A general integrative framework for modelling woody biomass production and carbon
sequestration rates in forests
David A. Coomes, Robert J. Holdaway, Richard K. Kobe, Emily R. Lines & Robert B. Allen
Journal of Ecology, doi: 10.1111/j.1365-2745.2011.01920.x
Appendix S1. Supporting Information
DEFINITIONS
The carbon stock of a forest is the total mass of carbon held within plants, coarse woody debris, litter
and soil organic matter. Carbon sequestration is the rate of change of that carbon stock. The regional
carbon balance of forests can be measured through the “statistical book-keeping” approach (sensu
Zaehle et al. 2006): tracking carbon stocks in above-ground biomass. This involves monitoring the
growth, death and recruitment of individual stems within inventory plots distributed across the region
(e.g. Nabuurs et al. 2003), and is sometimes supplemented by repeated measurements of soil carbon
and coarse woody debris volume (e.g. Coomes et al. 2002).
Only a fraction of the carbon absorbed by trees ends up sequestered in recalcitrant organic
compounds. Gross primary production (GPP) is the total quantity of carbon fixed by photosynthesis
over a year. Some of the organic carbon is returned to the atmosphere almost immediately through
plant respiration (Rplant) and the remainder comprises net primary production (NPP = GPP – Rplant).
Carbon sequestration in a forest is less than its NPP – often considerably so – because some of the
fixed carbon is returned to the atmosphere within days to months (e.g. microbial respiration of carbonbased root exudates and litter decomposition) or years (e.g. decomposition of dead wood).
Ecosystem modelling approaches quantify the important fluxes of carbon between different
pools. GPP is impossible to measure, and must be calculated from NPP and plant respiration (Rplant).
However, measurement of forest NPP itself is difficult (Clark et al. 2001) for two reasons. Firstly, a
large portion of NPP does not accumulate in woody tissue and has to be accounted for separately.
This includes plant organs with short lifespans (e.g., leaves, fine roots) and those lost to the
environment (e.g., rhizosphere exudates, volatile organics, above- and below-ground herbivory) (Fig.
S1). Secondly, mass allocation to roots (particularly coarse roots) is not well characterized and often
has to be modelled as a constant fraction of above-ground mass (Cairns et al. 1997, Kobe et al.,
unpubl. ms). Net ecosystem production (NEP) is a useful framework for taking into account rapidly
cycling carbon pools, and especially plant, animal and microbe respiration (NEP = GPP – Rplant + animal +
microbe) (Chapin et al. 2006). It involves measuring CO2 fluxes into and out of forests using eddy
covariance techniques, and assumes that negligible amounts of carbon are lost in water.
Fig. S1 Components of the carbon cycle of forest ecosystems
H0 RELATIONSHIP BETWEEN HEIGHT, DIAMETER AND MASS
Permanent plot network design: Landscape-level estimates of carbon sequestration were obtained
from 246 plots within the distributed network. The plots were established systematically along 98
compass lines during the austral summers of 1970/71 and 1972/73 (henceforth the starting year of an
austral summer is given) with line origins located randomly along stream channels (30–1000 m apart),
and aligned along a random compass direction (Harcombe et al. 1998). Plots were then located at 200
m intervals along each line until the tree line was reached, giving rise to lines containing between 1
and 8 plots (mean = 2.6).
Estimating biomass: Methods adapted from Harcombe et al. (1998) were used to estimate aboveground biomass for each tree. Based on diameter and bark thickness data for 753 trees, Harcombe et
al. (1998) estimated diameter under bark DUB (in cm):
𝐷𝑈𝐵 = 0.945D − 0.218
1
3
They then calculated stem-wood volume (Vol in m ) as a function of DUB and tree height (H in m):
ln(𝑉𝑜𝑙) = 0.9602 × ln(DUB 2 H) − 9.762
2
Using equation 2, they calculated Vol by back transforming to arithmetic units, applying the
Baskerville correction factor (ems/2, where ems = mean square error = 0.0012).
𝑉𝑜𝑙 = exp(0.9602 × ln(D𝑈𝐵 2 H) − 9.762) + ems
3
They multiplied by 0.514 (wood density of mountain beech) to get stem wood mass, in tonnes. We
multiplied by 1.35 to convert stem-wood mass into above-ground biomass (M), as recommended by
Harcombe et al. (1998), and multiplied by 0.5 ×1000 to calculate biomass in units of kg C.
𝑀 = 0.5 × 0.514 × 1.35 × 1000 × (exp(0.9602 ∗ ln(DUB 2 H) − 9.762) + 0.0012)
4
Inserting the DUB formula and simplifying gives:
𝑀 = 0.0179 (D − 0.231)1.9204 H 0.9602 + 0.416
5
We measured stem diameter (D, in cm) and tree height (H, in m) for a total of 201 stems within the
Craigieburn study area, selected haphazardly to encompass the full range of diameters present at a
range of different altitudes (700–1400 m a.s.l.). Diameters at breast height were measured with
diameter tapes. Using a vertex hypsometer (Haglöf, Sweden), two perpendicular measurements of
crown width W were taken (North–South and East–West directions). We modelled tree height (H) as
a function of stem diameter (D) and altitude. We explored a range of functional forms using the nonlinear least squares regression (the nls function in R), and selected the best fitting model based on
goodness of fit (r2) and visual assessment of predicted vs. observed values for misspecification. The
best fitting model took the form:
log(𝐻 − 1.35) = log(16.7) + log(1 − 0.076𝐴𝐿𝑇) + log(1 − exp(−0.059𝐷1.21 ))
7
where ALT is scaled altitude, calculated from the minimum altitude (640 m):
𝐴𝐿𝑇 = (𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 − 640)/100
8
Tree height was then predicted by back transforming to arithmetic units, applying the Baskerville
correction factor (ems/2, where ems = mean square error = 0.050) (Fig. S2, r2 = 0.86).
Fig. S2. Observed and predicted relationship between stem diameter and height of 201 Nothofagus trees sampled from a
range of altitudes. Green triangles show data from low altitudes (640–800 m), red circles from mid altitudes, and blue
squares from high altitudes (1100–1380m). Model predictions for the midpoints of each altitudinal range are drawn in the
same colour as the corresponding data points.
H1: TESTING FOR NUTRIENT AND HYDRAULIC LIMITATION
A summary of canopy properties in young, intermediate and old stands in the stand-development
sequence are provided in Table S1.
Table S1. Storage of N and P in the canopies, and leaf area index, of young, intermediate and old
stands of mountain beech sampled from a stand-development sequence (n = 4).
Total N in canopy (g cm-2 of ground)
Total P in canopy (g cm-2 of ground)
Leaf Area Index
Young
6.2 ± 0.49
0.64 ± 0.09
5.86 ± 0.54
Intermediate
9.86 ± 0.40
1.10 ± 0.10
7.32 ± 0.29
Old
9.03 ± 0.34
0.88 ± 0.04
5.53 ± 0.46
The best-supported model of diameter growth for trees in the distributed plot network was:
Annual Growth = 0.0638 𝐷0.66 exp(−0.014𝐷)
9
calculated for trees without taller competitors at ALT =0 (see section H4). This function is much
more strongly supported than a power function (ΔAIC = 243). It predicts similar diameter growth to a
power function when trees are small (D < 25 cm) but much lower growth rates for larger trees (e.g.
40% less when D = 50 cm; Fig. S3).
Fig. S3. Predicted growth rate of Nothofagus trees, estimated using a power function (dotted line) and a modified power
function with an exponential multiplier (solid line).
H2: TESTING THE CANOPY OPTIMIZATION HYPOTHESIS
Field measurements
Leaf angle profiles were sampled during January–March 2008. Within each stand, the canopy was
divided into four or five height tiers, and within each height tier four branches containing 200–400
leaves were randomly selected and removed using orchard cutters. Each branch was suspended just
off the ground using a clamp and stand, and oriented in such a way to ensure that the angle of the cut
was identical to that when it was in the canopy. For each branch, the leaf inclination angle (θ) was
recorded for 50 leaves using a protractor and hanging weight, with horizontal leaves having θ = 0 and
vertical leaves θ = 90. Twigs containing 5–10 leaves were selected at random from the branch and all
the leaves within each twig were measured until a total of 50 leaves had been measured. A selection
of leaves was removed from each branch; these were pooled among branches within each height tier,
dried at 60ºC and analysed for leaf nitrogen and phosphorus concentrations using the acid digest and
colorimetric methods described in Blakemore et al. (1987). Total canopy N and P were estimated by
multiplying the leaf nutrient content (per unit leaf area) of each tier by the leaf area index of that tier,
and then summing across all tiers in the canopy.
Whole-stand light interception was measured from January to March 2008 using paired quantum
sensors to record total photosynthetically active radiation (PAR) at wavelengths of 400 to 700 nm
(Skye Instruments, UK). One sensor was mounted 4–5 m above the surrounding canopy on a tower
located 50–500 m away from the stands (most stands were within 200 m). Another sensor was used to
simultaneously record understory light levels below the lowest leaf in the canopy (typically at a height
of 30–50 cm above the ground). A hundred measurements were taken systematically throughout the
understory of each stand within two hours of the solar noon on uniformly overcast days.
Modelling canopy carbon uptake
To take into account the effect of variation in leaf angles on light transmission within the canopy, an
adjusted leaf area index (Ladj) was calculated to represent the midday projected leaf area index of each
̅̅̅̅̅̅, where LAI is the one-sided leaf area index of the stand, and θ is the leaf
stand 𝐿𝑎𝑑𝑗 = 𝐿𝐴𝐼𝑐𝑜𝑠𝜃
inclination angle.. Assuming a random distribution in leaf azimuth, within-canopy light profiles were
then modelled for each stand using Beer-Lambert law (Monsi and Saeki 1953): 𝐼 = 𝐼0 exp(−𝐾𝐿𝑎𝑑𝑗 )
where I is the light level measured underneath the canopy, I0 is the incoming light levels at the top of
the canopy (both in units of μmol m-2 s-1 PAR), and K is the extinction co-efficient. Using measured
values for I, I0, and LAI, the extinction coefficient K was calculated for each stand for both cloudy and
sunny conditions. These K values, along with the adjusted leaf area index profiles, were used to model
the light levels at the base of each height tier (I(z)) as follows: 𝐼(𝑡) = 𝐼0(𝑧) exp(−𝐾𝐿𝑎𝑑𝑗(𝑧) ), where I0(z)
is the light at the top of tier (z) (set at 1500 μmol m-2 s-1 for the top of the canopy), Ladj(z) is the
projected leaf area index of tier (z), and K is the canopy-level extinction co-efficient. The amount of
light (PAR) per unit leaf area for a given height tier (Iadj(z)) was calculated by multiplying the
geometric mean of the light levels at the top and bottom of that tier by the mean (cos(θ)). The
geometric mean was used to account for the exponential decline in light within a tier. The
relationship between maximum photosynthetic capacity (Amax) and leaf nitrogen concentration per unit
area (Narea) for mountain beech was constructed using data from Hollinger (1989) who reported
canopy profiles of both Amax (μmol m-2 s-1) and Narea (g m-2). This relationship took the form: Amax =
2.118Narea + 0.41, and we used this to estimate the Narea values for sun and shade leaves in relation to
their reported Amax (Benecke and Nordmeyer 1982; see next paragraph). Uptake rates of CO2 per unit
leaf area (Aarea) were then calculated for each height tier: 𝐴𝑎𝑟𝑒𝑎(𝑧) = 𝑎 + 𝑏𝑒𝑥𝑝(−𝑐𝐼𝑎𝑑𝑗 ) where the
parameters a, b, and c were empirically derived as linear functions of Narea. We then multiplied Aarea(z)
by the tier LAI, and summed over all height tiers to give a measure of instantaneous net canopy-level
CO2 uptake (Acan) for each stand 𝐴𝑐𝑎𝑛 = ∑𝑛𝑧=1 𝐴𝑎𝑟𝑒𝑎(𝑧) 𝐿𝐴𝐼𝑧 . This estimate was then used to calculate
stand-level light use efficiency (LUE, canopy CO2 uptake per unit intercepted PAR), nitrogen use
efficiency (NUE, canopy CO2 uptake per unit nitrogen), and leaf area efficiency (LAE, canopy CO2
uptake per unit leaf area).
Light response curves for “sun” and “shade” mountain beech leaves were taken from Benecke
and Nordmeyer (1982) (Fig. S4). Assuming all the variation between sun and shade leaves was due to
variation in leaf nitrogen content (Hollinger 1989), a general model for light response curves was
developed in the form y = a + bexp(-cIpar ), where Ipar is the flux of photosynthetically active radiation,
and a, b, and c are constants that were expressed as functions of leaf nitrogen concentration per unit
leaf area (Nleaf), such that: a = 2.118(Nleaf ) + 0.41, b = -0.735(Nleaf ) - 2.19, c = -0.00207(Nleaf ) +
0.0088.
Fig. S4. Photosynthetic light response curves for sun and shade leaves of mountain beech (adapted from
Benecke and Nordmeyer 1982). Equations of the lines are: CO2 uptake (sun leaves, solid line) = 5.136 –
5.612exp(-0.00417Ipar); and CO2 uptake (shade leaves, dashed line) = 3.513 – 3.939exp(-0.00576Ipar).
H3: TESTING THE CANOPY-PACKING HYPOTHESIS
Theoretical model:
Crown form is highly influenced by competition with neighbouring trees, which can lead to selfpruning of shaded lower branches, but this factor is not included in MSTF (Makela & Valentine
2008). Trees direct resources towards branches positioned in sunshine and away from those
positioned in shade, resulting in the eventual death of shaded branches (see Henriksson 2001; Sprugel
et al., 2002; Strigul et al. 2008). Self-pruning acts to reduce the depth of crowns and is highly
dependent on growing conditions. Trees growing in open fields have canopies that resemble volumefilling fractals, but in dense plantations the crown ratio (canopy depth to height ratio) can be very low
and canopies are more like discs on top of a pole. The scaling of leaf mass to above-ground biomass
depends on crown depth and is less than M3/4 in self-pruned trees (Makela & Valentine 2006).
Imagine a scenario in which saplings of identical size and genotype are planted in equally spaced
rows, such that no tree can gain sufficient height advantage to overtop its neighbours and outcompete
them through asymmetric competition. These conditions are often encountered in forestry
plantations. As trees grow their canopies expand until they start to intermingle; during this phase
MSTF predicts that diameter increment scales as D1/3. Once all space is occupied, self-pruning keeps
each canopy at a constant area; during this phase the net productivity of each tree is constant (because
its crown and leaf areas are both constant). Assuming that the mass–diameter allometry is unaffected
8
by self-pruning (i.e. that the MSTF allometric rule 𝑀 ∝ 𝐷 3 still holds), diameter increment scales as
𝑑𝐷
𝑑𝑡
∝ 𝐷 −8/5 . Thus diameter increment increases until the point of canopy closure then declines as the
crown-area to biomass ratio falls.
Field measurements and analyses
We measured stem diameter (D), crown width (W) and depth (B) on a total of 201 Nothofagus
solandri var cliffortiodes stems within the distributed plot sampling area (the same trees used for
height estimation). Trees were haphazardly selected to encompass the full range of diameters present
at a range of different altitudes (700 to 1400 m a.s.l.). Using a vertex hypsometer two measures of
crown width were taken perpendicular to each other (in North-South and East-West directions) and
the average of these measurements was used to represent crown width.
SMA line-fitting is not an appropriate method for predicting crown widths and depths from
diameters (Warton et al. 2006), so we fitted lines using non-linear least-square regression. We
explored various functional forms and chose the ones that gave unbiased predictions (based on
inspecting residuals). Crown width (W, in m) was modelled as a non-linear function of diameter (cm)
and normalised altitude (Fig. S5):
log(W) = log(0.21) + log (1 − 0.068 ∗ 𝐴𝐿𝑇)+0.953 log(𝐷 + 3.41)
r2 = 0.81
10
Which transforms to W = 0.21 (1 − 0.068 ∗ 𝐴𝐿𝑇)(𝐷 + 3.41)0.953 + 0.07 , where 0.07 is the
Baskerville correction. Crown depth (B, in m) was modelled as a non-linear function of tree height
(m) and normalised altitude (Fig. S5):
log(B) = log(1.28) + log (1 − 0.096 ∗ 𝐴𝐿𝑇) + 0.67(𝐷 + 0.19))
r2 = 0.81
Which transforms to B = 1.28 (1 − 0.096 ∗ 𝐴𝐿𝑇)(𝐷 + 0.19)0.670 + 0.12, where 0.12 is the
Baskerville correction.
11
SMA line-fitting is the preferred method for ascertaining the slopes of allometric relationships
(Warton et al. 2006). In order to determine the slope of the crown-area vs diameter relationship, it
was first necessary to add a constant to each stem diameter, because the power function would
otherwise predicts that a tree of zero diameter at breast height has zero canopy area, which is not
correct. We resolved a similar problem when modelling crown width and depth by adding a
correction factor to D, as shown in equations 10 and 11. We used least-squares regression to estimate
the correction factor for crown-area vs diameter relationships (it was 3.20 cm), added that value to all
D values, then used SMA line-fitting to estimate the relationship between corrected diameters and
crown areas. The same procedure was used when relating crown volume to diameter (correction
factor = 2.0).
Fig. S5. Observed and predicted relationships between stem diameter and crown width (a), and crown depth and tree
diameter (b), for 201 mountain beech trees sampled from a range of altitudes, and allometric relationship between crown
area and corrected stem diameter (c). Green triangles show data from low altitudes (640–800 m), red circles from mid
altitudes, and blue squares from high altitudes (1000–1400 m). Predictions for low-altitude samples (700 m), mid-altitude
samples (1100 m) and high-altitude samples (1400 m) are shown by green, red, and blue lines, respectively. In (c), the solid
line is the fitted SMA regression with an exponent of 2.16; the dashed blue and dotted red lines have the predicted slopes of
2 and 4/3 respectively, both being fitted through the centroid of the data.
H4 FITTING GROWTH, RECRUITMENT AND MORTALITY FUNCTIONS
We used an adaptive MCMC Metropolis algorithm to estimate parameters and credible intervals (CIs)
for models of individual annual growth and annual probability of mortality. We fitted several different
functional forms for each model and compared them using information criteria (discussed below). The
MCMC algorithm compares parameter values using the log-likelihood of the data given the model. At
each iteration the algorithm selects a parameter to alter and recalculates the likelihood. If the new
parameter improves the likelihood then it is accepted by the algorithm. If not, it is accepted with
probability of the ratio of the new and old likelihoods. In this way it returns not only a best-fit value
for each parameter given the data but also estimates its distribution. For a set of starting parameter
values 𝜃 for each model M tested, the algorithm calculated the predicted growth or probability of
mortality and then the log-likelihood of the data (X) given the model and parameters.
To model growth we used stem diameter (D), altitude and competition at the first survey (𝑑1 ),
to predict D at the second survey 19 years later (𝑝𝑑19 ). This was then compared to the observed D at
the second survey 𝑑19 using the following model form:
d19 ~N(f(d1 , θ), σ2 )
12
where the function f(𝑑1 , θ) was a discrete-time model for annual growth rate. Since growth was sizedependent we therefore compounded growth rate to find 𝑝𝑑19 :
𝑝𝑑19 = 𝑝𝑑18 + 𝐺𝑅(𝑝𝑑18 ) = 𝑝𝑑18 + 𝐺𝑅(𝑝𝑑17 + 𝐺𝑅(𝑝𝑑17 ))
13
We modelled the standard deviation as increasing with dbh:
𝜎 = 𝜌0 + 𝜌1 𝑑1
14
The predicted growth therefore had corresponding likelihood
(𝑑19 −𝑝𝑑19 )2
1
)
exp
(−
)}
2σ2
√2π σ
𝑙(𝑋|𝑀, 𝜃) = ∑i ln {(
15
We modelled annual probability of mortality for each individual tree i, as P(mortality, i). Since
P(mortality, i) must lie between 0 and 1, we used a logistic transformation
𝑃(mortality,𝑖) = 1⁄(1 + exp(−𝑘𝑖 ))
16
where ki (which can vary from ± ) is a function of the predictor variables. This had corresponding
likelihood
[1 − 𝑃(mortality,𝑖)]19 if tree 𝑖 survived
𝑙(𝑋|𝑀, 𝜃) = {
17
1 − [1 − 𝑃(mortality,𝑖)]19 if tree 𝑖 died
The individual annual growth and mortality rates that we tested are shown in Tables S2 and S3.
The algorithm has two periods: burnin and sampling. During the burnin period (we used
750,000 iterations of the algorithm) the algorithm alters the search range ("jumping distance") of each
parameter value to achieve an optimal acceptance ratio of 25% (Gelman et al. 1999). After the burnin
period, the jumping distance is fixed (separately for each parameter). During sampling (250,000
iterations), parameter values are recorded every 100 iterations and the resulting parameter samples are
taken as samples from the distribution of each parameter. The resulting 2500 samples are then used to
calculate mean and 95% confidence intervals for each parameter. We used non-informative uniform
priors on all parameters so the MCMC algorithm (see below) needed to refer to the log-likelihood
only. For both growth and mortality we rescaled altitude values so that the minimum was 0. All
parameters were constrained to take values within (-100, 100) apart from the parameter 𝜌0 (eqn 14)
which was constrained to be positive. All models were fitted using an adaptive Metropolis algorithm
written in C (complied using MS Visual Studio 2008).
We used Akaike Information Criterion (AIC; Akaike 1974) to test several different model
forms for both annual growth (Table S2) and annual probability of mortality (Table S3), using stem
size (D), altitude (ALT, standardized so that the minimum was 0 using 𝐴𝐿𝑇 = (altitude(m) −
640)/100) and the crown area of trees taller than a specified height relative to the canopy height of
the target tree (𝐶𝐴𝐼ℎ ). We tested all model forms using basal area of the plot instead of 𝐶𝐴𝐼ℎ as a
measure of competition but all models with 𝐶𝐴𝐼ℎ were better fits. For initial model fits we used 𝐶𝐴𝐼ℎ
calculated at h = H – aV, where a is 0.5 and V is the crown depth, i.e. 𝐶𝐴𝐼ℎ representing the summed
projected crown area of trees taller than the mid-point of the crown of the target tree. We also
compared model fits for 𝐶𝐴𝐼ℎ values calculated at different heights, for a = 0 (i.e. only trees taller
than the target tree), a = 0.1, a = 0.25 and a = 1 (i.e. bottom of canopy). We fitted the three different
annual growth and mortality rates for each of these indices (Tables S4 and S5). We selected the best
models for growth and mortality (Fig. S7) to use in the PPA analysis (for growth the eighth model in
Table S4, for mortality the third model in Table S5).
Table S2. Comparison of different annual growth models, showing the different functional forms
tested. Models are compared using the Akaike Information Criterion (AIC). The model with the
lowest AIC is best supported, and all other models are compared with it using ΔAIC; alternative
models with ΔAIC < 4 are also considered to be well supported.
Annual growth model
Max log
Par
AIC
likelihood
𝜌4
exp(𝜌5 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌4
𝜌2 (1 + 𝜌6 𝐴𝐿𝑇)𝐷𝜌3 ⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌4
𝜌2 𝐷 𝜌3(1+𝜌6𝐴𝐿𝑇) ⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌
4
𝜌2 𝐷 𝜌3 ⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ + 𝜌6 𝐴𝐿𝑇)]
𝜌2
𝜌4
𝜌2 𝐷 𝜌3 ⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ + 𝜌6 𝐴𝐿𝑇 × 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌4
𝜌2 𝐷 𝜌3 exp(𝜌7 𝐷)⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ + 𝜌6 𝐴𝐿𝑇 × 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌4
𝜌2 𝐷 𝜌3 (1 + 𝜌8 𝐴𝐿𝑇)exp(𝜌7 𝐷)⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ + 𝜌6 𝐴𝐿𝑇 × 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌4
𝜌2 𝐷 𝜌3 (1 + 𝜌6 𝐴𝐿𝑇)exp(𝜌7 𝐷)⁄[1 + exp(𝜌5 𝐶𝐴𝐼ℎ )]
𝜌2
𝜌2 𝐷 𝜌3 ⁄[1 +
∆AIC
AIC
Rank
-29763.2
6
59538.39
7
2820.6
-28534.9
7
57083.74
2
366.0
-41164.7
7
82343.32
8
25625.5
-28540.1
7
57094.22
3
376.4
-28754.7
7
57523.45
6
805.77
-28710.5
8
57437.05
5
719.3
-28349.9
9
56717.78
1
0
-28553.6
8
57123.15
4
405.4
Table S3. Comparison of different models for annual probability of mortality, showing the different
functional forms tested. D is stem diameter, ALT is altitude, rescaled as (altitude – 640)/100, and
CAIh is the sum of the crown area of taller trees. τ0 – τ6 are parameters that were estimated by the
MCMC algorithm. Models are compared using the Akaike Information Criterion (AIC). The model
with the lowest AIC is best supported, and all other models are compared with it using ΔAIC;
alternative models with ΔAIC < 4 are also considered to be well supported.
Logit(Annual probability of mortality)
Max log
Par
AIC
likelihood
AIC
∆AIC
Rank
𝜏0 + 𝜏1 𝐷 exp(𝜏2 𝐷) + 𝜏3 𝐶𝐴𝐼ℎ
-10519.4
4
21046.8
6
316.5
𝜏0 + 𝜏1 𝐷 exp(𝜏2 𝐷) + 𝜏3 𝐶𝐴𝐼ℎ + 𝜏4 𝐴𝐿𝑇
-10433.2
5
20876.4
4
146.1
𝜏0 + 𝜏1 𝐷exp(𝜏2 𝐷)(1 + 𝜏4 𝐴𝐿𝑇) + 𝜏3 𝐶𝐴𝐼ℎ
-10480.9
5
20971.9
5
241.6
𝜏0 + 𝜏1 𝐷exp(𝜏2 𝐷(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼ℎ
-10519.2
5
21048.5
7
318.2
𝜏0 + 𝜏1 𝐷exp(𝜏2 𝐷) + 𝜏3 𝐶𝐴𝐼ℎ (1 + 𝜏4 𝐴𝐿𝑇)
-10423.5
5
20857.0
3
126.7
𝜏0 + 𝜏1 𝐷 exp(𝜏2 𝐷(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼ℎ + 𝜏5 𝐴𝐿𝑇
-10365.8
6
20743.6
2
13.3
𝜏0 + 𝜏1 𝐷 exp(𝜏2 𝐷(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼ℎ + 𝜏5 𝐴𝐿𝑇 + 𝜏6 𝐶𝐴𝐼ℎ × 𝐴𝐿𝑇
-10358.1
7
20730.3
1
0.0
Table S4. Comparison of three alternative annual growth models (M1, M2 and M3 as given at bottom
of table) calculated using five alternative CAIh indices, giving a total of 15 alternative models. CAIh
was calculated as the crown area of trees taller than the specified proportion of height below the top of
the target tree. These were calculated at heights 0 (top of the tree), 0.1, 0.25, 0.5 and 1 (bottom of
crown). Models are compared using the Akaike Information Criterion (AIC). The model with the
lowest AIC (in bold) is best supported, and all other models are compared with it using ΔAIC;
alternative models with ΔAIC < 4 are also considered to be well supported.
Growth
CAIh
model*
Max log
Par
AIC
likelihood
AIC
∆AIC
Rank
M1
0
-33200.8
7
66415.7
15
9882.5
M2
0
-28416.5
9
56850.9
4
317.8
M3
0
-33179
8
66374.0
14
9840.8
M1
0.1
-28637.6
7
57289.2
9
756.0
M2
0.1
-28402.1
9
56822.1
3
289.0
M3
0.1
-28599.1
8
57214.3
8
681.1
M1
0.25
-28517.6
7
57049.1
5
516.0
M2
0.25
-28257.6
9
56533.1
1
0
M3
0.25
-33160.5
8
66337.0
13
9803.8
M1
0.5
-28595.5
7
57205.1
7
671.9
M2
0.5
-28350.2
9
56718.4
2
185.2
M3
0.5
-28551.6
8
57119.2
6
586.0
M1
1
-28783.4
7
57580.9
12
1047.7
M2
1
-28640.1
9
57298.3
10
765.1
M3
1
-28684.6
8
57385.2
11
852.1
* 𝑀1 =
𝑝2 (1+𝑝6 𝐴𝐿𝑇)𝐷 𝑝3
, 𝑀2 =
𝑝
1+ 4exp(𝑝5 𝐶𝐴𝐼ℎ )
𝑝2
𝑝2 (1+𝑝8 𝐴𝐿𝑇)𝐷𝑝3 exp(𝑝7 𝐷)
𝑝
1+ 4 exp(𝑝5 𝐶𝐴𝐼ℎ +𝑝6 𝐴𝐿𝑇×𝐶𝐴𝐼ℎ )
𝑝2
, 𝑀3 =
𝑝2 (1+𝑝6 𝐴𝐿𝑇)𝐷 𝑝3 exp(𝑝7 𝐷)
𝑝
1+ 4exp(𝑝5 𝐶𝐴𝐼ℎ )
𝑝2
Table S5. Comparison of three models of annual probability of mortality (k1, k2, k3) calculated using
five different CAIh indices, giving rise to 15 alternative models. CAIh was calculated as the crown
area of trees taller than the specified proportion of height below the top of the target tree. These were
calculated at heights 0 (top of the tree), 0.1, 0.25, 0.5 and 1 (bottom of crown). The best supported
model (lowest AIC) is shown in bold.
CAIh
Max log
Par
AIC
AIC
likelihood
∆AIC
Rank
k1
0
-10448.4
5
20906.9 11
513.4
k2
0
-10202.7
6
20417.4 2
23.9
k3
0
-10189.7
7
20393.5 1
0.0
k1
0.1
-10459.1
5
20928.1 12
534.6
k2
0.1
-10225.5
6
20463.0 4
69.5
k3
0.1
-10217.2
7
20448.4 3
54.9
k1
0.25
-10485.2
5
20980.5 13
587.0
k2
0.25
-10288.4
6
20588.9 6
195.4
k3
0.25
-10278.75
7
20571.5 5
178.0
k1
0.5
-10519.2
5
21048.5 14
655.0
k2
0.5
-10365.8
6
20743.7 8
350.2
k3
0.5
-10358.2
7
20730.4 7
336.9
k1
1
-10552.5
5
21114.9 15
721.4
k2
1
-10443.9
6
20899.7 10
506.2
k3
1
-10440.3
7
20894.6 9
501.1
Annual probability of mortality P(mortality)=1/(1+exp(-k)) where:
𝑘1 = 𝜏0 + 𝜏1 𝑑 exp(𝜏2 𝑑(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼,
𝑘2 = 𝜏0 + 𝜏1 𝑑 𝑒𝑥𝑝(𝜏2 𝑑(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼 + 𝜏5 𝐴𝐿𝑇,
𝑘3 = 𝜏0 + 𝜏1 𝑑 𝑒𝑥𝑝(𝜏2 𝑑(1 + 𝜏4 𝐴𝐿𝑇)) + 𝜏3 𝐶𝐴𝐼 + 𝜏5 𝐴𝐿𝑇 + 𝜏6 𝐶𝐴𝐼 × 𝐴𝐿𝑇.
The parameter estimates for the best fitting models (Bayesian means) and their 95% credible intervals
are given in Table S5.
Table S5.Parameter values for the best fit models for annual growth (eighth model in table S3) and
annual probability of mortality (third model in table S4) showing Bayesian mean and 95% credible
interval (CI).
Annual growth model parameters
Bayesian
mean
95% CI
𝑝0
𝑝1
𝑝2
𝑝3
𝑝4
𝑝5
𝑝6
𝑝7
𝑝8
0.871
0.038
0.080
0.612
0.009
1.213
0.167
-0.013
-0.079
(0.896,
0.847)
(0.040,
0.036)
(0.093,
0.066)
(0.671,
0.552)
(0.0140,
0.004)
(1.366,
1.060)
(0.190,
0.144)
(-0.010,
-0.017)
(-0.076,
-0.082)
𝜏5
𝜏6
Annual probability of mortality model parameters
𝜏0
Bayesian
mean
95% CI
𝜏1
𝜏2
𝜏3
𝜏4
-3.859
-0.429
-0.091
0.693
-0.070
0.222
0.098
(-3.605,
-4.113)
(-0.400,
-0.457)
(-0.086,
-0.097)
(0.847,
0.540)
(-0.062,
-0.077)
(0.273,
0.170)
(0.133,
0.062)
The models provide unbiased predictions of the data, as evident from Fig. S7.
Fig. S7. Observed and predicted growth over 19 years (a) and annual mortality rates (b) plotted against initial stem diameter
for the best supported models (model 7 in Table S4 and model 2 in Table S5). Observed mortality rates were calculated by
binning the data and taking an average total probability of mortality divided by 19 (total survey interval); the average
predicted probability of mortality is shown for the same binned data.
We note that growth and mortality analyses of mountain beech data from the Craigieburn study area
have been published by Coomes & Allen (2007a&b) and Hurst et al. (2012). Previous analyses were
based on measurements taken in 1974, 1983 and 1993 in 250 permanent plots. The current analyses
are based on 246 plots measured over a longer period (1974-2004). The reason for the discrepancy in
plot number is that four plots were obliterated by the 1994 earthquake. The previous growth and
mortality analyses are similar to the ones presented here except that (a) we used summed canopy area
in our competition models rather than summed basal area; (b) we included random plot effects in
previous papers but not here; and (c) we allowed residual errors in the growth model to vary with tree
size. Several corrections have been made to the dataset; for instance, some trees recorded as dead in
earlier census were found to be alive in subsequent surveys. Despite these differences in methods and
data, the results used to develop simulation models are similar to those published previously.
H4 EXPLORING THE ROLE OF FOREST DYNAMICS: SIMULATING STAND
PRODUCTIVITY USING THE PPA MODEL
We used the crown allometry, stand biomass, stem growth and mortality relationships described
above to implement a modified version of the PPA model (Purves et al. 2008, Caspersen et al. in
press) to simulate stand productivity changes from 1974 to 1993 in the 208 permanent plots from the
mountain beech dataset. Initial stand composition was set based on 1974 plot data. For each time-step
and each tree, we then calculated 𝐶𝐴𝐼ℎ (using a = 0.25 for growth and a = 0 for mortality). We then
used the best fitting growth model (model 7 in Table S4) to calculate stem diameter growth,
calculated the probability of mortality (using model 6 in Table S5), and used the runif function in R to
determine if the tree died or not. The number of new recruits (with diameter of 2.5 cm and height of
1.35m) was then calculated based on the total CAI of the stand using the following function:
𝑅𝑒𝑐𝑟𝑢𝑖𝑡𝑠 = 𝑎𝑒𝑥𝑝(−𝑣 ∗ 𝐶𝐴𝐼)
18
where the parameters a and v were estimated using the recruitment data over the period from 1974 to
1993 (Fig. S8, a = 0.83 and v = 1.438). Examination of the data from 1974–1993 showed that
recruitment was low in nearly all plots, and that new recruits accounted for a very small fraction of the
observed biomass changes during this period. The PPA model simulations gave very similar results
with or without the inclusion of recruitment.
At the end of each time-step (i.e. annually), we calculated ProdM (the biomass growth of
trees alive at both the start and end of the time period, summed across each plot), LossM (the biomass
of trees that died during that time period, summed across each plot), RecrM (the biomass of new
recruits, summed across each plot), and SeqM (the net change in biomass for each plot). We ran the
model for 19 years, simulating 100 replicates of each of the 208 plots.
The annual ProdM, LossM, RecrM and SeqM values were averaged across all simulations
and across all years for comparison with the observed values (see main text for details). Plot-level
predicted and observed values for ProdM, LossM and SeqM are shown in Fig. S9. At the plot-level,
ProdM was more accurately predicted than LossM or SeqM (ProdM r2 = 0.46, LossM r2 = 0.13 ,
SeqM r2 = 0.09). This is because of the stochastic nature of tree mortality, since death of a single large
tree can have a significant influence on plot-level biomass. Predicted biomass in 1993 was close to the
observed biomass (Fig. S10, r2 = 0.68), however the model slightly over-estimated 1993 biomass in
low-biomass plots, and under-estimated it in high-biomass plots.
To validate our model’s ability to predict carbon sequestration patterns in independent data,
we fitted the growth, mortality, and recruitment functions (Tables S2 & S3) to a random subset of 180
of the 208 plots. We then ran PPA simulations using these parameters to predict carbon sequestration
patterns for the 21 independent validation plots (Fig. S11 and S12). For the validation plots, the
average predicted values (in MgC ha-1 ± SEM) for ProdM (1.18 ± standard error of 0.07), LossM (1.19 ± 0.15) and SeqM (-0.002 ± 0.18) were not significantly different from the observed values for
ProdM (1.07 ± 0.07), LossM (-1.08 ± 0.15) and SeqM (-0.002 ± 0.19).
Annual recruits from 73-94
15
10
5
0
0.5
1.0
1.5
2.0
2.5
Total CAI of stand in 1974
Fig. S8 Observed stand-level stem recruitment for individual plots from 1974–1993, plotted against total CAI of the stand.
Fitted line shows predicted values using eqn. 18.
Fig. S9 Predicted and observed ProdM, LossM, and SeqM for the 208 thinning plots, based on the PPA simulations using
the best fitting growth and mortality models.
Fig. S10 Predicted and observed stand biomass in 1993 for the 208 thinning plots, based on the PPA simulations using the
best fitting growth and mortality models (r2= 0.68).
Fig. S11. Predicted and observed ProdM (r2 = 0.52), LossM (r2 = 0.08), and SeqM (r2 = 0.11), for the independent validation
plots.
Fig. S12. Predicted and observed stand biomass in 1993 for the independent validation plots (r2 = 0.70).
When modelling CWD dynamics in the permanent plot network, initial CWD stocks in 1974 were
generated (independently of stand biomass) from a log-normal distribution with mean and standard
deviation taken from a subset of plots from the distributed plot network for which CWD stocks had
been measured (N=19 thinning plots measured in 1998). Using data from Richardson et al. (2009) and
wood density decay class values from Clinton et al. (2002) we were able to estimate the fraction of
CWD pool in each decay class (i = 10%, ii = 65%, iii = 25%), and the average age of each pool in
years (i = 9 years, ii = 18 years, iii = 29 years). At each time-step we calculated the biomass loss due
to decay processes, and the CWD biomass inputs from tree mortality. All FWD (branches <10 cm)
and litter are assumed to decompose entirely within the first year. We therefore divided total aboveground biomass of dead trees by 1.35 to get stem wood biomass, and then applied the decay function
to these CWD biomass stocks in subsequent years.
H5 CLIMATE CHANGE
Long-term data from a meteorological station at the Craigieburn study indicates no trend in mean
annual temperature, or in growing season temperature, but 0.18 oC increase in mean winter
temperature each decade.
Fig. S12. Long-term trends in mean annual temperature (diamonds), mean growing season temperature (solid circles) and
mean winter temperatures (open circles) measured at 914 m elevation in the Craigieburn study area. There was a significant
upward trend in winter temperature (3.62 + 0.018 × year since 1964, F1,44 = 6.6, P = 0.014), but no trend in summer or
mean temperatures. The growing season of year x extended from October in year x to April in the following year (mean
monthly temperatures exceeded 5oC in these months), the winter season from May to September in year x, and the whole
year ran from July in year x to June in the following year.
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