Tree Physiology 21, 505–512
© 2001 Heron Publishing—Victoria, Canada
Estimates of the distributions of forest ecosystem model inputs for
deciduous forests of eastern North America
PHILIP J. RADTKE,1 THOMAS E. BURK2 and PAUL V. BOLSTAD3
1
Department of Forestry (0324), Virginia Tech, Blacksburg, VA 24061, USA
2
Department of Forest Resources, University of Minnesota, 1530 Cleveland Ave N., St. Paul, MN 55108, USA
3
Department of Forest Resources, University of Minnesota, 1530 Cleveland Ave N., St. Paul, MN 55108, USA
Received August 18, 2000
Summary Techniques for evaluating uncertainties in process-based, computer simulation models are evolving in response to the proliferation of such models and the demand for
their use in the management of forest ecosystems. Many evaluation techniques require precise statements of the uncertainties
associated with each model input. Statements of uncertainty
are typically formulated as probability density functions
(pdfs). Here, pdfs are developed for 29 inputs of the process-based, forest ecosystem, computer simulation model
PnET-II, many of which are inputs to other well-known forest
ecosystem models. The inputs considered describe vegetation
characteristics of forests typical of the Eastern Deciduous Forest biome of North America. Data were compiled largely from
published literature to estimate pdfs. The compiled distributions can be used to conduct various model evaluations including uncertainty assessment, calibration, and sensitivity
analysis.
Keywords: Bayesian melding, BGC, big leaf model, calibration, model evaluation, PnET, process model, sensitivity analysis.
Introduction
The use of computer models to simulate complex ecological
systems is becoming increasingly common as the power and
speed of computers increase. Despite the proliferation of models and their applications, tools to evaluate the quality of
model predictions have evolved slowly. Compared with the
many techniques for evaluating predictions derived from traditional statistical models, few techniques are available for
evaluating predictions derived from process-based, computer
simulation models. What tools are available are seldom applied to practical modeling applications.
Evaluation is a process that investigates whether a model is
suitable for its intended purpose. Many criteria may be used to
evaluate model suitability, and these criteria vary with the intended use of the model. For many modeling applications, the
uncertainty associated with model predictions is of fundamental concern. Efforts are increasingly focused on evaluating uncertainty associated with computer model predictions (Cipra
2000, Mäkelä et al. 2000). The results of our study are presented in the context of uncertainty assessment, although they
are readily applicable to various evaluation procedures including model calibration and sensitivity analysis (Radtke 1999).
Probability is a primary tool for quantifying uncertainty, the
starting point for most uncertainty assessments. Defining
probability distributions is a basic requirement for analyses
that assess how uncertainties associated with model inputs affect model predictions (Gertner et al. 1996, Green et al. 1999,
MacFarlane et al. 2000). The goal of the present study was to
quantify the uncertainty associated with the inputs of a process-based, forest ecosystem, computer simulation model,
PnET-II (Aber et al. 1995), by means of statements of the
probability distributions for the inputs. The probability statements are intended to characterize the main features of each
distribution: location, spread and shape. In addition to statements of probability for individual inputs, we quantified correlations observed between pairs of model inputs. Information
on correlations can be incorporated into many procedures for
model evaluation (Guan 2000). A number of PnET-II inputs
are similar or identical to inputs used by other process-based,
forest ecosystem, computer simulation models (e.g., Running
and Coughlan 1988). It follows that the results presented here
can be applied to evaluations of these models as well.
The model PnET-II requires a set of more than 40 input values that describe various site and vegetation characteristics of
the ecosystem to be modeled. Gathering information on all the
inputs of a model is an essential but laborious step in the assessment of model uncertainties. Yet, this is a step that is often
overlooked by modelers. A significant body of information is
available about the PnET-II model and the parameters needed
to run it. For example, point estimates for model inputs needed
to simulate deciduous forest types were compiled by Aber et
al. (1995, 1996). These values, excluding site-specific (climate and soil) inputs, are listed in Table 1. Undoubtedly, it required considerable effort to compile such a list; yet, the point
estimates fail to provide necessary information for evaluating
model uncertainties. Having recognized the need for precise
statements of probability, and building on the resources gathered by the model authors, we compiled a comprehensive set
506
RADTKE, BURK AND BOLSTAD
Table 1. Inputs needed to run the PnET-II model for deciduous forest types (point estimates of model inputs from Aber 1995, 1996).
Name
Definition (units)
Value
AmaxA
AmaxB
AmaxFrac
BaseFolRespFrac
CFracBiomass
DVPD1
DVPD2
FolMassMax
FolMassMin
FolNCon
FolRelGrowMax
GDDFolEnd
GDDFolStart
GDDWoodEnd
GDDWoodStart
GRespFrac
HalfSat
k
MinWoodFolRatio
PlantCReserveFrac
PrecIntFrac
PsnTMin
PsnTOpt
RespQ10
RootAllocA
RootAllocB
RootMRespFrac
SenescStart
LWADel
LWAMax
WoodMRespA
WueConst
Intercept of relationship between foliar N and maximum photosynthetic rate (Amax )
Slope of Amax (mmol CO2 g –1 leaf s –1) versus N relationship
Daily Amax as a fraction of early morning instantaneous rate
Respiration as a fraction of maximum photosynthesis
Carbon as a fraction of tissue mass
Coefficients for photosynthesis reduction (DVPD) caused by vapor pressure deficit
(VDP, kpa) in the power function DVPD = DVPD1 × VPDDVPD2
Site specific maximum summer foliage biomass (g m –2 )
Site specific minimum winter foliage biomass (g m –2 )
Foliar nitrogen (%)
Maximum relative growth rate for foliage (% per year)
Growing degree days (GDD) at which foliar production ends
GDD at which foliar production begins
GDD at which wood production ends
GDD at which wood production begins
Growth respiration, as a fraction of allocated carbon
Half saturation PPFD (mmol m –2 s –1)
Canopy light extinction constant
Minimum ratio of carbon allocation to wood and foliage
Fraction of plant carbon held in reserve after allocation to bud carbon
Fraction of precipitation intercepted and evaporated
Minimum temperature for photosynthesis (°C)
Maximum temperature for photosynthesis (°C)
Q10 value for foliar respiration
Intercept of relationship between foliar and root allocation
Slope of relationship between foliar and root allocation
Ratio of fine root maintenance respiration to fine root biomass production
Day of year after which leaf drop can occur
Change in leaf weight per area (LWA) with accumulated foliar mass above (g m –2 g –1)
LWA at the top of canopy (g m –2 )
Wood maintenance respiration as a fraction of gross photosynthesis
Coefficient in equation for water-use efficiency (WUE) as a function of VPD
–46
71.9
0.76
0.1
0.45
0.05
2
300
0
1.9–2.4
30
900
100
900–1600
100–900
0.25
200
0.58
1.5
0.75
0.11
4
24
2
0
2
1
270
0.2
100
0.07
10.9
of information for the PnET-II inputs specific to deciduous
forest types. The results presented here can be used for specific model evaluations such as Bayesian melding (Green et al.
2000, Poole and Raftery 2000). They may also serve as a pattern for model builders in the future, who might wish to report
the distributions of their models’ inputs to facilitate independent model evaluation and testing.
In addition to vegetation characteristics, PnET-II requires a
set of inputs that describe soil characteristics and climate conditions. These inputs are necessarily site or time-specific, or
both; therefore, input distributions depend on the spatial and
temporal context of the simulation to be conducted. We
avoided this detail by focusing only on inputs that describe the
vegetation characteristics of a generic forest type, broadleaved deciduous forests typical of eastern North America.
Quantification of site and climate-specific input distributions
for PnET-II is addressed elsewhere (Radtke 1999).
Materials and methods
Probability distributions were estimated based on empirical
evidence where possible, and otherwise were derived from
theoretical considerations. Where little or no information was
available, bounds were identified and a uniform distribution
was assumed. In two cases, inputs were fixed at constant values, thereby eliminating uncertainty by way of assumption.
This was done for model inputs DVPD2 and FolMassMin,
which were fixed at the values listed in Table 1 and effectively
removed from the list of inputs for which uncertainty was to be
quantified.
Data
Data from published studies were the primary source of empirical evidence for defining the probability distributions in this
study. The population of interest was defined as the Eastern
Deciduous Forest (EDF) biome of eastern North America. The
EDF ranges from the northern coast of the Gulf of Mexico and
northern Florida to northeastern Ontario, Canada, and west as
far as central Minnesota and Arkansas. Taxa typical of the
EDF include oak (Quercus spp.), maple (Acer spp.), hickory
(Carya spp.), birch (Betula spp.) and aspen (Populus spp.).
Much of the previous work with PnET-II involved predictions
within the EDF, so a considerable body of data applicable to
this biome was available. Data on the characteristics of plant
species within the EDF were obtained from the sources listed
in Table 2.
TREE PHYSIOLOGY VOLUME 21, 2001
DISTRIBUTIONS OF FOREST ECOSYSTEM MODEL INPUTS
Estimation
A subjective assessment was made as to which family of density functions best represented the empirical distribution of
each input. The normal (Gaussian) probability density function (pdf) was adopted for inputs that demonstrated unimodal,
symmetric distributions with no evident minimum or maximum values, or for inputs whose distributions were estimated
by regression analysis. Lognormal or Weibull pdfs were
adopted for inputs whose distributions showed natural minima
or skewness. Beta pdfs were adopted for most inputs bounded
between [0,1]. As noted previously, uniform densities were
assigned to inputs about which little was known.
To avoid confusion, we distinguish between two meanings
of the term parameter as used in this study. The term model parameter refers to an input quantity given to the model to describe some aspect of the population being simulated. For
example, HalfSat is a parameter of the PnET-II model. The
term pdf parameter refers to a quantity used to describe the
distribution of a model input. For example, if an input is observed to be normally distributed, then its mean and standard
deviation are parameters that describe the normal pdf.
The goal of pdf parameter estimation was to characterize the
location, spread and shape of the distributions of interest. Depending on the distribution family, one of several techniques
was used to estimate the pdf parameters. Consideration was
given to estimation techniques that could be conveniently im-
507
plemented; however, techniques that give estimates with desirable statistical properties were used where practical. Maximum likelihood estimates (MLE) were used where they could
be obtained readily using available software or with minimal
effort to program customized algorithms. Estimators based on
the method of moments (MOM) were used as an alternative to
MLE, and a percentile-based estimator was used where no
MOM or MLE alternative was convenient. Plausibility of specific density functions was tested by the Kolmogorov-Smirnov (KS) goodness of fit test (Darling 1957,
Lindgren 1993). Fits of the estimated pdfs were confirmed by
inspection of density plots and the guidelines listed above.
For some inputs, pdfs were estimated directly from the compiled data sets. For others it was necessary to make various assumptions, compute ratios, standardize data with respect to a
reference value, perform regression analysis, or use observed
data in combination with prediction models. In some instances
two or more of these techniques were applied in combination.
Reference to the methods involved is given in Table 3. A brief
explanation of some details of the estimation methods is made
in the following paragraphs.
Direct estimation The inputs AmaxFrac, k and MinWoodFolRatio serve as examples of how pdfs were derived from direct examination of data. For AmaxFrac, the mean and variance of compiled observations were calculated and used to
estimate the parameters of the beta density by the method of
Table 2. Sources and numbers of observed data compiled for estimating distributions of PnET-II inputs. Abbreviation: n/a = not available.
Input
Data source
n
AmaxA, AmaxB
AmaxFrac
BaseFolRespFrac
CfracBiomass
DVPD1, DVPD2
FolMassMax, FolMassMin
Reich et al. 1995
Aber et al. 1996
Walters et al. 1993
K. Mitchell, University of Minnesota, unpublished data
Ellsworth and Reich 1992, Kruger and Reich 1993
Whittaker 1966, Whittaker et al. 1974, Crow 1978, Aber et al. 1993,
Yin 1993, Fassnacht and Gower 1997, Martin and Aber 1997
Yin 1993, Reich et al. 1995, Martin and Aber 1997, Mitchell et al. 1999
n/a
Farnsworth et al. 1995, Goulden et al. 1996, Bassow and Bazzaz 1998,
Gill et al. 1998, P. Reich, University of Minnesota, unpublished data
Same as GDDFolStart, GDDFolEnd
Williams et al. 1987
Landsberg 1986, Sullivan et al. 1996
Jarvis and Leverenz 1983, Bolstad and Gower 1990, Ellsworth and Reich 1993, Vose et al. 1995
Same as FolMassMax, FolMassMin, excepting Martin and Aber 1997
n/a
Helvey and Patric 1965, National Climate Data Center (NCDC)
Dougherty et al. 1979, Jurik et al. 1988, Harley and Baldocchi 1995
Bolstad et al. 1999
Raich and Nadelhoffer 1989
n/a
Bassow and Bazzaz 1998, Gill et al. 1998, P. Reich, unpublished data
Hutchison et al. 1986, Jurik 1986, Ellsworth and Reich 1993
Same as LWADel, also Aber et al. 1990 and Mitchell et al. 1999
Walters et al. 1993
Baldocchi et al. 1987
28
14
19
423
28
87
FolNCon
FolRelGrowMax
GDDFolStart, GDDFolEnd
GDDWoodStart, GDDWoodEnd
GrespFrac
HalfSat
k
MinWoodFolRatio
PlantCReserveFrac
PrecIntFrac
PsnTMin, PsnTOpt
RespQ10
RootAllocA, RootAllocB
RootMRespFrac
SenesceStart
LWADel
LWAMax
WoodMRespA
WueConst
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–
19
–
14
12
17
54
–
859
104
51
14
–
15
120
30
15
23
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RADTKE, BURK AND BOLSTAD
Table 3. Distributions of PnET-II inputs over the range of the EDF and reference to the method of estimation.
Input
Family
Estimation
AmaxA
AmaxB
AmaxFrac
BaseFolRespFrac
CFracBiomass
DVPD1
FolMassMax
FolNCon
FolRelGrowMax
GDDFolEnd
GDDFolStart
GDDWoodEnd
GDDWoodStart
GrespFrac
HalfSat
k
MinWoodFolRatio
PlantCReserveFrac
PrecIntFrac
PsnTMin
PsnTOpt
RespQ10
RootAllocB
RootMRespFrac
SenesceStart
LWADel
LWAMax
WoodMRespA
WueConst
Normal
Normal
Beta
Beta
Weibull
Normal
Lognormal
Lognormal
Uniform
Normal
Normal
Normal
Normal
Beta
Weibull
Normal
Lognormal
Uniform
Weibull
Normal
Normal
Weibull
Normal
Uniform
Normal
Normal
Normal
Beta
Normal
Regression
Regression
Direct
Direct
Direct
Regression (standardized)
Direct
Direct
Assumption
Direct
Direct
Direct with assumptions
Direct with assumptions
Direct
Direct with assumptions
Direct
Direct ratio
Assumption
Modeled
Regression (standardized)
Regression (standardized)
Direct
Regression
Assumption
Direct
Regression
Direct
Direct
Regression
moments. The mean and standard deviation of 17 published
values were calculated as estimates of the parameters of the
normal pdf for the light extinction coefficient (k). For
MinWoodFolRatio, a set of 54 wood-to-foliage production ratios were computed and used to obtain MLE of the lognormal
pdf parameters.
Direct estimation with assumptions Some pdfs could be estimated directly, but only after making significant assumptions
about the distribution. These inputs are labeled “Direct with assumptions” in Table 3. The following paragraphs summarize
the key assumptions made in estimating these distributions.
For GDDWoodEnd and GDDWoodStart, it was assumed
that ring-porous species (e.g., Quercus spp., Fraxinus spp.,
Carya spp. and Ulmus spp.) produce springwood at the same
time they produce foliage, whereas diffuse-porous species
(most other hardwoods) produce springwood after foliage
production (Aber et al. 1995). Forest composition of 50%
ring-porous and 50% diffuse-porous was assumed for the
EDF. A further assumption was made that the duration of
wood production is the same as the duration of foliage production. Following these assumptions, GDDWoodStart and
GDDWoodEnd were formulated as linear combinations of
GDDFolStart and GDDFolEnd, so their pdf parameters were
derived from the means, variances and covariances from the
Distribution
µ = –12.2, σ = 15.4
µ = 62.7, σ = 7.0
α = 23.2, β = 7.30
α = 6.0, β = 37.3
β = 0.494, γ = 35.4
µ = –0.043, σ = 0.011
µ = 5.76, σ = 0.349
µ = 0.74, σ = 0.26
α = 20, β = 50
µ = 898, σ = 189
µ = 379, σ = 75
µ = 1158, σ = 238
µ = 639, σ = 132
α = 8.2, β = 23.0
β = 299.9, γ =1.87
µ = 0.56, σ = 0.12
µ = 0.46, σ = 0.42
α = 0.05, β = 0.95
α = 0.11, β = 0.05, γ = 1.8
µ = 0.68, σ = 0.64
µ = 22.2, σ = 0.33
α = 1.9, β = 0.60, γ = 1.87
µ = 2.52, σ = 0.214
α = 0.5, β = 2.0
µ = 260, σ = 13
µ = –0.155, σ = 0.014
µ = 87.2, σ = 18.8
α = 12.4, β = 99.8
µ = 10.9, σ = 0.875
KS test P-value
–
–
0.56
0.94
0.07
–
0.51
0.21
–
0.95
0.90
–
–
0.50
0.26
0.95
0.47
–
0.56
–
–
0.24
–
–
0.52
–
0.99
0.99
–
GDDFolStart and GDDFolEnd observations. The derived
distributions for GDDWoodStart and GDDWoodEnd were
consistent with the start and end dates of cambial activity of
hardwood species recorded by Ahlgren (1957) between 1951
and 1956 in northeastern Minnesota.
Half-saturation irradiances (HalfSat) were estimated from
published, photosynthetic light-response curves (Table 2).
Mean photosynthetic rates of leaves measured in high light (>
1000 PPFD) by Sullivan et al. (1996) were halved to arrive at
the photosynthetic rate at half-saturation irradiance (Asat /2).
HalfSat observations were computed by inverting the light-response curves and solving for the 12 recorded (Asat /2) values.
Estimation by regression Estimation of pdf by regression
analysis was relatively straightforward. Functional forms of
regression models were dictated by the submodels within
PnET-II. Regression models were fitted to data pairs and the
regression coefficients and standard errors were used to specify the parameters of normal pdfs. For example, data pairs for
maximum, instantaneous, mass-based net photosynthetic rate
(Amax) and leaf nitrogen concentration (N) of broadleaf species
were used to fit a simple linear regression model of Amax versus
N (Reich et al. 1995). It was then assumed that the intercept
(AmaxA) and slope (AmaxB) were distributed normally with
means defined by the regression coefficients and variances de-
TREE PHYSIOLOGY VOLUME 21, 2001
DISTRIBUTIONS OF FOREST ECOSYSTEM MODEL INPUTS
fined by the square of standard errors of the regression coefficients.
In some cases, data were standardized relative to some maximum observed value (DVPD1, PsnTMin, PsnTOpt; see Table 3). Where necessary, nonlinear regression was used to
estimate coefficients and standard errors. In the case of
WueConst, a theoretical mean value was supported by regression analysis and the standard error of the mean was estimated
from weighted least-squares regression based on data of
Baldocchi et al. (1987).
Model-based estimation The PnET-II input PrecIntFrac describes the fraction of precipitation intercepted within the forest canopy and returned to the atmosphere directly through
evaporation. Helvey and Patric (1965) defined the components
of gross precipitation (P) in terms of canopy interception loss
(C), litter interception loss (L), stemflow (S) and throughfall
( T ). They developed prediction equations for stemflow and
throughfall for growing season and dormant season precipitation events in broad-leaved deciduous forests based on the
amount of rainfall. Litter interception loss was estimated as
2.5% of annual gross precipitation, based on published values
of 2% (Blow 1955) and 3% (Helvey 1964) for eastern deciduous forests.
Annual stemflow and throughfall accumulations were predicted from the Helvey and Patric (1965) equations using daily
National Climate Data Center (NCDC) precipitation records
(http://www.ncdc.noaa.gov/) from 22 climate stations selected across the geographic range of the EDF (Table 4). The
period of recorded data was typically 40–50 years (data collected between 1949 and 1998), excluding years for which
daily climate records were missing. From accumulated P, S
and T values, PrecIntFrac was estimated for each year in the
climate record (n = 859) with Equation 1, where the fraction
L /P was set to 0.025 as noted above:
PrecIntFrac = (P − S − T + L) P .
(1)
509
Table 4. City and state of 22 NCDC climate stations within the range
of eastern deciduous forests, sorted by latitude.
City
State
Latitude (°N)
Tallahassee
Meridian
Augusta
Anniston
Little Rock
Asheville
Oak Ridge
Paducah
Roanoke
Lexington
Parkersburg
Frederick
Harrisburg
Altoona
Worcester
Concord
Watertown
Burlington
Plattsburgh
Wausau
Caribou
International Falls
FL
MS
GA
AL
AR
NC
TN
KY
VA
KY
WV
MD
PA
PA
MA
NH
NY
VT
NY
WI
ME
MN
30.4
32.3
33.4
33.6
34.7
35.4
36.0
37.1
37.3
38.0
39.4
39.4
40.2
40.3
42.3
43.2
44.0
44.5
44.7
44.9
46.9
48.6
Relationships between inputs
Relationships between some vegetation inputs were observable from the data used to obtain the pdf parameters (Table 5).
Correlation between FolNCon and FolMassMax was directly
estimated based on 31 data pairs compiled by Yin (1993) for
deciduous forests (excluding data from Alaska, Utah, Washington and Wyoming). The other correlations listed in Table 5
involved inputs obtained from regression relationships, so the
correlation estimates were obtained from regression parameter covariance matrices.
Site-specific inputs
Results
Although results involving site-specific inputs are not reported in detail here, some of the vegetation inputs described
Distributions
A summary of the distributions and pdf parameters for
PnET-II vegetation inputs is listed in Table 3. Many of the
pdfs were estimated directly from compiled data sets. Where
significant assumptions were made, it is noted in the column
that refers to the methods of estimation. The KS test results are
listed for pdfs estimated directly from data. With the exception
of the PnET-II input CfracBiomass (P = 0.07), none of the KS
tests indicated any significant inconsistencies between the observed and estimated densities. Visual inspection of the
CfracBiomass histogram showed that the estimated Weibull
pdf characterized the location and spread of the data well, with
a slight difference in the shapes of the observed and estimated
distributions (Figure 1). None of the other distribution families studied were capable of accounting for the CfracBiomass
distribution any more satisfactorily.
Figure 1. Histogram and estimated beta density for 423 CFracBiomass observations.
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510
RADTKE, BURK AND BOLSTAD
varied significantly across sites within the EDF. FolNCon and
latitude were positively correlated and FolMassMax and latitude were negatively correlated (r = 0.59 and –0.32, respectively; Yin 1993). Based on the Helvey and Patric (1965)
models and NCDC climate records, a nonlinear relationship
between PrecIntFrac and annual precipitation was observed
(Figure 2A). Because annual precipitation and latitude are
negatively correlated within the EDF (r = –0.69), a nonlinear
relationship also exits between PrecIntFrac and latitude (Figure 2B). The distributions of LWAMax and LWADel appeared to differ across the study sites from which the data were
compiled (Figure 3), but it was not possible to determine
whether the variability could be accounted for by measurable
climate or site factors. Thus, the distributions for LWAMax
and LWADel given in Table 3 were estimated based on all of
the compiled data (see Table 2) in order to encompass the
range of values these model parameters might be expected to
take throughout the EDF.
Discussion
The goal of estimating distributions for PnET-II inputs was to
characterize uncertainty by the explicit formulation of pdfs.
The results are suitable for use in model evaluation procedures
that require a set of pdfs describing the uncertainty of model
inputs. Among these procedures are uncertainty assessment,
calibration, and sensitivity analysis (Radtke 1999, Green et al.
2000). With few exceptions, the pdfs developed here characterized the main features of each of the input distributions: location, spread and shape. For some inputs it may be desirable
to formulate alternative density functions in order to investigate the effects of choosing one distributional assumption over
another in subsequent analyses. Such comparisons are often of
interest during model evaluation (Givens et al. 1994). Correlations between model inputs were quantified to facilitate methods for uncertainty analysis that account for known relationships between model inputs (Guan 2000).
Development of pdfs that describe PnET-II vegetation inputs was a relatively straightforward exercise, although the
demands of data compilation and pdf estimation were considerable. Because PnET-II is a generalized model, many of its
inputs apply to broad classes of vegetation, such as the
broad-leaved deciduous canopies of the EDF. As such, most of
the pdfs developed here are applicable to this important class
of forests. Although PnET-II has been used extensively to
Table 5. Correlation coefficients for pairs of PnET-II vegetation parameter inputs.
Variable pair
Correlation
AmaxA, AmaxB
FolNCon, FolMassMax
PsnTMin, PsnTOpt
LWAMax, LWADel
–0.97
–0.49
0.67
0.73
model canopies of the EDF, it has also been used to model
needle-leaved canopies and forests in distinct geographic regions (McNulty et al. 1996, Goodale et al. 1998, Ollinger et al.
1998). To conduct model evaluations for simulations involving these ecosystems, it will be necessary to estimate pdfs relevant to the specific applications.
Distributions for site-specific inputs may be estimated on a
case by case basis, depending on the population targeted for
modeling. Some of the vegetation inputs examined here are
sometimes assumed to vary with site-specific conditions,
e.g., FolMassMax, FolNCon, GDDFolEnd, GDDFolStart,
LWADel, LWAMax and SenesceStart (Yin 1993, Aber et al.
1996). Data examined here were consistent with this assumption for FolMassMax, FolNCon, LWADel and LWAMax. In
addition, our analyses demonstrated that PrecIntFrac varies
with latitude within the EDF (Figure 2). By ignoring site-specific factors in favor of the regional approach, we estimated
marginal distributions that apply to the entire EDF biome. Although marginal distributions may be useful for some applications, e.g., for use as Bayesian prior distributions, additional
research would be required to develop distributions for
site-specific applications. Rather than develop site-specific
distributions on a case by case basis, it may be possible to develop conditional distributions that express the distribution of
one input (e.g., PrecIntFrac) conditional on the value of another (e.g., latitude; cf. Figure 2B). Development of such conditional distributions assumes that the correlation structure is
known or can be modeled empirically. We found this to be
practical for many of the PnET-II climate inputs for the EDF
because long-term averages in precipitation, temperature and
solar radiation change predictably with latitude across the
range of the EDF (Radtke 1999).
Although distributions of model outputs were not addressed, they are relevant to some model evaluation tech-
Figure 2. PrecIntFrac modeled from
historical climate records versus annual precipitation (A) and latitude
(B) for climate stations across the
range of the EDF.
TREE PHYSIOLOGY VOLUME 21, 2001
DISTRIBUTIONS OF FOREST ECOSYSTEM MODEL INPUTS
511
evaluation techniques in order to further our understanding of
the performance of process-based, forest ecosystem models.
References
Figure 3. Leaf weight per area (LWA) versus cumulative foliar mass
above various positions in the canopy derived from three published
data sets (data estimated from published figures).
niques (Green et al. 2000, Poole and Raftery 2000). When
required, it will be necessary to develop pdfs for output distributions that are specific to the model simulations being evaluated. In some cases, marginal distributions may suffice;
otherwise, careful consideration of the site-specific distributions of model outputs will be required.
A concern that might arise from this study is whether the
distributions estimated from the various sources of published
data accurately represent the best possible understanding of
uncertainty related to model inputs. A key element of this concern is whether the data studied were representative of the
population of interest. In addition, concerns might arise about
the assumptions made in arranging or deriving some of the
data. The key to addressing these concerns is that the estimated distributions can be updated relatively easily, as additional information becomes available. Through the processes
of review and discussion, it should be possible to update these
distributions so that they represent the best available information on the PnET-II inputs. The International Whaling Commission (1992) and Raftery et al. (1995) describe how reasonable distributions of simulation model inputs and outputs were
attained by the comprehensive assessment of the best available scientific information and expert opinions about the
quantities of interest. Presumably, the same result can be
achieved for quantities of interest in process modeling of forest ecosystems.
A central idea of this research was that certain evaluation
techniques require information on the distributions of model
inputs and outputs. We have presented a comprehensive set of
pdfs for PnET-II inputs that describe vegetation characteristics of the EDF. Many of these inputs are relevant to other ecosystem models as well. As new information on these distributions is brought to light, it can be applied to emerging model
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