the univariate analysis mode corresponds and respectively
bivariate, if by sampling strategy in the structure of a
network also uniform, are distributed.
Directly and indirectly submited to Fourier
transformation, the heights values lead to the obtaining of
spectral estimates, which in ensemble form the power
spectrum. The magnitude of spectrum values render the
energetlical levels corresponding to height samples,
therefore implicitly to terrain variations, offering in a
first variant the possibility of this characterising by
" power " ( emergy ). Consequently, ordered according to
their frequency in a spectrogram ( periodogram ), will be
the frequency content descriptor and at the same time, an
efficient and objective means for different types of the
forms existing in the studied terrain pointing out.
2.1 Power Spectrum Estimation
At present, the spectral analysis uses two methods to
obtain the power spectrum. The indirect method or the
"standard " method ( Blakman - Tukey ) conceived on the
basis of Wiener - Hinchine relations, which express the
property that correlation function and power spectrum
form Fourier paires, and the direct method ( Cooley -
Tukey ). Im the first case the spectral estimation result
indirectly, through Fourier transformation applied to the
correlation function values, and the second case as a result
of Fourier transformation application starting directly
from the measured Zi heights samples. It is worth
mentioning that in both cases, the use of fast Fourier
transformation ( FFT ) procedure has a prevailing
importance.
The direct method was chosen in the experiment, aiming
to the easiest calculation effort. Its implementation was
done in a program of processing which uses a fast Fourier
transformation subroutine.
A brief description of this method presents the following
characteristics. The input data are represented by terrain
heights reduced to a trend function (Zn = Zn - T ) for the
spectrum values calculus or for the spectral density
function. Then according to the overall strategy the Xf)
amplitude spectrum is determined and subsequently the
power spectrum:
1 1
GG) - — I X' G0 * XGU. 17 —1 XGO |? (1)
L L
Smoothing opperations are required because the
estimations of amplitude spectrum will be afiected by
errors due {0 truncate effect ( the terrain profile have a L
limited lenght ).The smoothness can be performed at a
level of amplitude spectrum or at the power spectrum
level. One of the weighting methods which operates
spatially or frequentially is used in this situation.
lí Z(s) is a terrain profile with L lenght, sampled a As
interval, process that issues Zu (n=0, N-1 ) row of heights,
after their reduction to the trend function, the spectral
lines are determined as follows:
XGq = Aly) - jig: Cq= 0, Q-1) (2)
vhere X(jq) is the spectral amplitude density, q the order
of speciral line, Alg) the real component and Ig) the
imaginary component. According (o the sampling theorem
the integral frequency content is obtained only when fe
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
sampling frequency is conforming with the condition:
fe > f max (3)
Taking into account Che relation; L= (N- 1) As= (N- 1)
/£ e, respectively f max = (@ - I) Af, in which @
represents the total number of spectral lines, and f= 1/ L,
the spectrum resolution, results that the Q 7 N/2 relation
exists at the limit of the fulfilment of ( 3 ) condition,
between N-the number of height samples and Q9. Thus
the periodogram must include N / 2 spectrum lines to
describe the frequency range as accurately as possible.
It was already mentioned that the spectrum estimations
obtained im the first phases are affected by errors
consequent to the use of finit profile. The limitation at a
finit interval of variable which represents in this case the
terrain haights is equal with a filtration in the space
domain. Thus the terrain profile is assimilated with a
truncate signal representing the product between Z(s) real
signal and spatial filler. The physical process, through
which adjacent spectral lines values, interveme for a
spectral line due to truncation in calculated spectrum,
having as effect the introduction of an error, is called in
technical literature leakage ( Davenport. W.B, Root. W.L,
1958 ). The solution of convolution with Hemming
weighting window dm) =0.5 * [1 + cos (2mwN) Is m=
0, N-1 applied to frequency, was used im order do
eliminate. According to addopled solution the components
of raw spectral lines which have following expresion:
N-1
Ag) = As Y Zn) cos (TGn/Q)
n=0
(4)
N-1
Big) = As X Zw) sin (Tqn/Q)
n=0 (q=0,@-1 )
are ajusted by relations:
A(0) = 0.5A0) + 0.5A(1)
B(0) - 9.510) * 9.500)
A(q) 7» 0.25A(q-1) * 0.5A(q) + 0.25A(q+1)
B(q) » 0.25A(q-1) * 0.5B(q) * 0.25B(q* 1) (5)
A(Q-1) - 9,5A(Q-2) * 0.5A(Q-1)
B(9-1) - 0.5A(9-2) * 0.5B(Q-1)
Then, the spectrum values are obtained using the
relations:
1
Gp) = — LA? (4) + B° (q) I : (q70, Q-1) (6)
L
The periodogram curve is drawn by using them, out of
wich (ir) frequency level corresponding (o roughness is
selected. This will be the main parameter or (fc) cut-off
frecvency used during the filtering process through which
the high frequency relief components are separated from
the other terrain forms.
2.2 Roughnes Components Filtering
Filtering is a commonly used procedure in signals
processing techniques. Kt comsist mainly im retaining
344
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