A Guide to SExI-FS Calibration

Version: 1.0.0

Abstract

This note aims at providing a guide for the collection of tree data to be used to calibrate the SEXI-FS model, with emphasis on the STReTCH module (crown deformation).

Allometric Data
PSP data analysis (repeated measurements)
1. Below ground crowding index (BGCI)
Cited references

1. Allometric Data

The purpose is here to define relationship between various tree dimensions and how those allometric relationships are affected by tree environment

1.1 Tree selection

Trees from the following three categories are purposefully sampled over the whole range of diameter of interest (e.g. 5 to 50 cm dbh); all trees should have a CF score >=3.

The three categories considered are

Isolated trees
Co-dominant trees in dense stands (usually pure stands) i.e. CP>=4
Suppressed trees (CP<=2)

1.2 Tree parameters to be measured:

Tree height (h)
Height of crown base (hcb)
Height of max. crown width (hmcw).
Height of maximum crown width may coincide with height of crown base. The height of maximum crown width (a shape parameter) is used to compute crown volume.
Crown width
Crown diameter is measured in two perpendicular directions. Crown projection diameter is first measured along maximum crown width axis and then perpendicularly to this first direction.

The average is used for crown width.

1.2.1 Crown Position (CP)

The crown position index, which depends on the relative position of the crown within the canopy, reflects the light conditions prevailing at a particular moment. Crown Position scale is defined as follows (Alder and Synnott 1992):

5 = Emergent: Crown plan exposed vertically and free from lateral competition at least within the 90º inverted cone subtended by the crown base.

4 = Full overhead light: Crown plan fully exposed vertically but adjacent to other crowns of equal or greater height within the 90º cone.

3 = Some overhead light: Crown partially exposed vertically but partly vertically shaded by other crowns.

2 = Some side light: Crown plan entirely vertically shaded but exposed to some direct light due to a gap or edge of overhead canopy.

1 = No direct light: Crown plan entirely shaded vertically and laterally.

1.2.2 Crown Form (CF)

The Crown Form index tries to capture the photosynthetic potential of a tree. It is an architectural characteristic and will tend to reflect the development history of the tree. Crown Form scale is defined as follows (Alder and Synnott 1992):

5 = Perfect. The best size and development generally seen, wide, circular in plan, symmetrical

4 = Good: Very near ideal, sylviculturally satisfactory, but with some slight defect of symmetry or some dead branch tips.

3 = Tolerable. Just sylviculturally satisfactory, distinctly asymmetrical or thin, but apparently capable of improvement if given more space.

2 = Poor: Distinctly unsatisfactory, with extensive dieback, strong asymmetry and few branches but probably capable of surviving.

1 = Very Poor: Definitely degenerating or suppressed, or badly damaged, and probably incapable of increasing its growth rate or responding to liberation.

1.2.3 Crown porosity (isolated, dominant, co-dominant trees)

Crown "porosity" to light is defined as the percentage of sky visible from below the crown and is simply assessed using sub-vertical photographs towards the sky.

A: Pterospermum javanicum , B: Shorea javanica Koord. et Valeton
C: Parkia speciosa Hassk., D: Lansium domesticum Correa

Best time to take good quality photographs is early morning or under heavily overcast skies (no direct sunlight). Low branches can make pictures of the entire crown difficult or impossible, as one can’t move back far enough to capture the whole crown. In that case it is recommended to beginners to take a series of pictures of parts of the crown, in a systematic pattern. Once experienced, selection of a representative part of crown in the field is a more efficient way of doing.

In most cases selection of a representative portion of crown (which can be the entire crown once it has been delineated on the photograph but is more commonly restricted to half a crown excluding the tree trunk) will be done by cropping part of the digitized image on the computer.

Once a representative portion of the crown has been selected and cropped the picture is converted into black and white bitmap format in order to assess the percentage of visible sky. Image thresholding (deciding which level of grey defines the limit between black and white i.e. between tree parts and the sky) is the critical step. Most image processing software offer facilities that allow instant comparison between the original image and the classified image which provide some control over the quality of the thresholding step.

Note: crown porosity cannot be measured on trees growing in the understorey. This may be problematic as there are indications that tree porosity is responsive to tree growth environment and may be significantly lower in shaded trees than trees fully exposed to light.

1.2.4 Tree growth environment

When relating tree dimensions to its growth environment care should to be taken in making sure that the current environment does reflect the growth environment of the tree (which may have changed over time through self thinning, tree fall creating gaps, differential growth rates in height affecting CP, etc…).

Local density and local basal area are recorded by measuring the trees growing in the vicinity of the target tree. A tree is recorded if its dbh is >= 5 cm.
A circular plot with radius rmax around the target tree is defined with rmax =max(r1,r2)
Where r1 is defined as the maximum crown width of target tree and r2 equals the distance to the furthest tree in physical contact with target tree.
If rmax=r1 then local density is simply computed as total number of trees divided by plot area (Pi*rmax^2) by and local basal area as the sum of all cross sectional areas of individual trees divided by plot area.
If rmax=r2 then local the furthest tree (which defines the plot radius) is counted as half inside and hal outside the plot and hence given a weight of 0.5 both when computing density and basal area.

Note in case of regular planting (which for example may be the case for rubber plantation) the elementary plot may be delineated as a rectangle (which is quicker in the field) including all 8 "neighboring" trees (two on the line and the three trees on each neighboring planting line). In that case the plot area is simply defined as 9 x average planting distance.

For all trees within a plot centered on the following three variables are recorded: tree species, tree diameter, whether the tree neighboring tree crown is in contact with target tree crown is (Boolean)

1.3 Data Format

The following format for storing data is suggested (Access database)

TargetTree Table

Field Name	Type	Definition
targetID	Number (Long)	Tree ID (automatically assigned)
speciesID	Number (Long)	Species ID (from species table)
girth	Number (Double)	In cm
CP	Number (Double)	Crown Position Index
CF	Number (Double)	Crown Form Index
height	Number (Double)	Total tree height (m)
hcb	Number (Double)	Height of crown base (m)
hmcw	Number (Double)	Height of maximum crown width (m)
crown_width_a	Number (Double)	Maximum crown projection width (m)
crown_width_b	Number (Double)	Crown projection width perpendicular to crown_width_a
isolated	Yes/No	“Yes” if the tree grows isolated “no” neighbors in its immediate vicinity
plot_typeID	Number (Byte)	Plot type ID (from Plot_type table)
plot_area	Number (Double)	Area of plot defining target tree neighborhood
neighbors	Number (Long)	Number of neighbors in tree neighborhood
contacts	Number (Long)	Number of trees in contact with target tree

Neighbors Table

Field Name	Type	Definition
targetID	Number (Long)	Unique target tree ID to which the Neighboring tree is related
speciesID	Number (Long)	Species ID (from table species) species table
girth	Number (Double)	cm
distance	Number (Double)	Distance to target tree (m)
contact	Yes/No	Whether the tree is in contact with target tree

Species Table

Field Name	Type	Definition
speciesID	Number (Long)	species ID
Local_name	Text	Species local name

Plot_Type Table

Field Name	Type	Definition
plot_tipeID	Number (Byte)	Plot type ID 1, 2 or 3
name	Text	Plot type (circular, rectangular or other=isolated tree)

Interface for data entry

1.4 Data processing

1.4.1 Dbh-crown diameter

Dbh and crown diameter are related by linear regression. Data from the various groups are pooled to establish this relationship. It is useful however to check that groups do not differ significantly (biologically meaningfully rather that statistically).

1.4.2 Dbh-Crown surface

Assuming a half-ellipsoid approximation of the crown profile we then compute the approximate crown surface as

If cd > cw Then e = Sqr(cd ^ 2 - cw ^ 2) / cd crown surface = pi * cw * (cw + cd * Arcsin(e) / e) If cd <= cw Then e = Sqr(cw ^ 2 - cd ^ 2) / cw crown surface = pi * cw * (cw + cd * Arcsinh(cw * e / cd) / (cw * e / cd))

where cd stands for crown depth (total height – height of crown base), cw crown width (see http://www.physik-astro.uni-bonn.de/~dieckman/SurfaceEllipsoid/SurfEll.html for derivation of the formula of surface area of an ellipsoid.

Then estimated surface (or volume) is fitted to dbh; a loglinear fit is usually satisfactory (as total leaf area is expected to scale linearly with stem cross sectional area e.g. Morataya et al. 1999).

Again we expect this relationship to vary little between groups (which can be tested by ANCOVA) and data for the various groups should be pooled for this adjustment to increase robustness of parameters estimates.

Note: multilayer trees (sensu (Horn 1971) which rare in our data sets) are likely to show a more consistent linear fit between crown volume and dbh rather than crown surface. This may be explored using the estimated volume of crown computed as
2/3*Pi*cwa *cwb*CD (half ellipsoid).

1.4.3 Estimation procedure of the flexi parameter in SExI-FS

Objectives

We are interested in assessing the change in the slope (derivative) of the height-dbh relationship observed in trees of various species when grown either isolated or in dense stands. In the SExI-FS model, this corresponds to the flexi parameter (precisely the ratio of the derivatives is equal to flexi +1)

Data

We assume we have two tree population samples measured in contrasted conditions (i.e. isolated or in dense stands). We further assume that the “dense stand” sub-population may be considered representative of the most extreme conditions, i.e. we capture most of the species possible range of growth conditions.

In case the height-dbh relationship of either of the two subpopulations shows a strong dispersion an envelop curve analysis could be used (e.g. stochastic frontier functions may be used instead of standard regression; see free software at http://www.uq.edu.au/economics/cepa/software.htm) but has not used in the present study.

We use the data collected by Jasnari and co-workers by mid 2004 for 6 species for which sample size seems suitable (Lansium domesticum, Hevea brasiliensis, Durio zibethinus, Archidendron jiringa, Alstonia angustiloba, Paraserianthes falcataria).

Methodology

See actual analyses in companion Systat output file HeightDbhAlteration.syo

Step 1: graphical analysis and data transformation
Assuming that the dbh-height relationship may be correctly described using the following relationship: height=a*dbh^b, data are first log transformed and plotted using linear smoother. We visually check that the log transformation is OK (graphical analysis of residuals may help pinpoint possible problem such as heteroscedascity or unwished pattern in the residuals).

Step 2: first parameter estimates.
For most species it can be seen that the regression lines of the two sub-populations are almost parallel and we therefore choose to analyze data using a GLM where subpopulation differ only in terms of their intercept (i.e. assuming homogeneous slope). In one case (Paraserianthes) this assumption appears to be clearly violated and in another case (Alstonia) it appears to be doubtful this will be re-examined under step 3.
From the model above we estimate three parameters for each species (a1, a2 and b).
And the ratio of the derivatives is equal to the ratios of the a1 and a2 parameters if b is identical leading to the estimates for the different species reported in table 1

Step 3: Refining the estimates.
We may first want to relax the assumption of constant b parameter and rerun a model with unequal slopes.
Then the ratio of the derivatives is dependant on dbh and equal to
(dh1/ddbh1)/(dh2/ddbh2)=a1b1*dbh^(b1-1)/a2b2*dbh^(b2-1)This function may be plotted and its maximum visually assessed (alternatively the maximum maybe computed by finding the value for which the second derivative of the function equals 0)

The following procedure was run for Paraserianthes and the resulting function is plotted below. Note that experimental data are limited to the diameter range 15-70cm and the early shape of the height-dbh curve is virtually unknown and may be poorly estimated

The case of Paraserianthes (a –single? - very dense plot of equal age and similar height of trees) may reflect limitations of the data. And I would suggest to stick to the previous estimate (using equal slope regressions) which does capture the high elasticity of the species.

The case of Alstonia may also reflect data set limitations as two extreme observations (one in each sub-population) appear to have high leverage; once those two observations are discarded then the slopes appear again nearly identical in both subpopulations. One outlier is the smallest tree in the isolated subpopulation which is especially short: it was measured just before the extension of another growth unit on the trunk and may be considered as an outlier indeed. The other extreme point is the largest tree of the pooled population (80 cm dbh) with no tree nearly as big in the isolated population.
Rerunning the equal model without those two outliers yielded similar estimates for sensi. (Marked with an asterisk in table below)

Table 1: estimates of flexi parameter for 6 species used in SExI-FS model

Species	group	Log(a)	a	Sensi + 1
Lansium	isolated	0.582	1.78961408
	dense plot	0.932	2.53958327	1.42
Durio	isolated	0.399	1.49033362
	dense plot	0.799	2.2233165	1.49
Archidendron	isolated	0.84	2.31636698
	dense plot	1.41	4.0959554	1.77
Hevea	isolated	0.963	2.61954333
	dense plot	1.275	3.57870141	1.37
Alstonia	isolated	-0.066	0.93613086
	dense plot	0.229	1.25734204	1.34
Sengon	isolated	0.853	2.34667633
	dense plot	1.441	4.22491862	1.80
Alstonia*	isolated	-0.006	0.99401796
	dense plot	0.268	1.30734714	1.32

1.6 Note on crown deformation parameterization

Crown asymmetry resulting from neighborhood competition is commonly observable and has been measured ,e.g.(Brisson 2001). However we have not yet attempted to directly measure the parameter governing the ability of a crown to adjust to lateral anisotropy of resources due to difficulties involved in standardizing such measures. One favorable situation which may occur with planted species would make use of crown deformation response of trees growing under different planting patterns (i.e. inter-row, and on the row inter-tree distances).

Rather, we make the assumption that flexibility in tree height adjustment (ratio of k value in the height-dbh relationship under contrasted vertical gradient) is a good proxy for the ability of a species to adjust its crown expansion under lateral anisotropic distribution of light.

2 PSP data analysis (repeated measurements)

Permanent sample plot data are used to derive the following parameters

Species potential growth function (site specific)
Species sensitivity to shading
Species sensitivity to tapping
Species influential zone (determining BGCI)

Standard procedures are used to analyze data from PSP (see for example Alder and Synnot 1992, Vincent et al 2001 for an introduction to such methods. Predictors used in the GLM include size, crown indices (and tapping regime). Rare species (< 10 individuals monitored) are grouped into a miscellaneous grey species for the data analysis purpose. Once factors effect are estimated, potential growth is computed after correcting for CF, CP, Tapping index (and possibly BGCI).

Corrected dbh increment is used to adjust the dbh_inc=f(dbh) function using a Chapman Richard function using standard non linear regression procedures.

Using precisely the method described above on PSP sample plot for rubber and comparing the growth rate as a function of size obtained from Sembawa plantings one can observe that the patterns are not consistent. Essentially, data from PSP provide an estimate of maximum potential growth which is strictly decreasing with tree size whereas data from Sembawa density trial indicate that maximum growth rate may be attained later in case of low density (6x6 planting pattern).

PSP standardized increment data (CF 5, CP 3, no tapping)

Density trial (annual dbh increment, plot average values, tapping starts around 0.15 cm dbh)

This strictly decreasing growth rate with size found in analyzing the PSP data (instead of the expected typical increase and decrease in growth rate) is probably at least partly due to the fact that the monitoring starts at about the size the rubber reaches its maximum growth. Early growth (needed in the model if one wants to simulate growth starting at diameters less than 0.05 m) cannot be directly estimated from PSP data but need to rely on additional measurements, this was done by using a SRAP plot and data from a gap planting to yield the default values for rubber in the current library.

Why should “maximum potential growth” decrease faster in PSP - even after increments have been corrected for CF, CP and tapping - than what is observed in low density plantation trials? There are at least two possible explanations. The first one is that below ground competition (which we have not corrected for) is stronger in PSP (mature agroforest) than in young plantations where it is minimal during the earlier stages. A similar conclusion, i.e. that below ground competition most probably limits early growth of rubber saplings grown in rubber agroforest was reached after careful comparison of growth of rubber plants under artificial shading and under live canopy. However, such an explanation is only partly satisfactory as high below ground competition should mostly likely translate into a sustained lower growth rate over the whole period of early growth and cannot be unequivocally related to a shift in maximum dbh growth rate. Another, possible explanation, is that the difference observed between rubber agroforest and young plantation reflects the fact that dbh increment in young trees growing under strong light gradient may be reduced as a consequence of accelerated height growth which occurs under limited light and which correlatively limits diameter increment. To test this hypothesis one can test for dbh*CP interaction using the same PSP data as above. It turns out that the interaction between both predictors is statistically highly significant and that smaller trees are indeed more sensitive than larger trees to suboptimal CP scores.

Note that the above procedure may eventually yield robust estimates only for abundant species. Hence it is preferable whenever possible to develop potential growth curve by repeated measurement of isolated trees (or low density stands).

Experience also indicates that sensitivity to shading is poorly captured in PSP data (often there is no clear species specific response) indicating that additional information should be used to estimate/check Minilum and Optilum parameter values (minimum and optimum light levels for growth).

For lesser abundant species, one option is to repeatedly measure purposefully sampled trees. Sample should whenever possible include open grown trees (Crown Position=5). Sample should only include trees with optimal or near optimal crown shape (CF>=4) and cover a range of diameters. Trees should be sampled in similar edapho-climatic environment. If a decent sample of trees is available across a range of CP classes shade response (CP effect on growth) can be meaningfully estimated.

Alternative/complementary options include using –scarce - published literature and local ecological knowledge about the species of interest. The latter may notably yield useful ranking between species (both in term of growth rate and shade tolerance).

2.1 Below ground crowding index (BGCI)

Usually below ground crowding index is correlated to above ground indices (CP and CF) and in species rich PSP it may be difficult to show statistically significant growth reduction not yet captured by CP and CF indices.

In some particular cases however (e.g. limited number of species and expected contrasted competitiveness for below ground resources e.g. water) it may be possible to actually estimate BGCI from repeated measurements.

What we try to estimate (and which is supposedly different between species) in the present case is the influential zone of each species. In other words we assume equal sensitivity to resource shortage but differential resource capture efficiency represented by a relatively larger or smaller influential zone. Thus the basic idea is to explore for the different species a range of species specific (and size dependent) influential zones.

The general model to be fitted for each species is

DBH_Inc=Pot_inc+CP+CF+tapping+BGCI

In the most general case, assuming that only CF is species independent fitting the above model for a particular target species requires estimating 3 (pot inc) + 5 (CP as categorical) + 1 (tapping) + n (lambda, species specific IZ scaling factor)= 8+ n parameters. In addition CF (common to all species) needs to be estimated too.

Although this is certainly feasible it cannot be done using procedures available in standard statistical packages but requires the development of a global optimization algorithm (see Canham et al 2004 for such an example) which we have not done yet.

3 Cited references

Alder, D. and T. J. Synnott (1992). Permanent sample plot techniques for mixed tropical forest. Oxford, Oxford Forestry Institute -Department of Plant Science. 25: 81-83.

Brisson, J. (2001). "Neighborhood competition and crown asymmetry in Acer saccharum." Can. J. For. Res./Rev. Can. Rech. For. 31(12): 2151-2159.

Canham, C. D., P. T. LePage and K. D. Coates (2004). "A neighborhood analysis of canopy tree competition: effects of shading versus crowding." Canadian Journal of Forest Research Revue Canadienne De Recherche Forestiere 34(4): 778-787.

Horn, H. S. (1971). The adaptive geometry of trees. Princeton, Princeton University Press.

Morataya, R., G. Galloway, et al. (1999). "Foliage biomass-sapwood (area and volume) relationships of Tectona grandis L.F. and Gmelina arborea Roxb.: silvicultural implications." Forest Ecology and Management 113(2-3): 231-239.

Vincent, G., H. De Foresta, et al. (2002). "Predictors of tree growth in a Dipterocarp based agroforest: a critical assessment." Forest Ecology and Management 161(1-3): 39-52.