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A topic of minor interest has been the interdependence of
the height errors of DEM points, especially in close neigh-
bourhood, which yields a large impact on the accuracy of
all those values that are derived from the elevation data
through local operations. Due to measurement techniques
and interpolation, there is a positive correlation of the ele-
vation errors within a certain neighbourhood of DEM-
points. That correlation of the height errors may show a
dependency not only from the point distance but also from
the slope itself (Steinmetz, 1992).
In this paper the correlation between the height values of
a DEM-point and its neighbours - according to their relative
position as seen from the DEM-point - are estimated from
a test data set. The DEM is obtained from photo-
grammetric imagery.
2. THE DIGITAL ELEVATION MODEL
2.1 The Square Grid Model
For the following investigation a grid DEM with square grid
is used. This kind of model is ideally suited for statistical
analyses. The consequently regular structure (in the pro-
jection plane) allows for a lot of simplifications in the algo-
rithms. Furthermore some results, especially the correlation
function between the height errors of neighbouring points,
may be generalized to irregular models.
It has widely been outlined that the square grid DEM has
a lot of disadvantages compared to other types of elevation
models, e.g. triangulated irregular networks (TIN-models)
or grid models with additional vector information (Hutchin-
son, 1989; Mark, 1979; Késtli, Sigle, 1986; Toomey, 1988,
just to mention some). Especially Mark demands that the
phenomenon should be the driving criterion for the repre-
sentation of data rather than computational considerations
(Mark, 1979). The author fully agrees with these opinions.
In the case of this paper, however, the nature of correlation
between data points shall be estimated, which can only be
done in a statistically relevant way, i. e. through a large
number of points. Irregular point distributions, however,
yield much more complex approaches, and furthermore the
derivation of elevation data is frequently done through
measurement of square grids (Krzystek, 1991; Theobald,
1989), thus these data can be directly tested, without any
additional interpolation to a square grid.
In the square grid some very simple relations are valid.
The following local point indices are defined:
12
81417
9|1]013111
5|2|6
10
Tab. 1: Relative point indices of neighbouring points (1-12)
to any point (relative index O) in the square grid
These indices are used throughout the paper in order to
address the respective neighbouring points to any one grid
point. It is assumed that those neighbouring points exist,
Which is realized if all operations are restricted to the "inner
zone" of the square grid model (not' using the outermost
two gridlines all around the rectangular data set).
691
2.2 The Data Sets for the Test
For the test two data sets are used: A reference data set
with superior accuracy which has been plotted from large
scale photographs, and a small scale test data set on
which the test takes place. In the actual case a scale factor
of 4 has been used between the two data sets, a factor
which allows to nearly neglect the influence of the errors in
the reference data set. Thus the height values H,. (row, co-
lumn) of the reference data set are assumed to be "true
values", while those of the test data set, h,. are assumed
to be erroneous (for the random parts of the errors the
influence of the large scale model errors can actually be
neglected, while for the systematic parts the estimated er-
ror values may be wrong by about 30 percent). The data
sets are described as the following matrices, with nc as the
number of columns and nr as the number of rows:
H, Hi ... His
H = Hy, H,, gd Fos (1)
H ri Ho ses He
is the reference data set and
Ay Ay h; 3
Da ho h uno
the test data set. Both matrices are equally dimensioned
and positioned. The neighbouring points of any grid point
H,, or h,, according to the local point indices as defined in
Tab. 1, are addressed as H,.; respectively ^, ; or, for better
readability and for simplicity reasons, as H; respectively h;
in most cases where the indices r and c do not point to a
specific point.
2.3 Errors in the Elevation Data
There is a lot of error sources present in the whole process
of obtaining a DEM from photogrammetric material. Gener-
ally spoken, three types of errors can be found, defined
through their frequency in the spectral space, expressed in
relation to the grid width - as the smallest unit - and the
whole DEM-area - as the largest unit -, and their impact on
the resulting height errors:
2.3.1 Errors with low frequency: These are mainly errors
resulting from the photogrammetric orientation process.
These processes yield errors that apply systematically on
at least a whole photogrammetric model. In terms of their
frequency spectrum they are either linear or show wave
lenghts of about twice the size of a single photogrammetric
model (with model orientation) or even larger (with bundle
or model blocks). Since these errors do not change sig-
nificantly between neighbouring grid points, they have only
little influence on the derivation of all those data that are
obtained by local height differences. They are critical main-
ly to the absolute height values.
2.3.2 Errors with medium frequency: These errors show
a local systematic behaviour in the sense that they are
highly correlated within a few grid widhts. They may result
from the measurement itself (inertia of the operator, espe-
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
