International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004
3.6 The horizontal curvature of slope lines
The horizontal curvature of the lines of greatest slope can be
calculated in the same way as the horizontal curvature of the
contour lines. The line /p=0 is of particular interest as is that of
lines /Zh=0 because it defines the topographic surface zones
e the left-hand zones where /p<0, for which an observer
who follows a slope line towards the valley constantly
moves diagonally towards his right,
e the right-hand zones where /p>0, for which an observer
who follows a slope line towards the valley constantly
moves diagonally towards his left.
It can be expressed as (7) :
2 2
d(a -b )-*ab(e-c)
T (7)
€
These geomorphometric parameters must lead to the description
of how the morphogenic systems function even if other factors
such as vegetation or soils intervene in the interrelations
between shapes and processes. In addition, the establishment of
more complex adapted indicators becomes possible to describe
the topographic surface.
4. REPRESENTATION OF COMPLEMENTARY
GEOMORPHOMETRIC VARIABLES
4.1 Other various curvatures
Several magnitudes indicate the manner in which the
topographie surface bends in vertically, but this time
independently from the line of the greatest slope. The results
obtained with these indicators may even contribute information
to a geological study of the area.
e The Laplacien's equation can be expressed as
AH =c+e,
e The average quadratic curvature is equal to
C -NXc «2d «e ,
my
^ 2
e Thetotal curvature is C, 2 d' —ce,
e Altitude magnitude, sometimes called relief, is the
difference between the highest altitude and the lowest
altitude in the 3*3 mobile. It is assigned to the central node
. This indicator is very close to the slope: it is simply more
concrete.
e In a 3*3 convolution window, the entrenchment is the sum
weighted by the distance in levels between a grid link and
its eight neighbors ; its expression is
8 ;
Entrenchment — Y (H l^ H 0 ) xd j^
i=1
A geomorphological typology can thus be defined within a
family of terrain shapes : horizontal plains (without significant
relief), inclined slopes, entrenched sites (valleys, depressions,
basins), dominant sites (ridges, summits), passes.
4.2 Indicators derived from slopes
The study of this slope value graph reveals significant
indicators :
814
e Minimum, maximum, average and absolute magnitude
(max-min) slopes
e Quantiles Q5, OI0,... and relative magnitudes (Q95-Q5,
Q90-Q10,...)
e Graph symmetry and histogramm curtosity.
For example, these indicators have been used to compare two
types of relief: one of recent erosion (“young” relief in
Briançon, the other, older, in the Vosges (figure 7). A study of
the function of slope distribution can also be carried out, as was
the case for altitude.
|
4 | | P^
M ^
D Nw y ]
A i
= rs s ; ; 5 |
- 4
=
Figure 7 : slope graph : DTM of Briangon (left), data from
IGN’s BD TOPO® 2003 ; DTM of the Vosges (right), data
from IGN's BD ALTI®, 2003 - X-axis scale : from 0? to 60°,
Y-axis scale : from 0% to 2% of the number of points in the
DTM
4.3 Indicators derived from orientation
4.3.1 Average slope orientations N/ NE/ E/SE/S/SW/
W / NW : As an example, average slope orientations indicate
the following : during the ice ages, water froze on the slopes of
the relief. The ubac (slope face with northern exposure) is not
subject to the warmth of the sun and the water remains frozen.
Inversely, the adret (slope face with southern exposure) is
reheated every day by the sun, which leads to a daily
freeze/thaw cycle. During the thaw, water flows off; it digs at
the slope which increases the inclination. The slopes with
northern exposure are thus not as steep as slopes with southern
exposure in the countries with temperate climates located in the
northern hemisphere. This phenomenon is similar to slopes with
western exposure, heated by the afternoon sun, more eroded
than the slopes with eastern exposure, heated by the cooler
morning sun.
4.3.2 Orientations spectrum : The analysis of the
orientations spectrum of a zone is used to assess the manner in
which they are distributed. This indicator was tested on a zone
in the Vosges (figure 8). The pronounced dissymmetry of a
spectrum can be explained by the fact that certain zones of the
relief resemble a rooftop : western slope faces are « longer »
than eastern slope faces, which explain why there are more
points with western exposure than with eastern exposure. The
spectrum is thus typical of the particularity of the zone.
Furthermore, average SE slope orientations are much less sharp,
but the dissymmetry stops there.
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