The correction vector A with dimensions 3nxl is 
defined by two components: 
Ad with dimensions 6pxl and 
AE with dimensions (3n-6p)xl 
The coordinate corrections of the points included 
in Ad vector involve the contribution of observa- 
tions of geodetic distances to photogrammetric 
observations of the end points of distance "p", 
whereas the Af vector is concerned with (n-2p) 
points which were observed only photogrammetrical 
ly. 
In the case of correlated observations, Nd isa 
full matrix which implies to invert a (6px6p) ma- 
trix and a number of n-2p matrices with (3x3) di- 
mensions. 
If the geodetic distance observations are consi- 
dered as uncorrelated, then one can simplify the 
inversion of N + NJ matrix. This operation is re- 
duced to the inversion of a number of p matrices 
with dimensions (6x6). 
eo =~ — 
By virtue of the structure of D, Ad , W and L, 
the normal equations can be expressed in the ex- 
panded form: 
Y= Disc (RL, Bi. seeee Nido) 
(6x6) (6x6) (eat) or 
tis Goon ns CUT 
y [fa cde se] (4) 
Wi Dias €. W4, WO, senescence Wp) 
(pxp) (ixi) (ixl) (1x1) 
in which: 
8m £d D 
od ) pc? (1x1) (1x6) 
= aq — 
CL = DR We Lp 
(6x1) (6x1) (1x1) (1x1) 
Wo =U E200 
The solution of the equation system is given by: 
a A» - A -4 
A. = fin +) AB TFT Th 
(6mx1) (6mx6m) (3nx3n) (3nx6m) 
Q A E (5) 
*| (CK) - N. GRAY (CH) 
(6mx1) (6mx3n)  (3nx1) 
Once the vector of exterior orientation parame- 
ters A has thus been obtained, the vector of 
ground poînt coordinates can be computed from: 
À Ges na ^S" A 
3nx1l) (3nx3n) |(3nx6m) (6mx1) 
(6) 
A =| N+Nd 
(3nx1) |(3nx3n) 
By virtue of the block diagonality of N + N and 
N + NJ matrices, each distance "d$ and each point 
"j' can be treated independently after the evalu- 
ation of A vector: 
S--4N (7) 
A = LHN 4 [6c 
(6mx1) [(6mx6m) (6mx1) 
For each distance "1" the ÆAdevector solution is: 
Adt= Ris A 
( 
T (8) 
{(6x6m) (6mx1) 
C+Cd | - [HN 
(6x1) |(6x6) t 
n (6x6) 
832 
For each point "j", the Aj vector solution is: 
Ki. o2 a. 
AN; 537% À (9) 
for 3j.” (2ptl, 2pt2, ......n) 
where: 
P n 
Se 2 2 Bolg pit S S 
(6mx6m) 1=1 (6mx6m) j=2p+1 (6n án) 
^ p n ^. 
CT > Cal. 4, 24 + = Cy 
(6mxl) 1-1 (6mxl) j=2p+1 (6mx1) 
Because each distance "dQ" is defined by two end 
points: 21-1 and 21, it results the following 
normal coefficient equation matrix: 
E (Np. +N Nat 120 +Nclp| Na fay? 
ate of VEU. Com tra 5) NC 
(10) 
1t, m (C 24-140) = 
(6mx1 ) (6mx1) 
À + NdR [| Carat + EdL 
(6 Reto [es 628" «i (6x6 Taste e 
and for each stereotriangulated point (j=2p+1, 
2pt2, .....n), the corresponding matrices: 
A 
Sj = Nj = Rj 3 Rj=Nj-0 rm 
—-{ AT A, A Ti, 
Q 78S NM: 8976-3 
Finally, to evaluate the À dfcorrection vector 
which corresponds to end points of each geodetic 
measured distance df, it is necessary to consider 
the following matrices and vectors: 
N el à 7 +Ndf 
(6x6) d (6x6) 
with: _ 0 
N 91.4 22]= Diag (N N 
| exes I DA 
and ir. _ = (12) 
C * C [7| *|C (* Cd 
p | [Ead at ex 
with : q 
C 24-124 7 = 
(25 [3385 Ed 
A computer program called DISTANCE was developped 
and its formulation is based on the principle of 
olservation equations, as described in the above 
paragraph. 
THE MATHEMATICAL MODEL OF DEM 
The development of mathematical models that pre- 
dict soil erosion and deposition on three-dimen- 
sional catchments also requires techniques that 
can measure distributed soil movement in order to 
test and to verify the model. In the case of wa- 
ter erosion risk, the most popular predictive mo- 
del is the universal soil loss equation (USLE): 
= RKLSCP (13) 
where: 
A is the average annual soil loss per unit area 
(in tonnes/ha), 
R is a measure of the erosivity of the rain fall 
or the surface runoff in a given region,