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
The umbra length distribution function n
U
l Fig. 6 is not continuous, especially in the long umbra interval
E
UL
, which indicates that the 12-m scan length is not adequate for obtaining a complete picture of umbra
statistics. Small umbrae dominate in all layers and the maxi-
mum of n
U
l corresponds to l ≈ 0. In the short umbra interval, F
U
l decreases rapidly with increasing l and is approximated by a straight line.
It is somewhat unexpected that α
US
has a distinct minimum at τ = 3–4. A possible explanation for this
is that branches with small leaves dominate in up-
J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 97
Fig. 6. Umbra length distribution function n
U
l left-hand axis and cumulative umbra fractional area function F
U
l right-hand axis calculated from the data in Fig. 4a and b. See Fig. 4 caption for information on scan characteristics. The length interval for n
U
l is 0.7 cm. E
US
— short umbra interval, E
UM
— medium-length umbra interval, E
UL
— long umbra interval, l
US
— maximum length of short umbra, l
UM
— maximum length of medium-length umbra, l
U
max
— maximum umbra length.
per coppice layers, causing the occurrence of a large number of small umbrae. Further down, overlapping
shadows from different shoots lead to reduction in the number of small umbrae. In deeper layers, large um-
brae dominate and n
U
decreases more slowly with τ , slowing down decrease in F
U
l. For medium-length umbrae, F
U
l is also approxi- mated by a straight line.
Ross et al. 1998 have shown that in a S. vimi- nalis coppice, relative mean PAR irradiance above the
canopy Q
r U
τ = Q
U
τ Q ,where Q
is global PAR irradiance above the canopy and Q
U
t is mean
98 J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102
PAR irradiance in umbra at the depth t, can be approx- imated by the formula Q
r U
τ = 0.16 exp−0.34τ , R
2
= 0.71. At τ = 0, Q
r U
τ = 0.16, which corre- sponds to the average fraction of diffuse sky PAR in
global PAR. Fast increase in umbra fractional area with increas-
ing τ in middle and deeper layers is caused by coop- erative shadowing, as a result of which large areas are
created where the sun is completely shaded by phy- toelements.
The number of umbrae increases rapidly at small τ and has a maximum at τ ≈ 3 for S. viminalis and at
τ = 4–5 for S. dasyclados Fig. 7. The data are fitted by the regression function
N
U
= aτ
3
+ bexpc √
τ , where a, b, c are regression constants, given in Table 3
together with the square of the correlation coefficient R
2
and standard error S
t
. It is obvious that when τ = 0, no phytoelements can
occur between the sun and the sensor, and the number of umbrae as well as umbra fractional area must be
zero. However, as in the upper canopy layer the leaf area density u
L
z is very small and leaf dimensions are finite, the number of leaves that can shade the sun
is small, and statistical methods prove unreliable. So
Fig. 7. Number of umbrae per 1 m scan length, N
U
, inside a willow coppice as a function of optical path length τ and fitted regression curves N
U
= aτ
3
+ bexpc √
τ see Table 3 for the values of constants. ⋄, solid line — S. viminalis; ×, dashed line — S. dasyclados.
the regression functions used in this study were chosen keeping in mind the asymptotic case where τ → 0, but
are applicable only when τ 0.5 for S. viminalis and τ 3 for S. dasyclados.
Smaller and narrower leaves of S. viminalis gen- erate more small umbrae compared with larger and
wider leaves of S. dasyclados Fig. 8. The differ- ence becomes less significant in deeper canopy lay-
ers where the overlapping shadows of separate shoots generate large umbra areas. For both subspecies, the
number-fraction of short umbrae i.e. the fraction of short umbrae in the total number of umbrae is larger
than 80 in all coppice layers, and short umbrae dom- inate in upper layers where long umbrae do not exist.
The number-fraction of short umbrae decreases lin- early for both subspecies, R
2
= 0.53 and R
2
= 0.43 for S. viminalis and S. dasyclados, respectively Fig. 8.
For S. dasyclados, the number-fraction of medium- length umbrae depends weakly on τ ; the square
of the correlation coefficient of the linear regres- sion shown in Fig. 8b, R
2
= 0.15. For S. vimi- nalis, the number-fraction of medium-length um-
brae increases noticeably with τ , R
2
= 0.53 Fig. 8a. As distribution density for medium-length um-
brae does not depend on τ , the increase must be caused by increase in l
UM
, which depends on canopy architecture.
J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 99
Table 3 Results of fitting N
U
by regression function see text a
b c
R
2
S
t
Salix viminalis 218.9
0.014 −3.38
0.812 2.10
Salix dasyclados 56.0
−5.57 −2.88
0.460 1.86
The maximum length of umbrae in the uppermost layers τ = 0.5 is about 5 cm and increases linearly
up to 140 cm at ground level. Larger leaves of S. dasyclados enhance the frac-
tion of large umbrae more rapidly with τ compared with narrow leaves of S. viminalis. However, as this
increase is concurrent with the constancy of the number-fraction of medium-length umbrae, it may
indicate more pronounced clumping at shoot level.
For mean umbra length, there is no major differ- ence between S. viminalis and S. dasyclados Fig. 9.
Data are fitted by the function hl
U
i = a + bτ
2
, the results of the fitting are given in Table 4. To describe
the variability of umbra length, we use the maximum lengths of short l
US
and medium-length l
UM
umbra intervals.
The maximum lengths of short and medium-length umbra intervals as the functions of optical path length
τ can be fitted by linear regression. There is no ma- jor difference between S. dasyclados and S. viminalis
in the short umbra region, l
S
increases with τ from
Fig. 8. Number-fraction of short N
US
N
U
, ⋄, solid line and medium-length N
UM
N
U
, +, dashed line umbrae as the functions of optical path length τ , linear regression, S. viminalis.
Table 4 Results of fitting hl
U
i by regression function see text a
b R
2
S
t
Salix viminalis 1.25
0.18 0.777
3.15 Salix dasyclados
2.72 0.11
0.763 2.20
2–5 cm in the upper layer to about 10 cm near the ground.
The maximum length of medium-length umbrae at the bottom of a S. viminalis coppice is about 30 cm
larger than at the bottom of a S. dasyclados coppice. The situation is similar for long umbrae: at ground
level maximum umbra length is about 50 cm larger for S. viminalis.
Division of umbrae into three length intervals is somewhat unreliable in upper coppice layers where
mutual shadowing by shoot cylinders is not present. The umbra fractional area k
U
, i.e. the total length of umbra per unit scan distance, increases rapidly with τ
until τ ≈ 6 Fig. 10. In deeper layers, umbra makes up 80–90 of total area. In middle layers τ = 3–5
k
U
for S. viminalis is twice as large as for S. dasycla- dos, probably due to different leaf area density dis-
tribution as a function of z, whereas in upper layers the difference almost disappears. However, the
fraction of umbra is smaller in S. dasyclados, indicat- ing more pronounced clumping.
100 J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102
Fig. 9. Mean umbra length hl
U
iand fitted regression curves hl
U
i = a + bτ
2
see Table 4 for the values of constants vs optical path length τ . ⋄, solid line — S. viminalis; ×, dashed line — S. dasyclados.
Experimental data have been fitted by the function k
U
τ = τ
2
a + τ
2
, where the value of the constant a = 5.90, R
2
= 0.97 for S. viminalis and a = 20.5, R
2
= 0.86 for S. dasyclados. Despite the large number of small umbrae, the tran-
sect fraction they occupy does not exceed 50–60
Fig. 10. Umbra fractional area k
U
inside a willow coppice as a function of the optical path length τ and fitted regression curves k
U
τ = τ
2
a + τ
2
. ⋄, solid line — S. viminalis; ×, dashed line — S. dasyclados.
Fig. 11 and decreases with τ to 10. Correlation between k
US
k
U
and τ is weak. The share of medium-length umbrae is 15–25 of
the total number of umbrae, but their fractional area varies greatly, from a few per cent to 70.
The role of medium-length umbrae in middle layers seems to be greater for S. viminalis, related probably
to the straight stems of the subspecies, which form
J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 101
Fig. 11. Ratio of short umbra fractional area k
US
to total umbra fractional area k
U
, k
US
k
U
vs optical path length τ fitted by a linear regression. a S. viminalis; b S. dasyclados.
a more expressed cylindrical shoot structure in upper canopy layers. Also, leaf area density function has a
wider maximum for this subspecies compared with S. dasyclados, which indicates taller shoots.
Another indicator of different canopy architecture is the ratio of the area of long umbrae to total umbra
area. Despite the fact that the average length of umbra is smaller in S. dasyclados, the species produces more
long umbra in deeper canopy layers 50–60 of total umbra area compared with S. viminalis less than 50
of total umbra area.
5. Concluding remarks
