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The transformation involves scale change, rotation, and translation. The mathematical details of the transformation, 
including the derivation of the transformation equations, are outside the scope of this presentation. Nevertheless the 
transformation equation is given below (equation 1) and this was used to compute the ground coordinates of each point 
from its measured photocoordinates. 
E = ax-by-* T, (1a) 
N = aytbx * T, (1b) 
Where 
r = Eastings Coordinate of point in the ground system 
N = Northings Coordinate of point in the ground system 
X = X - Coordinate of point measured on the photograph 
y = Y - Coordinate of point measured on the photograph 
a, b, 
TT, 3 = Transformation parameters 
For the two dimensional comformal coordinate transformation a minimum of two control points are required; but where more 
than two control points exist (as in the present case) then a "least squares" solution could be obtained, yielding more 
accurate values of the transformation parameters as well as offer a check on observational errors. First the transformation 
parameters were computed using the known ground coordinates and measured photocoordinates of the control points. 
The transformation parameters were then used to compute the ground coordinates (E, N) of all other points defining the 
outline of the ravine, from their measured photocoordinates (x, y). 
5.2 Computation of Areal Extent of the Ravine 
After the transformation, ground coordinates were obtained for all points defining the outline of the ravine. The ground 
area covered by the ravine for each of the years was accordingly computed from these coordinates using the following 
formulae (equation 2). The computations could be performed and the same result obtained using the procedure shown in 
table 5. 
ENEN-+EN E N'EN-C Qa) 
NE +NEANE ..N EINE =C (2b) 
QC CEA Qc) 
Where: 
A = The required area 
E = Easting Coordinate of points 
N Northing Coordinate of points 
The area covered by the portion ofthe ravine under study in 1969 was 5.94 hectares; in 1978, 11.76 hectares; and in 1988, 
17.03 hectares. 
5.3 Computation of Rates of Erosion 
By differential analysis the total land area encroached upon by the ravine, and the annual rate of encroachment between 
1969 and 1988 could be obtained. Thus by simple subtraction it was deduced that a total land area of 5.82 hectares was lost 
to that portion ofthe ravine between 1969 and 1978; this gave an average rate of 0.582 hectares per annum. By differenti- 
ating the area with respect to the length the total distance encroached upon between 1969 and 1978 was obtained; and from 
this a linear rate of 26.80 metres per annum was computed. 
h the 1988 computations; this is because one ofthe control points (Point B) had been 
removed by erosion and its location now formed part of the ravine. However a total area of 17.03 hectares was obtained for 
the ravine's extent. The net area of about 5.27 hectares was lost at an average rate of 0.480 hectares per annum, between 1978 
and 1988. The linear rate for the same period was calculated as 20.87 metres per annum. 
The situation was slightly different wit 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 393