IF(k) | 2 is the power spectrum of f(x) as obtained from the discrete 
Fourier transform given n = L/ Ax values of f(x); the discrete power 
spectrum or the amplitude spectrum !F(k)! is defined at frequencies 
Y= k/L, if the profile has a record length L. 
The transfer function expresses the m.s.e. of the curve which is ob- 
tained by ES of a single sinusoid p.sin(27V x + ¢) and subsequent 
interpolation: 
2 
.P- 9e 
Hu) L-——. 
Gn P 
; u = vAX 
An ideal reconstruction of the sinewave means € = 0, thus H = 1, i.e., 
100% fidelity. The phase angle ¢ does not influence the r.m.s.e. € if 
the record length L of the sinusoid is sufficiently long, i.e., if nd x 
is equal to an integer multiple of the period length A = 1/v, and if 
A is not smaller than 24x. 
Knowing what the result is of sampling and interpolation of a sinusoid 
allows deduction of the response of the system to a linear combination 
of sinusoids. If a profile f(x) can be represented by a superimposition 
of sinusoids (Fourier series) with Ai » 2 Ax, then the power spectrum 
carries the necessary information about f(x), i.e., amplitudes versus 
frequency, to compute the mean square error of its digital model. 
Since terrain profiles, in general, can be only approximated by a finite 
Fourier series, the above formula for Og provides only an estimate. By 
applying some simple corrections to the raw estimate as calculated from 
transfer function and power spectrum (also including, for example, the 
influence of the spectrum computation from a sample corrupted by a ran- 
dom measuring error) /2/, estimates were obtained which, for the cases 
examined to date, did not deviate from the actual r.m.s.e. by more than 
5%. The approach is also attractive from the point of practical 
application: there are no check measurements required, the discrete 
spectrum can be computed by the Fast Fourier Transform which is a stan- 
dard algorithm, and the transfer function can be calculated by using the 
DEM interpolation program with a sinusoid as input which is sampled at 
fixed intervals. The transfer function of a particular interpolation 
method has to be computed only once and can be retrieved later from a 
library for every new task of accuracy estimation. 
3. PROGRESSIVE SAMPLING 
PS, as conceived by Makarovic /5/, starts with the zero-sampling-run, 
where elevation is measured at the intersection points of a regular 
grid. Subsequently, the second differences of the measured values are 
analyzed along the x- and the y-grid lines. Finer sampling should be 
done only in those areas where the second difference exceeds a pre- 
specified threshold (see figure la). 
= 655"