A formula for average foliage density

1967 ◽  
Vol 15 (1) ◽  
pp. 141 ◽  
Author(s):  
JB Miller
Keyword(s):  
Fourier Analysis ◽  
Density Function ◽  
Frequency Function ◽  
Contact Frequency ◽  
Foliage Density ◽  
Point Quadrat

Point quadrat readings, taken to assess density of foliage in a plant, determine a contact frequency function f. A formulais given expressing the average foliage density of the plant in terms of f. This is compared with the work of Philip, who determined from f using a foliage angle density function g and an approximate Fourier analysis.

The distribution of foliage density with foliage angle estimated from inclined point quadrat observations

1965 ◽  
Vol 13 (2) ◽  
pp. 357 ◽  
Author(s):  
JR Philip
Keyword(s):  
Integral Equation ◽  
Standard Deviation ◽  
Leaf Angle ◽  
Frequency Data ◽  
Sampling Errors ◽  
Contact Frequency ◽  
Foliage Density ◽  
The Mean ◽  
Point Quadrat ◽  
The Given

Estimation of the distribution of foliage density with foliage angle from contact frequency data for a number of quadrat inclinations involves solution of a Fredholm integral equation of the first kind. The kernel is known from the work of Warren Wilson and Reeves, and the observed contact frequencies constitute the given function f(β). The solution is g(α), the foliage angle density function. f (β) is known at only a finite number of points, and each value contains inevitable sampling errors. The structure of the solution is such that g(β) is consequently subject to serious errors. A technique involving smoothing of the data is developed with the aim of minimizing this difficulty. The technique is critically discussed and applied to observations of Warren Wilson on lucerne leaves. The analysis indicates that the distribution of leaf angle is roughly symmetrical about the mean angle, with a standard deviation of about 15°.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

Linear Functionals of the Foliage Angle Distribution as Tools to Study the Structure of Plant Canopies

1984 ◽  
Vol 32 (2) ◽  
pp. 147 ◽  
Author(s):  
RS Anderssen ◽  
DR Jackett ◽  
DLB Jupp
Keyword(s):  
Synthetic Data ◽  
Angle Distribution ◽  
Linear Functionals ◽  
General Transformation ◽  
Area Index ◽  
Plant Canopies ◽  
Contact Frequency ◽  
Numerical Experimentation ◽  
Actual Evaluation ◽  
Foliage Density

In 1967, Miller showed how average foliage density could be computed from contact frequency data. It formalized mathematically the idea posed earlier by Warren Wilson of estimating the leaf area index as a linear combination of measured values of the contact frequency. Recently, it has been shown that Miller's result is a special case of a general transformation that allows linear functionals defined on the (generally unknown) foliage angle distribution (foliage angle functionals) to be evaluated as linear functionals defined on the (measured) contact frequency (contact frequency functionals). This result has important consequences for the use of foliage angle functionals in the study of the structure of plant canopies. For example, it allows Warren Wilson's idea to be extended to the evaluation of such functionals, and thereby simplifies greatly their actual evaluation. In this paper, we first motivate and review the use of foliage angle functionals in the study of plant canopies; then we introduce new functionals (the segmented foliage density and the moments); and finally, we use numerical experimentation with synthetic data to illustrate the advantages of having formulas for the foliage angle functionals of interest that are defined explicitly in terms of the (measured) contact frequency.

The use of pont quadrats, with specieal reference to stem-like organs.

1966 ◽  
Vol 14 (1) ◽  
pp. 105 ◽  
Author(s):  
JR Philip
Keyword(s):  
Integral Equations ◽  
Sampling Error ◽  
Practical Implementation ◽  
Relative Standard Error ◽  
Sampling Errors ◽  
Relative Variance ◽  
Axial Angle ◽  
Relative Standard ◽  
Contact Frequency ◽  
Foliage Density

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.

The LDE-Type Flare Occurrence (1969-1993)

1994 ◽  
Vol 144 ◽  
pp. 279-282
Author(s):  
A. Antalová
Keyword(s):  
Time Series ◽  
Fourier Analysis ◽  
Sunspot Cycle ◽  
List Type ◽  
Type Number ◽  
Solar Cycles ◽  
Type Iv ◽  
Long Duration ◽  
Cycle Maximum ◽  
Flare Index

AbstractThe occurrence of LDE-type flares in the last three cycles has been investigated. The Fourier analysis spectrum was calculated for the time series of the LDE-type flare occurrence during the 20-th, the 21-st and the rising part of the 22-nd cycle. LDE-type flares (Long Duration Events in SXR) are associated with the interplanetary protons (SEP and STIP as well), energized coronal archs and radio type IV emission. Generally, in all the cycles considered, LDE-type flares mainly originated during a 6-year interval of the respective cycle (2 years before and 4 years after the sunspot cycle maximum). The following significant periodicities were found:• in the 20-th cycle: 1.4, 2.1, 2.9, 4.0, 10.7 and 54.2 of month,• in the 21-st cycle: 1.2, 1.6, 2.8, 4.9, 7.8 and 44.5 of month,• in the 22-nd cycle, till March 1992: 1.4, 1.8, 2.4, 7.2, 8.7, 11.8 and 29.1 of month,• in all interval (1969-1992):a)the longer periodicities: 232.1, 121.1 (the dominant at 10.1 of year), 80.7, 61.9 and 25.6 of month,b)the shorter periodicities: 4.7, 5.0, 6.8, 7.9, 9.1, 15.8 and 20.4 of month.Fourier analysis of the LDE-type flare index (FI) yields significant peaks at 2.3 - 2.9 months and 4.2 - 4.9 months. These short periodicities correspond remarkably in the all three last solar cycles. The larger periodicities are different in respective cycles.

The structure of amorphous and polycrystalline materials using energy-filtered RDF analysis

1992 ◽  
Vol 50 (2) ◽  
pp. 1190-1191
Author(s):  
David Cockayne ◽  
David McKenzie
Keyword(s):  
Electron Diffraction ◽  
Diffraction Pattern ◽  
Density Function ◽  
Electron Diffraction Pattern ◽  
Polycrystalline Materials ◽  
Nearest Neighbour ◽  
X Ray ◽  
Selected Area Electron Diffraction ◽  
Reduced Density ◽  
System F

The technique of Electron Reduced Density Function (RDF) analysis has ben developed into a rapid analytical tool for the analysis of small volumes of amorphous or polycrystalline materials. The energy filtered electron diffraction pattern is collected to high scattering angles (currendy to s = 2 sinθ/λ = 6.5 Å-1) by scanning the selected area electron diffraction pattern across the entrance aperture to a GATAN parallel energy loss spectrometer. The diffraction pattern is then converted to a reduced density function, G(r), using mathematical procedures equivalent to those used in X-ray and neutron diffraction studies.Nearest neighbour distances accurate to 0.01 Å are obtained routinely, and bond distortions of molecules can be determined from the ratio of first to second nearest neighbour distances. The accuracy of coordination number determinations from polycrystalline monatomic materials (eg Pt) is high (5%). In amorphous systems (eg carbon, silicon) it is reasonable (10%), but in multi-element systems there are a number of problems to be overcome; to reduce the diffraction pattern to G(r), the approximation must be made that for all elements i,j in the system, fj(s) = Kji fi,(s) where Kji is independent of s.

FOURIER ANALYSIS OF HYSTERETIC DAMPING MECHANISMS

1985 ◽  
Vol 46 (C10) ◽  
pp. C10-171-C10-173 ◽  
Author(s):  
D. N. BESHERS ◽  
V. F. CORONEL
Keyword(s):  
Fourier Analysis ◽  
Hysteretic Damping
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