Statistical treatment Directory UMM :Data Elmu:jurnal:A:Agricultural & Forest Meterology:Vol100.Issue2-3.Febr2000:

J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 95 Table 2 Coppice height h c , leaf area index L, and layer thicknesses on selected days Salix dasyclados Salix viminalis Date 30 June 97 21 July 97 h c cm 475 610 L 5.2 5.2 Layer thickness cm i 200 240 ii 65 120 iii 210 250 2. The middle foliage layer can be modelled as a horizontally homogeneous turbid plate medium where individual branches are indistinguishable. 3. The lowest foliage layer is almost leafless and contains mostly cylindrical, nearly vertical stems. Some phytometrical characteristics of S. viminalis and S. dasyclados coppices, measured on selected days, are given in Table 2. The vertical distribution of leaf area density u L z for both subspecies shows that in the upper layer with a thickness of about 200 cm, leaf area density increases almost linearly, accounting for 60–80 of total leaf area. Compared with S. dasyclados, S. viminalis leaves are longer and narrower. The mean area of a S. vim- inalis leaf is about 4.0 cm 2 , that of a S. dasyclados leaf, about 4.9 cm 2 . The variability of leaf dimensions is large for both subspecies and can be approximated by lognormal distribution Ross and Ross, 1998. The stems of a S. viminalis shoot can be modelled as nearly vertical cylinders, unlike the stems of S. dasy- clados which are less vertical and more convex.

3. Statistical treatment

Let us assume some horizontal area A at the depth L. This area can be divided into three parts: A S — sunfleck area, A P — penumbra area, and A U — um- bra area, their fractional areas being k S L,h = A S A, k P L,h = A P A, and k U L,h = A U A, respectively. The measured area of an umbra caused by the shadow of a single leaf Fig. 1 depends on the following factors: 1. angular diameter of the solar disc 32 ′ ; 2. leaf shape and area characterized by its effective diameter; 3. leaf orientation characterized by the normal of leaf surface R L = θ L ,ϕ L , where θ L and ϕ L are the inclination angle and the azimuth angle, respectively; 4. direction of sunrays R S = θ S ,ϕ S , where θ S and ϕ S are the solar inclination and azimuth angles, re- spectively. The leaf area S L , projected onto a sur- face perpendicular to sun’s rays, is determined by the cosine of the angle between the two directions: cos ∠R S , R L = cos θ S cos θ L + sin θ S sin θ L ×cos ϕ S ndash ϕ L ; 5. distance between the leaf and the sensor, z L − z, which determines the effective angular diameter of the leaf area projection S ′ L . In deeper layers, total shading may be caused by phytoelements located at different distances from the sensor. This cooperative shadowing effect depends on canopy architecture, 3D distribution of branches and stems, size and distribution of individual leaves on shoot cylinders, etc. These factors cause an extremely great variability of umbra area and make it impossible to calculate it analytically. Therefore a statistical treatment of umbra in a plant canopy is needed. Let N U L,h be the number of umbrae per 1 m of scan at the depth L and solar elevation h; these um- brae have different lengths varying from 0 to l U max , and their lengths are statistically distributed over the interval [0,l U max ]. Let n U l be umbra length distribu- tion function, i.e. the number of umbrae with a length between l and l + 1l per 1 m of scan, and l Umax X l=0 n U l = N U . 1 In our case, the distance between two consecutive mea- surements, 1l = 0.7 cm. Mean umbra length is hl U i = P l Umax l=0 l n U l N U 2 and the fractional area of umbra inside coppice equals umbra length per 1 m of scan: k U = l Umax X l=0 l n U l. 3 From 2 and 3 it follows that k U = hl U iN U . 96 J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 For analysis of umbra length variability, the umbra length distribution function n U l and the cumulative umbra fractional area function F U are used. F U l de- notes the fraction of a transect occupied by umbrae larger than l, or the fractional area of umbrae with a length from l to l U max : F U l = l Umax X l=0 ln U l or in the integral form F U l = Z l Umax l ln ∗ U ldl, 4 where n ∗ U l = lim 1l→0 n U 1l . From 3 and 4 it follows that F U 0 = k U , F U l U max = 0. 5 Analysis of F U l and n U shows that it is reasonable to divide the interval between 0 and l U max into three parts: Fig. 6a,b 1. E US , short umbra interval [0,l US ], k US L,h = F U 0–F U l US ; 2. E UM , medium-length umbra interval [l US ,l UM ], k UM L,h = F U l US –F U l UM ; 3. E UL , long umbra interval [l UM ,l U max ], k UL L,h = F U l UM . Fractional umbra area for each of the intervals k US , k UM , k UL is defined so that k U = k US + k UM + k UL . 6 It should be noted that the location of points l S and l M separating the intervals and, consequently, frac- tional areas k US , k UM , and k UL are somewhat arbi- trary and subjective. In umbra length intervals E US and E UM , F U l is approximated by a straight line us- ing the value of F U l at the endpoints of the section. The slope of the line in the short umbra interval α US and in the medium-length umbra interval α UM can be calculated from the formulae α US = F U l US − k U l US 7 α UM = F U l UM − F U l US l UM − l US . 8 The slope of F U l is associated with the number of umbrae in that particular interval, differentiaton of 4 yields d dl F U l = −ln ∗ U l ≈ − 1 1l ln U l. 9 Typically, the umbra length distribution function n U l decreases with l, while the decrease is maximum in the small umbra interval E US and, according to definition, the cumulative umbra fractional area function F U l decreases continuously. Therefore both α US and α UM are negative and α US α UM . It is appropriate to assume that short umbrae are caused by individual leaves or a few leaves with um- brae being separated by rays passing through shoots. Hence their lengths are determined by the effective di- ameter of a leaf, while the lengths of medium-length umbrae are determined by the dimensions of shoot cylinders. Large umbrae occur only in deeper canopy layers as a result of cooperative shadowing by shoots and individual leaves. Analysis of experimental data shows that umbra length characteristics are largely determined by three factors: solar elevation h, depth of the measurement level characterized by Lz, and leaf, plant and canopy architectural characteristics. The first two factors de- termine the optical path length of direct solar radiation in plant canopy τ = Lsin h, expressed in the units of L. Hereafter umbra length characteristics will be con- sidered as the functions of the optical path length τ , assuming that it is the key factor which determines the absorption pattern of direct sunlight.

4. Results and discussion