1.4.2 Sampling and the Sampling Theorem
As we actually work with a discrete representation h n = h(t n ). The function
h(t) is said to be band limited if its Fourier transform H(f) = 0 for jfj > f c
where f c is a nite \critical" frequency. In seismic case all signals are band limited
due to a limited source spectrum, and are almost always treated with an analogue
\anti-alias" lter to ensure this property before sampling.
The sampling theorem states that a band-limited function h(t) is completely
specied by the sampled values f n (t n ), provided that the sampling interval, t
satises
f c 1 N y
= 1
2t
(1.9)
The frequency N y is known as the Nyquist frequency for the given sampling interval
t.
Beyond the Nyquist frequency, the periodicity and the conjugate symmetry
of the DFT (for a real valued time series) causes the highest frequencies to be
\wrapped" around the frequency axis and to be aliased as lower frequencies. This
theorem is important if we are attempting to construct an un-aliased frequency
spectrum from a time series.
A similar sampling theorem is relevant ifweare trying to reconstruct a time
series from a limited number of samples of the frequency spectrum, as is the case for
frequency domain modelling. We must sample the frequency spectrum suciently
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