DIGITAL ELEVATION MODEL FOR PHOTOGRAMMETRIC MEASUREMENTS OF SOIL EROSION 
Dr. Julian C. Barbalata, professor 
Dr. Roland Lebel, professor 
Université de Moncton, ESF 
165, Boulevard Hébert, Edmundston 
New Brunswick, E3V 2S8, Canada. 
Commission IV 
ABSTRACT 
A mathematical model was developped for both stereophotogrammetric measurements and digital elevation 
model, to monitoring soil erosion. The model is based on projective methods and simultaneously adjusts 
ground distances and image plate coordinates by the least squares method. This formulation provides 
complete flexibility in the weighting of the photogrammetric observations and on the type of control 
necessary in the scaling and the orientation of the model. The end product of the photogrammetric process 
is a list of coordinates which define the spatial position of a finite number of discrete points. An 
interpolation procedure was used for modeling functions of two independent variables with irregular con- 
trol distribution points. Practical experiments were carried out with UMK-1318 camera in clear cutting 
areas of an experimental forest. The photogrammetric measurements were performed on a Wild Aviolyt BC2 
plotter and were processed by the author's programs DISTANCE and SURFACE. 
KEY WORDS : Analytical Photogrammetry, Theory, DEM, Change Detection. 
The observation equations which are appended to 
INTRODUCTION photogrammetric observation equations, are gene- 
rated by a number "p" of slope distances "d", 
Soil erosion which leads to a decrease in soil measured by precise surveying methods between the 
productivity, is a major problem in forestry points which appear on the photographs. The equa- 
areas, especially after a clear cutting. tion had to be transferred into a  Cartesian coor- 
(Sneddon &Jordan,1983). dinate system and are given in linearized form 
(Barbalata, 1980) by the equation: 
Microtopography and movement of soil can be de- 2 
termined by measuring the elevation above a datum v+DAd=L (1) 
of a series of points defining a surface. If two 
measurements are made at different times, the where: 
change in elevation indicates whether erosion or Vv = residual vector in slope distance "d" equa- 
deposition is occuring as well as the volume of a tions, 
of soil moved (Barbalata, 1972; Frasier & Hooper, D = coefficient matrix of the Jacobians in slope 
1983; Jackson & Ritchie, 1988; Lyon & All, 1986) distance equations, 
B= correction vector of coordinates of points 
However, topographic methods of measurement which define the ends of each distance "d". 
(transit or level surveys and erosion pins) are L 7 vector of discrepancy in slope distance equa- 
cumbersome to implement and often of insufficient tions. 
frequency or accuracy to detect small changes re- 
sulting from erosion process.  Close- range photo- The complete mathematical model is obtained by 
grammetry, on the other hand, offers a means of combining the photogrammetric linearized  obser- 
obtaining accurate measurements of eroded sur- vation equations with equation (1): 
faces and the differencies between these surfaces 
at different times, can give an indication of v B B 0 A E 
erosion or of deposition patterns and volumes 9 9 - 
(Welch & Jordan, 1983). v |+ |B 0 0 AH |= |6 (2) 
> e 
For these reasons, close-range photogrammetry was v 0 0 Di Ad L 
used to measure soil erosion resulting from fo- 
restry activities and a specific analytical pho- where: 
togrammetric model was developped to determine v = residual vector in collinearity equations, 
the ground coordinates of a grid points. Based on Ÿ = residual vector in constraint equations, 
this three-dimensional coordinate file, a digital B = Jacobians in collinearity equations for orien 
elevation model was developped and used to define = tation parameters, 
the surface at a certain time. B = Jacobians in collinearity equations for 
ground object points, 
THE MATHEMATICAL MODEL OF COMBINED PHOTOGRAMME- B = Jacobians in constraint equations. 
TRIC AND GEODETIC ONSERVATIONS À = correction vector of the orientation parame- 
- ters, 
The idea of the bundle adjustment is to use the 4^7 correction vector of the ground object coor- 
well known collinearity equations in order to dinates. 
obtain a unique solution for the system of obser- 
vation equations by the least squares method (Ar- The normal equations for a least squares solution 
menakis & Faig, 1988; Barbalata, 1979, 1988-b, are then given by the following expression: 
1988-c, 1990; Brown,1976). ° ^A A 
A mathematical model is presented in this paper, NN N A C+C 
involving the bundle adjustment associated with ^T =. -€—I-—l. . (3) 
A^ 
geodetic distance measurements. N N + NJ C +Cq 
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