EE EZ 
140 
separate kinematic adjustment. 
The weights of the observations are equal to the ones of the kinematic adjustment, while the pseudo-observations have the same 
weights assumed equal to the variances of the residual noises derived from separated spatial analyses. 
The use of both observation and pseudo-observation equations forces to introduce a hybrid norm: 
aul O0 15 S iA 
1/2! s'n! | ^55 Fr AGS -5—20))7 min (10) 
| | 0 s 0) 
where P is the weight matrix and À is a vector of Lagrange multipliers. According to the expressions (9) and (10) the estimates for 
the signal and the noise become: 
$2 C,B'(BC4B! - o2p!yla, (11) 
ñ = a,- B8 (12) 
The computation of these quantities requires the solution of a system with dimension m, equal to the number of observations. Since 
it is much more convenient to have analogous expressions which need the solution of a system with dimension n « m, equal to the 
number of parameters, the previous ones (11) and (12) can be transformed, by using two theorems of linear algebra, into the 
following ones: 
3 -(B'PB) ! B' Pa, - o;|(B PB )C,,(B‘PB) + o2(B' PB J| B'Pag : (13) 
ñ = a, - BS (14) 
By starting from these new expressions, the law of covariance propagation lets to compute the expression of the corresponding 
covariance matrices: 
Cz; - o2(B'PB)! - o (B'PB)C,, (B'PB) + ABER) (15) 
Ci =04[P" -B(B'PB)'B']+ o$B|(B'PB)C.(B' PB) + o4(B'PB)]  B" (16) 
where é —5s —$ represents the estimation error of the signal. 
Note that, if the matrix (B'PB) is sparse and the covariance matrix C,. is sparse too, when it is built up by finite covariance 
functions, the product (B'PB) C. (B'PB) is again sparse and avoids heavy computations. 
4. BLOCK ADJUSTMENT AND DIGITAL MODELS 
( Ammannati, et al., 1983, Cunietti, et al., 1984; Colombo, et al., 1986a ). 
An example broadly studied during the eighties in Italy, is newly shown in order to prove the validity of the method. 
In December 1982, a large scale landslide (-220 ha) interested the district of Ancona (Italy), involving also some residential 
' quarters of the city. A working group consisting of geologists, geophysicists, geotechnicians and geodesists was set up in order to 
analyze the causes, to control the ground settling and to decide on the consequent destination of the area. 
Inside this working group, the geodesists performed the control surveying by topographic and photogrammetric instruments. The 
former was set up to control the settlement of the terrain after the landslide, the latter to study the deformation of the landscape as 
derived from photogrammetric material taken before and after the landslide occurrence. 
The photogrammetric models, of a simultaneous block adjustment of photograms taken before and after the landslide, were 
observed in order to achieve a regular grid of points on the ground. In this way it was possible to compute the earth settlements, to 
evaluate the volumes involved, to analyze the shape and the pattern of the break lines and to point out the landslide boundary. 
A levelling network was established, to control the settlement of the terrain after the landslide, and eleven campaigns were 
executed during a period of 2.5 years, while every campaign lasted 15-20 days. The lasting of the observations being comparable 
with the time interval, it was considered suitable to perform the kinematic adjustment of the observations. Successively, the 
kinematic parameters were spatially analysed. 
The photogrammetric work consisted of a simultaneous block adjustment of images taken at two different times (Summer 1980 and 
December 1982, the day after the sliding) and at two different heights. The higher strip (scale — 1: 11000) was the old one (4 
models) and the three lower strips (scale — 1: 6000) were the new ones (19 models), as shown in figure 1. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995