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LAI theory
Document code: SS1-UM-1.05
The results were then analysed in terms of
L
a
, the LAI of a canopy of black leaves
that would give the same transmission as a canopy of LAI
L
assuming incomplete
absorption, all other factors being equal.
L a
.
L (
)
1
g(
)
1
a
L
is the "true" LAI,
L
a
is the LAI that when used in the black leaf model, gives the
same transmission as
L
used in the complete model.
a
is the leaf absorptivity in the
PAR band.
The function
g
varied with all the other parameters in a complex way, but most
strongly with
x
, the leaf angle distribution parameter, and with solar
zen
ith angle for
the direct beam. The following equations represent quite a crude approximation to
the full model, but give satisfactory results for most situations. If any given
transmission fraction is inverted using the approximation, the LAI calculated is
within ±10% ±0.1 of the "true" LAI indicated by the full model, except for
x
near 0
(extreme vertical leaves) and
zen
ith angle > 60
°
(strong low sun).
For diffuse light:
For direct beam:
where:
x
is the ellipsoidal leaf angle distribution parameter
zen
is the solar zenith angle in radians.
The full equation thus becomes:
This looks hard to invert to get LAI from
ττττ
, but an iterative solution is fairly
straightforward given the computing power, and is much simpler than the full
numerical solution.
Calculating zenith angles
Zenith angles are calculated from latitude, longitude, and local time using standard
astronomical equations as given in
Practical Astronomy.
These give zenith angles
accurate to better than 0.1° and times of sunrise or sunset to within a few seconds.
Summary
A computer model has been created which calculates accurately the transmitted light
below the canopy based on the assumptions given. This has been run over the whole
range of each of the different variables, i.e. Direct beam angle, Direct beam fraction,
Leaf Angle Distribution, Leaf Absorption and Leaf Area Index. The results of these
runs, taking many hours of computer time, have been collected and functions found
to fit them.
These approximating functions are used in the SunData software to predict LAI from
the measured inputs in the field. The LAI values calculated by the SunData software
are within
±
10%
±
0.1 of the LAI that would have been calculated by the full
model.
Scientific references
Campbell G S (1986).
Extinction coefficients for radiation in plant canopies using
an ellipsoidal inclination angle distribution.
Agric. For. Meteor., 36:317-321.
g diff 0.5
g dir
.
exp(
)
.
1.5 x
0.2
.
0.7 zen
2
.
0.2 zen
5
0.3
τ
+
...
.
f b exp
.
.
K (
)
,
x
θ
1
.
g dir (
)
1
a
L
.
1
f b
exp
L a
.
.
A ( )
x
L a
3
exp
.
B ( )
x
L a
C( )
x
Diffuse
Direct part
