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Causal integration
is represented in the time domain
by convolution with a step function.
In the frequency domain this amounts to multiplication by .Integrating twice amounts to convolution by a ramp function,
, which in the Fourier domain is multiplication by
.Integrating a third time is convolution with
which in the Fourier domain is multiplication by
.In general
| |
(1) |
Proof of the validity of equation (1) for integer values of n
is by repeated indefinite integration which also indicates
the need of an n! scaling factor.
Proof of the validity of equation (1) for fractional values of n
would take us far afield mathematically.
(Fractional values of n, however,
are exactly what we need to interpret Huygen's secondary wave sources in 2-D.)
The factorial function of n in the scaling factor becomes a gamma function.
The poles suggest that a more thorough mathematical study of convergence
is warranted, but this is not the place for it.
A common application is when n=1/2 and
(ignoring the scale factor)
equation (1) becomes
| |
(2) |
It is well known that
| |
(3) |
A product in the frequency domain corresponds
to a convolution in the time domain.
A time derivative is like convolution with a doublet .Thus, from
equation (2) and
equation (3)
we obtain
| |
(4) |
Next: HANKEL TAIL
Up: Claerbout: Hankel tail
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Stanford Exploration Project
11/17/1997