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Linear functionals of foliage angle density

Published online by Cambridge University Press:  17 February 2009

R. S. Anderssen  and
D. R. Jackett
D. R. Jackett
Affiliation:
Division of Mathematics and Statistics, CSIRO, P.O. Box 1965, Canberra City, A.C.T. 2601.
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Abstract

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Knowledge about the foliage angle density g(α) of the leaves in the canopy of trees is crucial in foresty mangement, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency f(β) by solving a version of the first kind Fredholm integral equation derived by Reeve (Appendix in Warren Wilson [22]). However, for inference purposes, the practitioner uses functionals defined on g(α), such as the leaf area index F, rather than g(α) itself. Miller [12] has shown that F can be computed directly from f(β) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on g(α). The key is the existence of an alternative inversion formula for the integral equation to that derived by Miller [11].

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 4 , April 1984 , pp. 431 - 442
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions (Dover, New York, 1965).Google Scholar
[2]Anderssen, R. S. and Bloomfield, P., “A time series approach to numerical differentiation”, Technometrics 16 (1974), 69–75.CrossRefGoogle Scholar
[3]Anderssen, R. S., “Stable procedures for the inversion of Abel's equation”, J. Inst. Math. Appl. 17 (1976), 329–342.CrossRefGoogle Scholar
[4]Anderssen, R. S., “On the use of linear functionals for Abel-type integral equations in applications”, in The application and numerical solution of integral equations (eds. Anderssen, R. S., de Hoog, F. R. and Lukas, M. A.), (Sjthoff and Noordhoff, The Netherlands, 1980).CrossRefGoogle Scholar
[5]Anderssen, R. S., Jackett, D. R. and Jupp, D. L., “On the use of linear functionals of the foliage angle distribution in the study of hte architecture of plant canopies”, submitted to Austral. J. Bot.Google Scholar
[6]de Hoog, F. R., “Review of Fredholm equations of the first kind”, in The application and numerical solution of integral equations (eds. Anderssen, R. S., de Hoog, F. R. and Jukas, M. A.), (Sijthoff and Noordhoff, The Netherlands, 1980).Google Scholar
[7]de Wit, C. T., “Photosynthesis of leaf canopies”, Argiculture Research Report No. 663, Centre for Argicultural Publications and Documentation, Wageningen, The Netherlands (1965).Google Scholar
[8]Erdélyi, A. et al., Tables of integral transforms–Bateman manuscript project (Vol. 2, McGraw-Hill, New York, 1954).Google Scholar
[9]Golberg, M. A., “A method of adjoints for solving some ill-posed equations of the first kind”, Appl. Math. Comp. 5 (1979). 123–130.CrossRefGoogle Scholar
[10]Lukas, M. A., “Regularization” in The application and numerical solution of integral equations (eds. Anderssen, R. S., de Hoog, F. R. and Lukas, M. A.), (Sijthoff and Noordhoff, The Netherlands, 1980).Google Scholar
[11]Miller, J. B., “An integral equation from phytology”, J. Austral. Math. Soc. 4 (1963), 397–402.CrossRefGoogle Scholar
[12]Miller, J. B., “A formula for average foliage density”, Austral. J. Bot. 15 (1967), 141–144.CrossRefGoogle Scholar
[13]Philip, J. R., “The distribution of foliage density with foliage angle estimated from inclined point quadrat observations”, Austral. J. Bot. 13 (1965), 357–366.CrossRefGoogle Scholar
[14]Philip, J. R., “Some integral equations in geometrical probability”, Biometrika 53 (1966), 365–374.CrossRefGoogle Scholar
[15]Ross, J., The radiation regime and the architecture of plant stands (Junk, W., The Hague, 1981).CrossRefGoogle Scholar
[16]Smith, J. A. and Oliver, R. E., “Effects of changing canopy directional reflectance on feature selection”, Appl. Optics 13 (1974), 1599–1604.CrossRefGoogle ScholarPubMed
[17]Smith, J. A., Oliver, R. E. and Berry, J. K., “A comparison of two photographic techniques for estimating foliage angle distribution”, Austral. J. Bot. 25 (1977), 545–553.CrossRefGoogle Scholar
[18]Sneddon, I. N., Mixed boundary value problems in potential theory (North-Holland, Amsterdam, 1966).Google Scholar
[19]Suits, G. H., “The calculation of the directional reflectance of a vegetative canopy”, Rem. Sens. Environ. 2 (1972), 117–125.CrossRefGoogle Scholar
[20]Wahba, G., “Practical approximate solutions to linear operator equations when the data are noisy”, SIAM J. Numer. Anal. 14 (1977), 651–667.CrossRefGoogle Scholar
[21]Wahba, G., “Ill-posed problems: numerical and statistical method for mildly, moderately and severely ill-posed problems with noisy data”, University of Wisconsin-Madison, Department of Statistics, Technical Report # 595 (1980).Google Scholar
[22]Wilson, J. Warren, “Inclined point quadrats” (with Appendix by J. E. Reeve), The New Phytologist 59(1960), 1–8.CrossRefGoogle Scholar
[23]Wilson, J. Warren, “Estimation of foliage denseness and foliage angle by inclined point quadrats”, Austral. J. Bot. 11 (1963), 95–105.CrossRefGoogle Scholar