GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-1071-2015Crop physiology calibration in the CLMBilionisI.DrewniakB. A.ConstantinescuE. M.emconsta@mcs.anl.govMathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, USAEnvironmental Science Division, Argonne National Laboratory, Argonne, IL, USAE. M. Constantinescu (emconsta@mcs.anl.gov)15April2015841071108318June201414October20144March201524March2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.geosci-model-dev.net/8/1071/2015/gmd-8-1071-2015.htmlThe full text article is available as a PDF file from https://www.geosci-model-dev.net/8/1071/2015/gmd-8-1071-2015.pdf
Farming is using more of the land surface, as
population increases and agriculture is increasingly applied for non-nutritional
purposes such as biofuel production. This agricultural expansion
exerts an increasing impact on the terrestrial carbon cycle. In
order to understand the impact of such processes, the Community Land
Model (CLM) has been augmented with a CLM-Crop extension that
simulates the development of three crop types: maize, soybean, and
spring wheat. The CLM-Crop model is a complex system that relies on
a suite of parametric inputs that govern plant growth under a given
atmospheric forcing and available resources. CLM-Crop development
used measurements of gross primary productivity (GPP) and net ecosystem
exchange (NEE) from AmeriFlux sites to choose parameter values that
optimize crop productivity in the model. In this paper, we calibrate
these parameters for one crop type, soybean, in order to provide a faithful projection in terms of
both plant development and net carbon exchange.
Calibration is performed in a Bayesian framework by developing a
scalable and adaptive scheme based on sequential Monte Carlo (SMC).
The model showed significant improvement of crop productivity with the new calibrated parameters.
We demonstrate that the calibrated parameters are applicable across alternative years and different sites.
Introduction
Development of Earth system models (ESMs) is a challenging process, involving
complex models, large input data sets, and significant computational
requirements. As models evolve through the introduction of new processes and
through improvement of algorithms, the ability of the models to accurately
simulate feedbacks between coupled systems improves, although results may not
have the desired impact on all areas. For example,
estimate that changes to the hydrology parameterization may be responsible
for the warm bias in high-latitude soils in the Community Land Model (CLM)
version 3.5 switching to a cold bias in CLM4.0. Although testing of ESMs is
extensive, ensuring after new developments are merged that the model can
still perform with limited (if any) degradation, on rare occasions, model
behavior can be negatively affected. The strong nonlinearity of such models
also makes parameter fitting a difficult task, and as global models are
developed by several different user groups simultaneously, combinations of
multiple alterations make identifying the specific cause that leads to a new
model output challenging. The CLM has been augmented with a CLM-Crop
extension that simulates the development of three crop types: maize, soybean,
and spring wheat . The CLM-Crop model is a complex
system that relies on a suite of parametric inputs that govern plant growth
under a given atmospheric forcing and available resources. CLM-Crop
development used measurements of gross primary productivity (GPP) and net
ecosystem exchange (NEE) from AmeriFlux sites to choose parameter values that
optimize crop productivity in the model.
Global climate models have historically been tuned or calibrated to meet
certain requirements, such as balancing the top of the atmosphere radiation
budget . Various techniques
have been applied to models to adjust parameters, including using data
assimilation , applying an ensemble Kalman filter (EnKF)
, and using a sampling
algorithm such as multiple very fast simulated annealing (MVFSA)
. Most calibration strategies can be traced to a Bayesian
approach that in most cases is simplified (e.g., MVFSA) or augmented with
assumptions that make the problem tractable (e.g., EnKF). The tuning
parameters that are not directly observed may be stated as an inverse problem
. Inverse problems are, in general, very
challenging, especially when the data are sparse, the models are complex, and
the state space is large. This is the case for the CLM-Crop model, as well as
for ESMs.
Our goal is to calibrate some of the CLM-Crop parameters in order to improve
model projection of plant development and carbon fluxes. To this end, we
follow a Bayesian approach . We
start by summarizing our initial state of knowledge in a prior probability
distribution over the parameters we wish to calibrate. After making some
observations, our updated state of knowledge is captured by the posterior
distribution. Since the posterior is not analytically available, we attempt
to approximate it using an ensemble of particles (samples) from it. To
construct this particle approximation, we employ ideas from sequential Monte
Carlo (SMC) . Basically, we define a one-parameter family
of distributions of increasing complexity that starts at the prior and ends
at the posterior. Starting from a particle approximation of the prior, we
gradually move it toward the posterior by sequentially applying importance
sampling. The scheme is highly parallelizable, since each particle of the
approximation can be computed independently. The way we move the particle
approximation towards the posterior is adjusted on the fly using the ideas
developed by and . Each
intermediate step of our scheme requires Markov chain Monte Carlo (MCMC)
sampling of the intermediate distributions. One of the
novelties of this work is the automatic construction of MCMC proposals for
those intermediate steps using Gaussian mixtures
. The result is an algorithmic framework
that can adjust itself to the intricacies of the posterior. As demonstrated
by the numerical examples, our scheme can perform model calibration using
very few evaluations and, by exploiting parallelism, at a fraction of the
time required by plain vanilla MCMC.
We present the results from a twin experiment (self-validation) and
calibration results and validation using real observations from two AmeriFlux
tower sites in the midwestern United States, for the soybean crop type. The
improved model will help researchers understand how climate affects crop
production and resulting carbon fluxes and, additionally, how cultivation
impacts climate.
The CLM-Crop model
CLM-Crop was designed and tested in the CLM3.5 model version
and in CLM4 . The crop model was
created to represent crop vegetation similarly to natural vegetation for
three crop types: maize, soybean, and spring wheat. The model simulates GPP
and yield driven by climate, in order to evaluate the impact of climate on
cultivation and the impact of agriculture on climate. Crops are modeled
within a grid cell sharing natural vegetation; however, they are independent
(i.e., they do not share the same soil column). This approach allows
management practices, such as fertilizer, to be administered without
disturbing the life cycle of natural vegetation. For a full description of
the crop model, see the study by ; the harvest scheme is
described by .
Crops are modeled similarly to natural vegetation, with the main exception of
how allocation is defined via a different growing scheme, which is separated
into four phases: planting, emergence, grain fill, and harvest. Each phase of
growth changes how carbon and nitrogen are allocated to the various plant
parts: leaves, stems, fine roots, and grain. During planting, carbon and
nitrogen are allocated to the leaf, representative of seed. This establishes
a leaf area index (LAI) for photosynthesis, which begins during the emergence
phase. The emergence phase allocates carbon and nitrogen to leaves, stems,
and roots using functions from the Agro-IBIS model .
During the grain fill stage, decreased carbon is allocated to leaves, stems,
and roots in order to fulfill grain requirements. When maturity is reached,
harvest occurs: all grain is harvested, while leaves, stems, and roots are
turned over into the litter pool. Residue harvest is not active in the model.
The allocation of carbon to each plant part is driven largely by the
carbon–nitrogen (CN) ratio parameter assigned to each plant segment. CLM
first calculates the potential photosynthesis for each crop type based
on the incoming solar radiation and the LAI. The total nitrogen needed to
maintain the CN ratio of each plant part is calculated as plant demand. If
soil nitrogen is sufficient to meet plant demand, potential photosynthesis is
met; however, if soil nitrogen is inadequate, the total amount of carbon that
can be assimilated is downscaled.
Prior information on the parameters.
NameDescriptionConstraintsMinMaxDefaultxlLeaf/stem orientation indexNone-0.400.60-0.40slatopSpecific leaf area at top of canopyNone0.000.150.07leafcnLeaf CNone5.0080.0025.00frootcnFine root CNone40.00100.0042.00livewdcnLive wood (phloem and ray parenchyma C)None10.00100.0050.00grpercGrowth respiration factor 1None0.100.500.25grpnowGrowth respiration factor 2None0.001.001.00graincnOrgan carbon nitrogen ratioNone20.00100.0050.00fleafcnFinal leaf carbon nitrogen ratio> leafcn10.00100.0065.00fstemcnFinal stem carbon nitrogen ratio> livewdcn40.00200.00130.00
During the grain fill stage, a nitrogen retranslocation scheme is used to
fulfill nitrogen demands by mobilizing nitrogen in the leaves and stems for
use in grain development. This scheme uses alternate CN ratios for the leaf
and stem to determine how much nitrogen is transferred from the leaves and
stems into a retranslocation storage pool. The total nitrogen transferred at
the beginning of the grain fill stage from the leaf and stem is represented
by
retransnleaf=Cleafleafcn-Cleaffleafcn,retransnstem=Cstemlivewdcn-Cstemfstemcn.Cleaf and Cstem are the total carbon in the leaf and
stem, respectively; leafcn and livewdcn are the pre-grain fill CN ratios for
the leaf and stem; and fleafcn and fstemcn are the post-grain fill CN ratios
for the leaf and stem. All of the CN ratios are fixed parameters, which vary
with crop type; default values are reported in Table .
In addition to the above, CLM-Crop has a fertilizer application and soybean
nitrogen fixation, described by . Planting date and
time to maturity are based on temperature threshold requirements
. For the calibration procedure, we used the actual
planting date reported for the Bondville site for the year 2004. Crops are
not irrigated in the model, nor do we consider crop rotation. Although
rotation will have an impact on the carbon cycle both above- and
below-ground, the CLM does not support crop rotation at this time.
The version of CLM-Crop detailed by was calibrated
against AmeriFlux data for both the Mead, NE, and Bondville, IL, sites' plant
carbon measurements, for both maize and soybean, using optimization
techniques to fit parameters. When available, parameter values were taken
from the literature or other models. Remaining parameters were derived
through a series of sensitivity simulations designed to match modeled carbon
output with AmeriFlux observations of leaf, stem, and grain carbon at the
Bondville, IL, site and total plant carbon at the Mead, NE, (rainfed) site.
When CLM-Crop was ported into the CLM4.5 framework, the parameter values
were no longer optimized as a result of various changes in model
processes that affected how crops fit into the model
framework. In addition, a new below-ground subroutine of carbon and nitrogen
cycling is included in CLM4.5 , which has a strong influence on
crop productivity. Therefore, we needed to retune the model parameters that
represented crops with a more sophisticated approach described later
in this paper.
Parameters affecting the crops
Over 100 parameters are defined in CLM4.5 to represent crops. Many of these
parameters are similar to those that govern natural vegetation, but some are
specific to crops. These parameters define a variety of processes, including
photosynthesis, vegetation structure, respiration, soil structure, carbon
nitrogen dynamics, litter, mortality, and phenology. To add further
complication, parameters are assigned in various parts of the model; some
parameters are defined in an external physiology file, some are defined in
surface data sets, and others are hardcoded in the various subroutines of
CLM4.5.
Performing a full model calibration for all parameters would be a monumental
task, so we began our calibration process by narrowing down the parameters
that are used only in crop functions or might have a large influence on crop
behavior. Of this list, parameter values can be fixed across all vegetation
types (or crop types), vary by crop type, or vary spatially and by crop type.
We chose to limit the parameters to those that are either constant or vary
with crop type.
Crop parameters are taken from the literature (when available) and used to
determine a range of values appropriate for each crop type. When parameters
are not available, optimization techniques are used to estimate parameter
values based on CLM performance. Determining a full range of acceptable
values was difficult for several parameters, and in some cases not possible.
Of the full list of parameters in need of calibration, we began our approach
with the ten parameters listed in Table that may have
a large influence on crop productivity and have the greatest uncertainty
because the values are based on optimization from a previous model version.
Six of the parameters are the carbon–nitrogen (CN) ratios for the various
plant parts (leaf, stem, root, and grain). Since the leaf and stem account
for nitrogen relocation during grain fill, they are represented by two
separate CN ratios, to separate pre- and post-grain fill stages of plant
development. They influence how carbon and nitrogen are allocated, thereby
affecting growth, nutrient demand, photosynthesis, and so on, and are
included as part of the physiology data file. Four additional parameters are
included in the calibration process. The leaf–stem orientation is used to
calculate the direct and diffuse radiation absorbed by the canopy, the
specific leaf area at the top of the canopy is used with the leaf CN ratio to
calculate the LAI, and the growth respiration factors determine the timing
and quantity of carbon allocated toward respiration of new growth.
Description of the observational data set
We used observations from the Bondville, IL, AmeriFlux tower located in the
midwestern United States (40.01∘ N, 88.29∘ W) using an
annual no-till corn–soybean rotation; a full site description is given by
. The site has been collecting measurements since 1996 of
wind, temperature, humidity, pressure, radiation, heat flux, soil
temperature, CO2 flux, and soil moisture. Soybeans were planted in
2002 and 2004, and corn was planted in 2001, 2003, and 2005. We used daily
averaged eddy covariance measurements of NEE and derived GPP in our model
calibration procedure, which are categorized as Level 4 data published on the
AmeriFlux site, and gap filled by using the Marginal Distribution Sampling
procedure outlined by . GPP is derived as the
difference between ecosystem respiration and NEE, where ecosystem respiration
is estimated by using the method of . In addition,
biomass information (which we convert to carbon assuming half of the dry
biomass is carbon) and LAI have been collected for years 2001–2005 for the
various plant segments, including leaf (LEAFC), stem (STEMC), and grain
(GRAINC), which are reported on the AmeriFlux website
(http://public.ornl.gov/ameriflux). The frequency of biomass
measurements is generally every 7 days, beginning a few weeks after planting
and continuing through the harvest. We chose to calibrate against the
Bondville AmeriFlux site because of the availability of unique biomass data
collected. By performing the calibration against site data that include crop
rotation, we hope to indirectly include the effects of crop rotation on GPP
and NEE in the model. Finally, in order to assess the transferability of the
calibrated parameters across sites, we perform one more validation experiment
using observations from the Mead, NE, AmeriFlux site, located at
41.1741∘ N, -96.4396∘ W, similar to the Bondville, IL, site
growing a corn–soybean rotation under no-till conditions. Although there are
three fields, only one is under rain-fed conditions. The site was initialized
in 2001; a description can be found in . The data
collected are the same as the Bondville, IL, AmeriFlux site. Soybeans were
planted in 2002 and 2004.
The time-dependent observations are denoted by z(t)={z1(t),…,z6(t)}, where the indices correspond to GPP, NEE, GRAINC, LEAFC, STEMC,
and TLAI. Because of uncertainties in fertilization use and measured data, we
focused on the peak observed values, as well as the growth slope for GPP,
NEE, LEAFC, and STEMC. To remove the atmospheric-induced noise in the NEE and
GPP measurements, we filtered the time series by applying a moving average
operator with a width of 30 days. These operations are denoted by the map
y=[y1,…,y10]T=[max(z1‾(t)),slope(z1‾(t)),max(abs(z2‾(t))),slope(z2‾(t)),max(z3(t),maxz4(t),slopez4(t),max(z5(t),slope(z5(t)),max(z6(t)]T,
where z‾ represents the filtered z and the slope is calculated
in the beginning of the plant emergence phase, resulting in one maximum and
one slope per variable per year. The observed GPP and NEE slopes were
computed as the slope between the 208th day and 188th day for 2002 and
between the 180th day and 160th day for 2004. The observed LEAFC and STEMC
slopes were computed based on observed values on 16 July–13 August and
23 July–10 September for 2002, and on 8 June–27 July and 8 June–10 August
for 2004, respectively.
Initial conditions and spin-up
CLM requires a spin-up to obtain balanced soil carbon and nitrogen
pools, which are responsible for driving decomposition and
turnover. A global spin-up of the model is provided with the model,
using the below-ground biogeochemistry and spin-up method provided
by . Crops are then interpolated to a higher resolution
over the Bondville, IL, site.
The meteorological forcing data used for the calibration procedure (post
spin-up) are from the Bondville, IL, flux tower site. The atmospheric data
cover the years 1996–2007, but we focus on 2002 and 2004 for this
experiment. The model is run in point mode, meaning only one grid cell is
simulated at a resolution of roughly 0.1∘×0.1∘.
Calibration strategy
We represent the CLM-Crop model output relevant to Eq. () by
f(θ)=(f1(θ),…,fq(θ)), where θ=(θ1,…,θd) are the d time-independent parameters that we
wish to calibrate and q=10 is the number of outputs. The slopes estimated
from numerical simulations were computed as the variable slopes between the
date when the fraction of growing degree days to maturity reaches 0.3 and 20
days prior to this point, where growing degree days are accumulated each day
by subtracting the minimum temperature for growth (10 ∘C for soybean)
from the average daily temperature; see .
We consider a set of ten calibration parameters that were indicated by the
model as being highly uncertain. This set consists of xl, slatop, leafcn,
frootcn, livewdcn, grperc, grpnow, graincn, fleafcn, and fstemcn. See
Table and Sect. for details.
The model calibration strategy aims to merge model predictions that depend on
parameters θ with observational data sets. We assume that the
relationship between observation data and the true process follows
a relationship of type
Y=f(θ*)+ε,
where θ* are the perfectly calibrated parameters and ε
represents the observational errors. This holds under the assumption that the
model is a perfect representation of reality . The
problem statement can be extended to account for imperfect models, but then
the statistical description of ε tends to become much more
complicated. Therefore, for this study, we start by considering a perfect
model assumption.
Following a Bayesian approach, we assume a prior distribution on the
calibration parameters:
p(θ)∝C(θ)∏i=1d1[θi,min,θi,max](θi),
where
θi,min and θi,max are the minimum and maximum allowed
values for the parameter θi, respectively; 1A(x) is the
indicator function of a set A (i.e., 1A(x) is one if x∈A and zero
otherwise); and C(θ) models any physical constraints that are known a
priori. For the parameters we are considering, the constraints are as
follows:
C(θ)=1[θfleafcn>θleafcn](θ)1[θfstemcn>θlivewdcn](θ).
We define the
likelihood as
p(y|θ)∝Ny|f(θ),Σobs,
where N(x|μ,Σ) is the Gaussian probability density with
mean μ and covariance matrix Σ. The covariance matrix
Σobs is taken to be diagonal, namely, Σobs=diagσobs,12,…,σobs,q2,
with each diagonal component σobs,i2 being the square of
10 % of the corresponding observed value. This choice of
Σobs is equivalent to a priori assuming 10 % observational
noise.
Our state of knowledge about the parameters θ after observing y (see
Sect. ) is captured by the posterior distribution:
p(θ|y)∝p(y|θ)p(θ).
Approximating the posterior
We are going to construct a particle approximation of
Eq. () w(i),θ(i)i=1N, in the sense that
p(θ|y)≈∑i=1Nw(i)δθ-θ(i),
where ∑i=1Nw(i)=1, and δ(⋅) is Dirac's delta
function. This is achieved by using a combination of MCMC and SMC methodologies . For more
details on the methodological aspects, we refer the reader to the work of
, and .
Here, we present the material briefly, focusing only on the novel aspect of
our approach that concerns automatically tuning the MCMC proposals.
Let us define a sequence of bridging distributions:
p(θ|y,γt)∝p(y|θ)γtp(θ)=:πt(θ),
where 0=γ0<γ1<…<γt<…≤1. Notice
that for γt=0 we obtain the prior and for γt=1 the
posterior. The key idea of SMC is to start from a particle representation of
the prior (γt=0), which is easy to obtain, and gradually increase
γt until it reaches 1, adjusting the weights along the way. We will
show later how this sequence can be determined on the fly by taking into
account the degeneracy of the particle representations.
Sequential importance sampling
Let wt(i),θt(i)i=1N be a
particle representation of p(θt|y,γt),
p(θt|y,γt)≈∑i=1Nwt(i)δθ-θt(i),
with the weights being normalized (i.e., ∑i=1Nwt(i)=1).
We now examine how this particle representation can be updated to a particle
representation corresponding to γt+1>γt. Toward this goal,
we introduce a fictitious probability density on the joint space of
θt and θt+1 by
qt(θt,θt+1)=p(θt+1|y,γt+1)Lt(θt|θt+1)∝πt+1(θt+1)Lt(θt|θt+1),
where Lt is a backward transition density (i.e.,
Lt(θt|θt+1) is the probability of θt given
θt+1) properly normalized, that is, ∫Lt(θt|θt+1)dθt=1. In addition, we introduce an
importance sampling density,
ηt(θt,θt+1)=p(θt|y,γt)Kt(θt+1|θt)∝πt(θt)Kt(θt+1|θt),
where Kt is a forward transition density (i.e., K(θt+1|θt) is the probability of θt+1 given θt) properly
normalized, that is, ∫K(θt+1|θt)dθt=1.
Notice that
p(θt+1|y,γt)=∫p(θt+1|y,γt+1)L(θt|θt+1)dθt=∫qt(θt,θt+1)dθt=∫qt(θt,θt+1)ηt(θt,θt+1)ηt(θt,θt+1)dθt=∫qt(θt,θt+1)ηt(θt,θt+1)p(θt|y,γt)Kt(θt+1|θt)dθt≈∑i=1Nwt(i)qtθt(i),θt+1ηtθt(i),θt+1Ktθt+1|θt(i).
This observation immediately suggests that to move the γt particle
representation of Eq. () to a γt+1 representation
wt+1(i),θt+1(i)i=1N,
p(θt+1|y,γt+1)≈∑i=1Nwt+1(i)δθ-θt+1(i),
with ∑i=1Nwt+1(i)=1, we must sample
θt+1(i)∼Kt(θt+1|θt(i)),
compute the incremental weights,
w^t+1(i)=πt+1θt+1(i)Ltθt(i)|θt+1(i)πtθt(i)Ktθt+1(i)|θt(i)∝qtθt(i),θt+1(i)ηtθt(i),θt+1(i),
get the unormalized γt+1 weights
Wt+1(i)=wt(i)w^t+1(i),
and get the normalized γt+1 weights
wt+1(i)=Wt+1(i)∑j=1NWt+1(i).
Convenient choices for Lt and Kt
The preceding remarks hold for any backward and forward transition densities
Lt and Kt, respectively. We now seek a convenient choice that will
simplify the form of the incremental weights given in
Eq. (). Suppose for the moment that Kt is given and
let us look for the optimal choice of Lt. Since qt is the target
distribution and ηt is the importance sampling density, the best choice
of Lt is the one that attempts to bring the two densities as close
together as possible. This is easily seen to be the conditional of ηt
on θt; in other words, the optimal choice is
Lt*(θt|θt+1)=η(θt,θt+1)∫η(θt′,θt+1)dθt′=πt(θt)Kt(θt+1|θt)∫πt(θt′)Kt(θt+1|θt′)dθt′.
From a computational point of view, however, it is more convenient to work
with the suboptimal choice,
Lt*,s(θt|θt+1)=πt+1(θt)Kt(θt+1|θt)∫πt+1(θt′)Kt(θt+1|θt′)dθt′,
which is motivated by the expectation that consecutive densities are similar
(i.e., πt≈πt+1). For this choice, the incremental weights
of Eq. () become
w^t+1(i)=πt+1θt+1(i)πt+1θt(i)πtθt(i)∫πt+1θt′Ktθt+1(i)|θt′dθt′.
To get rid of the integral in the denominator, we pick Kt to be invariant
with respect to πt+1:
∫πt+1(θt′)Kt(θt+1|θt′)dθt′=πt+1(θt+1).
This can always be achieved with a suitable choice of a Metropolis–Hastings
transition kernel (see below). For this case, the incremental weights
simplify to
w^t+1(i)=πt+1θt(i)πtθt(i)=py|θt(i)γt+1-γt.
Metropolis–Hastings-based Kt
As shown in the previous paragraph, it is convenient to select Kt to be
invariant with respect to πt+1. The easiest way to achieve this is to
associate Kt with one or more steps of the Metropolis–Hastings algorithm.
Let ht(θ′|θ) be any proposal density (e.g., a simple random
walk proposal). The single-step Metropolis–Hastings forward transition
density is
Kt1(θt+1|θt)=ht(θt+1|θt)a(θt+1,θt),
where
a(θt+1,θt):=min1,πt+1(θt+1)ht(θt|θt+1)πt+1(θt+1)ht(θt+1|θt).
Samples from Eq. () may be obtained by performing one
step of the well-known Metropolis–Hastings algorithm. The forward kernel
corresponding to M>1 Metropolis–Hastings steps is given recursively by
KtM(θt+1|θt)=∫KtM-1(θt+1|θ′)Kt1(θ′|θt)dθ′.
The number of Metropolis–Hastings steps, M, at each γt is a
parameter of SMC. This is the forward kernel we use in all numerical
examples. Theoretically, M=1 is enough, since the number of particles
N→∞. Therefore, we will use M=1 in our numerical examples.
Resampling
As SMC moves to higher values of γt, some of the particles might find
themselves in low probability regions. Consequently, their corresponding
weights will be small. This degeneracy of the weights can be
characterized by the effective sample size (ESS) metric, defined by
ESS(γt)=1∑i=1Nwt(i)2.
Notice that ESS is equal to N when the particles are equally important
(i.e., wt(i)=1/N) and equal to 1 when only one particle is important
(e.g., wt(1)=1 and wt(i)=0 for i≠1), and in general
takes values between 1 and N for arbitrary weights. Resampling is triggered
when the ESS falls below a pre-specified threshold (12N in our
numerical examples). The idea is to kill particles that have very small
weights and let the particles with big weights replicate. This process must
happen in a way that the resulting particle ensemble remains a valid
representation of the current target probability density. This can be
achieved in various ways. Perhaps the most straightforward way is to use
multinomial sampling. Let the resulting particles be denoted by
w̃t(i),θ̃t(i)i=1N. In multinomial resampling,
the final weights are all equal:
w̃t(i)=1/N.
The sequence θ̃t(i)i=1N is found by
sampling a sequence of integers ji⊂{1,…,N} with
probabilities {wt(1),…,wt(N)} and by setting
θ̃t(i)=θt(ji),
for i=1,…,N.
Choosing γt+1 on the fly
We note that the incremental weights of Eq. () do not
depend on the γt+1 samples obtained in Eq. ().
They depend only on the likelihood of the γt samples. In this part,
we exploit this observation in order to devise an effective way of selecting
γt+1 based on the ESS. The idea is to pick the new γt+1
so that the resulting particles do not become too degenerate. Their
degeneracy is characterized by ESS(γt+1) given in
Eq. (). From Eqs. (),
(), and (), evaluation of
ESS(γt+1) does not require any new likelihood evaluations.
We select the new γt+1 by requiring that
ESS(γt+1)=ζESS(γt),
where ζ is the percentage of the degeneracy we are willing to accept
(ζ=0.99 in our numerical examples). It is fairly easy to show that
ESS(γt+1) is a strictly decreasing function for
γt+1∈(γt,1]). Therefore, Eq. () has a
unique solution that can be easily found by using a bisection algorithm.
Adapting the KtM on the fly
Based on the discussion above, we expect that p(θt|y,γt)
should be similar to p(θt+1|y,γt+1). To exploit this fact,
we pick the proposal ht(θ′|θ) required by the
Metropolis–Hastings kernel KtM given in Eq. () to be a
mixture of Gaussians that approximates p(θt|y,γt). In
particular, we pick
ht(θ′|θ)=∑i=1LciNθ′|μt,i,Σt,i,
where the non-negative coefficients ci sum to 1, μi,i∈Rd, and Σt,i∈Rd×d are covariance
matrices. The number of components L as well as all the parameters of the
mixture are fitted to a resampled version of the particle approximation of
p(θt|y,γt) (see Eq. ) using the
procedure of as implemented
by .
Parallelization
SMC is embarrassingly parallelizable. Basically, each CPU can store and work
with a single particle. Communication is required only for normalizing the
weights (see Eq. ), finding γt+1 (see
Eq. ), and resampling. The first two have a negligible
communication overhead and can be implemented easily. Implementation of the
resampling step is more involved and requires more resources. However, the
cost of resampling is negligible compared with the evaluation of the forward
model.
The final algorithm
We now collect all the details of SMC discussed above in a single algorithm
for convenience: Algorithm . Our implementation is in Python and
is provided at https://github.com/ebilionis/pysmc.
Sampling
from the posterior using sequential Monte Carlo.
Results
In this section, we present our calibration results for the parameters
described in Sect. by using the observations detailed in
Sect. . In this study, we focus only on the parameters affecting
the soy crop and restrict our calibration to year 2004. With these calibrated
parameters, we perform a validation experiment by using the data from year
2002. In addition, we forecast 2004 through a 2002–2003–2004 simulation. We
recognize that the Bondville observations include crop rotation during 2003
that will influence the sequence of output, but since the model does not
support crop rotation, we plant soybean during 2003. The role of the latter
experiment is to demonstrate the robustness of the proposed calibration
scheme. Moreover, we perform a twin experiment that consists of generating
artificial data by using some control parameter values, then applying the
calibration strategy to recover the control parameters.
In all our numerical examples, we fix the planting and harvest days. This
approach is essential in order to avoid overfitting the physiological
parameters due to offsets in the growing seasons. The planting dates for 2002
and 2004 are 2 June and 7 May, respectively. The harvest day is controlled
via input variable hybgdd (growing degree days for maturity), where growing
degree days are defined in Sect. . The values of hybgdd that
give the right harvest days for 2002 and 2004 are 1474 and 1293,
respectively.
The number of particles we use is N=1280. Each particle is assigned to a
different computational core; i.e., we use 1280 computational cores. A
simulated year takes about 2 min to complete if data localization is used.
Calibration requires approximately 100 000 simulations and completes in
about 6 h.
Validation of the method
We begin the twin experiment with the aim of validating the proposed
calibration strategy. We generate artificial observations by randomly
sampling θ from its prior Eq. (). We apply the
calibration strategy to the artificial observations to see whether the method
can recover the ground truth. The adaptively selected γt sequence is
shown in Fig. a. In Fig. , we compare the
posterior of each parameter with the prior. The true parameters are indicated
by red dots. The fit to the artificial outputs is shown in
Fig. . The parameters that are not specified precisely are
parameters that have a small (if any) effect on the observed outputs.
Adaptation of γt for the twin experiment (a) and
the calibration (b). The small jumps indicate the locations where
resampling occurs.
Calibration using real data
In our next experiment, we calibrate the parameters listed in
Table . The observational operator (Eq. )
is defined by taking the annual maximum of the absolute value of LEAFC, LAI,
GRAINC, STEMC, GPP, and NEE; and the slope of LEAFC, STEMC, GPP, and NEE as
described in Sect. . In Fig. , we compare the
posterior we obtain with the prior. The default parameters are indicated by
red dots. The fit to the artificial outputs is shown in
Fig. . The adaptively selected γt sequence is shown
in Fig. b. In Table , we summarize
our findings, by showing the median and the p=0.05 and p=0.95 quantiles
of each calibrated parameter.
Twin experiment: comparison of the posterior with the prior. The red
dot indicates the true parameter value. The figure is continued on the next
page.
Twin experiment: comparison of the true model output with samples
from the posterior of the calibration.
Calibration experiment (Bondville, IL): comparison of the posterior
with the prior. The red dot indicates the default parameter value. The figure
continues on the next page.
Validation of real data results
To validate the generalization potential of our calibration, we perform
a one-way validation. We use the calibrated parameters to predict the
observables in 2002. In Fig. , we plot the median and
95 % error bars of the calibrated time series and we compare the results
with observations and the default parameter output of 2002. We observe a
notable improvement in the ability of the model to explain the observations.
One of the most important improvements is related to LAI calculations, which
comes from improvements to the leaf CN ratio and the specific leaf area. The
timing of maximum LAI is important for the carbon allocation; when the crops
in CLM4.5 reach peak LAI, carbon allocation shifts from above- and
below-ground to strictly below-ground (roots). With the default parameter
values, peak LAI occurred early in the growing season, resulting in large and
unrealistic allocation of carbon to roots and insufficient carbon to leaves,
stems, and ultimately grains. The large increase in stem carbon and the
slower rate of growth and peak of GPP are clear indications that the shift in
allocation to roots no longer occurs with the new calibrated parameter
values. The grain carbon is still low, however, a result of the low leaf
carbon and the overestimation of stem carbon, which increases the amount of
carbon allocated to maintenance respiration at the expense of new growth for
this year. The increase in uncertainty is likely a result of a limitation in
nitrogen availability in some scenarios. When CN ratios are low, a higher
demand of nitrogen from plants contributes to an increase in competition for
resources with below-ground decomposition processes. When the nitrogen demand
from the two sources exceeds availability, the amount of carbon that can be
assimilated is downscaled, resulting in a lower GPP, increase in NEE, and so
on. We continue the simulation through 2003 to 2004 and compare the
calibrated time series with the observations and the default parameter output
in Fig. . The differences between this plot and
Fig. are due to differences in the below-ground conditions
of carbon and nitrogen that drive the dynamics for plant competition with
below-ground decomposition processes. Crops in CLM4.5 tend to be sensitive to
variation in carbon and nitrogen pools, and since we ran the calibration over
1 year and did not consider variability in previous years' carbon and
nitrogen pools, demand for nitrogen is likely different when the model is run
for multiple years. Since the change in pools is minor, the resulting change
in output by the model is also small. This is also likely responsible for the
increased uncertainty in GPP and NEE, which occurs from competition for
resources as discussed above. Finally, Fig. shows the
results of the same validation experiment at the Mead, NE, site.
Calibration experiment (Bondville, IL): comparison of the observed
data for 2004 with calibrated outputs. The grey areas correspond to 95 %
confidence error bars.
Validation experiment (Bondville, IL): comparison of the observed
data for 2002 with the model. The grey areas correspond to 95 % confidence
error bars.
Discussion
In this paper, we sought to improve CLM-Crop model performance by parameter
calibration of a subset of model parameters governing, mostly, the carbon and
nitrogen allocation to the plant components. By using a Bayesian approach, we
were able to improve the model-simulated GPP, NEE, and carbon biomass to
leaf, stem, and grain with the new parameter values. In addition, we
demonstrated that the calibrated parameters are applicable across alternative
years and not solely representative of 1 year.
Validation experiment (Bondville, IL): comparison of the observed
data for 2004 with the model started in 2002. The grey areas correspond to
95 % confidence error bars.
Validation experiment (Mead, NE): comparison of the observed data
for 2002 with the model (top). Comparison of the observed data for 2004 with
the model (bottom).
This study does have a few limitations stemming from a lack of observation
data. Currently our results are suitable at one site across multiple years;
testing at multiple sites would give a better indication of how well the
model can perform globally or even across a region. However, the limited data
over agricultural sites constrain our ability to determine parameter values
that are relevant at a global scale. In addition, our use of actual planting
dates is not a typical approach with CLM4.5, which generally uses temperature
thresholds to trigger planting. Thus, the model may plant earlier or later
compared with observations, which, if significant, could influence the growth
cycle and resulting carbon fluxes. In addition, CLM-Crop does not have crop
rotation, which is common across agricultural landscapes, including in the
observation data set. Crop rotation can modify below-ground carbon and
nitrogen cycling that would have an impact on crop productivity through
nutrient availability as well as NEE. While we would like to include crop
rotation, CLM does not currently have the capability to support this
function. Therefore, we tried to include the effects indirectly by
calibrating against data that include crop rotation. As more sophisticated
crop representation is introduced into the model, we will revisit the
calibration to improve model parameters. Moreover, we considered the initial
litter, carbon, and nitrogen pools fixed by the values of the prior
parameters because a direct spin-up calculation would have made sampling
prohibitively expensive. We will address this issue in a future study by
including these pools in the calibration procedure.
Our approach has focused on one crop type, soybean, with the intent of
determining the effectiveness of the proposed calibration method. We consider
the results promising and, as part of future work, hope to expand this
research to additional years, crop types, and other parameters. Many other
variables are of interest, including fertilization rate, timing of the growth
stages, and a few other parameters related to photosynthesis. As the model
continues to evolve with the addition of new or improved processes, we also
may need to revisit the parameter choices and evaluate their appropriateness.
Moreover, a calibration procedure carried for such complex models with
relatively few data and a few calibration parameters has the potential to
lead to overfitting. To assess this effect, we performed a validation
experiment, which provides good confidence in, albeit not proof of, a robust
calibration of the parameters. Richer data sets will likely sharpen the
results and enhance the confidence intervals.
The introduction of new data sets documenting agriculture productivity or
carbon mass will also allow us to determine the applicability of our new
parameter values across regions. In general, the calibration results depend
on an accurate specification of the observational errors. In this study, we
did not have access to any information regarding the measurement process and,
therefore, assumed a certain observational noise. These calibration results
can be sharpened by annotating the observational data with levels of
confidence. The calibration strategy presented in this study has the
potential to improve model performance by helping modelers define parameters
that are not often measured or documented.
Acknowledgements
This work was supported by the Office of Biological and Environmental
Research, US Department of Energy, under contract DE-AC02-06CH11357. We
gratefully acknowledge the computing resources provided on Fusion, a 320-node
computing cluster operated by the Laboratory Computing Resource Center at
Argonne National Laboratory. Edited by:
S. Arndt
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