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Vittorio Casella
where the first equation simply demonstrates that our estimation is correct. About the second one, let' s now assume that
the variances of the two planimetric components coincides; let's indicate their common value with s, . Now we can
write the final expression for the variance
; si (9)
2 Si
Therefore, the dispersion of the A variable is the sum of two components: the pure height error committed by the laser
sensor, s7, and the induced error, s, p^ , which is a function of the planimetric error and of the slope of the ramp.
Indeed the first of the (9) formally demonstrate what can be intuitively thought: that, on flat areas, planimetric error has
no effect.
Now if s; has already been estimated on some flat areas; if one ramp is very precisely known; if this ramp has been
scanned by a laser sensor, then the total height error can be easily estimated simply forming the differences indicated in
(7). Therefore the planimetric error can be calculated by means of the second of (9).
All what we have said could be repeated and reformulated for a general surface: we could speak of local slope, rather
than, simply, of surface’s slope. But planar surfaces make everything easier and, provided that our towns offer many
pure ramps, we will use them, exclusively. Besides, equations (9) shows why it is necessary to look for the steepest
available ramps: because the induced altimetric error is proportional to the slope.
Our formulas allow also to detect systematic errors, due for instance to non perfect sensor orientation. Indeed let’s sup-
pose that the Z measurements are affected by a bias A, . This implies that the first of (8) become
m, = Z, + A, — (X,Psina +Y, pcosa +c)=
(10)
= A, #0
Let's suppose again that one planimetric component, X, is affected by a bias A, . We can argue
m, 2 Z, -((X, A, )psina +Y, p cosa +c)= iD
— —A, psina x0
So if the estimated mean of the A random variable is significantly different from zero, we must think there are system-
atic errors. But since we can always suppose that systematic height errors have been discovered and corrected during
the first step of our procedure, we come to the conclusion that (10) will never happen, and relations (11) can be used to
detect planimetric systematic errors.
4 ESTIMATION OF THE ALTIMETRIC PRECISION
This section illustrates some first results on altimetric precision which we obtained on a flat check-area. We think they
are interesting even if they don't constitute a full and complete application of our proposed methodology. The estima-
tion has been worked out exploiting a tennis court whose ellipsoidal height is now known very precisely. We selected
the raw-data coming from both sensors which are inside the court, and then we made a comparison of the height given
by the lasers and of the height directly measured on the ground.
It is remarkable that we have in Pavia a permanent GPS station, which is part of our geodetic network and that both the
companies used data produced by our station to reference their products. Therefore the products delivered by both com-
panies are, from this point of view, perfectly homogeneous between each other and in respect with the local GPS net-
Work, that we used to make the control measurements.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 161
