al Wi=2T 1, e2-2111,
ral gig 2 7 PAT 2201; aa
Substituting the values of w;,w,.
0, 44070 2—— LS - £2 i p - "AA
Fy 17749 12 3 TII F( l 9 2 ) (15)
Equation (15) represents the relationship between the dis-
crete and continuous spectra of a two dimensional process
(Dudgeon , 197 ) . From this relationship it can be shown
that the continuous data can be recovered from the discrete
data only if the surface of the continuous spectrum ap-
proaches zero when y, and W, approach the Nyquist frequen-
cies .
FeCG0 9) =0 for y>T/A and y > 1/52
It should be noticed in eqn.(15) as well as in eqn.(7) that
x and y &re integer variables that represent the position
of the point rather than its distance from the two axes :
distances are represented by A, x and A23. .
As the results obtained in the bivariate case are the same
as those obtained in the univariate case , the method emp-
loyed in this paper for determining the sampling density
of profile data can readily be used to determine the two
dimensional sampling density for rectangular grid DEMs.
5. RESULTS AND CONCLUSION
Synthetic data are used to represent different types of
terrain profiles . For each terrain type different sampling
intervals are used , and in each case the discrete Spectrum
for the range 0 < w <T is computed . Elevations are
generated in profiles using trigonometric polynomials of
sine and cosine terms . In order to obtain an unbiased es-
timate of the discrete spectrum , a large number of data
points must be used . In the present investigation 100
elevations are used for each profile . This number of points
is considered suitable in normal cases of DEM processing
because sampling density may differ from one set of points
to the other.
The algorithm used in this investigation does not employ
& Fast Fourier Transform program , instead , two matrices
of sines and cosines were computed and stored for use as
data for the computation of the spectrum . In this way
the loops of the algorithm include only multiplications
and additions , which saves considerable computational
time . However , fast Fourier transforms should be used
in practice for much more efficiency and speed.
