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The foliage density equation revisited

  • John Boris Miller (a1)

Abstract

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 f, a function of angle estimated from measurements, to the unknown function g. The explicit formula for g is known and depends upon f and its first three derivatives; the operator f →, g is unbounded, and the problem is ill posed.

In this paper we give the form of g when f 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 f from numerical data. To assess the reliability of the method we discuss the convergence of Fourier series representations of f and g.

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Copyright

References

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[1]Anderssen, R. S. and Jackett, D. R., “Computing the foliage angle distribution from contact frequency data”, in Computational Techniques and Applications: CTAC 1983 (eds. Noye, J. and Fletcher, C.), (North-Holland, Amsterdam, 1984), 863872.
[2]Anderssen, R. S. and Jackett, D. R., “Linear functionals of foliage angle density”, J. Austral. Math. Soc. Ser. B 25 (1984), 431442.
[3]Anderssen, R. S., Jackett, D. R. and Jupp, D. L. B., “Linear functionals of the foliage angle distribution as tools to study the structure of plant canopies”, Aust. J. Botany 32 (1984), 147156.
[4]Erdélyi, A.et al., Higher transcendental functions, Vol. 2 (Bateman Manuscript Project, McGraw-Hill, 1953).
[5]Miller, J. B., “An integral equation from phytology”, J. Austral. Math. Soc. 4 (1964), 397402.
[6]Miller, J. B., “A formula for average foliage density”, Aust. J. Botany 15 (1967), 141144.
[7]Philip, J. R., “The distribution of foliage density with foliage angle estimated from inclined point quadrat observations”, Aust. J. Bot. 13 (1965), 357366.
[8]Wilson, J. Warren (with Appendix by J. E. Reeve), “Inclined point quadrats”, The New Phytologist 59 (1960), 18.
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