Agricultural and Forest Meteorology 100 2000 89–102
Statistical treatment of umbra length inside willow coppice
J. Ross
∗
, M. Mõttus
Tartu Observatory, Tõravere, 61602 Tartumaa, Estonia Received 31 March 1999; received in revised form 7 October 1999; accepted 13 October 1999
Abstract
Measurements of the statistical characteristics of umbra and sunflecks at different depths inside a willow coppice — Salix viminalis and Salix dasyclados — were carried out at Tartu Observatory, Tõravere, Estonia, in 1997. A new instrument, the
sunfleck indicator, constructed by M. Sulev, was used. This instrument, moving perpendicularly to rows in the horizontal direction, counts the number and length of sunflecks and umbrae at a level where downward cumulative leaf area index is
L. During statistical data processing, several umbra characteristics — umbra length distribution function, mean number of umbrae, mean umbra length, fractional area of umbra, etc. — were calculated at different measurement heights as the functions
of the optical path length τ = Lsin h, where h is the solar elevation.
The number of umbrae N
U
increases rapidly at small τ , has a maximum at τ ≈ 3–4 and decreases slowly with further increase in τ . This interrelationship was fitted by an exponential function. Umbra length distribution function can be divided
into three regions: small umbrae 0–10 cm in length, medium-length umbrae 10–20 cm and long umbrae up to 100 cm. At all depths the number of small umbrae exceeds the number of medium-length and long umbrae by 3–10 times. The fractional
area of umbra k
U
τ increases with τ and was approximated by a rectangular hyperbola. In lower layers τ = 8–12 k
U
τ reaches 0.85–0.90 and these layers are dominated by umbra. ©2000 Elsevier Science B.V. All rights reserved.
Keywords: Willow coppice; PAR variability; Statistical umbra characteristics
1. Introduction
Inside a plant canopy, PAR is characterized by great spatial and temporal variability Ross, 1981; Chazdon,
1988; in some middle canopy layer, PAR global irra- diance may vary about 50-fold with a rapid sequence
of sunflecks, penumbra and umbra.
Statistical treatment of umbra and sunflecks is closely related to the progress of new comprehen-
sive computer models of canopy photosynthesis and evapotranspiration, which require detailed 3D char-
∗
Corresponding author. Tel.: +372-7-410-265; fax: +372-7-410-205.
E-mail address: rossaai.ee J. Ross.
acteristics of different radiation types for input. Since the interrelationship between PAR and photosynthesis
is nonlinear, mean values of radiation do not yield correct results. In our opinion, for further calculation
of canopy photosynthesis leaves in plant canopies should be divided into three groups according to
the type of direct sunlight they receive: sunflecks, umbra and penumbra; and photosynthesis should be
calculated separately for each group. Therefore, it is necessary to know the area of umbra, its vertical
distribution and temporal dynamics so that different data for umbra, penumbra and sunflecks can be used
in modern radiation transfer models.
The second possible application area of umbra statistics is the study of understorey growth and
0168-192300 – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 9 9 0 0 1 4 3 - 4
90 J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102
Fig. 1. The shadowing effect of a single leaf. h
c
— coppice height, z
L
— leaf height, z — measurement height, R
S
— direction of sunrays, R
L
— direction of leaf normal.
renewal of vegetation. In understorey growth, umbra length distribution, and temporal dynamics play a
certain role. According to Oker-Blom 1984, Myneni and Im-
pens 1985 and Ross et al. 1998 sunfleck, penumbra and umbra can be defined as follows Fig. 1:
1. sunfleck is an area inside plant canopy where the sun’s disc is not shaded by any phytoelements. In
sunflecks, direct solar irradiation Sh, which de- pends on solar elevation h, is a nonrandom quan-
tity, i.e. Sh = S h, where S
h is direct solar irradiation above the canopy;
2. penumbra is an area inside plant canopy where the sun’s disc is partly covered by phytoele-
ments. In penumbra, Sh is a random quantity and 0 Sh S
h; 3. umbra is an area inside plant canopy where the
sun’s disc is fully covered by phytoelements and Sh = 0.
1.1. Literature overview In 1971, Miller and Norman developed a mathe-
matical model for penetration of sunlight into plant canopy Miller and Norman, 1971a, b. In the first
part of the paper, sunfleck size distribution was cal- culated proceeding from the probability that a probe
line with the length l would fall completely into sun- light if the sun was a point source at infinity and if the
canopy consisted of identical leaves at a fixed distance from flat ground. In the second part, finite dimen-
sions of the solar disc were taken into account and radiation intensity distribution was estimated in
penumbra. These two results were combined in Norman et al. 1971, where theoretical results were
found to be in good agreement with actual measure- ments in sumac and sunflower canopies.
Several formulae for canopy gap fraction were re- viewed by Nilson 1971: the most common Poisson
law, and positive and negative binomial distributions. Nilson also proposed a new formula for gap frequency,
based on Markov chains. In Markov models, differ- ent canopy layers are not considered independent, and
with proper selection of parameters the model can fit well with experimental data.
Another way to calculate the distribution of solar radiation intensity is computer simulation Stenberg,
1995; Oker-Blom, 1984, 1985. Monte Carlo meth- ods can yield an estimate of radiation distribution in
penumbra for more complicated canopy structures and have shown that penumbra makes a considerable con-
tribution to total photosynthesis, especially in lower canopy layers.
Chen and Black 1992 and Chen and Cihlar 1995a, b have developed a method for estimating leaf area
index from direct solar radiation measurements. They attempt to account for foliage clumping and nonran-
dom gaps in the canopy using the gap size distribu- tion proposed by Miller and Norman 1971a and the
formula for fractional sunfleck area proposed by Nil- son 1971. Due to foliage clumps, effective leaf area
intercepting solar radiation is reduced by about 30, while a single leaf may not be the basic foliage ele-
ment responsible for radiation interception, especially in conifers. A method for calculation of the character-
istic width of a basic foliage element is developed in Chen and Black 1992.
Ross et al. 1998 proposed a statistical treatment of PAR variability and its application to willow coppice.
In this treatment, statistical distribution of PAR global irradiance, obtained with a LI-COR quantum sensor,
was approximated by normal distribution in sunfleck and umbra areas, and the fractional area of penumbra
was studied as a function of canopy depth and solar elevation h. Umbra and sunfleck length were not ex-
amined.
Usually, PAR horizontal variability has been inves- tigated on ground surface beneath the canopy. Studies
of PAR variability at different canopy depths have not been extensive.
J. Ross, M. Mõttus Agricultural and Forest Meteorology 100 2000 89–102 91
Fig. 2. Sunfleck indicator.
The objective of this paper is to study different statistical parameters of umbra length distribution,
umbra fractional area, number of umbrae, mean um- bra length, etc. in different canopy layers as the func-
tions of the optical depth τ = Lsin h for both Salix viminalis and Salix dasyclados coppices. For this pur-
pose, a new instrument, the sunfleck indicator Fig. 2, constructed by M. Sulev, was used. Moving in the
horizontal direction at different canopy depths, this instrument measures the length of sunflecks and
umbrae.
2. Materials and methods
