The foliage density equation revisited

John Boris Miller
1986 The Journal of the Australian Mathematical Society Series B Applied Mathematics
The foliage density equation is the means by which the foliage density g in a leaf canopy, as a function of the angle of inclination of the leaves, is to be estimated from discrete data gathered using photometric methods or point quadrats. It is an integral equation relating /, a function of angle estimated from measurements, to the unknown function g. The explicit formula for g is known and depends upon / and its first three derivatives; the operator / •-» g is unbounded, and the problem is
more » ... posed. In this paper we give the form of g when / is a trigonometric polynomial, extending earlier results due to J. R. Philip. This provides a means of estimating g without directly estimating the derivatives of / from numerical data. To assess the reliability of the method we discuss the convergence of Fourier series representations of / and g. with the rather complicated kernel function K given in (5) below. The equation relates two functions / and g, / being deemed known and g unknown. The function /, called the contact frequency function, is estimated from numerical data obtained from a specified spatial region of the canopy of leaves of a plant (for example, using point quadrats); the function g, called the foliage density function, describes the foliage density distribution: g(a)da is the contribution to foliage density due to foliage inclined at angles between a and a + da to the horizontal. The form of K is due to J. Warren Wilson and J. E. Reeve [8], the integral