Vittorio Casella 
  
Laser ray 
(X, Y,Z) 
  
  
Z = aX+bY+c 
Figure 1 — Heuristic explanation of the origin of the altimetric error induced by planimetric error 
As the drawing suggests, the total estimated error, e, , equals the sum of the pure altimetric error e, with the induced 
error, e, , that is, the projection of the planimetric error, e,, , onto the vertical plane 
e =2Z—(aX +bY +c) 
=e, + ©, tan a 
(2) 
where a indicates the inclination of the ramp. The order of magnitude of the errors of laser scanners can be assumed to 
be 30-40 cm for X and Y, and 15-20 cm for Z. Since the slope p =tana of a court’s ramp can easily reach the value of 
20% or more, it can be argued that the induced altimetric error easily assumes values around 6-8 cm. Therefore, this 
component of the total error is clearly recognizable and significant, provided that the ramp has been surveyed with high 
precision methods, whose uncertainty is not greater than 1-2 cm. So if the pure altimetric error has been previously 
estimated and precisely known ramps are used to estimate the total altimetric error, the component of the latter induced 
by planimetric error can be estimated by subtraction; the pure planimetric error can be finally quantified. 
Our method is based on three steps. First: estimation of the pure altimetric accuracy of the laser scanner by means of 
flat surfaces. Second: very accurate measurement of the equation of some court ramps by means of GPS and land sur- 
veying measurements. Third: estimation of the total height error and then of the laser planimetric error as the difference 
between the former and the pure laser altimetric error. Due to the high density of the laser data we have, there some tens 
or even some hundreds of laser points available for each test site, and this greatly increases the reliability of our estima- 
tions. It is remarkable that, using some ramps with different orientations, it could be also possible, in principle, to dis- 
criminate different values for e, and e, . Finally our approach, besides allowing the estimation of the planimetric ran- 
dom error, easily allows us also to discover any systematic error, as we will show in a further section of the paper. 
This paper will give a description of the test site (Section 2) that has been made in Pavia, Northern Italy, in the frame of 
a national research project. We will also describe some of the check-areas we have measured inside the site; by check- 
areas, we mean small parts of the territory such as tennis courts, court ramps or underground parking ramps, which we 
have measured with high precision methods. Then we will expose, in some detail, the basic formulas (Section 3) of our 
approach, explaining how we propose to estimate the unknowns. Finally, we will give the initial but interesting results 
for one flat check-area (section 4). Some conclusions and a quick sketch of our future activities will conclude the paper. 
  
158 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.