(extreme vertical leaves) and

The results were then analysed in terms of La, 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, La 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 zenith 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

zenith angle > 60 ° (strong low sun).

For diffuse light:

g diff

 

 

0.5

 

1.5.x ).

 

 

 

 

 

0.7.zen2

 

 

 

0.2.zen5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For direct beam:

g dir

 

 

exp(

 

 

0.2

 

 

 

 

 

 

 

 

 

0.3

 

 

 

 

 

 

 

 

 

 

 

 

 

where: x is the ellipsoidal leaf angle distribution parameter zen is the solar zenith angle in radians.

The full equation thus becomes:

τ

 

f

.exp

 

 

 

K ( x , θ ).

 

1

 

g

.( 1

 

a )

 

.L

 

...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dir

 

 

 

 

 

 

 

 

 

 

+

1

 

f

 

 

.

exp

 

 

 

L

 

 

 

 

 

A ( x ).L

3.exp

 

 

 

B ( x ).L

C( x )

 

 

 

 

 

 

 

 

b

 

 

 

 

 

 

a

 

 

 

 

 

a

 

a

Direct part

Diffuse

This looks hard to invert to get LAI from τ , 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.

62 • LAI theory

Document code: SS1-UM-1.05

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Delta Electronics SS1-UM-1.05 user manual Scientific references, Calculating zenith angles, Summary
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SS1-UM-1.05 specifications

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