Models are an important tool to predict Earth system dynamics. An accurate
prediction of future states of ecosystems depends on not only model
structures but also parameterizations. Model parameters can be constrained
by data assimilation. However, applications of data assimilation to ecology
are restricted by highly technical requirements such as model-dependent
coding. To alleviate this technical burden, we developed a model-independent
data assimilation (MIDA) module. MIDA works in three steps including data
preparation, execution of data assimilation, and visualization. The first
step prepares prior ranges of parameter values, a defined number of
iterations, and directory paths to access files of observations and models.
The execution step calibrates parameter values to best fit the observations
and estimates the parameter posterior distributions. The final step
automatically visualizes the calibration performance and posterior
distributions. MIDA is model independent, and modelers can use MIDA for an
accurate and efficient data assimilation in a simple and interactive way
without modification of their original models. We applied MIDA to four types of ecological models: the data assimilation linked ecosystem carbon (DALEC)
model, a surrogate-based energy exascale earth system model: the land
component (ELM), nine phenological models and a stand-alone biome
ecological strategy simulator (BiomeE). The applications indicate that MIDA
can effectively solve data assimilation problems for different ecological
models. Additionally, the easy implementation and model-independent feature
of MIDA breaks the technical barrier of applications of data–model fusion in ecology. MIDA facilitates the assimilation of various observations into
models for uncertainty reduction in ecological modeling and forecasting.
Introduction
Ecological models require a large number of parameters to simulate
biogeophysical and biogeochemical processes (Bonan,
2019; Ciais et al., 2013; Friedlingstein et al., 2006) and specify model
behaviors (Luo et al., 2016; Luo and Schuur, 2020). Parameter values in
ecological models are mostly determined in some ad hoc fashions (Luo et al., 2001), leading to considerable biases in predictions (Tao et al.,
2020). The situation becomes even worse when more detailed processes are
incorporated into models (De Kauwe et al., 2017; Lawrence et al., 2019). Data assimilation (DA), a statistically rigorous method to integrate observations and models, is gaining increasing attention for parameter estimation and uncertainty
evaluation. It has been successfully applied to many ecological models (Fox
et al., 2009; Keenan et al., 2012; Richardson et al., 2010; Safta et al.,
2015; Wang et al., 2009; Williams et al., 2005; Zobitz et al., 2011).
However, almost all those DA studies require model-dependent, invasive
coding (Walls et al., 2005). This requires a DA algorithm to be programmed
for a specific model. Such model-dependent coding creates a large technical
barrier for ecologists to use DA to solve prediction and uncertainty
quantification problems in ecology. Thus a model-independent DA toolkit is
required to facilitate the use of DA technique in ecology.
DA is a powerful approach to combine models with observations and can be
used to improve ecological research in several ways (Luo et al., 2011). First, DA can be used for parameter estimation (Bloom et al., 2016; Hararuk et al., 2015; Hou et al., 2019; Ise and Moorcroft, 2006; Ma et al., 2017; Ricciuto et al., 2011; Scholze et al., 2007). It
enables the optimization of parameter values across sites, time and
treatments (Li et al., 2018; Luo and Schuur, 2020). For example, Hararuk and his colleagues applied DA to a global land model and substantially improved the explainability of the global variation in soil organic carbon (SOC) from
27 % to 41 % (Hararuk et al.,
2014). When DA was combined with deep learning to improve spatial
distributions of estimated parameter values, for example, the Community Land
Model version 5 (CLM5) predicted the SOC distribution in the US continent
with much higher R2 of 0.62 than CLM5 with default parameters (R2=0.32) (Tao et al., 2020). Second, DA can be used to select alternative model structures to better represent ecological processes (Liang
et al., 2018; Van Oijen et al., 2011; Shi et al., 2018; Smith et al., 2013;
Williams et al., 2009). In the study by Liang et al. (2018), DA was used to
evaluate four models. And a two-pool interactive model was selected after DA
to best represent SOC decomposition with priming. Additionally, DA can be
applied to locate the most informative data to reduce uncertainty, thus
guiding the sensor network design (Keenan et al., 2013; Raupach et al., 2005; Shi et al., 2018; Williams et al., 2005). One DA study at Harvard Forest (Keenan et al., 2013) indicated that only a few data sources contributed to the significant reduction in parameter uncertainty. In spite of powerful applications of DA to ecological research,
computational cost is a major hurdle, especially with complex models. Fer et
al. (2018) developed a Bayesian model emulation to reduce the time cost of
DA from 112 to 6 h with the simplified Photosynthesis and Evapotranspiration
model. Overall, DA is essential for ecological modeling and forecasting (Jiang et al., 2018) and is helpful for evaluation of different inversion methods (Fox et al., 2009).
Applications of traditional DA to ecological research require highly
technical skills of users. A successful DA application usually involves
model-dependent coding to integrate observations into models. This requires
users to have knowledge about model programming. For example, if a complex
model (e.g., the community land model) is used in DA, users need to know the
programming language (e.g., Fortran) of the model and its internal content
to write DA algorithm into the model source code before DA can be conducted.
The learning curve for model programming is steep for general ecologists.
Furthermore, users often need to update the programming knowledge when a
different model is used in DA. For example, scientists who implemented the
DA algorithm coded in MATLAB (Xu et al., 2006) to an ecosystem carbon cycle model programmed in Fortran (e.g., TECO) need to understand both MATLAB and Fortran (Ma et
al., 2017). Moreover, DA often involves reading observation files about a
specific study site. As a result, users usually have to update the codes of
model-dependent DA to read new observations from every new study site.
Comparison among MIDA and available DA tools.
GlobalPosteriorDA toolAgnosticDA algorithmsoptimadistributionVisualizationCCDASNoAutomatic differentiation from TransformationNoNoNoof Algorithms in Fortran (TAF)CARDAMOMNoMarkov chain Monte CarloYesYesNoEcoPADNoMarkov chain Monte CarloYesYesYesOpenDANoEnKF, ensemble square-root filter, particle filterYesYesNoDARTYesEnKFYesYesNoPDAFYesEnKFYesYesNoPESTYesLevenberg–Marquardt methodRely on initialNoNoparameter valuesMIDAYesMarkov chain Monte CarloYesYesYes
A number of tools have been developed to facilitate DA applications (Table 1) but many of them are model dependent, such as the Carbon Cycle Data
Assimilation Systems (CCDAS) (Rayner et al., 2005; Scholze et al., 2007), the Carbon Data Model Framework (CARDAMOM) (Bloom et al., 2016), the
Ecological Platform for Assimilating Data (EcoPAD) into model (Huang et al.
2019) and Predictive Ecosystem Analyzer (PEcAn)
(LeBauer et al., 2013). These
tools combine DA algorithms with a specific model. For example, CCDAS
specified the DA algorithm to the Biosphere Energy Transfer Hydrology
(BETHY) model (Rayner et al., 2005). The hardcoding feature of
aforementioned tools make them inflexible to be applied to different models.
There are some model independent DA tools that are not tailored to a
specific model, such as Data Assimilation Research Testbed (DART)
(Anderson et al., 2009), the open Data Assimilation
library (openDA) (Ridler et al., 2014),
the Parallel Data Assimilation Framework (PDAF) (Nerger and Hiller, 2013) and Parameter Estimation & Uncertainty Analysis software suit (PEST)
(Doherty, 2004).
However, these model-independent tools suffer from some limitations for a
general and flexible DA application. For example, openDA requires users to
code three functions to initialize a Java class
(Ridler et al., 2014) (Table 1). DART enables incorporating a new model through a range of interfaces (Anderson et al., 2009). It has been successfully applied to atmospheric and oceanic models with currently available interfaces (Anderson et al., 2009; Raeder et al., 2012) and recently to the community land model (Fox et al., 2018). It is
likely that users may need to prepare new interfaces for new ecological
models to use DART. DART and PDAF adopted the Ensemble Kalman Filter (EnKF)
method (Evensen, 2003), which may makes it difficult
to obey mass conservation for biogeochemical models. This is because the
parameter values estimated by EnKF change each time when new data sets are
assimilated (Allen et al., 2003; Gao et al., 2011; Trudinger et al., 2007). The sudden changes in estimated parameter values at time points when data are assimilated by EnKF usually do not reflect reality of biogeochemical cycles in the real world. PEST utilizes the Levenberg–Marquardt method (Levenberg, 1944), which is a local optimization method for parameter estimation. If the
relationship between simulation outputs and parameters is highly nonlinear,
which is common in ecological models, this method may trap into a locally
optimization solution (Doherty, 2004).
In this work, we developed a model-independent DA module (MIDA) to enable a
general and flexible application of DA in ecology. MIDA was designed as a
highly modular tool, independent of specific models, and friendly to users
with limited programming skills and/or technical knowledge of DA algorithms.
Additionally, MIDA implemented advanced Markov chain Monte Carlo (MCMC)
algorithms for DA analysis which can accurately quantify the parameter
uncertainty with informative posterior distribution. The anticipated user
community in this initial phase of MIDA development is the biogeochemical
modelers who are looking for appropriate parameter estimation methods. In
the following Sect. 2, we first introduce the development details of MIDA
and its usage. In Sect. 3, we demonstrate the application of MIDA to four
different types of ecological models. In Sect. 4, we discuss the strengths
and weaknesses of MIDA in ecological modeling, and lastly we give our
concluding remarks in Sect. 5.
Model-independent data assimilation (MIDA)Bayes' theorem and DA
Based on Bayes' theorem, DA is a statistical approach to constrain parameter
values and estimate their posterior density distributions through
assimilating observations into a model. The posterior density distributions
p(C|Z) of parameters C for a given observation Z can be obtained
from prior density distributions p(C) and the likelihood function p(Z|C):
p(C|Z)∝p(Z|C)p(C).
The prior density distribution p(C) is assumed as a uniform distribution over
the parameter range. And the likelihood function is negatively proportional
to a cost function, J, as
p(Z|C)∝exp(-J).
The cost function measures the misfit between simulation outputs and
observations and is described in more detail in Sect. 2.4. The posterior
density distribution p(C|Z) is estimated from sampling parameter
values to maximize the likelihood function p(Z|C) or minimize the
cost function J. DA usually uses a sampling technique, such as Markov chain
Monte Carlo (MCMC) in this MIDA. The MCMC algorithm successively generates a
new set of parameter values from the prior parameter ranges and requires a
model run with these new parameter values. Then the cost function is
calculated to determine whether this new set of parameter values will be
accepted or not according to the Metropolis–Hastings criterion (see more
description in Sect. 2.4). All accepted parameter values are used to
generate posterior distributions where the distinctive mode indicates the
parameter uncertainty is well constrained. Meanwhile, we derive maximum
likelihood estimates (MLEs) of parameters from the posterior density
distributions.
MIDA realizes model-independent Bayesian-based DA to estimate posterior
density distributions and MLEs of parameters via data exchanges between a
given model and DA algorithm.
The three-step workflow of the Model Independent Data
Assimilation (MIDA) module. The workflow includes data preparation,
execution of data assimilation (DA), and visualization. The data preparation
step is to provide all the formatted essential data for DA via user input.
The execution step is to calibrate parameter values towards a constrained
posterior distribution with the fusion of observations. The visualization
step is to diagnose the effects of DA. The rhombus in orange represents
user-input data. The rectangle represents procedures, and document/multidocument
shape is for data files in computers. Dashed lines indicate locations of
data. Solids lines indicate data flow pathways. With the three-step
workflow, DA is agnostic to specific models, and users will be released from
technical burdens.
An overview of MIDA
MIDA is a module that allows for automatic implementation of data
assimilation without intrusive modification or coding of the original model
(10.5281/zenodo.4762725, Huang, 2021). Its
workflow includes three steps: data preparation, execution of data
assimilation, and visualization (Fig. 1). Step 1 (data preparation) is to
establish the standardized data exchange between the DA algorithm and the model.
Step 2 (execution of data assimilation) is to run DA as a black box
independent of the model. Step 3 (visualization) is to diagnose parameter
uncertainty after DA. The modularity of the three-step workflow is designed to
enable MIDA for a rapid DA application and adaption to a new model. In the
following, we introduce the three-step workflows of MIDA, its technical
implementation, and its usage in detail.
Step 1: data preparation
Step 1 is designed to initialize data exchange to transfer parameter values,
model outputs, observations, and their variances between the DA algorithm and the
model to be used. Four types of information are required either from
interactive input or by modifying the “namelist.txt” file (Fig. 1). The
first type is about DA configuration, including the number of sampling
series in DA and the working path where the outputs of DA will be saved. The
number of a sampling series is essential in a DA task to define how many
times parameter values are sampled to run the model. The second type of
information is about parameter ranges and their covariance. The third is the
model executable file. Finally, the fourth type is an output configuration
file which contains the file paths of model outputs, observations, and their
variance. This file also instructs how to read model outputs and compare
each output with corresponding observations.
Traditional DA requires users to modify the code of the model to incorporate the
process of data exchange between the DA algorithm and the model. Therefore, the
program of data exchange in traditional DA is model-specific, and users need
to repeat such a program when a new model comes. In MIDA, the process of data
exchange calls a model executable file which hides the details of the model
code. When applied to a new model, MIDA only requires users to provide a
different model executable file in the namelist.txt file and does not
involve any additional coding in either the model or MIDA. Thus, MIDA lowers
the technical barrier for general ecologists to conduct DA.
Traditional DA usually presets the number of parameters and the model outputs
according to a specific model before initializing the data exchange. This is
because data exchange between the DA algorithm and model uses memory to transfer
items such as parameter values. Instead, MIDA organizes items in data
exchange using different files. Items in data exchange are decided by the
data file loaded when MIDA is running. The number of parameter values, for
example, will be decided after the file of parameter range is read in MIDA.
Through modifying files, MIDA allows efficient choices about the
model-related items in data exchange to be made. Thus, MIDA is highly flexible and
modular for DA with different models.
Traditional DA also presets observation types in the data exchange according
to a specific study before the data exchange. For example, if the
traditional DA uses carbon flux observation, it cannot switch to satellite
remote sensing products without additional coding. MIDA uses the concepts of
object-orient programming (Mitchell and Apt, 2003) and dynamic
initialization (Cline et al., 1998) in computer science to
provide a homogenous way to create various observation types from a unified
prototype class. A prototype class includes variables to store observations
and their variance and functions (e.g., read from observation files). The
values in variables are dynamically decided after the observation files are
loaded when MIDA is running. Different observation types derive from the
prototype class with a high degree of reusability of most functions. In such a
way, MIDA only requires users to provide different filenames of the
observations to be integrated in DA. Therefore, MIDA is highly flexible and
modular for DA to assimilate various observations.
Step 2: execution of data assimilation
After the establishment of the standardized data exchange (step 1), step 2
is to run DA as a black box for users without knowledge of DA itself.
Notwithstanding the black-box goal, this section provides a general
description of DA below.
Data assimilation as a process integrates observations into a model to
constrain parameters and estimate parameter uncertainties. Data assimilation
usually uses some types of sampling algorithms, such as Markov chain Monte
Carlo (MCMC), to generate posterior parameter distribution under a Bayesian
inference framework (Box and Tiao, 1992). As mentioned in
Sect. 2.1, DA with a MCMC algorithm estimates the posterior density
distributions through sampling to maximize likelihood function p(Z|C)
or minimize the misfit J between simulation outputs and observations. This
version of MIDA uses the MCMC algorithm implemented by the Metropolis–Hastings
(MH) sampling method (Hastings, 1970; Metropolis et al., 1953). The future version of MIDA could incorporate other data assimilation algorithms. Each iteration in the Metropolis–Hastings sampling includes a proposing phase and a moving phase. The proposing phase generates a new set of parameter values based on the starting point for the
first iteration or current accepted parameter values in the following
iterations. If parameter covariance (covparam) is specified in step 1 on data preparation, this proposing phase will draw new parameter values (Cnew) within the prior ranges from a Gaussian distribution
N(Cold,covparam), where Cold is the predecessor set of parameter values. Without parameter covariance, a new set of parameter values will be generated from a uniform distribution within the prior ranges (Xu et al., 2006).
The moving phase first calculates mismatches between observations and the
model simulation with the new set of parameter values as a cost function
(Jnew in Eq. 3) (Xu et al., 2006):
Jnew=∑i=1n∑t∈obsZiZi(t)-Xi(t)22σi2,
where n is the number of observations, Zi(t)
is the ith observation at time t, Xi(t) is the
corresponding simulation, and σi2 is the variance of the
observation. The error is assumed to independently follow a Gaussian
distribution. This new set of parameter values will be accepted if Jnew is smaller than Jold, the cost function with the previous set of accepted parameter values, or the value, exp-JnewJold, is larger than a random number selected from a uniform distribution from 0 to 1 according to the Metropolis
criterion (Liang et al., 2018; Luo et al., 2011; Shi et al., 2018; Xu et al., 2006). Once the new set of parameter values is accepted, Jnew becomes Jold. Those two phases of sampling will be iteratively executed until the number of sampling series set in step 1 on preparation of DA is reached. Finally, the posterior density distributions can be generated from all the accepted parameter values.
MIDA realizes the execution of data assimilation according to the procedure
described above. First, MIDA uses a “call” function to execute model
simulations to get values of Xi(t). Observations
Zi(t) and their variance σi2 are
already provided via the standardized data exchange as described in step 1.
Then, MIDA calculates Jnew according to Eq. (3) to decide the
acceptance of the current parameter values used in this simulation. If
accepted, MIDA saves this set of parameter values and associated Jnew values in Caccepted and Jaccepted array, respectively, and triggers a new proposing phrase based on this set of accepted parameter values. If not, MIDA discards this set of parameter values and generates another new set of parameter values. MIDA saves the new parameter values generated in the proposing phrase to “ParameterValue.txt”, from which the model reads
before execution of the next model simulation. MIDA repeats the proposing
and moving phases until the number of sampling series is reached. At the
end, MIDA selects the best parameter values through maximum likelihood
estimation and runs the model again using this set of values to get optimized
simulation outputs Xi(t). Then MIDA saves the arrays of
accepted parameters, associated errors, maximum likelihood estimates (MLEs),
and optimized state variables Xi(t) to four files,
“parameter_accepted.txt”, “J_accepted.txt”, “MLE.txt”, and “OptimizedSimu.txt”, respectively.
This execution of the DA algorithm in MIDA enables users to conduct DA as a
black box and is independent of any particular model.
Step 3: visualization
Step 3 is to visualize the results of DA in step 2. The end products of DA
are accepted parameter values, their associated Jnew values, the
maximum likelihood estimates, and optimized simulation results as saved in
the output files. MIDA enables visualization of parameter posterior density
distributions with a Python script. In the script, MIDA first read accepted
parameter values from the parameter_accepted.txt file. Then,
MIDA generates a posterior probabilistic density function (PPDF) for each
parameter via the “kdeplot” function in the “seaborn” package. The maximum
likelihood estimates of parameters correspond to the peaks of PPDF. The
distinctive mode of PPDF indicates how well the parameter uncertainty is
constrained. Finally, MIDA visualizes the PPDF for all parameters in a
figure using the “matplotlib” package.
Implementation and architecture of MIDA
MIDA is equipped with a graphical user interface (GUI), and users can easily
execute it through an interactive window. Users can also run MIDA as a
script program without the GUI. MIDA is written in Python (version 3.7). For
the GUI-version, all relevant Python packages used in MIDA are compiled
together; thus users do not need to install them by themselves. For the
non-GUI version, users need to install Python 3.7 and relevant packages
(i.e., numpy, pandas, shutil, subprocess, matplotlib, math, os, and
seaborn). MIDA is compatible with model source codes written in multiple
programming languages (e.g., Fortran, C/C++, C#, MATLAB, R, or
Python). It is also independent of multiple operation systems (e.g.,
Windows, Linux, MacOS). In addition, MIDA is also able to run on
high-performance computing (HPC) platforms via task management systems
(e.g., Slurm).
The architecture of MIDA is class-based, and each class is designed to
describe an object (e.g., parameter, observations) with variables and
operations. Five classes are defined in MIDA: parameter, observation,
initialization, MCMC algorithm, and the main program. The main program is the
start of MIDA execution. It calls functions from all other classes to finish a
three-step workflow. As described in Sect. 2.2, parameter and observation
classes contain variables to be transferred in data exchanges via file I/O
operations. These operations are implemented using the “numpy” package. The
initialization class is to read namelist.txt in step 1 on data preparation and to assign values for the variables in all other classes. Then the class
of MCMC algorithm conducts DA as described in step 2. In this step, the
simulation operation uses a call function in the “subprocess” package to call the model executable file. At the start of model simulation, MIDA writes new
parameter values to the “ParameterValue.txt” file in the “working path”
directory specified in step 1 on data preparation. Then model executable reads parameter values from the ParameterValue.txt file and run. After
model simulation, the DA algorithm can read the model outputs by the output
filenames indicated in the output configuration file. After DA, step 3
executes an additional Python script to read accepted parameter values and
plot the posterior density distributions of parameters. The plotting
operations use the matplotlib and seaborn packages. The implementation of
GUI uses the pyQt5 toolkit to support interactive usage of MIDA. Users can also
run MIDA in a non-interactive way with a “main.py” script to trigger the
three-step workflows.
User information of MIDA
In order to use MIDA, users need to prepare data and a model. The data to be
used in MIDA are prior ranges and default values of parameters, parameter
covariances, output configuration file, observations, and their variances.
They are organized in different files. Before running MIDA, users need to
specify their filenames as suggested in step 1. When users want to use
different data sets in DA, they can simply change filenames with the new
data sets via GUI or in the namelist.txt file. Figure C1 is an example of
the namelist.txt file for a data assimilation study with the DALEC model.
The model to be used in MIDA should have those to-be-estimated parameter
values not fixed in model source code rather than changeable through
ParameterValue.txt file. MIDA writes new parameter values in each
proposing phase during DA to the ParameterValue.txt file, from which the
model reads the parameter values to run the simulation.
To calculate the cost function, J, we have to have a one-to-one match
between observations and model outputs. For example, phenology models in one
of the application cases of MIDA below generate discrete dates of leaf
onset, which is a one-to-one match to the observations of spring leaf onset.
In this case, observation Zi(t) and model
output Xi(t) to be used in calculation of J are
straightforward. In the application case for dynamic vegetation, the data to
be used are leaf area in six layers in a forest 302 years old, whereas the
model simulates leaf areas in eight layers from 0 to 800 years. To match
observation, the model generates outputs of leaf areas in six layers when
simulated forest age reaches 302 years. This requires users to prepare an
output configuration file to instruct MIDA to read model outputs and
re-organize their outputs to match observation. The output configuration
file starts with a single line listing an observation filename and its
corresponding output filenames. Content after the directories in the output
configuration file are instructions to map model outputs with the
observation signified in the first line. Each instruction is to match one or
continuous elements in observation with elements in outputs with the same
length. A blank line means there are no further instructions. Then a new
matching between another observation and model outputs starts. An example of
output configure file is available in Appendix B.
Once MIDA finishes the execution of data assimilation, users may need basic
knowledge to assess the performance of DA. For example, the acceptance rate,
which is given by MIDA, is the fraction of proposed parameter values that is
accepted. Ideally, the acceptance rate should be about 20 %–50 % (Xu et al., 2006). A very low acceptance rate indicates that many new proposed parameter values
(Cnew) are rejected because Cnew jumps too far away from the
previously accepted parameter values (Robert and
Casella, 2013; Roberts et al., 1997). In this case, users are suggested to
reduce a jump scale in the proposing phase. On the other hand, a very high
acceptance rate is likely because Cnew moves slowly from the previously accepted parameter values. Users may increase the jump scale.
In addition, DA usually requires a convergence test to examine whether
posterior distributions from different sampling series converge or not.
A convergence test requires running DA parallelly or multiple times with
different initial parameter values. MIDA provides a Gelman–Rubin (G–R) test
(Gelman and Rubin, 1992) for this purpose. To use the G–R test,
users need to prepare a file containing initial parameters values in
different sampling series and indicate its filename in the namelist.txt
file as described in step 1. If the G–R statistics approaches 1, the
sampling series in DA is converged. When the sampling series is converged, all
accepted parameter values are used to generate the posterior distributions.
There are three types of posterior distributions: bell shape, edge hitting,
and flat. The bell-shaped posterior distributions indicate that these
parameters are well constrained. Their peak values are the maximum
likelihood estimates of parameter values. The flat posterior distributions
suggest that the parameters are not constrained due to the lack of relevant
information in data. The edge-hitting posterior distributions result from
complex reasons, such as improper prior parameter range. Users may change
the prior ranges to examine whether those posterior distributions can be improved
or examine correlations among estimated parameters.
The GUI-MIDA window includes two panels. The upper panel
is to set up a data assimilation task. Inputs can be loaded and applied to step 1 on data preparation for DA. The lower panel is to run DA as
described in step 2 and visualize the posterior distributions of parameters
in step 3.
Applications of MIDA
We applied MIDA to four groups of models, which are an ecosystem carbon
cycle model, a surrogate-based land surface model, nine phenology models,
and a dynamic vegetation model. These four cases demonstrate
that MIDA is effective for stand-alone DA, flexible to be applied to
different models, and efficient for multiple model comparison.
Case 1: independent data assimilation with DALEC
The first case study is to demonstrate that MIDA can be effective for
independent data assimilation with the data assimilation linked ecosystem
carbon (DALEC) model (Lu et al., 2017). DALEC has been used for data
assimilation in several studies (Bloom et al., 2016; Lu et al., 2017; Richardson et al., 2010; Safta et al., 2015; Williams et al., 2005). Previous studies all incorporated data assimilation
algorithms into DALEC, which requires invasive coding. This case study is
focused on reproducing the data assimilation results as in the study
by Lu et al. (2017) but with MIDA.
The version of DALEC used in this study is composed of six submodels (i.e.,
photosynthesis, phenology, autotrophic respiration, allocation, litterfall,
and decomposition) to simulate the carbon exchanges among five carbon pools
(i.e., leaf, stem, root, soil organic matter, and litter) (Ricciuto et al., 2011). There are 21 parameters in DALEC, of which 17 parameters are derived from the six
submodels and four parameters serve to initialize the carbon pools. Table 2
summarizes the names, prior ranges, and nominal values of these 21
parameters. The observation is the Harvard Forest daily net ecosystem
exchange (NEE) from the years 1992 to 2006. DALEC is coded in Fortran. In a Windows
system, a gfortran compiler converts the model code to an executable file
(i.e., DALEC.exe).
A summary of 21 parameters to be calibrated in the DALEC model. The
default parameter value and prior parameter range are shown.
ParameterDescriptionUnitDefaultRangeGDDminGrowing degree day threshold for leaf out∘C d100[10, 250]GDDmaxGrowing degree day threshold for maximum LAI∘C d200[50, 500]LAImaxSeasonal maximum leaf area index–4[2, 7]TleaffallTemperature for leaf fall∘C5[0, 10]KleafRate of leaf falld-10.1[0.03 0.95]NUEN use efficiency–7[1, 20]ResgrowthGrowth respiration fraction–0.2[0.05, 0.5]ResmBase rate for maintenance respiration×10-4µmol m-2 d-11[0.1,100]Q10mrTemperature sensitivity for maintenance respiration–2[1, 4]AstemAllocation to plant stem pool–0.7[0.1, 0.95]τrootRoot turnover time×10-4 d-15.48[1.1,27.4]τstemStem turnover time×10-5 d-15.48[1.1,27.4]Q10hrTemperature sensitivity for heterotrophic respiration–2[1, 4]τlitterBase turnover for litter×10-3µmol m-2 d-11.37[0.548,5.48]τsomBase turnover for soil organic matter×10-4µmol m-2 d-19.13[0.274,2.74]KdecompDecomposition rate×10-3 d-11[0.1,10]LMALeaf mass per areag C m-280[20, 150]XsteminitInitial value for stem C pool×103 g C5[1,15]XrootinitInitial value for root C poolg C500[100, 3000]XlitterinitInitial value for litter C poolg C600[50, 1000]XsominitInitial value for soil organic C pool×103 g C7[1, 25]
Figure 2 is the GUI window of MIDA. We first set up a DA task as described
in step 1 using the upper panel. In this application, the number of sampling
series is set as 20 000. Once users click the “choose a directory” or
“choose a file” button, a new dialog window will pop up and users are able
to choose the directory or load files interactively. As describe in step 1
on preparation of DA, the working path is where the outputs of DA and
ParameterValue.txt are saved (e.g., C:/workingPath). After the output
configuration file is loaded, the filenames of model outputs, observations,
and their variance will be displayed in the window automatically. This
application only uses a “NEE.txt” observation file. Similarly, after users
load the parameter range file (e.g., a file named “ParamRange.txt” contains
three rows which are minimum, maximum, and default values of parameters), the
content in this file is displayed as well. To replace the current parameter
range file loaded, users can simply upload another file. In this
application, the executive model file is “DALEC.exe” with a Fortran compiler
in a Windows system. Because we do not have parameter covariance information,
this input is left blank. After “save to namelist file” is clicked, a
namelist.txt file containing all the inputs will be generated in the
working path.
After the DA task setup, we load the namelist.txt file and click the “run data assimilation” button in the lower panel to trigger step 2 on execution
of DA. A new dialog will pop up to show the acceptance rate information and
notify the termination of DA. Then we will click the “generate plots” button
to visualize the posterior distributions of 21 parameters as described in
step 3.
Comparison between the simulated daily net ecosystem
exchange (NEE) by DALEC and the observed NEE at Harvard Forest from 1992 to
2006. Red circles represent modeled NEE with the optimized parameter values,
and green circles represent simulated NEE with the original parameter
values. Simulations of DALEC are substantially improved after data
assimilation in comparison with those before data assimilation.
Figure 3 shows that the simulation outputs using the optimized parameter
values from MIDA better fit with the observations than those using default
parameter values. Figure 4 depicts posterior distributions of the 21
parameters estimated from MIDA. More than half of the parameters are
constrained well with a unimodal shape. Xsteminit and
Xrootinit have a wide occupation of the prior range, indicating that the observation data do not provide useful information for them. The constrained posterior distributions in this study are similar to those from Lu et al. (2017). Note that
MCMC estimates have a large variance and a low convergence rate, especially
in high-dimensional problems; with a finite number of samples it is not
expected that two simulations would give exactly the same results.
Case 2: application of MIDA to a surrogate land surface model
This case study is to examine the applicability of MIDA to a surrogate-based
land surface model. The original model is Energy Exascale Earth System
Model: the Land Component (ELM) (Ricciuto et al., 2018). As ELM is
computationally expensive (one forward model simulation takes more than 1 d), a sparse-grid (SG) surrogate system was developed to reduce the
computational time (Lu et al., 2018). The
forcing data for the surrogate model is half-hourly meteorological
measurements at the Missouri Ozark flux site from 2006 to 2014. The observations
that were used for optimization are annual sums of net ecosystem exchange
(NEE), annual averages of total leaf area index, and latent heat fluxes from
2006 to 2010. The eight parameters selected (Table 3) are the most important
parameters for the variations in outputs (Ricciuto et al., 2018). The model is written in Python. A “pyinstaller” library packages the model code into an
executable file. The iteration number in MIDA is 20 000.
A summary of eight parameters to be calibrated in the surrogate-based
ELM model. The default parameter value and prior parameter range are shown.
ParameterDescriptionUnitDefaultRangecrootRooting depth distribution parameterm-12.0[0.5,4]SLAtopSpecific leaf area at canopy topm2 g C-10.03[0.01,0.05]NleafFraction of leaf N in RuBisCO–0.1007[0.1,0.4]CNrootFine root C : N ratio–42[25,60]Ar2lAllocation ratio of fine root to leaf–1.0[0.3,1.5]ResmBase rate for maintenance respiration×10-6µmol m-2 s-12.525[1.5,4]tleaffallCritical day length for senescence×104 s3.93[3.5,4.5]GDDonsetAccumulated growing degree days for leaf out∘C d800[600,1000]
Figure 5 shows posterior distributions of calibrated parameters. croot, SLAtop, tleaffall, and GDDonset are constrained well with a unimodal distribution. However, the distribution of the other four parameters (i.e., Nleaf, CNroot, Ar2l, and Resm) clusters near
the edge. These results match well with the study by Lu et al. (2018). As shown in Fig. 6, the calibrated parameters induce a performance improvement in simulating total leaf area index and NEE. For latent heat, both the default and optimized simulation obtain good agreement with the observation. These conclusions are also similar to those in Lu et al. (2018).
MIDA hides the detailed differences between models. For example, the DALEC model
in case 1 is a process-based model to simulate the ecosystem carbon cycle while
surrogate-based ELM in case 2 is an approximation of a land surface model.
They are also different in programming language, simulation time, forcing
data, etc. MIDA is able to deal with models with so many different
characteristics and hides these differences from users. Users only need to
indicate the filenames of the model to be used, its parameter range, the
output configuration file, etc. in the namelist.txt file. Thus, MIDA
simplified the DA applications using different models.
Comparison between posterior distributions (red line) and
default values (gray dash line) of the 21 parameters in DALEC. The peak in
posterior distribution is the constrained parameter value from the maximum
likelihood estimation. This distinctive mode and its divergence from the
default value indicates the effects of DA. Most parameters are well
constrained, and some are far different from the original values.
Comparison between posterior distributions (red line) and
default values (gray dash line) of the eight parameters in surrogate-based
ELM. The peak in posterior distribution is the constrained parameter value
from maximum likelihood estimation. This distinctive mode and its divergence
from the default value indicate the effects of DA. Most parameters are well
constrained, and some are far different from the original values.
Case 3: evaluation of multiple phenological models
This study case uses nine phenological models (Yun et al., 2017) to demonstrate the applicability of MIDA in model comparison. Five out of the nine models predict phenological events, such as the day of leaf onset, using growing degree days, which are calculated as temperature accumulation above a base temperature. The other four models consider two processes: chilling effects of cold temperature on dormancy before budburst and forcing effects of warm temperature on plant development. Each model uses different response
functions to represent chilling and forcing effects. The detailed model
descriptions and associated parameter information are in the Supplement table.
Data are from the Spruce and Peatland Responses Under Climatic and
Environmental Change experiment (SPRUCE) (Hanson et al.,
2017) located in northern Minnesota, USA. The experiment consists of
five-level whole-ecosystem warming (i.e., +0, +2.25, +4.5, +6.75,
+9 ∘C) and two-level elevated CO2 concentrations (i.e.,
+0, +500 ppm). Dates of leaf onset were observed with PhenoCam
(Richardson et al., 2018) for tree species Picea mariana and
Larix laricina. For the sake of demonstration of MIDA application, we only show DA results for Larix laricina with +9 ∘C warming treatment and +0 ppm CO2 treatment from 2016 to 2018.
MIDA was used to compare performances of the nine models in reference to the
same observations of leaf onset dates after DA. We as users changed
filenames of model executable files (i.e., PhenoModels.exe), defined
parameter ranges, and assigned the directory of working path for each model.
MIDA then estimated the optimized parameters and saved the corresponding best
simulation outputs to the working path for each of the nine models. Figure 7
shows the best simulation output of these nine models. The simulation output
of the sixth, seventh, eighth, and ninth models better fits the
observation than the other models. It demonstrates that models that consider
both chilling and heating effects can achieve good simulations of the leaf
onset dates.
Comparison between the simulated NEE, total leaf area
index, latent heat flux by surrogate-based ELM, and the observed ones at
Missouri Ozark flux site from 2006 to 2014. The blue lines indicate the
observations, and their 95 % confidence interval is in the shaded area.
The green and red lines indicate the simulations with default parameter
values and optimized values, respectively. Simulations are generally improved
after DA for all three variables.
Comparison between the simulated growth date by nine
phenology models after DA and the observed growth date for Larix laricina with
+9∘ treatment at the SPRUCE site from 2016 to 2018. The colored number
indicates different models, and shape represents different year. Overall,
models 6, 7, 8, and 9 achieve better performance after DA.
Case 4: supporting data assimilation with a dynamic vegetation
model
This case study is to examine the efficiency of MIDA to integrate remote
sensing data into a dynamic vegetation model. The model used in this study
is Biome Ecological strategy simulator (BiomeE) (Weng et al., 2019). This model simulates vegetation demographic processes with individual-based competition for light, soil water, and nutrients. Individual trees in BiomeE model are represented by cohorts of trees with similar sizes. The light competition among cohorts is based on their heights and crown areas according to the rule of perfect plasticity approximation (PPA) model (Strigul et al., 2008). Each cohort has seven pools:
leaves, roots, sapwood, heartwood, seeds, nonstructural carbon, and nitrogen.
After carbon is assimilated into plants via photosynthesis, the assimilated
carbon enters the nonstructural carbon pool and is used for plant growth
(i.e., diameter, height, crown area) and reproduction according to empirical
allomeric equations (Weng et al., 2019). In this application, two parameters to be constrained (Table 4) are annual
productivity rate and annual mortality rate of trees.
A summary of two parameters to be calibrated in the BiomE model.
The default parameter value and prior parameter range are shown.
ParameterDescriptionUnitDefaultRangeVannualAnnual productivity per unit leaf areakg C yr-1 m20.4[0.2,2]McanopyAnnual mortality rate in canopy layeryr-10.02[0.01,0.08]
Observations to be used in DA are leaf area indexes in six vertical heights
(i.e., 0–5, 6–10, 11–15, 16–20, 21–25, and 26–30 m) at the Willow Creek study site, Wisconsin, USA. The forest at the site is an upland deciduous broadleaf forest around 302 years old. The observations were from Global Ecosystem Dynamics Investigation (GEDI) acquired by a light detection and ranging (lidar) laser system, which is deployed on the International Space Station (ISS) by NASA in 2018 (Dubayah et al., 2020). The observations were first averaged from three footprints, and then leaf area indexes in the six canopy layers were standardized to be summed up as 1.
To use MIDA, we reorganized the simulation outputs to match observations as
suggested in Sect. 2.6. The BiomeE model simulates leaf areas in eight
layers (i.e., 0–5, 6–10, 11–15, 16–20, 21–25, 26–30, 31–35, and
36–40 m) from 0 to 800 years. An output configuration file was provided to
post-process model outputs of leaf area indexes in six layers to match
observations at the forest age of 302 years. These simulated leaf area
indexes in the six canopy layers were also standardized to match
standardized observations of leaf area indexes. The observations and
post-processed simulation outputs were saved to “LAI.txt” and
“simu_LAI.txt” files, respectively. The two files are used in
MIDA for data assimilation to generate posterior distributions of the two estimated
parameters as shown in Fig. 8. The optimized parameter values through
maximum likelihood estimation are different from their default values.
Figure 9 compares the simulation outputs with optimized parameters estimated
by MIDA to those with default parameter values. After DA with GEDI data in
MIDA, the simulation accuracy of leaf area index is substantially improved,
especially in middle (16–20 m) and highest (26–30 m) layers.
Discussion
This study introduced MIDA as a model-independent tool to facilitate the
application of data assimilation in ecology and biogeochemistry. The
potential user community is ecologists with limited knowledge of model
programming and technical implementation of DA algorithms. Several
model-independent DA tools have already been developed, such as DART
(Anderson et al., 2009), openDA (Ridler et al., 2014), PDAF
(Nerger and Hiller, 2013), and PEST (Doherty, 2004), mainly for applications in the research areas of hydrology, atmosphere, and remote sensing. These DA tools either use the gradient descent method, such as the Levenberg–Marquardt algorithm in PEST, or Kalman filter methods, such as EnKF in DART, openDA, and PDAF. The Levenberg–Marquardt algorithm is a local search method, for which it is hard to find a
global optimization solution for highly nonlinear models. EnKF updates state
variables and parameter values each time when observations are sequentially
assimilated, resulting in discrete values of estimated parameters. Jumps in
estimated parameter values by EnKF make it very difficult to obey mass
conservation in biogeochemical models (Gao et al., 2011). In this study, we
used the MCMC method in MIDA to generate parameter values and their
posterior distributions. MCMC is a widely used method in many DA studies
with biogeochemical models but has been applied to individual models with
invasive coding (Bloom et al., 2016; Hararuk et al., 2015; Liang et al., 2018; Luo and Schuur, 2020; Ricciuto et al., 2011). Compared to the other model-independent DA tools mentioned above, MIDA is the first tool that uses the MCMC method for DA.
Comparison between posterior distributions (red line) and
default values (gray dash line) of the two parameters in BiomeE. The peak in
posterior distribution is the constrained parameter value from maximum
likelihood estimation. This distinctive mode and its divergence from the
default value indicate the effects of DA. All parameters are well
constrained and different from their original values.
Biogeochemical models are incorporating more detailed processes related to
carbon and nitrogen cycles (Lawrence et al., 2020). Complex biogeochemical
models yield predictions with great uncertainty (Frienlingstein et al., 2009, 2014). Data assimilation has been increasingly used to estimate
parameter values against observations and reduce uncertainty in model
prediction (Luo et al., 2016; Luo and Schuur, 2020). However, current
applications of DA are almost all model dependent. This requires ecologists to
write code to integrate the DA algorithm into models. The coding practice is a
big technical challenge for ecologists with limited programming ability. The
distinct advantage of MIDA is to enable ecologists to conduct model-independent DA. MIDA streamlines workflow of the three-step procedure for DA
to enable users to conduct DA without extensive coding. Users mainly need to
provide numerical and character values for data exchanges to transfer data
(i.e., parameter values, simulation outputs, observations) between the model
and MIDA by a file named namelist.txt or by interactive inputs via a GUI
window (Fig. 2).
Comparison between the simulated leaf area index (LAI) by
BiomeE and the observed NEE at Willow Creek. Circles represent modeled NEE
with the optimized parameter values, and triangles represent simulated NEE
with the original parameter values. Simulations of LAI are substantially
improved after data assimilation in comparison with those before data
assimilation.
We tested MIDA in four cases for its applicability to ecological models. The
first case is applied to the DALEC model, which has been used in several data
assimilation studies (Bloom et al., 2016; Lu et al., 2017; Safta et al.,
2015; Williams et al., 2005). The previous DA studies all used invasive
coding to incorporate the DA algorithm into models. As demonstrated in this
study, MIDA was applied to DALEC without invasive coding but by providing
the directory to save DA results and filenames of DALEC model executable file,
parameter prior range, and output configuration files through the
namelist.txt file or interactive inputs in the first preparation step of
the workflow. Then, MIDA runs DA as a black box with DALEC before visualizing
the DA results. Next, we tested the applicability of MIDA, a surrogate-based
ELM model, and a dynamic vegetation model BiomeE. To switch the test case
from DALEC to the surrogate-based ELM model and the BiomeE model, we changed
the filenames of the model executable file, parameter prior range, and output
configuration file in the namelist.txt file for MIDA. This flexibility of
MIDA in switching models for DA makes it much easier for model comparisons.
We tested this capability of MIDA with nine phenological models to compare
alternative model structures. Similarly, MIDA enables efficient switches of
observations to be assimilated into models. Users only need to change
filenames of observations in the output configuration file. This feature of
MIDA makes it easier to utilize abundant trait databases such as TRY
(Kattge et al., 2020), FRED (Iversen et al., 2017), etc. Moreover, this feature of MIDA also helps evaluate the relative information content of different observations for constraining model parameters and prediction (Weng
and Luo, 2011). Consequently, MIDA can facilitate selection of the most
informative observations and then better guide data collections in field experiments. Ultimately, MIDA can aid ecological forecasting and help reduce
uncertainty in model predictions (Huang et al., 2018; Jiang et al., 2018).
Although MIDA helps users to get rid of model detail, users may still need
basic knowledge about the model outputs to prepare the output configuration
file which is to match model outputs to observations one by one (see Sect. 2.6). This effort of preparing the correspondence between model outputs and
observations for MIDA is not that difficult because users are reading or
writing a text file, and most model developers will provide reference to help
understand observations or model output files.
Generally, MIDA requires longer time to run DA than the embedded DA
algorithm, because MIDA calls model simulation as an external executable file
rather than a function embedded. Thus, we recommend MIDA for beginners of DA
users with models that are less complex. Besides, the current version of
MIDA only incorporates the Metropolis–Hastings sampling approach. More MCMC
methods (e.g., Hamiltonian Monte Carlo) may be incorporated into MIDA in the
future.
Conclusions
We developed MIDA to facilitate data assimilation for biogeochemical models.
Traditional DA studies require ecologists to program codes to integrate DA
algorithms into model source codes. The easy-to-use MIDA module enables
ecologists to conduct model-independent DA without extensive coding, thus
advancing the application of DA for ecological modeling and forecasting. We
demonstrated the capability of MIDA in four cases with a total of 12
ecological models. These cases showed that MIDA is easy to perform for a
variety of models and can efficiently produce accurate parameter posterior
distributions. Moreover, MIDA supports flexible usage of different models
and different observations in the DA analysis and allows a quick switch from
one model to another. This capability enables MIDA to serve as an efficient
tool for model intercomparison projects and enhancing ecological
forecasting.
Nine phenological modelsGrowing degree (GD)
The growing degree (GD) model is one of the most widespread phenological
models to simulate the date of leaf onset (D^). In this study, the
timescale is limited to daily based on observation records. The kernel of
GD is to calculate the growing degree days (GDD,
∑d=DsD^-1Δd), which is the heat
accumulation above a base temperature (Tb). For simplicity, the daily temperature (Td) can be approximated by the average of daily maximum and minimum temperatures. The heat accumulation starts at day Ds, which is empirically estimated, and ends when GDD reaches a forcing requirement threshold (Rd). Two parameters to be constrained are base temperature (Tb) and the forcing requirement (Rd). Their default values and prior range are listed in Table A1.
A1Δd=Td-TbifTd>Tb0otherwiseA2∑d=DsD^-1Δd<Rd≤∑d=DsD^Δd
Sigmoid function (SF)
Compared to the linear response function of GDD in the GD model, the sigmoid
function (SF) model provides a non-linear function to better represent the
non-linearity of the growth response to heat accumulation. Three parameters
to be constrained in DA are base temperature (Tb), the forcing
requirement (Rd), and temperature sensitivity (St). Their default values and prior range are listed in Table A1.
A3Δd=11+eStTd-TbA4∑d=DsD^-1Δd<Rd≤∑d=DsD^Δd
Beta function (BF)
In reality, the plant growth rate, as described with Δd, gradually
increases up to a specific temperature and then rapidly declines to a
supra-optimal level. Such a response can be well described by a beta function
with uni-modality and non-symmetrical shape. Three parameters are involved
in DA: minimum temperature (Tn), optimal temperature (To), and forcing requirement (Rd). The other parameter values are fixed with empirical values. For example, maximum growth rate (Rx) is set to 1, and maximum temperature (Tx) is assumed to be 45.
A5rd=RxTx-TdTx-ToTd-TnTo-TnTo-TnTx-ToA6Δd=rdifrd>00otherwiseA7∑d=DsD^-1Δd<Rd≤∑d=DsD^Δd
Days transferred to standard temperature (DTS)
According to Arrhenius law, the relationship between growth rate and daily
temperature Td can be interpolated by Eq. (A8)
(Ono and Konno, 1999). With a factor weighted
by standard temperature, the equation for DTS (Eq. A9) can better represent
growth rate dependent on temperatures. Three parameters considered in DA
are temperature sensitivity rate (Ea), standard temperature (Ts), and forcing requirement (Rd).
A8k=e-EaR⋅TdA9Δd=eEaTd-TsR⋅Td⋅TsA10∑d=DsD^-1Δd<Rd≤∑d=DsD^Δd
Thermal period fixed model (TP)
The difference between GD and TP models is that heat accumulation occurs in a
fixed time period (Dn). The day of leaf onset is the last day
(Ds^+Dn) when the accumulated heat reaches the forcing
requirement. The start day (Ds^) of heat accumulation begins on day one and moves 1 d forward each time to estimate Eq. (A12). Three
parameters are involved in DA: the base temperature (Tb), the period length (Dn), and the forcing requirement (Rd).
A11Δd=Td-TbifTd>Tb0otherwiseA12Rd≤∑d=Ds^Ds^+DnΔd
Chilling and forcing (CF)
Compared to GD, there is another distinctive chilling period for dormancy.
The CF model sequentially calculates two accumulations in opposite directions:
chilling accumulation and anti-chilling accumulation. The start day of
chilling accumulation (Ds) is implicitly set as 273.0, which is 1 October. The end day of chilling accumulation (D0) is the beginning of
anti-chilling accumulation. Three parameters are considered in DA: the
chilling requirement (RdC), the forcing requirement
(RdF), and the temperature threshold (Tc).
A13Δd=Td-TcifTd≥0-TcotherwiseA14ΔdC=ΔdifΔd<00otherwiseA15ΔdF=ΔdifΔd>00otherwiseA16∑d=DsD0-1ΔdC>RdC≥∑d=DsD0ΔdCA17∑d=D0D^-1ΔdF<RdF≤∑d=D0D^ΔdF
Sequential model (SM)
The difference between CF and SM models is that SM used a beta function (Eq. A18) for the calculation of chilling accumulation and adopted a sigmoid
function (Eq. A20) for anti-chilling accumulation. The detailed descriptions
of these two functions can be referred to in the introductions of the BF model and
CF model. The maximum temperature is empirically set as 13.7695. Six
parameters are constrained in DA: minimum temperature (Tn), optimal
temperature (To), temperature sensitivity (St), forcing base temperature (Tb), chilling requirement (RdC), and forcing requirement (RdF).
A18rd=Tx-TdTx-ToTd-TnTo-TnTo-TnTx-ToA19ΔdC=rdifrd<00otherwiseA20ΔdF=11+eSt(Td-Tb)A21∑d=DsD0-1ΔdC>RdC≥∑d=DsD0ΔdCA22∑d=D0D^-1ΔdF<RdF≤∑d=D0D^ΔdF
Parallel model (PM)
The critical difference between PM and the above two-step models is that the
chilling and anti-chilling accumulations happen simultaneously
(Fu et al., 2012). In the earlier dates during the chilling period, only a small fraction (Kd) of forcing (Eq. A25) will be accumulated. The maximum temperature is empirically set as 15.3. Seven parameters will be considered in DA: minimum temperature (Tn), optimal temperature (To), temperature sensitivity (St), forcing base temperature (Tb), chilling requirement (RdC), forcing requirement (RdF), and a forcing
weight coefficient (Km).
A23rd=Tx-TdTx-ToTd-TnTo-TnTo-TnTx-ToA24ΔdC=rdifrd<00otherwiseA25Kd=Km+(1-Km)∑i=DsdΔiCRdCif∑d=DsD0-1ΔdC>RdC1otherwiseA26ΔdF=Kd1+eSt(Td-Tb)A27∑d=DsD0-1ΔdC>RdC≥∑d=DsD0ΔdCA28∑d=D0D^-1ΔdF<RdF≤∑d=D0D^ΔdF
Alternating model (AM)
The AM fixes the start date of the chilling period (DsC) as 1 November and the start date of anti-chilling period (DsF) as 1 January. The difference between the AM and the other models above is that the forcing requirement is not a parameter value but is decided by the length of chilling days (Fu et al., 2012). Five parameters to be constrained in DA are chilling temperature (Tc), forcing base temperature (Tb), and three
coefficients (a,b,c) in calculation of the forcing requirement.
A29ΔdC=1ifTd≤Tc0otherwiseA30ΔdF=Td-TbifTd>Tb0otherwiseA31RdC=∑i=DsCdΔiCA32RdF=a+b⋅e-c⋅RdCA33∑d=DsFD^-1ΔdF<RdF≤∑d=DsFD^ΔdF
A summary of parameters to be calibrated in nine phenological
models. Their default parameter value and prior parameter range are shown.
ModelParameterDescriptionUnitDefaultRangeGDTbBase temperature∘C10[-5, 25]RdForcing requirement∘C d35[0, 200]SFTbBase temperature∘C-1.5[-10, 25]RdForcing requirement∘C50[0, 500]BFToOptimal temperature∘C15[10, 35]TnMinimum temperature∘C0[-10, 5]RdForcing requirement∘C d11[0, 50]DTSEaTemperature sensitivity rate–250[1, 1500]TsStandard temperature∘C10[-30, 40]RdForcing requirement∘C d50[1, 200]TPTbBase temperature∘C12.5[0, 30]DnPeriod lengthd25[0, 50]RdForcing requirement∘C d20[0, 150]CFRdCChilling requirement∘C d-124[-300, 0]RdFForcing requirement∘C d120[0, 300]TcChilling base temperature∘C5[0, 30]SMTnMinimum temperature∘C-20[-80, 0]ToOptimal temperature∘C0[-26, 10]StTemperature sensitivity–-1.8[-5, 0]TbForcing base temperature∘C5[-5, 35]RdCChilling requirement∘C d20[0, 80]RdFForcing requirement∘C d20[0, 80]PMTnMinimum temperature∘C-20[-80, 0]ToOptimal temperature∘C0[-26, 10]StTemperature sensitivity–-0.6[-1, 0]TbForcing base temperature∘C5[-5, 35]RdCChilling requirement∘C d11.35[0, 80]RdFForcing requirement∘C d44.01[0, 80]KmForcing weight coefficient–0.2[0, 1]AMTcChilling base temperature∘C4.6[-10, 10]TbForcing base temperature∘C5[-5, 35]aCoefficient for forcing adjustment–11.51[0.01, 15]bCoefficient for forcing adjustment–88[0, 200]cCoefficient for forcing adjustment–-0.01[-1, -10-4]An example of the output configuration file
The output configuration file (e.g., config.txt) is to indicate the directories
of observations and simulation output files as well as how they map to each
other. Figure B1 is an example of the output configuration file. There are
three blocks of functions to map simulation outputs to observed gross primary production (GPP), respiration (RE), and net ecosystem exchange (NEE). The blocks of mapping functions are separated by a blank line. Each
mapping block starts with the directories of one observation, its
observation variance, and model outputs, which are separated by a hash key.
If there is no observation variance available, users can ignore this
directory. If multiple simulation outputs are used to correspond to one
observation, the directories of simulation outputs are separated by a comma.
The rest of the mapping block describes how to map simulation outputs to
observations. The simu_map variable is simulation output
after mapping. The simuList variable saves the simulation outputs specified
in the first line. Taking the third mapping block in Fig. B1 as an example,
simuList[0] saves contents in simuNEE_1.txt, and
simuList[0][0:365] saves the first 365 elements in this file.
An example of the output configuration file.
An example of the namelist.txt file
Figure C1 shows an example of the namelist.txt for the first study case
with the DALEC model. Users need to prepare the namelist.txt before
execution of data assimilation (DA) either manually or via GUI. Below
describes the content in the namelist.txt. Detailed explanation and tutorials
are available in the Zenodo repositories at the end of the appendixes.
“workpath” is the directory where the MIDA executable files are saved. “nsimu” is
the number of iterations in execution of data assimilation.
“J_default” is the default mismatch (i.e., cost function) to
be compared in the first moving phase of data assimilation.
“ProposingStepSize” controls the jump scale in the proposing phase of data
assimilation. Users can increase or decrease this value to adjust the
acceptance rate to be in a range from 0.2 to 0.5. “paramFile” is the
directory of a csv file saving parameter-related information such as
parameter range. “obsList” saves the directories of observations. Multiple
observations are separated by semicolon. Similarly, “obsVarList” saves the
directories of observation variance in the same order as that of obsList.
“simuList” saves the directories of simulation outputs corresponding to the
observations. With GUI, MIDA reads directories in the output configuration
file (e.g., config.txt), which users provide and assign values for
obsList, obsVarList, and simuList in the namelist.txt automatically. In this case, if the directories of observations change, users only need to modify the output configuration file and generate the namelist.txt again with GUI-based MIDA.
“paramValue” is the directory of a txt file where MIDA writes out a new set of parameter values for model execution in each iteration of data assimilation.
Its default value is ParameterValue.txt under the workpath specified in
the first line of the namelist.txt. “model” saves the directory of model
executable files. “nChains_convergeTest” indicates whether to
conduct a Gelman–Rubin (G–R) convergence test or not. If the G–R test is used, its
values are the number of multiple MCMC chains. If not, its value is zero.
“convergeTest_startsFile” is the directory of a csv file
saving default parameter values as the start points in multiple MCMC chains.
“outConvergenceTest” saves the results of the G–R test. If
“nChains_ConvergeTest” is zero, both values of
“convergeTest_startsFile” and “outConvergenceTest” are empty.
“DAresultsPath” is the directory saving the results of DA whose directories
are also listed in the following six lines: “outJ” for the accepted
mismatches, “outC” for the accepted parameter values, “outRecordNum” for the number of accepted parameter values, “outBestSimu” for the best simulation outputs with the optimal parameter values, and “outBestC” for the optimal parameter values. For MIDA without GUI, “display_plot”
indicates whether or not to visualize the posterior distributions after DA.
An example of the namelist.txt file. In order to use
MIDA, users need to prepare data and a model and specify their file names
and directories in the namelist.txt file.
Code and data availability
The code of MIDA is available at the Zenodo repository
10.5281/zenodo.4762725 (Huang, 2021a). Data used in
this study are available at 10.5281/zenodo.4762779 (Huang, 2021b). A
comparison of the time cost using the embedded DA algorithm and MIDA is
available at the Zenodo repository 10.5281/zenodo.4891319 (Huang, 2021c).
Video supplement
Tutorial videos of how to use MIDA are available at
10.5281/zenodo.4762777.
Author contributions
XH, IS, and YL designed the study. XH built the workflow of MIDA and
tested its capability in four cases. DL, DMR, and PJH provided data and
models for the first and second test cases. XL prepared models, and ADR
provided observations for the third case. EW and SN helped to prepare data
and models for the fourth case. XH, LJ, EH, and YL analyzed the results. All
authors contributed to the preparation of the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support
This work was funded by subcontract 4000158404 from Oak Ridge National
Laboratory (ORNL) to the Northern Arizona University. ORNL is managed by
UT-Battelle, LLC, for the U.S. Department of Energy under contract
DE-AC05-00OR22725. Ensheng Weng is supported by the NASA Modeling, Analysis, and
Prediction Program (NNH16ZDA001N-MAP).
Review statement
This paper was edited by Hisashi Sato and reviewed by two anonymous referees.
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