In the geomorphological literature, glacial valleys are often described as parabolic or U-shaped. Accordingly, the author has given a methodReference Svensson 1 for computing the parabola which gives the best approximation for the cross-section of any glacial valley. However, as far as is known to the author, no attempt has been made to determine the type of curve which best represents the cross-section of a glacial trough, and this paper presents some points of view on this subject.
If we start at the lowest point of a uniform glacial valley, without hiding the underground deposits too much, we can express the increasing height of the valley wall as a function of the horizontal distance from this lowest point. These values for the height increase steadily, and form a monotonic series of figures. If we suppose the cross-section of the glacial valley to be approximately of the form y=ax b (each half of the section being treated separately with x measured as positive from the central line of the valley), then the question is, in each separate case, to determine the coefficient a and the exponent b in such a way that the approximation will be the best possible. The exponent b determines whether the cross-section of the valley is parabolic or not.
The above power law can be written
If we put Y = log y and X = log x and A = log a, then the equation connecting Y and X becomes
To compute the constants A and b in this linear relation, the method of Ieast squares can be applied. By this method the sum of the squares of the differences between the observed values of Y and those calculated from the formula is minimized. This minimizing of the sum of squares of the residuals gives a unique solution for A and b:
where n is the number of observations.
The author has determined the parameters for the glacial valley Lapporten in northern Sweden using the above method. In order to eliminate local post-glacial unevennesses of the slope as far as possible, the x values are the averages for three cross-sections, taken at a short distance from each other.Footnote *
The calculation resulted in the following equations:
From the equations deduced, it can be seen that the shape of the trough is not quite symmetrical. The close agreement with a parabola is, however, evident. Particularly in the first equation, the deviation of the exponent from the value 2 is only slight.
It would not be appropriate to draw a general conclusion for all glacial valleys from this example. It is to be hoped that a number of valleys could be examined and the results correlated, from which the question of their type could be discussed.
It might seem unimportant to find out the exact shape of the well-known glacial valley; but it is the author’s opinion that if anything is to be revealed from the shape of the glacial rock surface concerning ice dynamics at the bottom of glaciers and ice sheets—and knowledge on this subject is still incomplete—discussions must proceed on the basis of accurately expressed observations.
MS. received 28 November 1958
