eesSchleip2015

eesSchleip2015
Environ Ecol Stat (2016) 23:23–41
DOI 10.1007/s10651-015-0321-z
Modeling change in forest biomass across the eastern US
Erin M. Schliep1 · Alan E. Gelfand1 ·
James S. Clark1,2 · Kai Zhu2,3,4
Received: 2 February 2015 / Revised: 7 June 2015 / Published online: 27 June 2015
© Springer Science+Business Media New York 2015
Abstract Predictions of above-ground biomass and the change in above-ground biomass require attachment of uncertainty due the range of reported predictions for forests.
Because above-ground biomass is seldom measured, there have been no opportunities
to obtain such uncertainty estimates. Standard methods involve applying an allometric
equation to each individual tree on sample plots and summing the individual values.
There is uncertainty in the allometry which leads to uncertainty in biomass at the tree
level. Due to interdependence between competing trees, the uncertainty at the plot
level that results from aggregating individual tree biomass in this way is expected to
overestimate variability. That is, the variance at the plot level should be less than the
sum of the individual variances. We offer a modeling strategy to learn about change
in biomass at the plot level and model cumulative uncertainty to accommodate this
dependence among neighboring trees. The plot-level variance is modeled using a parametric density-dependent asymptotic function. Plot-by-time covariate information is
introduced to explain the change in biomass. These features are incorporated into a
hierarchical model and inference is obtain within a Bayesian framework. We analyze
data for the eastern United States from the Forest Inventory and Analysis (FIA) Program of the US Forest Service. This region contains roughly 25,000 FIA monitored
Handling Editor: Pierre Dutilleul.
B
Erin M. Schliep
erin.schliep@duke.edu
1
Department of Statistical Sciences, Duke University, Durham, NC, USA
2
Nicholas School of the Environment, Duke University, Durham, NC, USA
3
Department of Global Ecology, Carnegie Institution for Science,
Stanford, CA, USA
4
Department of Biology, Stanford University, Stanford, CA, USA
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plots from which there are measurements of approximately 1 million trees spanning
more than 200 tree species. Due to the high species richness in the FIA data, we combine species into plant functional types. We present predictions of biomass and change
in biomass for two plant functional types.
Keywords Allometric equations · Bayesian hierarchical model · Cumulative
uncertainty · Forest biomass
1 Introduction
Forests play an important role in the global carbon cycle (Pan et al. 2011). Given
the range of predictions reported for carbon sinks in forests, estimates of associated
uncertainty are critical. For eastern North America, predictions of the annual carbon
sink range from 0.21 to 0.25 petagrams of carbon per year (Pg C/yr) (Pan et al. 2011).
Unfortunately, there has been no way to assign statistical model-based uncertainty
to these predictions because biomass per unit area is not directly measured. Instead,
allometric equations are applied to the diameter (and sometimes height) of each tree
on a plot to obtain predictions of above-ground biomass at the tree level, resulting
in uncertainty in tree-level biomass. At the plot level, predictions of above-ground
biomass are obtained by summing the independent tree-level predictions of biomass.
Summing the tree-level variances to obtain variance in total biomass ignores the fact
that trees interact. That is, due to the interdependence and crowding between competing trees, the sum of the variances will not equal the variance of the sum. One
approach to this problem is to model the cumulative variance as density-dependent.
The variance model should allow for dependence as a function of density in the form
of diminishing returns; the plot-level variance is something less than the sum of the
individual variances.
Understanding forest biomass change is essential for human society to cope with
global climate change (Barford et al. 2001; Schimel et al. 2001; Wright 2005; Susan
2007). This has led to substantial literature providing modeling efforts for both biomass
and the change in biomass. For example, McMahon et al. (2010) model biomass using
the Monod function which describes the increase in biomass of forests during recovery.
Their approach focuses on patterns of resource use and limitation and is a function of
stand age and the age at half-saturation. Vayreda et al. (2012) use principal component
analysis to model change in carbon, as well its components, growth rate and mortality
rate. This approach has the disadvantage of making interpretation of model parameters
difficult.
We offer a species-level modeling strategy to quantify forest biomass change that
accounts for density and species differences as they vary geographically. Forests are
made up of a mix of tree species with varying functional traits and growth patterns as
a response to light, moisture, and nutrients. In the eastern United States, forests have
experienced dramatic change due to human disturbance for centuries (MacCleery
1993). In the process of recovery, forests are changing their species compositions over
time, a process referred to in forest ecology as succession. Thus, biomass change of
eastern US forests depends heavily on successional status of species and management
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practices across geography. To accurately quantify forest biomass change, we take
into account species, environment, geography, shade tolerance, and seral stage.
Allometric equations are available for common tree species regionally and globally
[e.g., Jenkins et al. (2003)]. Uncertainty at the tree level has been investigated in
terms of measurement error, the specification in the allometric equation, the sampling
protocol of stems in a plot, and the representativeness of small plots for a forest
landscape (Chave et al. 2004, 2014). However, we are interested in above-ground
biomass, referred to as biomass hereafter, at the plot level. Therefore, our estimates of
uncertainty should also be at the plot level.
Our data come from the Forest Inventory and Analysis (FIA) Program of the US
Forest Service and include species, size, and health of trees, as well as tree growth,
mortality, and removal by harvest. We analyzed data obtained from the eastern United
States. This region contains roughly 25,000 FIA monitored plots from which there
are measurements of roughly 1 million trees spanning more than 200 tree species.
Between 1997 and 2011, each plot in the region is surveyed twice. Due to the high
species richness in the FIA data, we combine species into plant functional types (as
described below) and model at that level.
To illustrate the nature of diminishing returns in the uncertainty of biomass at
the plot level, we predict tree-level biomass for late successional hardwoods in the
FIA data across the eastern US using species-specific allometric equations and associated errors. That is, for each late successional hardwood on the plot we predict
biomass using species-specific allometric equations and the tree’s diameter at breastheight. Using the estimates of uncertainty in the allometric equations, we repeat this
for 10,000 iterations to obtain a distribution of predicted biomass for each tree. For
each plot, we sum the tree-level predictions to obtain a distribution of predicted plotlevel biomass. We compute standard deviations of these samples for each plot to
obtain estimates of uncertainty in plot-level biomass. Working with plots of size
0.067 ha, Fig. 1 (left) shows boxplots of the standard deviation of plot-level biomass binned according to plot density. Initially, uncertainty in plot-level biomass is
increasing as a function of the number of trees. When the plot has more than 20
trees, however, uncertainty saturates; additional trees do not add to the uncertainty
Fig. 1 Standard deviation (left) and coefficient of variation (right) of the distribution of predicted biomass
at the plot level as a function of the number of trees
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in plot-level biomass. Also shown in Fig. 1 (right) are boxplots of the coefficient
of variation (CV) by plot density, where CV is computed as the ratio of the standard deviation to the mean. CV decreases as a function of the number of trees
indicating, again, that the variance of biomass at the plot level reaches saturation.
Chave et al. (2014) report a similar decrease in CV with increasing plot density
for tropical and sub-tropical forests and woodland savannas. We argue that the CV
decreases with the number of trees and it results from dependence among individuals.
We are interested in predicting biomass with uncertainty at the plot level and,
therefore, we model at the plot level. Additionally, we want predictions of the change
in biomass at the plot level. As a result of the FIA data collection protocol, small trees
are measured only on a subset of a plot while larger trees are measured on the entire
plot. This means that some trees will not have been included in a previous sample. For
this reason, we model as separate responses plot-level biomass of saplings and trees.
Due to the sparse surveying of plots, we model plot-level biomass statically at each
survey time with change in biomass induced by differencing. Plot-by-time covariate
information is introduced, as well as plot-level covariates, to explain change. A benefit
of our model is that we have an explicit conditional distribution for the rate of change
of biomass given current biomass for each plot and plant functional type. We propose
a parametric functional specification for cumulative uncertainty at the plot level that
results from aggregating individual-level biomass to plot-level biomass. All of these
features are incorporated into a hierarchical model and we implement inference within
a Bayesian framework.
The plan of the paper is as follows. In Sect. 2 we describe the FIA data, how
and where they are collected and the species and plant functional types observed.
Allometric equations for computing individual-level biomass are described in Sect. 3.
We show how biomass is aggregated to the plot level where total biomass is defined
as the summation of sapling biomass and tree biomass. We define Δ-biomass as the
annual rate of change in total biomass. In Sect. 4, we outline the models for sapling
and tree biomass. In Sect. 5, we apply the model to two plant functional types, late
successional hardwoods and southern pines. The paper concludes with a summary and
suggestions for future work in Sect. 6.
2 The FIA data
FIA applies a nationally consistent sampling protocol using a quasi-systematic design
covering all ownerships across the United States resulting in national sample intensity of one plot per 2428 ha (Bechtold and Patterson 2005). Within the eastern
US, the FIA surveys roughly 25,000 plots (Fig. 2, left). Data obtained for each
plot include forest type, site attributes, tree species, tree size, and overall tree condition. We included only non-disturbed plots in this analysis. Between the years
1997 and 2011, each plot is surveyed twice and the time between surveys ranges
from 1 to 12 years (Fig. 2, right). The inventory includes 218 species of trees. Due
to the large number of different species and the rarity of some species across the
region, we group species into 11 plant functional types (PFTs) (Dietze and Moorcroft
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Fig. 2 Number of plots surveyed by the FIA across the eastern US (left) and years between FIA surveys
across plots (right)
Table 1 Summary of the 23,259 FIA plots by plant functional type
No.
PFT
Acronym
No. of plots
1.
Early successional hardwoods
ESH
16,364
2.
Evergreen hardwoods
Evergreen
1502
3.
Hydrics
Hydric
1213
4.
Late successional conifers
LSC
7627
5.
Late successional hardwoods
LSH
17,306
6.
Midsuccessional conifers
MC
3951
7.
Northern midsuccessional hardwoods
NMH
16,055
3531
8.
Northern pines
NP
9.
Southern midsuccessional hardwoods
SMH
10,810
10.
Southern pines
SP
4664
11.
Other
693
2011). The PFTs are listed in Table 1 along with the number of plots each PFT was
observed.
An FIA plot consists of four circular subplots arranged in the pattern shown in Fig.
3. The subplots each have a radius of 7.32 m and the distance between the subplot
centroids is 35.58 m. Measurements are taken of all trees within subplots where a
tree is classified as an individual with diameter greater than 12.7 cm. All saplings,
referring to individuals with diameters less than or equal to 12.7 cm, are measured
on the four microplots, each of which is a subset of a subplot. The radius of each
micro plot is 2.07 m. The total microplot area is 53.85 m2 and the total subplot area
is 673.34 m2 . Ecologically, the distance between plots in the FIA data is too large
to adopt spatial dependence between plots. Instead, we capture heterogeneity across
plots using random plot effects.
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Fig. 3 FIA sampling scheme for
subplots (grey) and microplots
(black) on an FIA plot
3 Individual and plot-level biomass
3.1 Allometric equations for individual biomass
Biomass for each individual is computed using an allometric regression equation that
converts diameter to above-ground biomass. The allometric equation proposed by
Jenkins et al. (2003) is
log(biomass) = β0 + β1 log(dbh)
(1)
where dbh is diameter at breast-height (cm) and biomass is measured in kilograms
(kg). The parameter values β0 and β1 are species-specific. There are numerous studies
on biomass equations for different species and regions [e.g., Jenkins et al. 2003;
Brown et al. 1999; Marklund 1988; Zianis et al. 2005]. With the exception of Chave
et al. (2004), Wutzler et al. (2008), and Stephenson et al. (2014), however, there is
much less in the literature about their uncertainty due the destructive sampling of
trees that is required. Chave et al. (2004) investigated the error associated with the
allometric equations in predicting biomass for tropical forests and reported the choice
of allometric equation contributed to error greater than 20 % of the above-ground
biomass Stephenson et al. (2014) found that Eq. (1) has a tendency to overpredict
biomass for larger trees for nine species from the temperate western USA. Using
a similar biometric equation, Wutzler et al. (2008) found confidence intervals for
biomass to be narrow where the coefficient of variation (CV) was 0.12 for an individual
at an average stand.
3.2 Sapling and tree plot-level biomass
Using (1) we obtain predictions of biomass (kg) at the individual level for each plot at
two survey times. Let s be an indicator for survey where s = 1 for the first survey and
s = 2 for the second survey. Let Yikjs denote the biomass (kg) of the jth individual
at plot i of PFT k at survey s. Due to the different sampling intensities within the
plot (i.e. subplot, microplot), we model total biomass in two components, (1) biomass
of saplings and (2) biomass of trees. Let dikjs denote the diameter of the individual.
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Individuals with dikjs ≤ 12.7 cm are classified as saplings at survey s and those with
dikjs > 12.7 cm are classified as trees at survey s. For plot i and PFT k we define
sapling biomass for survey 1 as
k
bi1
=
Yikj1 I[d k
i j1 ≤12.7]
j
and at survey 2 as
k
bi2
=
Yikj2 I[d k
i j2 ≤12.7]
j
k is in terms of kg per four microplots within the FIA plot for all i, s, and k.
where bis
We denote total biomass of trees at plot i of PFT k for survey 1 as
k
Bi1
=
Yikj1 I[d k
i j1 >12.7]
j
and at survey 2 as
k
Bi2
=
Yikj2 I[d k
i j2 >12.7]
j
k is in terms of kg per four subplots within the FIA plot for all i, s, and k.
where Bis
k and B k as “noisy” total sapling and tree biomass, respectively, due
We consider bis
is
to the error in the predictions of each Yikjs resulting from the allometry. That is, total
sapling and tree biomass is derived; it is never observed. However, with interest in
understanding the behavior of total biomass, below, we model the b’s and B’s.
Figure 4 gives histograms of the number of late successional hardwoods (LSH)
classified as saplings (left) and trees (right) observed at each plot during the first
survey. Also in this figure are histograms of total sapling and total tree biomass across
plots for LSH.
k denote total biomass at plot i and survey s of PFT k. Total biomass is the
Let T Bis
summation of sapling and tree biomass computed as
k
T Bis
=
k
bis
Bk
+ is
As
At
(2)
where As and At are the total area (ha) of the four microplots where all saplings are
k
observed and four subplots where all trees are observed, respectively. Therefore, T Bis
is in terms of kg per hectare (kg/ha). The rate of change in biomass, given in kg per
hectare per year (kg/ha/yr), referred to as Δ-biomass, for plot i and PFT k is
Δik =
k − T Bk
T Bi2
i1
ti2 − ti1
(3)
where ti2 − ti1 is the time between the two surveys for plot i. This assumes a constant
rate of change in biomass between ti1 and ti2 as opposed to what would result from
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Fig. 4 Histograms of the number of individuals (top) and total biomass (bottom) of saplings (left) and
trees (right) for LSH for survey 1. Total biomass is given in kg per four microplots (saplings) and subplots
(trees) on the FIA plots
rapid or slow onset disturbance. However, we are limited by having only two times
points; more complex modeling for the rate of change would require additional surveys.
4 The model
We define models for noisy total sapling and tree biomass in a hierarchical framework.
Noisy total sapling biomass, henceforth referred to as sapling biomass, is modeled as
k
k k
k
k
= n is
μis + g(n is
; φbk )is
bis
(4)
k is the number of saplings, μk is the average sapling biomass, and
where n is
is
k
k is first-stage measurement error. We assume k is independent error
g(n is ; φbk )is
is
k ; φ ) is a function of n k and a parawith variance σb2k for all i, s, and k, and g(n is
bk
is
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meter φbk . Similarly, we model noisy total tree biomass, henceforth referred to as tree
biomass, as
k
k
= Nisk θisk + g(Nisk ; φ B k )ηis
(5)
Bis
k
where Nisk is the number of trees, θisk is the average tree biomass, and g(Nisk ; φ B k )ηis
k
2
is first-stage measurement error where ηis is independent error with variance σ B k .
Again, g(Nisk ; φ B k ) is a function of Nisk and a parameter φ B k .
We assume the function g(m; φ) is an exponential asymptote function with parameter φ that is bounded above where
g(m; φ) =
1 − exp−m/φ .
(6)
This parametric functional form was motivated by Fig. 1 to govern cumulative uncertainty at the plot level that results from aggregating from individual-level biomass
to plot-level biomass. It assumes that measurement error increases as the number of
individuals on the plot increases while restricting total plot-level measurement error.
Here, φbk denotes the range parameter controlling the asymptote of sapling biomass
measurement error and φ B k denotes the range parameter controlling the asymptote of
tree biomass measurement error.
Next, we model average sapling and tree biomass as incorporated in (4) and (5). To
k and θ k using a tobit
ensure that average biomass is non-negative, we model both μis
is
k
k
model with latent random variables μis and θis , respectively. That is,
k
μis
=
and
θisk =
k
μis
k >0
μis
0
k ≤0
μis
θisk
θisk > 0
0
θisk ≤ 0.
(7)
(8)
In general, the tobit model is a more natural model than say, a log-normal, if we expect
many average sapling or tree biomasses to be small, and thus, don’t think the left tail
of the density should go to 0 at 0.
Both latent average biomass variables are specified through a linear mixed model
with normally distributed error. For PFT k, latent average sapling biomass is
k
k
= W is
α 1 + λik
μis
λik = X i α k2 + νik
(9)
where W is is a vector of survey year-specific covariates for plot i, α k1 is a vector of
coefficients, and λik is a plot random effect. The plot random effect λik is centered
with mean X i α k2 where X i is a vector of covariates for plot i and α k2 is a vector of
coefficients. Lastly, νik is normally distributed random noise with mean 0 and variance
τλ2k . The multilevel structure of the model is specified in the form of hierarchical
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centering1 which better identifies the model parameters and leads to better behaved
Markov chain Monte Carlo (MCMC) model fitting (Gelfand et al. 1995). Similarly,
latent average tree biomass is modeled as
k
θisk = W is
β 1 + γik
γik = X i β k2 + ζik .
(10)
Here, β k1 and β k2 are vectors of coefficients, γik is a plot random effect, and ζik is
normally distributed random noise with mean zero and variance τγ2k .
k and T B k ,
Given the model parameters, we have explicit distributions of both T Bi1
i2
k using (2) and (3) (See “Appendix”). From (4) and (5), total biomass
as well as Δik |T Bi1
might possibly be less than 0, in which case we set it to 0. Using these distributions
and composition sampling we are able to obtain draws from the posterior predictive
distributions at the plot level for total biomass and Δ-biomass.
We assign prior distributions to the model parameters defined in (4), (5), (9), and
(10) above as follows. At the data level, we assign diffuse, conjugate inverse-Gamma
distributions to the variance parameters, σb2k and σ B2 k . The range parameters, φbk and
φ B k , controlling the exponential asymptote functions are assigned truncated-normal
distributions constrained to be positive. We assign mean zero multivariate normal
distributions to the coefficient vectors α k1 and β k1 . These priors are not conjugate
k and θ k , are nonbecause both true average sapling and tree biomass parameters, μis
it
negative as defined by the tobit model in (7) and (8). Thus, sampling of these parameters
requires Metropolis-Hastings steps and the details of these algorithms are given in
“Appendix”.
The coefficients α 2 and β 2 are assigned noninformative conjugate multivariate
normal distributions. We include plot-level random effects, νik and ζik , to capture any
remaining heterogeneity in sapling and tree biomass across plots beyond that being
explained by the covariates. The random noise parameters νik and ζik are assumed i.i.d.
normal random variables with mean 0 and variances τλk and τγk , respectively, where
τλk and τγk have conjugate inverse-Gamma distributions.
5 Application: Modeling biomass in the eastern US
We model biomass for two plant functional types: late successional hardwood (LSH)
and southern pine (SP). There are 17,306 plots that contain at least one LSH and 4664
plots that contain at least one SP. The plots are mixed stands such that other PFTs may
also be present. LSH are found throughout the majority of the eastern US while SP
are concentrated in the southeast. Again, each plot is surveyed twice according to the
1 Hierarchical centering is a reparameterization technique for models with multiple levels of random effects.
It can be applied in its simplest form in the context of a standard ANOVA model where population means
are often expressed as a global mean and a population level deviation, e.g., μ + αi . Here, the data will
well-identify the sum but not as well the components. Hierarchical centering entails reparameterizing from
μ and αi [with prior π(μ)π(αi )] to ηi = μ + αi and μ and specifying the prior as π(ηi |μ)π(μ), i.e.,
centering the ηi hierarchically.
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sampling scheme outlined in Sect. 2. We use 80 % of the plots of each plant functional
type to fit the model and hold out the remaining 20 % to do out-of-sample prediction
for model evaluation.
As shown in Fig. 4, the average total biomass of LSH saplings per four microplots
is 36.6 kg and average total biomass of trees per four subplots is 2340 kg. The number
of LSH saplings observed on the four microplots ranges from 1 to 45 with a median
of 3. The number of LSH trees observed on the four subplots ranges from 1 to 64 with
a median of 6. For SP, the average total biomass of saplings is 27.5 kg and average
total biomass of trees is 2318 kg. The total number of SP saplings ranges from 1 to
126 with a median of 3 and the total number of SP trees ranges from 1 to 105 with a
median of 7 (Figures not included).
The survey year-specific covariates in each model include tree density and stand
age of the plot. Tree density is the total number of trees of all species observed in
the plot at the time of the survey. Other covariates in the model include temperature
and precipitation, both of which are centered and scaled, and indicator variables for
physiographic class code of available moisture in the soil. The three classes of moisture
availability are xeric, mesic, and hydric where xeric is low or deficient, mesic is
moderate and used as the base level, and hydric is abundant. These covariates are not
survey year-specific.
The prior distributions assigned to the parameters σb2k and φbk of the measurement error variance of sapling biomass are inverse-gamma and truncated normal,
respectively, where σb2k ∼ I G(3, 104 ) and φbk ∼ T N (0, 40, 0, ∞). The tree biomass measurement error parameters are assigned σ B2 k ∼ I G(3, 105 ) and φ B k ∼
T N (0, 40, 0, ∞). The priors for σb2k and σ B2k were both chosen to be diffuse. Since
tree biomass is larger than sapling biomass, we assume measurement error may also
be larger. Thus, the median of the distribution of σ B2 k is greater than the median of the
distribution of σb2k . The priors for φbk and φ Bk were chosen such that the distributions
k and N k .
spanned the observed n is
is
The coefficient vectors α k1 and β k1 have mean zero multivariate normal prior distributions with variance 106 I p1 × p1 where p1 = 2. The coefficient vectors α k2 and β k2
each contain an intercept term and are assigned mean zero multivariate normal prior
distributions with variance 106 I p2 × p2 where p2 = 5. Lastly, the variances τλ2k and
τγ2k are assigned I G(4, 4) prior distributions.
The model is fitted using R software (R Development Core Team 2007) running
on an Intel Core i7 processor with Scientific Linux 6.4. We use Markov chain Monte
Carlo (MCMC) to sample from the posterior distribution of the parameters given the
data. The MCMC algorithm was contrived specifically for this application and some
explicit details are included in 1. We run MCMC for 100,000 iterations. Convergence
was assessed by computing the Gelman and Rubin R statistic for each of the model
parameters using three chains with varying starting values. The upper 97.5 % bound
on the statistic was less than 1.10 for each of the parameters, indicating no issues
with convergence. We disregard the first 50,000 samples as burn-in and retain every
10th iteration for inference. Posterior median estimates and 95 % credible intervals
are given in Table 2.
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Table 2 The posterior medians and 95 % credible intervals for parameters of average sapling and tree
biomass
Parameter
Late successional hardwoods
Southern pines
α11 (Tree density)
0.14 (0.12, 0.17)
0.23 (0.21, 0.24)
α12 (Stand age)
0.07 (0.06, 0.09)
−0.05 (−0.07, −0.03)
α20 (Intercept)
6.28 (5.53, 7.00)
11.56 (10.78, 12.36)
α21 (Temperature)
−1.58 (−1.99, −1.19)
0.56 (0.05, 1.11)
α22 (Precipitation)
0.31 (−0.08, 0.69)
0.05 (−0.40, 0.49)
α23 (Xeric)
−1.12 (−2.03, −0.16)
2.13 (0.83, 3.41)
α24 (Hydric)
1.13 (−0.17, 2.48)
3.09 (0.39, 6.02)
−0.29 (−0.38, −0.21)
β11 (Tree density)
−3.30 (−3.51, −3.10)
β12 (Stand age)
3.36 (3.26, 3.46)
6.41 (6.19, 6.61)
β20 (Intercept)
158.6 (150.9, 168.5)
−6.42 (−17.62, 5.67)
β21 (Temperature)
−14.46 (−21.52, −7.83)
56.77 (47.96, 65.85)
β22 (Precipitation)
−4.81 (−11.40, 1.71)
−12.93 (−21.15, −4.75)
β23 (Xeric)
−131.2 (−149.3, −112.0)
−114.1 (−137.0, −91.62)
β24 (Hydric)
−12.72 (−34.03, 8.81)
−2.85 (−40.74, 36.02)
Bold indicates credible intervals not containing 0
The coefficient for tree density in average tree biomass is negative for both PFTs
indicating that plots with high tree density have lower average tree biomass. Interestingly, however, the same coefficient for average sapling biomass is positive. This is
because plots with high tree density tend to have fewer saplings, resulting in larger
values of average sapling biomass. Both average sapling and tree biomass for LSH
are increasing between the first and second survey indicated by positive coefficients
for stand age. Average SP tree biomass is increasing between the first and second
survey but average sapling biomass for SP is decreasing. Average LSH sapling and
tree biomass decreases with temperature while SP sapling and tree biomass increases
with temperature. Additionally, high soil moisture tends to increase average sapling
biomass and both low and high soil moisture tend to decrease average tree biomass.
We compute estimates of the measurement error variance of sapling biomass and
tree biomass using posterior estimates of σb2k , φbk , σ B2 k , and φ B k for LSH and SP.
We plot posterior median estimates and 95 % credible intervals of the variance as a
function of the number of saplings and trees in Fig. 5. The histograms in each figure
are of the the number of saplings (top) and trees (bottom) for each of the PFTs. The
k
measurement error of sapling biomass is σb2k (1 − exp−n is /φbk ) and the measurement
error of tree biomass is σ B2 k (1 − exp−Nis /φ B k ). The measurement error variance for
both sapling and tree biomass for LSH asymptotes within the range of the number of
individuals per plot that we observed. Tree biomass reaches an upper bound for SP but
sapling biomass does not. This is likely due to the majority of the plots having fewer
than 30 saplings and two plots in the region being outliers with more than 60. Changes
in the range parameter φbk can inflate the variance which explains the large values of
posterior variability seen in the measurement error for sapling biomass of SP.
k
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Environ Ecol Stat (2016) 23:23–41
35
Fig. 5 Posterior median estimates and 95 % credible intervals (kg2 ) of measurement error variances for
sapling (top) and tree (bottom) biomass for late successional hardwoods (left) and southern pines (right) as
a function of the number of individuals. The histograms in each figure are of the number of individuals on
the plot
Table 3 In-sample RMSE for total sapling and tree biomass (kg) at surveys 1 and 2 for late successional
hardwoods and southern pines
Model
b1
b2
B1
B2
Late successional hardwoods
15.66
15.07
266.02
266.73
Southern pines
24.72
23.46
240.02
236.19
In-sample root mean square error (RMSE) is reported in Table 3 for sapling and tree
biomass at each survey time for both PFTs. Samples from the posterior distribution
k and B k are obtained using posterior draws of the model parameters. The values
of bis
is
are similar for both survey times and functional type. We obtain posterior samples of
k and B k .
Δik using (2) and (3) and posterior draws of bis
is
Posterior median predictions of change in sapling and tree biomass per year are also
shown spatially for both LSH and SP in Fig. 6. We see that sapling biomass for LSH is
increasing in North Carolina, while changes in the rest of the eastern US are small. The
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Environ Ecol Stat (2016) 23:23–41
Fig. 6 Posterior median predictions of the change in biomass (kg/ha/yr) of saplings (top) and trees (middle)
for late successional hardwoods (left) and southern pines (right). The bottom panel give posterior median
predictions of Δ-biomass (kg/ha/yr) for the two PFTs
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37
Fig. 7 Standard deviations of the posterior distributions of Δ-biomass (kg/h/yr) for late successional
hardwoods (left) and southern pines (right)
mild increases in tree biomass across the eastern US result from growth and is driving
the positive predictions of Δ-biomass in the bottom panel of the figure. Decreases in
sapling biomass through time for SP is predicted in much of the southern states except
for regions in North Carolina and Alabama. Nearly all plots are seeing increases in
tree biomass of SP, many of which are large in comparison to the growth of LSH. This
is, in part, due to the rapid growth rate of SP and their high tree density in the region.
Posterior median predictions of Δ-biomass (kg/ha/yr) are shown in the bottom
panel of Fig. 6 for both functional types. Biomass is increasing in the upper midwest
and northeast, as well as North Carolina for LSH. Biomass is increasing throughout
the entire southeast for SP. The predominant contributor to positive predictions of
Δ-biomass for SP is the increase in tree biomass, or growth. Decreases in biomass
appear to be very localized events as there are no regions reporting clusters of negative
Δ-biomass. Large decreases in biomass are often the result of the mortality of a large
tree. One of the benefits of our model is that we also obtain estimates of uncertainty
for Δ-biomass. Show in Fig. 7 is the standard deviation of our posterior distribution
of Δ-biomass across space for both PFTs.
We predict biomass at hold-out plots for both PFTs. Table 4 reports 90 % empirical
coverage probabilities for total sapling and tree biomass for both survey times. Each
of the coverage probabilities is slightly greater than the nominal level indicating that
our prediction intervals for total sapling and tree biomass are conservative. We also
compute out-of-sample root mean square prediction errors (RMSPE) for the hold-out
plots (Table 5). RMSPE values are similar between the two survey times and PFTs.
6 Discussion
The interdependence between trees introduces challenges in estimating the uncertainty
of biomass at the plot level. Due to this dependence, the variance of plot-level biomass
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Table 4 Out-of-sample 90 % empirical coverage probabilities
Model
b1
b2
B1
B2
Late successional hardwoods
90.49
93.09
96.19
95.90
Southern pines
92.88
94.84
94.37
93.64
Table 5 RMSPE for total sapling and tree biomass (kg) at surveys 1 and 2 for late successional hardwoods
and southern pines
Model
b1
b2
B1
B2
Late successional hardwoods
46.55
45.65
1753.36
1885.80
Southern pines
52.15
47.15
1877.32
2381.18
should be less than the sum of the variances of the individual trees. Therefore, we
propose a parametric density-dependent asymptotic functional form for the plot-level
variance of biomass as motivated by our illustration.
We model biomass for each plant functional type by first defining total biomass
as the summation of sapling and tree biomass. Sapling and tree biomass are modeled in terms of the average biomass of an individual in each size class. Modeling in
terms of average biomass is advantageous due to the challenges of directly measuring change in biomass. We model sapling and tree biomass separately due to
the unequal sampling intensities between saplings and trees. The model is defined
specifically for the sampling scheme of the FIA data in the eastern US and would
therefore need to be modified to accommodate other sampling schemes. Predictions
of the rate of change in biomass, Δ-biomass, are computed as the difference between
total biomass at the two surveys divided by the time between surveys. We applied
the model to two plant functional types, late successional hardwoods and southern
pines.
Ongoing work includes scaling biomass predictions at the plot level to larger spatial
regions. Both regional and global predictions of biomass are of interest to management
as they assess the sustainability of biomass. Additionally, we plan to model more plant
functional types. In fact, if we model all the functional types we will be able to consider
the behavior of and change in total biomass across the region. The modeling challenge
then becomes to incorporate suitable dependence between PFT’s as we sum across
them in order to obtain appropriate estimates of uncertainty.
Further challenges include projecting biomass change under varying covariate scenarios to address the carbon cycling issue raised at the outset of the paper. Also, we
may be able to avail ourselves of additional data sources (e.g., National Ecological
Observatory Network (NEON) products) presenting the opportunity to implement data
fusion to enhance our understanding of the process.
Acknowledgments This research was supported by the National Science Foundation under grant numbers
EF-1137364 and CDI-0940671 and the Coweeta LTER. The authors would also like to thank Bradley
Tomasek for providing useful discussion on biomass allometry.
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39
Appendix
Implicit distributions of total biomass and Δ-biomass
k , T B k and Δk .
The model implies distributions relating to T Bi1
i2
i
k
k
2
2
k
Let Θs = {μis , θis , σbk , φbk , σ B k , φ B k } for s = 1, 2. Then
1 k k
1
1 2
k |Θ k ∼ N
k k 1 2
k
k
2
2
(a) T Bi1
1
As n i1 μi1 + At Ni1 θi1 , A2s σbk g(n i1 ; φbk ) + A2t σ B k g(Ni1 , φ B k )
1 k k
1
1 2
k |Θ k ∼ N
k k 1 2
k
k
2
2
(b) T Bi2
2
As n i2 μi2 + At Ni2 θi2 ,A2 σbk g(n i2 ; φbk ) + A2 σ B k g(Ni2 , φ B k ) .
s
t
k and T B k are conditionally independent given Θ k and Θ k .
Additionally, T Bi2
1
2
i2
T B k −T B k
i1
Furthermore, since Δik = ti2
,
i2 −ti1
1
1 k k
1
k θ k − 1 n k μk − 1 N k θ k ,
(c) Δik |Θ1k , Θ2k ∼ N ti2 −t
n
μ
+
N
i2
i2
i2
i2
i1
i1
i1
i1
A
A
A
A
t
s
t
i1
s
1
1 2
k ; φ )2 + g(n k ; φ )2
σ
(g(n
k
k
b
b
i1
i2
(ti2 −ti1 )2 A2s bk
1 2
k
k
2
2
+ A2 σ B k g(Ni1 , φ B k ) + g(Ni1 , φ B k )
t
1
1 k k
1
k , Θk, Θk ∼ N
k θ k − T Bk ,
(d) Δik |T Bi1
n
μ
+
N
1
2
i1
ti2 −ti1 As i2 i2
At i2 i2
1
1 2
k ; φ )2 + 1 σ 2 g(N k , φ )2
.
σ
g(n
k
k
2
2
2
k
k
b
B
i2
i2
(t −t )
A b
A B
i2
i1
s
t
k.
Therefore, given the model parameters, we have an explicit distribution for Δik |T Bi1
These distributions can be used in conjunction with prior samples through composition
sampling to obtain samples from the posterior predictive distributions at the plot level
for total biomass and Δ-biomass.
MCMC algorithm
The tobit latent variable approach to modeling average sapling and tree biomass
requires Metropolis-Hastings algorithms for iterative sampling of α k1 , β k1 , λk , and
γ k . As a proposal distribution for each of these parameters, we use the full conditional
k =
k and θ k = μis
θisk (i.e.,
distribution of the parameter under the assumption that μis
is
dropping the tobit models (7) and (8) for average sapling and tree biomass, respectively). For simplicity, let p1 = 1. Dropping k for ease of notation, the full conditional
distribution of α1 is
p(α1 |b1 , b2 , λ, β 2 , τλ2 , σb2 , φb ) ∝ p(b1 , b2 |α1 , λ, σb2 , φb ) p(α1 )
∝ p(b1 |α1 , λσb2 , φb ) p(b2 |α1 , λ, σb2 , φb ) p(α1 ).
k = k , this conditional distribution is normal with known mean and
μis
Letting μis
variance. We propose a candidate value from this distribution, denoted α1∗ . Letting α1c
c
∗ = W α∗ + λ , c
denote the current value of α1 , we compute μis
i μis = Wis α1 + λi ,
is 1
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Environ Ecol Stat (2016) 23:23–41
c ) for each plot i and survey s. The
∗ ), and μc = max(0, (μis )∗ = max(0, μis
μis
is
candidate value is accepted with probability
min
p(b1 |μ∗1 , σb2 , φb ) p(b2 |μ∗2 , σb2 , φb ) p(β ∗ ) p(b1 |
μc1 , σb2k , φb ) p(b2 |
μc2 , σb2k , φb ) p(β c )
p(b1 |μc1 , σb2 , φb ) p(b2 |μc2 , σb2 , φb ) p(β c ) p(b1 |
μ∗1 , σb2 , φb ) p(b2 |
μ∗2 , σb2 , φb ) p(β ∗ )
,1
which reduces to
min
p(b1 |μ∗1 , σ 2k , φbk ) p(b2 |μ∗2 , σ 2k , φbk ) p(b1 |
μc1 , σ 2k , φbk ) p(b2 |
μc2 , σ 2k , φbk )
b
b
b
b
,1
p(b1 |μc1 , σ 2k , φbk ) p(b2 |μc2 , σ 2k , φbk ) p(b1 |
μ∗1 , σ 2k , φbk ) p(b2 |
μ∗2 , σ 2k , φbk )
b
b
b
b
.
(11)
∗ are greater than or equal to zero,
Note that when all values of the latent variable μis
the candidate value will be accepted with probability 1. Similar proposal distributions
and algorithms are employed for β 1 , λ, and γ .
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Erin M. Schliep is a postdoctoral fellow at Duke University in the Department of Statistical Science.
Alan E. Gelfand is James B Duke Professor of Statistical Science and Professor of Environmental Sciences and Policy at Duke University.
James S. Clark is H.L. Blomquist Professor of the Nicholas School of the Environment, Professor of
Biology, and Professor of Statistical Science at Duke University.
Kai Zhu is a postdoctoral fellow at Carnegie Institution for Science in the Department of Global Ecology
and Stanford University in the Department of Biology.
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