The use of pont quadrats, with specieal reference to stem-like organs.
Authors: Philip JR 
Published: December 2005

Refinement of point quadrat techniques leads to three integral equations.
(A) relates the variation of contact frequency with quadrat angle, f(β), to the
distribution of foliage deiisity with foliage angle, &(α) (Philip 1965a). (B), app!icab!e
to stems or stem-like organs, relates f (β) to the distribution of foliage (stem surface)
density with axial angle h(γ); and (C) connects g(α) and h(γ). A trio of integral
equations analogous to (A), (B), (C) holds for any class of axisymmetrical organs
whose members are geometrically similar. The utility nf these equations in practice
depends on the differential order of their solutions: the higher the order, the
greater the amplification of errors. The order is 2½ for (A) and 3 for (B). Reliable
results on the distribution of stem axial angles thus require very accurate data
(and hence a great deal of labour).
The kernels entering (A), (B), and (C) are basic, not only to "integral
equation" studies of the problem, but also to less ambitious approaches. Data on
these kernels are therefore presented. They are used to illustrate the inherent
difficulties in estimating h(γ). Simple methods are developed for estimating foliage
density for stems from quadrat observations at one, two, or three angles. These
are appreciably more accurate than the similar formulae (for foliage in general)
developed by Warren Wilson (1960, 1963). The reason for this is indicated.
The latter sections of the paper deal with some statistical aspects of the use
of point quadrats. For a given "relative variance" the accuracy of any f(β) observation
depends solely on the number of quadrat contacts, N. The relative variance
is typically of order unity, and it follows that the relative standard error of f(β) is
of order N-½. The accuracy of f(β) observations may therefore be determined
a priori by fixing minimum contact numbers rather than by fixing quadrat numbers.
Practical implementation of procedures of this type is discussed.
Optimal strategies for simple estimates of foliage density are considered,
the criterion being maximum accuracy for a given quantity of observational labour.
Accuracy may be improved markedly by proper distribution of contact numbers
amongst the quadrat angles. The optimal distribution is indicated. A basis for the
choice between one-, two-, and three-angle formulae is developed. The accuracy of
alternative formulae depends on the total variance arising from (i) sampling error
in the observations, and (ii) intrinsic error in the formula. The method is arbitrary
in the sense that a ruie is required to distinguish between the labour needed to
observe a fixed total number of contacts at one, two, and three quadrat angles.
The approach is illustrated by applying it to Warren Wilson's formulae. It may
be used also for the corresponding "stem" formulae and for formulae involving f(0°)
and f(90°), which are better adapted to give estimates of "mean" foliage or axial
angle as well.
The errors in estimates of "mean" foliage and axial angles due to sampling
errors in f(0°) and f(90°) are examined. The determination of "mean" axial angle
(even if the assumption of a uniform angle were valid) is inherently rather inaccurate,
especially for small values of the angle.