Automatic generation of elevation data over Danish landscape

2 Methods of determining elevation data
� grid and unstructured raw elevation data from different sources/producers
A mathematical model must therefore be established which can handle the problem of historical, present
and future elevation data, captured from different sources/producers, with different accuracies and structures
(grid or raw elevation data). A solution to this problem, could be the ‘multi grid method’. The ‘multi
grid method’ is a highly effective iterative solution strategy which can solve large areas with thinly determined
points with the help of an adjustment system. The great advantage with the ‘multi grid method’ is
that the adjustment system is independent of the numbers of unknowns.
The starting point could be: two clouds of structured or unstructured points, where the elevation data is
independent and has different accuracies. Should one of the clouds of points consist of grid points, this
grid will be looked at, as a non-structural cloud of points.
This is a classic adjustment problem, the solution being, an adjustment is done by means of the least
squares measurement principle. (For the readers, who are familiar with least squares measurements,
skip the next 1½ pages). The principle of least squares measurement is, that the random values (residuals)
(vi) have to be minimum, which means:
v
m
2 2 2
2
2
� 1 v � 2 v � � � � �
3 v � m �v i
i�1
� min
When the observations are independent and have different weights the principle of least squares measurement
demands that the weight function minimises like:
� �
m
c � c v
2
� c v
2
� ��
��
��
� � c
2
� c v
2
� � min
2 2 2 3 3 m i i
i � 1
This function can be noted as matrix form:
�
�
�vvv���v� 1 2 3 m
�
�
�
� �
�
�
�
�
c1
c
2
c
3
�
c
� �
� �
� �
� � �
� �
� �
� �
� �
m
� 1
�
2�
�
3
�
� �
�
m�
v
v
v
This matrix form can be written more simply as:
T
� � v �C�
v
The equations for the observations can be written in matrix format as:
l � A � x
- v
The above equation can be rewritten as:
v � A � x
- l
T
and together with equations � � v �C�
v the equation will be:
� = (A�x - l) T �C � (A�x - l)
= (x T �A T �C - l T �C) � (A�x - l)
= (x T �A T - l T )�C�(A�x-l)
= x T �A T �C�A�x - l T �C�A�x - x T �A T �C�l + b T �C�l
v
37