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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
  
profiles are variable. 
The method presented here (Dufour, 1988a) is based on the 
hypothesis that the terrain can be modelled with the help of 
Taylor’s equations (1). It will be limited to the second order 
here to allow us to describe a sufficiently large number of 
products. This function, of the z=H(x.y) type has two branches : 
the first is linear, the second is quadratic. 
z=H(x,y)=h,+ax+by+ 1 (ey? +2dxy + ey? )+E (1) 
  
  
where x-X-X, and y=Y-Y, are small and 
2 9 9 
oH oH CH OH CH 
a=——,b=—,c= wid meg, pln 5 are the 
Ox Oy ax“ OxOy ay" 
interpolation coefficients to be determined. The equations to 
determine the polynom's coefficients depend on the grid chosen 
(figurel). 
mi) 
PO) 
(6-0, 
Figure 1 : around a current central position (X0, Y0) with an 
elevation of H0, the neighboring points, having altitudes H/, 
H2, ... H8 are numbered as shown 
The equations presented can be used and are fully applicable to 
the vicinity in which the calculations of the altitudes have been 
based. This method of calculating by « pieces » supplies the 
numerical values which thus lead to discontinuities ; each grid 
node in fact generates a surface with distinct coefficients. It is 
possible to extend the expression by a third cubic term or even 
others of higher degree. This hypothesis is justified by the fact 
that it can be used to calculate as many topographic surfaces in 
this case, as the number of terms increases. 
For this square configuration of 3 lines x 3 columns, the 
coefficients 7, a. b, c, d and e of the Taylor polynom calculated 
by least squares in this vicinity (2) (Dupéret, 1989). 
3H0. 2 / : 
hy tS Uma me), (mensus +H7) 
! I 
a- —(HI«H73 H8-H3-H4-H5), b- (HI H2« H3-H5-H6-H7) 
6 6. (2) 
! 2 ! 2 
=—(n8+54)-=(H2+H6)+—(1+H3+H5+H7)-= HO 
c ~ (#84114) Ur H6) ;Un IBS HIT )-— 
d x (Hi H5-H3-H7) 
4 
eec ns )- (is ns) nu 31S 17) 10 
3. THE REPRESENTATION OF BASIC 
GEOMORPHOMETRIC VARIABLES 
The quantitative description of a topographic surface requires 
taking into account at least 5 basic parameters which are: 
811 
altitude, slope, orientation and vertical and horizontal 
curvatures. Whenever necessary, an analytic expression will be 
given for each of them based on the terrain surface model 
presented at the beginning of this publication. 
3.1 The altitude 
The altitude is a fundamental variable upon which the analysis 
is derived. It is considered as a function Z(x,y) which presents a 
dual aspect, simultaneously random and structured. The altitude 
can be considered as a random variable whose behavior can be 
studied with familiar concepts. It can be assigned values in the 
zone under investigation, each of them corresponding to an 
event whose frequency of occurrence can be used to define a 
law of probability. Increases are directly associated with effects 
like a reduction in atmospheric pressure or temperature to 
which they are linked by a simple linear equation. It contributes 
towards the complex determination of thresholds between 
morphogenic systems, the staging of these latter not 
representing a mere succession of altitudinal strips. 
It is practical to represent altitude by shading, which 
corresponds to a scaled combination of a normal topographic 
surface vector and a light vector, the calculation capable of 
being parametered via a formula (3) linking the shading value 
to the azimuth Azi, and the zenith distance zi, from each of the 
light sources used. 
Hun o ( 
Shad. = k;(-asinzig < sinAzig - bsinzig * cosAzig + coszie ) 
j= J 
i= 
3) 
The most direct representation is the graph in which altitudes 
are represented in the X-axis while the totals in number of 
points or links appear in the Y-axis for each altitude value. This 
representation can be declined by replacing the totals in number 
of links (or pixels) by the corresponding surface covered (the 
surface of the DTM's link unit being known). If the criteria for 
the Y-axis is standardized, dividing by the total number of 
pixels (or by the total area of the zone), the function represented 
becomes the altitude probability density function, or more 
precisely, the probability distribution, since the function is 
discrete. In the case of a standardized representation in the Y- 
axis, the study of graph modes can be used to represent the 
altitudinal spectrum of the zone, in the probabilistic sense of the 
word. 
Generally, and especially when the zone studied presents 
remarkably terraced surfaces, their characteristics, identifiable 
in the altitudinal spectral modes, can be quantified. Each one of 
them has three characteristic level zones (figure 2) 
corresponding to a terrace : the upper zone presenting the 
proportion of surface elements being incorporated into the level 
studied, the actual level and the lower zone being incorporated 
or transferred into the lower terrace level. This study can only 
be carried out when the slope of the altitude density function is 
not too sharp in relation to the zone studied : in this case, the 
overlap is such that no differentiation is possible from the 
altitudinal spectrum.