We calibrated the JSBACH model with six different stomatal conductance formulations using measurements from 10 FLUXNET coniferous evergreen sites in the boreal zone. The parameter posterior distributions were generated by the adaptive population importance sampler (APIS); then the optimal values were estimated by a simple stochastic optimisation algorithm. The model was constrained with in situ observations of evapotranspiration (ET) and gross primary production (GPP). We identified the key parameters in the calibration process. These parameters control the soil moisture stress function and the overall rate of carbon fixation.

The JSBACH model was also modified to use a delayed effect of temperature for photosynthetic activity in spring. This modification enabled the model to correctly reproduce the springtime increase in GPP for all conifer sites used in this study. Overall, the calibration and model modifications improved the coefficient of determination and the model bias for GPP with all stomatal conductance formulations. However, only the coefficient of determination was clearly improved for ET. The optimisation resulted in best performance by the Bethy, Ball–Berry, and the Friend and Kiang stomatal conductance models.

We also optimised the model during a drought event at a Finnish Scots pine forest site. This optimisation improved the model behaviour but resulted in significant changes to the parameter values except for the unified stomatal optimisation model (USO). Interestingly, the USO demonstrated the best performance during this event.

Plants exchange carbon dioxide (

Soil water deficit and high water vapour pressure deficit can result in suppressed plant transpiration

Ecosystem and land surface models, describing the plant photosynthesis, transpiration and soil-hydrology-related processes, usually include descriptions and parameterisations for various stress effects. These parameters often lack a theoretical foundation

Stomatal conductance models describe the pathway of

In many other studies, where the aim has been to optimise land surface model parameters, the optimisation is based on estimating the gradient of the cost function:

APIS (adaptive population importance sampler) is a Monte Carlo (MC) method that can be run iteratively as presented by

In this study we apply the land surface model JSBACH for 10 boreal coniferous evergreen forest eddy covariance sites to examine the performance of different stomatal conductance models, and their effect on calibrated parameters related to photosynthesis, phenology and hydrology. First, we utilise APIS to sample the full parameter space with the different stomatal conductance formulations and to locate different modes of the target distributions (peaks of high probability). Second, using the distributions generated by APIS as the prior distributions, we optimise the parameters using a simple stochastic optimisation method. Finally, we assess the inter-site variability and the robustness of the calibrated parameters together with different stomatal conductance formulations. Optimised parameters for a specific drought are also investigated and compared with the parameters for the general optimisation.

We will next introduce the measurement sites, followed by the model and modifications made to it. Afterwards we will give a general overview of the simulations as well as the sampling process, the algorithms and methods used to analyse the results.

We use data from 10 FLUXNET

Based on the quality and quantity of their respective measurements, the sites were divided into calibration and validation sites. Essentially, if we have enough data from a site, it is used for both calibration and validation purposes. We required the site to have at least 8 years of measurements, where the first five were used for calibration, and the consecutive three for validation. Otherwise we used the site only for a 3-year validation. The FLUXNET datasets were missing both the long- and shortwave radiation for the two Russian sites – Fyodorovskoye (RU-Fyo) and Zotino (RU-Zot). These were generated from ERA Interim data. The soil types of all of these sites can mostly be identified as mineral soils with varying sand, clay and peat contents. Fyodorovskoye and Poker Flat (US-Prr) are natural peatlands, and Lettosuo (FI-Let) is a drained peatland site.

The measurement error in the EC flux data was separated into systematic and random errors. The main systematic errors (density fluctuations, high-frequency losses, calibration issues) were taken into account as part of the post-processing of the data, and the random errors tend to dominate the uncertainty of the instantaneous fluxes. The random error is often assumed to be Gaussian but can be more accurately approximated by a symmetric exponential distribution

Descriptions for the sites used in this study sorted by their FLUXNET identifier. The first six sites are used for both calibration and validation purposes, with the first 5 years of each site used for calibration. The last 3 years as well as the last four sites are used for validation only. The reported elevation is in metres above sea level, LAI is the one-sided leaf area index, and the average stand age is in years, along with average annual precipitation (

JSBACH

We focus only on the most essential parts of JSBACH relating to our work. A more complete model description with details on, e.g., soil heat transfer, water balance and coupling to the atmosphere can be found in

In JSBACH, the land surface is divided into grid cells, which are split into bare soil and vegetative areas. The vegetative area is further divided into tiles representing the most prevalent vegetation classes, called plant functional types (PFTs)

The predictions of phenology are produced by the Logistic Growth Phenology (LoGro-P) sub-model in JSBACH

Parameters detailing site-specific soil properties, such as soil porosity and field capacity, were derived from FLUXNET datasets and the references in Table

All parameters of interest, presented in Table

Descriptions of model parameters with default values, range of acceptable values and references to equations in the paper or in the appendices. Parameters in the same group were calibrated simultaneously.

The start of the growing season in the JSBACH model is defined by a “spring event” in LoGro-P (Appendix

However, coniferous evergreen trees do not shed all of their leaves for winter and the existing foliage enables them to quickly initiate photosynthesis in the following spring. The start of the photosynthetically active season in the model has been observed to occur too early in the boreal region by, e.g.,

The state of acclimation (

The JSBACH model was also modified to include altogether six different stomatal conductance formulations following

We have also included two additional parameters (

All of the stomatal conductance models contain an empirical water stress factor

In JSBACH, the stomatal conductance (

The water stress factor (

Stomatal conductance models with default values and range for

The

The site-level measurements, used as model inputs, are air temperature, air pressure, precipitation, humidity, wind speed and

The initial state of the JSBACH model can be generated from predefined values of state variables (usually empty initial storage pools) or the model can be restarted from a file describing the state of some previous run. Depending on the area of interest, a model spin-up may be required to bring the model into a steady state. In our simulations, some of the more slowly changing variables (e.g. soil water content and LAI) need to be equilibrated, so a spin-up is required. This can be achieved by running the model over a set of measurements multiple times, each time restarting from the final state of the previous run.

The calibration period consists of the first 5 years given for the calibration sites in Table

During the summer 2006, the Hyytiälä (FI-Hyy) measurement site suffered from a severe drought

We describe the modelling setup with the equation

Using Bayes' rule on conditional probability we can write the parameter posterior density (

Above

Above

The initialisation of a multichain MCMC sampler and APIS are very similar. In our simulations, APIS is set up as 40 simultaneous and independent importance samplers. This is similar to an independent 40-chain MCMC sampler. Each sampler or chain has a random starting location drawn from a uniform distribution defined by the parameter ranges, given in Table

In an MCMC setup, the model would be run once (for each chain) and evaluated and then the draw (parameter values) accepted or rejected accordingly. In APIS, instead of a single element (one run) we use a sample size of 50. This means that we draw 50 elements with each IS sampler (or “chain”) independently. These draws are then evaluated and reweighted as presented in Eq. (

The 50 reweighted draws (for each IS sampler separately) are used to calculate a new location for the sampling distribution. This location is automatically accepted (no rejection criteria), and we also adapt the shape of the distribution using the self-normalising estimator by

Additionally, all of the draws in APIS are used to calculate “global” estimates of the parameter expected values. This process utilises the deterministic mixture approach

MCMC chains track the evolution of single elements and occasionally adjust the sampling distribution. The sample size in APIS is larger (it is not a Markov chain method) and the focus is on the evolution of the locations of the sampling distributions, not on the individually drawn elements. These location parameters are expected to be around all the modes of the target and the deterministic mixture ensures the stability of the estimation of the (global) parameter expected values. As an importance sampler, APIS is also a variance reducing method.

Before taking a more detailed look at APIS, we make some further notes about the sampling process. The first element of the 50 draws (item 2 in the list above) is always fixed as the current mean. We run the spin-up (Sect.

Normally, only the location parameters of the IS proposals are adapted, but we also adapt the shape parameters using the self-normalising estimators by

In our simulations, APIS is formed of

The simple IS estimators alone are rarely sufficient if the target is even slightly complicated. One classical way of tackling this problem is to join multiple IS estimators together. The simplest approach is to calculate the weights for each of these estimators separately and to normalise the result by the combined sum of all weights. However, this leaves the estimators susceptible to “bad” proposals. APIS suppresses the bad proposals by utilising the deterministic mixture approach

The parameter expectation values and the normalising constant in Eq. (

The APIS algorithm is a rather robust method meant for examining the full target probability distribution and locating the modes of the target distribution. Adaptation in APIS utilises multiple draws simultaneously, which can easily lead to few parameters controlling this process (the marginal density of one or few parameters dominates the calculations). Since we also did not run the model spin-up for all drawn samples (although the discrepancies should be minimal), we utilise a simple custom stochastic optimiser to locate the optimal set of parameter values. This optimiser is run after the APIS calibration simulations and separately for the drought period. The optimiser utilises the exact same datasets (calibration, validation, observations, etc.) as APIS, the spin-up is generated for all drawn samples separately and the initial state of the algorithm is the mean value of the APIS final configuration (location parameters).

Our optimiser is a simple random sampler amplified by the “velocity” of the last jump (the idea is similar to Hamiltonian or Hybrid Monte Carlo by

The covariance matrix of the proposal distribution is recalculated at predefined intervals (for all parameters). Additionally, we utilise a subset sampling procedure, where the samples are first drawn from the full parameter space; in the next step they are drawn only from group I in Table

Examples of the evolution of the APIS algorithm from the Bethy calibration. The left panel is the kernel density estimate of the location parameters at the start of the process (black), after 20 iterations (blue) and after 100 iterations (green). The right panel shows the location parameters (grey), their mean (red) and 1 standard deviation (dashed) as well as the global estimate (yellow, calculated with the deterministic mixture approach) of the parameter expected value.

Even though APIS is not a Markov chain method, we can (naively) interpret the evolution of the location parameters of each IS sampler as chains. The resulting 40 chains have random starting positions, but they are relatively short (we present results from the Bethy calibration, where the chains were adjusted 100 times); hence we did not discard any of the samples. We test the convergence of these chains with the Gelman–Rubin diagnostic tests

In order to visualise the results, we have utilised a Gaussian kernel density estimation (KDE) to produce distributions from the APIS simulation location parameters. In practice, KDE places a Gaussian distribution centred at each sample, and the constructed composite distribution is an estimate of the underlying actual distribution. The bandwidth for the distributions is calculated using Scott's rule

The effectiveness of each parameter was calculated from the final state of each optimisation process. This was done by first setting all parameters to their optimised values. Then we (evenly) sampled each parameter separately from their range of acceptable values, given in Table

We report the slope of the regression line

The Bayesian framework requires a likelihood function that optimally combines pointwise model and observational errors. The JSBACH model error is unknown as is the (pointwise) observation error. We could use a general type of error estimate (such as that of 20 % of the flux value) for the observations but would have to include a minimal site and instrumentation-dependent precision. In this study, the full error is treated as Gaussian white noise. Because of these limitations, we are calling and defining our likelihood as a cost function. It is calculated with the same parameter values for each site, using site-specific daily measurements with the gap-filled, low-quality and winter (between the 315th and the 75th day of the year) values removed (resulting in

The cost function (Eq.

We also use a modified version of this cost function, where the NMSEs are weighted by factors based on coefficients of determination (

First we present the performance of the APIS algorithm and the parameters themselves, followed by site and stomatal conductance model-specific results and finally an examination of the Hyytiälä drought event in 2006. For simplicity, we use the name of the stomatal conductance model to refer to the JSBACH model utilising that stomatal conductance formulation.

The evolution of the APIS algorithmic process is presented in Fig.

We also report the results of the Gelman–Rubin

Parameter scale reduction

The results of the optimisation process are gathered in Table

Parameter default and optimised values for the calibration period with corresponding cost function value. The values written in boldface were the most effective and the italic values the least effective for the given experiment. Also presented are the fixed parameter values for the drought period optimisation, with “opt” referring to the use of the corresponding optimised value from this table.

Some of the parameters have converged to their limiting values, which can reflect deficiencies in the model structure or the preset parameter ranges. Convergence to the boundary can also be a problem in model calibration, but in this experiment, the algorithms were able to cope with the situation as APIS located the area of high probability and the optimiser located the maxima. The different parameter effectiveness levels reported in Table

We present the average annual cycles for the validation period and for all sites in Fig.

Validation period average annual cycles of evapotranspiration and gross primary production; observations (black) and the model using the Bethy stomatal conductance formulation with default (green) and optimised (blue) parameterisation. Also presented are daily model values cross plotted against observations with corresponding slope of the regression line (

Slope of the regression line (

The optimisation has improved the model bias and the correlation coefficients for the GPP in Fig.

The resulting parameter values, from the optimisation during the drought conditions in Hyytiälä (FI-Hyy) in the summer of 2006, are presented in Table

We can now compare the parameter values in Table

The changes these different parameterisations have on the model output are visualised in Fig.

The Bethy and the USO models demonstrate the most variability in the

Optimised parameter and corresponding cost function values with different stomatal conductance formulations for the extended dry period.

Hyytiälä site drought in summer 2006. The time series for evapotranspiration and gross primary production are 5 d running averages and for

Hyytiälä site water use efficiency for the Bethy and USO formulations. Scatter-plotted are the dry-period 5 d running averages of ET and GPP, coloured by the intensity of the drought (

Finally, we used both optimised parameter sets (Tables

We will first discuss the validity of our approach and the simulation setup, followed by an examination of the success of the modifications made to the model, and close with some further remarks on the parameter values.

Before we calibrated the model, we fixed the limiting value for LAI and adjusted the site-specific vegetative area fractions to reproduce the measured site-level maximum of LAI. In the simulations, we focused on boreal coniferous forests, where light penetration is deep and the light conditions are homogenous – consequently we could assume a homogenous leaf distribution. Furthermore, the JSBACH model takes into account leaf clumping, and we can assume the leaf orientation and shape to be similar throughout the study sites. Therefore, we argue that reproducing the site-level maximum of LAI is appropriate approach in this study. Together with parameter calibration, it has resulted in improved ET and GPP fluxes as can be verified from the

We encountered difficulties in reproducing the fluxes for the validation sites with low LAI (i.e. RU-Zot and US-Prr). This can be a consequence of the area scaling as the adjustment linearly changes the proportions between vegetative area and bare soil. Another reason is the lack of the site understorey in these simulations. For example, approximately half of the

There were no clear differences between sites dominated by pine or spruce. Neither did we notice any particular effect on the bias, NMSE or correlation coefficient that could be explained by geographical location, stand age,s or annual precipitation or temperature. We optimised the model for individual (calibration) sites as well (not shown). Mostly this changed the values of parameters (such as

The APIS performance tests (Gelman–Rubin and

We modified the JSBACH model by introducing the delayed effect of temperature for photosynthesis to restrain the respiration and photosynthesis of conifers in spring. The effect of this (delayed increase in GPP) is apparent in the annual GPP cycles of CA-Qfo, FI-Hyy, FI-Ken, FI-Sod and RU-Zot in Fig.

We examined the model behaviour with six stomatal conductance formulations, and the resulting

The model behaviour was also examined during the Hyytiälä drought of 2006. Some of the parameter values were kept fixed during these simulations; most of the fixed parameters should not affect the drought period calibration, but there are exceptions, such as the maximum carboxylation rate

The stomatal conductance function (

In general, the site-level estimates of (

Some of the parameters in this study have been calibrated before by, e.g.,

The exponential scaling factor

The values of soil water parameters are closely grouped in the optimisations except for the values of

APIS is a recent method, capable of estimating complicated multidimensional probability distributions using a population of different proposal densities. The algorithm was able to produce reasonably stable estimates for most parameters quickly. Prior to calibrating the model, we adjusted the site-specific vegetative area fractions to reproduce the measured site-level maximum of LAI. This practical approach resulted in improved ET and GPP fluxes, although we encountered difficulties in replicating these for sites with low LAI. The model parameters were optimised simultaneously for all sites without any additional site-level tuning. The parameters that were most effective in the optimisation processes were consistent for all stomatal conductance formulations.

The introduction of the

The optimisation improved the predictive skill of the model with all stomatal conductance formulations as was seen during the validation period. The Bethy, Ball–Berry, and Friend and Kiang versions were the most in agreement with the observations, although the differences between these and the other formulations were small. Most of the model versions had some problems during the extended dry period, and the best

The data required to calibrate and validate the model are originally part of the FLUXNET2015 dataset that can be accessed through the FLUXNET database (

In this Appendix we present the most relevant equations that are governed by the parameters in Table

The Farquhar model

Oxygenation of the Rubisco molecule reduces the carboxylation rate, which is given as

Here

Likewise, the light-limited assimilation rate can be expressed as a function on electron transport rate (

In JSBACH the soil water budget is based on several reservoirs (e.g. skin, soil, bare soil, rain intercepted by canopy), and the different formulations are plentiful. We present here only the most crucial of these. Changes in volumetric soil water (

The interception parameter (

Evaporation from wet surfaces (

Transpiration from vegetation (

Evaporation from dry bare soil (

The total evapotranspiration is a weighted average of

The parameters from the LoGro-P are mainly used to determine the spring and autumn events for JSBACH. To determine the date of the spring event, we first introduce a few additional variables, namely the heatsum

Heatsum

The number of chill days is calculated as the number of days when the mean temperature is below

The critical heatsum (

The autumn event requires the definition of one more variable, the (pseudo) soil temperature (

In this Appendix we present the stomatal conductance model formulations used in this study. In the original JSBACH formulation, the Baseline model

After accounting for soil water stress, the net assimilation rate (

In the Bethy approach

The potential (unstressed) transpiration rate (

The Ball–Berry variants relate the stomatal conductance (

The supplement related to this article is available online at:

The experiments were planned by JM, TA, TM and TT. JM ran the simulations and prepared the paper with contributions from co-authors. JK originally implemented the Ball–Berry type stomatal conductance formulations into JSBACH under SZ's supervision. JS maintained the framework for testing the algorithm. MA, AB, MH, AL, IM, HM and HK provided the site-level observations required in this study. TA, TM and TV extensively commented and revised the document.

The authors declare that they have no conflicts of interest.

This work used eddy covariance data acquired and shared by the FLUXNET community, including these networks: AmeriFlux, AfriFlux, AsiaFlux, CarboAfrica, CarboEuropeIP, CarboItaly, CarboMont, ChinaFlux, Fluxnet-Canada, GreenGrass, ICOS, KoFlux, LBA, NECC, OzFlux-TERN, TCOS-Siberia and USCCC. The FLUXNET eddy covariance data processing and harmonisation was carried out by the ICOS Ecosystem Thematic Center, AmeriFlux Management Project and Fluxdata project of FLUXNET, with the support of CDIAC, and the OzFlux, ChinaFlux and AsiaFlux offices.

This research has been supported by the Jenny ja Antti Wihurin Rahasto, the NordForsk (grant no. 57001), the Academy of Finland (grant nos. 295874 and 307331), the ICOS-Finland (grant no. 281255) and the EU-Life+ (grant no. LIFE12 ENV/FI000409).

This paper was edited by Hisashi Sato and reviewed by three anonymous referees.