Automatic generation of elevation data over Danish landscape

2 Methods of determining elevation data
These equations describe a vector for the unknown heights of the grid points.
u T �
part
�z(i
�1,
j �1),
z(i, j �1),
z(i �1,
j �1)
�
�
�
�
�
, z(i �1,
j), z(i, j), z(i �1,
j)
�
��
, z(i �1,
j �1),
z(i, j �1),
z(i �1,
j �1)
��
The residuals vxx(i,j), vxy(i,j) and vs for observations are presumed normally distributed.
The vectors ai for (i=2,3,4), which correspond to the nine terrain heights of the four grid meshes in figure
2.11, are multiplied by the coefficient matrices defined as:
�1
� 2
�T
1
a2
: � vector(
�
2 4
4 �
�
��
1 � 2
�
a
T
4
The final mathematical model is finished by setting up the stochastic model:
1 2
P( z
�
�
xx ) � Diag(
D ( � xx ))
1 2
P( z
�
�
yy ) � Diag(
D ( � yy ))
�
�1
2
P( z xy ) � Diag(
D ( � yx ))
�
�1
2
P( z)
� Diag(
D ( � ))
This stochastic model represents the weight matrix for the temporary observations and their
a priori standard deviations. Instead of eliminating the vector for the u unknown DEM heights by a linear
Gauss-Markoff process by maximising the function:
�
�
�
f ( u,
� , � , � ) � v
xx yy xy
z
z
the mathematical model is determined by introducing a weight function with approximated a priori weights
for the observations in the stochastic model with a value for their normalised residuals t. As suggested by
[Förstner 1989] and [Krarup et al. 1980], the weight functions are:
w
� 1
1
: � vector(
�
0
4 �
��
�1
2 ( t)
� e
These weight functions are used as the modification for the observation weight in a two-part iterative
Gauss-Markoff process by use of the relation:
P � P
wj(
t
( v�1)
( 0)
( v)
z
z z
P � P
2
�(
t)
wj(
t
( v�1)
( 0)
( v)
zxx zxx
zxx
Where: v = the iteration step.
z
z
0
0
0
)
)
T
and
1�
2
�
�
1��
P
T
�1�
0
�
�
1 ��
( v)
T
)
)
� � �
( z)
v � v
�
a
T
xx
T
3
P
P � P
wj(
t
( v�1)
( 0)
( v)
zyy zyy
zyy
P � P
( v)
w ( t)
�
1
�
( z
xx
wj(
t
�
) v
( v�1)
( 0)
( v)
zxy zxy
zxy
� 1
1
: � vector(
�
2
4 �
�
��
1
1
1�
( t)
xx
)
2
)
�
� v
T
yy
P
� 2
� 4
� 2
( v)
�
( z
yy
1 �
� 2
�
�
1 ��
�
) v
yy
T
)
�
� v
T
xy
P
( v)
�
( z
xy
�
) v
xy
33