T co
= | VA + S a (27 n -w) coswk dw
o t AX C AX
From which :
oo Go
2T n 27 n
20) = 7 Nem RU Po(-w+2 2) (6)
From this relationship it can be noticed that if the con-
tinuous profile contains no variations at frequencies
above the Nyquist frequency, such that PF cC? = 0 for
w>T/ ax , then :
Fa) = F_(w) .
In this case no information is lost by sampling . But more
generally , the effect of sampling will be that variations
at frequencies above the Nyquist frequency will be folded
back and produce an effect at a frequency lower than the
Nyquist frequency in the spectrum of the discrete data.
The frequencies w,-w+2MW/ax , w+ 2WAx, ..... are
called aliases of one another (Chatfield,1975)
3. ESTIMATING THE SAMPLING DENSITY OF DEM PROFILES
From the previous discussion it was shown that aliasing
will cause no troubles if Ax is chosen small enough to
ensure that F(w) =0 for w> T/ox.
In the case of DEM data processing it is not possible to
determine the continuous spectrum E (c) , therefore ,
it is only possible to assume a value for the sampling
interval Ax and compute the discrete spectrum of the
sampled data . If the resulting estimate of the discrete
spectrum approaches zero near the Nyquist frequency T/Ax ,
then our choice of Ax was sufficiently small and can be
used for the given profile . However , if the values of the
discrete spectrum does not reach zero near the Nyquist
frequency , then the chosen value of the sampling interval
was larger than that required for the given terrain and
should be reduced accordingly.
The criterion presented by the discrete spectrum approach
can be used in a progressive sampling process to determine
the most suitable sampling interval for a given terrain.
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