cially in dynamic profile measurement, or height values 
from one point that are propagated to the next point as 
initial height values) or from interpolation techniques ap- 
plied to the original data. In the last case, the correlation 
could be available - in theory. In practice, however, the 
covariance matrices of the interpolation are not available. 
These errors are the object of this paper. 
2.3.3 Errors with high frequency: These are well-known 
as "random errors". They may result from many sources 
such as the physical representation of colour in the images 
(granularity, irradation and so on) or the instability of the 
sensors used (e. g. scanners, but also the human eye). 
Since these errors are neither known in their absolute 
value nor in their direction, they can be treated statistically 
according to the general laws of random errors. 
2.4 Estimation of the Errors in Elevation 
The mean height error (RMS) m, of the test data set - 
which is the square root of the variance o, - and the cova- 
riance to the neighbouring point with index / (height value 
h,.) can be estimated according to 
  
nr nc 
S hs 7H? (3) 
2 . re c4 
mp = 09,5 
nr-nc - 1 
nr nc 
>> (hye = He) "(Ary = He) 4) 
RE i 
e, nr:nc - 1 
(Note that the range of the indices has to be slightly modi- 
fied, because of the matrix edges. In practice the best way 
to do is to limit the range to the interval 3 to nc-2 respec- 
tively nr-2.) 
For an infinitely large data set the following relations apply: 
0, 7105 027 04:95 7.0, Og =, 0g (5) 
0967 Go S49 7 go. 
With an increasing number of points involved these rela- 
tionships are fulfilled better. In practice they are fulfilled 
compared to the magnitude of the values and their accu- 
racy even for a few thousand points, as will be shown 
later. Thus the following relative point indices will be used 
for the covariances and correlation coefficients from now 
on instead of the numbers 0 through 12: 
  
2y 
e yid 
2x| x | h | x 2x 
  
  
  
diyje 
2y 
  
  
  
  
  
  
  
  
Tab. 2: Modified relative point indices of neighbouring 
points to any point in the square grid (relative index h) for 
the correlation values and covariances 
The correlation coefficients r; between a grid point and its 
neighbouring point / are estimated according to: 
692 
oO 
I 
n=—, 
Op 
ieix y d e 2x 2y) . (6) 
m, is composed of a random part r (also "noise") and a 
systematic part s ("'systematic" means here that there is a 
non-zero (normally positive) correlation between the height 
errors of a grid point and of any one of its neighbouring 
points up to a certain distance (i. e. a few grid widths)): 
a, =r+s. (7) 
3 THE DERIVED DATA 
3.1 Slope and Aspect 
The slope vector v(v,,v,) in any grid point can be calcu- 
lated from the height values of the neighbouring points, h, 
and the grid width A according to 
hy - h hy - h 
uA tA (8) 
The slope value s/ is the length of the slope vector: 
Sl = rv (9) 
y 
The slope direction o. (aspect) in any grid point, that is the 
direction of steepest slope, measured as the angle from 
the x-axis, can be calculated according to 
V, 
tano = —#, with v,#0 (10) 
V, 
X 
(Note: In the case of /v,/ smaller than /v,/ the slope direc- 
tion may better be calculated via the cotangent of the in- 
verse fraction of equation (10). In both cases the following 
considerations remain valid.) 
3.2 The Error of the Slope Vector 
The slope vector is computed according to equations (8) 
or, using the matrix nomenclature, as 
V = A . h (11) 
with 
v! - (v, vy, 
10-10 
2:A 01 0 -1 
The variance- / covariance-matrix of the slope vector com- 
ponents, Q,,, with the variances o,, and o,, for the compo- 
nents of the vector and the covariance o,,,, between the 
two components, results from the general law of error 
propagation with the variance- / covariance-matrix of the 
four involved elevation values h, to ^4, Q,,; 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996