6) 
a 
ht 
7) 
he 
m 
0) 
11) 
12) 
po- 
the 
ror 
the 
  
  
0, Og Oy Og 
Og," Op O04, 0 
e- Opn Og Op, 
Q,, = (13) 
dax 0g "Op 9 
Og 95y Op Op 
according to: 
= T. : 
Q, = A° Q,,A 
: Oy, Ouy, 1 (n o,-04) (14) 
= = ; 1 ¢ 
Ov, Oy, 2-A Og^Oq Op 05y 
  
3.3 The Error of the Slope Value 
The variance of the slope value can be estimated accord- 
ing to: 
o, = dsiT-Q@,, ds! (15) 
with ds/ being the vector of the partial first derivatives of 
the slope value in the two axis-directions: 
Vai Mi 
dsl” = [s d = e 5 - (cosa sina), (16) 
av, Ov, sl sl 
if o. is the slope direction. So o, results to: 
  
: Oy, Ov, COSa 
9, - (cosa sina): e en 
g g sina 
VyxVy j Vy 
0,7 (cos?a -o,, - cosa Sina (0, -0g) *sin?a o, ) 
2:A% 
In the case of 6, = o, i. e. the correlations in both diago- 
nals are equal, the components of the slope vector are 
independent from one another. For this case equation (17) 
reduces to: 
Op T (cos?a ay, + sin’a +0.) (18) 
2:4? 
  
Og = 
The special case of rotational symmetry in correlation fur- 
thermore simplifies equation (17), when 2d is used as the 
index for grid points in a distance of 2 grid widths along 
any one of the two axes: 
_ On-92g _ Op (1-1) (19) 
2-4? 2.4? 
  
Og 
log is the correlation coefficient between two grid points 
that are neighboured via two grid widths along any one 
axis-direction. The author has shown that for a constant 
principle distance of the camera of 15cm, a constant grid 
Width in the map of 3 to 5mm, and a scaling factor from 
the aerial photograph to the map of 3 to 5, above equation 
ylelds a constant error for the slope value of about 0.01 to 
0.02 (or, 1 to 2%), if an RMS height error of 0.015%Hg 
(flying height above ground) is assumed for each grid point 
(Rieger, 1992), a value which has earlier been found by 
Stechauner and Ehgartner in empirical analyses (Stechau- 
693 
ner et al., 1988). This would mean that slope values less 
than about 1 to 2% cannot be assumed to be significant. 
For this example, however, it is not taken into regard, that 
the local correlations in the height errors of medium fre- 
quency simulate a random error when the height values 
are checked over the whole area. Good interpolation me- 
thods may - and should - smooth flatter areas significantly, 
so that the slope value can still be significant even in very 
flat areas (Steinmetz, 1992). 
3.4 The Error of the Slope Direction (Aspect Ratio) 
The slope direction is computed according to equation 
(10). With the partial first derivatives of the aspect value in 
both axis-directions, 
da’ - | 2x Oa -[-% 5 (20) 
av, Ov, sP sP)’ 
its error is estimated according to 
0, = da”-Q,, da = 
  
9 A 0, v, -V 
atem Hom 
sf Ov, Ov, Vx 
= 0, ; Vy 05,*2 VV, (05 04) + V2-0p, 
2-sP-A? 2:s^- A? 
Analogous to equation (18), in the case of 6, = 6, equa- 
tion (21) reduces to: 
e 0, Vv; op, Vi oy, (22) 
2-87 -A? 2-sP-A? 
9, 
And in the special case of rotational symmetry in correla- 
tion equation (21) furthermore reduces to: 
  
d enl, O20 s On (1-20) s dgsl (23) 
* 2:sP-A7 2-sP-A? SP 
Kay is the correlation coefficient as in equation (19). 
4 THE EMPIRICAL TEST 
4.1 The Test Area 
The demands on the testing area were manifold: Since the 
correlation between the height errors shows dependency 
from the steepness of the terrain (Steinmetz, 1992), an 
area with differing slope values was necessary. There 
should have been as much free view to the ground as 
possible, so the testing area had to be mainly agricultural 
land. And last but not least aerial photographs had to be 
available in two scales, differing by a factor of larger than 
3, flewn as close in time as possible. Thus an area was 
chosen in the Northwestern part of the province of Upper 
Austria. The terrain is hilly with nearly no forest and hou- 
ses. There were available false colour infrared aerial pho- 
tographs in the scales 1:8.000 respectively 1:32.000, stem- 
ming from a project to examine the condition of the forests 
in Austria. The two flights were done within two hours, so 
that there was no change in vegetation height - a nearly 
ideal situation. The only disadvantage is the usage of a 
relatively large principal distance of 30cm for the image 
size of 23 x 23 cm?, so that the reachable height accuracy 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996