J. Ross et al. Agricultural and Forest Meteorology 101 2000 237–246 243
Fig. 5. Shape function Sz of stem foliage vertical distribu- tion throughout the growth period in 1998. Points correspond to
measured values, curves — to fitted polynomials.
The fitting polynomial corresponds to some ‘mean’ stem foliage vertical distribution averaged over the
whole shoot height range. During the growth period, maximum leaf area density is located at z=0.8, and
the maximum value of S fluctuates between 1.5 and 2.5.
Having a set of coefficients a
ik
and applying Eq. 4 to each day and each relative height z it is possible
to calculate the values of the shape function Sz, t. 4.1.4. Calculation of leaf area density and leaf area
index Finally, one can calculate the canopy leaf area
density, uz, t as the function of height and time. In accordance with Eq. 8 the leaf area density uz, t
is determined by three functions: distribution of stem height, interrelationship between stem height and
stem total leaf area, and the shape function of stem foliage vertical distribution.
The downward cumulative leaf area index Lz, esti- mated for different days of the growth period, is given
in Fig. 6.
5. Discussion
The presented interpolation method has several con- straints. In the first half of the growing season, stem
height distribution was approximated by the normal distribution, sample size being 250–350 stems. How-
ever, agreement was not the best see Table 2. Conjec- turing that the sample size was too small, we doubled
the number of samples during a special experiment, which still did not improve agreement essentially.
In the second half of the growing period, stem height distribution function has a tendency to transform from
unimodal to bimodal. This is caused by competition between stems and by development of stem hierarchy
e.g. suppressed and dominant stems. As we have demonstrated earlier for the second year of growth,
height distribution becomes bimodal and can be fitted by superposition of two normal distributions Ross and
Ross, 1998. Actually, the share of short stems in total canopy leaf area is not very large, and more careful
fitting is essential in the range of long stems.
Correlation between stem height and stem foliage area is good during the whole growth period. The
empirical formula S
P
= b
z
b P
used is a good approx-
244 J. Ross et al. Agricultural and Forest Meteorology 101 2000 237–246
Table 3 Empirical coefficients a
ik
in Eq. 6b for calculation of the shape function of the stem foliage vertical distribution Sz
∗ n
, t at different relative heights z
∗ n
z
∗ n
a
i 0
a
i 1
a
i 2
a
i 3
R
2
0.9 −
6.35×10
− 1
8.06×10
1
− 1.1×10
− 3
3.95×10
− 6
0.894 0.8
6.55×10
− 1
4.09×10
− 2
− 4.76×10
− 4
2.11×10
− 6
0.975 0.7
1.45 −
4.05×10
− 1
3.86×10
− 5
1.43×10
− 6
0.994 0.6
1.58 −
1.66×10
− 2
3.00×10
− 4
− 1.26×10
− 6
0.997 0.5
1.53 −
2.23×10
− 2
4.05×10
− 4
− 2.50×10
− 6
0.984 0.4
1.72 −
3.33×10
− 2
5.71×10
− 4
− 3.44×10
− 6
0.992 0.3
6.56×10
− 1
3.24×10
− 2
− 7.82×10
− 4
3.88×10
− 6
0.981 0.2
2.85 −
9.14×10
− 2
9.63×10
− 4
− 3.32×10
− 6
0.999
imation R
2
= 0.86÷0.98 and the seasonal change of
the empirical parameters b and b can be success-
fully fitted by polynomials, R
2
being 0.94 and 0.89, respectively.
Estimation of the shape function of the vertical distribution of the stem foliage is less exact. When
defining the shape function we assume that the shape function of the vertical distribution of the stem foliage
is similar for all stems. However, this assumption is not exact for the second half of the growth period.
Short branches with a length of 3–8 cm develop in the
Fig. 6. Downward cumulative leaf area index Lz vs height on different days of the growth period in 1998. 1 — 25 June, 2 —
3 July, 3 — 15 July, 4 — 28 July, 5 — 28 August, 6 — 14 September, 7 — 7 October.
upper part of long stems causing additional increase in leaf area density at the relative height of about
z=0.7−0.9. This may increase dispersion of the experimental
points in Fig. 4. One source of errors is the limited number of sampled stems. Practically, we were able to
measure leaf area distribution for 15–30 stems on sam- pling days, which should represent all height classes
of z
P
. However, in radiative transfer and photosynthe- sis, leaf area distribution in upper canopy layers plays
a more important role and hence the share of dominant trees is larger.
The success of the method depends also on the fitting accuracy of the temporal course of model pa-
rameters during the growing period. Increasing the number of sampling days improves the situation.
Abnormal weather conditions drought, late frosts, etc. may affect leaf and shoot growth and cause a
non-smooth change of model parameters. Under such conditions input parameters should be estimated at
shorter intervals.
It is difficult to analyse the accuracy of the proposed interpolation method. We made an attempt to test its
accuracy by using the data of phytometric measure- ments for 24 August, which were not included in the
calculation of the coefficients of the method. For this day the functions uz, Lz and the value of LAI were
estimated on the basis of phytometrical measurements as well as by using the interpolation method. The
estimated and calculated values of LAI were 3.6 and 4.1, respectively Fig. 7. In the upper part of
the canopy, which is more important considering light attenuation, the results obtained by interpolation
method are in good agreement with the direct mea- surements of LAI. Differences in the leaf area density
uz,t did not exceed 10–15.
J. Ross et al. Agricultural and Forest Meteorology 101 2000 237–246 245
Fig. 7. Vertical distribution of a the leaf area density uz and b the downward cumulative leaf area index Lz on 24 August 1998.
1 — experimental data, 2 — calculated using interpolation method.
6. Concluding remarks
