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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
This pure random approach is a "worst case uncertainty
assessment", as also suggested by Wechsler (2000).
34 Spatially-dependent Field — Filtered
In reality, elevation error is not purely random because of the
spatial autocorrelation nature of elevation and so is the error
(Hunter and Goodchild, 1997; Holmes, 2000; Wechsler, 2000).
34. Semivariograms: X Semivariogram analysis can
characterize spatial data by relating the semivariance between
two sample points to the distance that separates them. The
distance interval is called /ag. The important properties of
semivariogram for the characterization of spatial dependence
are: sill and range. Range is the value of lag (h) when it levels
off, which shows the distance that the data are spatially
dependent. Sill is the semivariance value in the maximum
distance of spatial dependence (range) (Figure 3a).
However, for topographic surface, it is very unlikely that the
semivariogram will reach sill, unless the strength of the
geologic properties limits the maximum elevation (Holmes,
2000). In the semivariogram of topographic data, very likely the
variability increases infinitely within the area under study
(Figure 3b).
= 1
mie SU
400,000- Ug
1
= 320,000 4 "d '
> 2400004 "d :
160,0004 yc ; Range
80,0004 , LA
o
0 1 2 2 4
distance between pauz
(a)
MATURE: SH PSE piper comente ettet pier SA SE CE e tert ute
Figure 3. (a) Range and Sill in a semivariogram; (b)
semivariogram with infinite increase of spatial variability
(picture courtesy of The Idrisi Project)
In the case of topographic data, directional semivariograms can
be calculated to obtain distance of spatial dependence. By
simultaneously plotting them, the distance over which all the
semivariograms show similar behavior can be used as the
distance of spatial dependence.
Semivariogram Analysis was conducted using Spatial
Dependence Modeller under Idrisi32. Both Upstream and
Downstream sites were analyzed and the directions applied
were omni directions, 45°, 90° and 180 °.
34.2 Weighted-mean filter. The perturbation layer developed
In this second approach is called “weighted-mean filter” which
Is developed following a method by Wechsler (2000). This
filler is composed of several layers, with the original values
being from the random field generated. Each layer is like a
‘square ring’ surrounding the central cell which uniformly
contains the values of ‘mean * weight’. The weight assigned to
each layer becomes decreasing as the layer moves farther away
from the central cell. The size of the filter comes from the
spatial dependence distance (SDD) obtained from
semivariogram analysis. For example, if the SDD is 50 m,
which means the SDD from the central cell is 50 m, then the
filter size is 100 m. The total weighted-mean values are
obtained as the sum of the total layers.
Figure 4. Example of weight layer, of 4*4 window
The formula for this method is as follows (Wechsler, 2000):
IL E 1 (1)
Weighted mean filter= > HC); m
izl i
i=l ^
where: u(L;)= mean of the values in layer i, i.e. mean of the
random field cells in the ring
i = layer number
TL = total number of layers
3.5 Slope as a DEM-derived Topographic Feature
The derived topographic parameter to be tested in this study is
slope. Slope is defined as the increase in vertical direction (dz)
per distance in horizontal direction (dx). The calculation of
slope takes the eight neighboring cells or 3x3 cell window. The
method follows ‘third-order finite difference’ by Horn (1981)
(PCRaster Environmental Software).
Error in elevation propagates to the slopes derived and the
uncertainty of the slope is observed by deriving the perturbed
DEM into slope grids and calculating the slope RMSE by the
end of the simulation.
3.6 Simulation Procedures
In conducting the simulation, the dynamic modeling tool of PC
Raster was used.
For each N, a random field of normal distribution (p= 0, c=
RMSE) was generated and was added to the original DEM, each
time with a non-repetitive random field perturbation. After Nept
times, a grid of elevation RMSE was obtained as the simulation
output. The same procedures were applied in assessing the
uncertainty of the derived product, in this case, slope. For each
N, after the perturbation, slope grid was calculated, and after
Nop times, the final slope RMSE grid was obtained.
To see the effects of resolution, the DEM was tested in three
cell sizes: 5, 10 and 20 m.
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