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it slope 
IR and 
bias is 
ition of 
the two DTMs with respect to the valley axis. A program to 
estimate biases between DTMs has been implemented to verify 
the existence of such translation. 
  
  
  
  
  
  
  
  
  
  
Class | Limits Percentage 
1 0 m € IAhl « 5 m 78.7896 
2 5m*zlAhl « 10m 13.80% 
3 10 m € IAhl « 20 m 5.99% 
4 20 m € IAhl « 50m 1.19% 
5 50 m € IAhl « 100 m 0.2296 
6 IA^hl 2 100 m 0.01% 
  
  
Table 5. Differences between LR Lombardy DTM and HR 
Lidar DTM 
= 
= 
20m 
5m 
2m 
  
Distribution of the elevation differences (absolute values) 
  
  
  
  
N a 
= we pP je 
E. | | | | L | | | | 9o 
i "Nr. m CN + © o oo r- e i 
© © © © © N Lo Lo WR u» Te) 
v < <r <r < <r <r <r <r < 
[.]Y 
Figure 5. Planimetric distribution of the height differences 
between HR and LR DTMs. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
46474 . 
46.465 
  
  
T T 
93 9.31 
  
Figure 6. HR-LR differences in the valley of S.Giacomo. The 
presence of systematic patterns is clear: biases with opposite 
signs that are present on the valley left and right slopes are 
hardly observable in the B/W figure. 
  
  
  
  
  
  
  
  
  
  
  
  
  
Class | Range Percentage 
1 Ah « -100 m 0 96 
2 -100m<Ah<-50m | 0.25% 
3 -50 m< Ah <-20 m 0.46 % 
4 220m <Ah<-10m 3.11 96 
5 -10m<Ah<-5m 15.74 % 
6 -5m<Ah<0m 39.98 % 
7 0mxAh«5m 31.19% 
8 Sm<Ah< 10m 5.98 % 
9 10m<Ah<20m 2.25 % 
10 20m<Ah<50m 0.99 % 
11 50m <Ah< 100 m 0.04 % 
12 Ah 2 100 m 0 96 
  
  
  
  
  
Table 6. HR-LR height differences in the valley 
The estimate of a translation and a bias between two DTMs is 
performed by Least Squares (Koch et al., 1987). The height of a 
point P is described by the following: 
h(P) = f,(x(P)) +V, 
h(P) = f,(x(P))+V, 
where f,,f, are respectively the height functions of DTMI (in 
our case, the HR DTM) and DTM2 (the LR DTM), v,,v;, are 
their observation noise, x-[N, E,] are the planimetric 
coordinates of P. In the absence of a translation and bias, clearly 
we have: 
f (x(P)) 2 f, (CP) 
Let suppose that a translation t — [t,, tl and a bias À exist 
between the two DTMs. The functional model becomes the 
following: 
f, (xXCP)) 2 f; (x(P) x0 * À 
It can be linearized by the following 
Af (x(P)) 2 fi (x(P)) - f, (xCP)) 2 Vf, (x(P) th 
where Vf is the gradient of the function. 
By all the observed differences Af(Dyizi2.. M. the 
translation and the bias can be estimated by LS. 
To fill the design matrix, the gradient of the LR DTM are 
computed by the usual numerical approximation. At the present, 
translations and biases have been estimated on the whole region, 
then they have been analyzed for the subgrids of the LR DTM. 
In Figure 7 the planimetric translations for each subgrid are 
depicted. The height biases estimates have values in the range