We use LAI data from the Moderate Resolution Imaging Spectroradiometer (MODIS) on board the Terra platform. We use the MODIS collection 5 product MOD15A, which is available at 1 km spatial resolution and an 8-day time step (https://lpdaac.usgs.gov/). The MODIS LAI is based on a reflectance algorithm, which uses the red and near-infrared bands and includes corrections for canopy structure and background soil reflectance . In cases where this main algorithm fails, a backup algorithm is used, which is based on an empirical relationship between LAI and NDVI (normalised difference vegetation index). We use the quality assurance flags provided with this product to filter pixels that were derived using the backup algorithm or which are classified as snow covered, as described in . We use data for the globe with a spatial resolution of 1 km, which we then aggregate to the GEOS-4 base resolution of 2∘ latitude by 2.5∘ longitude, by calculating the mean of all pixels within a grid cell. All pixels classified as cropland were excluded prior to averaging. The aggregation to the coarser resolution was done both to reduce computational effort and to obtain time series without the gaps resulting from our filtering procedure. The data were split into a training (2001–2005) and an evaluation (2006) data set.

To drive the model, we use temperature and photosynthetically active radiation (PAR) data from assimilated meteorological data products of the Goddard Earth Observing System (GEOS-4) , which is available at a spatial resolution of 2∘ latitude by 2.5∘ longitude and a temporal resolution of 3 h, which we average to a 1-day temporal resolution. The soil moisture data required in the model were obtained from the NCAR/NCEP reanalysis daily average surface flux data set (http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.surfaceflux.html) , which is provided at a 1-day temporal resolution and has been regridded to the GEOS-4 spatial resolution.

We use a global PFT map, which is used in the Integrated Biosphere Simulator Model (IBIS) . This differentiates between 13 different plant functional types based on general plant properties (trees vs. grasses), temperature tolerance (tropical vs. temperate), and leaf habit (deciduous vs. evergreen). The PFT data are provided at a 1 km spatial resolution, which we re-grid at the GEOS 4 native resolution based on majority land cover in each grid cell.

We use a global-scale mechanistic phenology model , which is based on a carbon benefit approach, so that leaf gains and losses are adjusted to achieve the optimal carbon assimilation at the canopy level. The phenological timing predicted by traditional models arises implicitly by predicting LAI values. At each time step t and for each location x, the model calculates leaf gain and loss, and hence overall change in LAI as dLAI(x,t)dt=P(I0(x,t),LAI(x,t-1))-∑a=0amaxL(x,t,a), where P refers to leaf production processes, which are calculated as a function of solar radiation I0 and the LAI at the previous time step LAI(x, t-1), and L refers to leaf loss summed over all groups of leaves of the same age a (see Table  for a full list of parameters). The model allows for continuous leaf gain and loss and does not include abrupt leaf on and leaf off thresholds, thus being able to capture irregular seasonal cycles .

To describe leaf gain, we define the concept of target LAI as the optimum number of leaf layers for a given light level at the top of the canopy I0 so that the bottommost leaf layer receives sufficient light for photosynthesis, that is light at the compensation point C (Wm-2). The target is calculated using Beer's law of light extinction and expressed as LAItarg=-1αln⁡CI0, where α is the canopy extinction coefficient calculated as a function of day of year and latitude . The solar radiation at the top of the canopy I0 is averaged over a number of p days. We calculate separate values for LAItarg for direct and diffuse radiation to account for the different response of photosynthesis to the two. The overall target is then calculated as the minimum of the two values. We choose to use a direct and diffuse compensation point as a simplified representation of light distribution in the canopy. At any time step, if the existing LAI is lower than the target value, new leaves are gained to reach the target LAI. We introduce a parameter gainmax to limit the new leaves that can be added at each time step to reflect the physiological limits to building new leaves. The gain at any time t and for all locations x is then calculated as P(x,t)=gainmax,LAItarg(x,t)-LAI(x,t-1),0, LAItarg(x,t)-LAI(x,t-1)>gainmax0<LAItarg(x,t)-LAI(x,t-1)<gainmaxLAItarg(x,t)-LAI(x,t-1)<0. To account for the effects of temperature, we set a threshold of 0 ∘C mean daily temperature under which no leaves are gained. Initial parameter optimisations where this threshold is a free parameter have shown that the model is not very sensitive to its value .

Following the optimality hypothesis, leaves are lost when their carbon assimilation is less than their respiration and maintenance costs, defined as the limit assimilation value Amin. We calculate the carbon assimilation as a linear function of PAR absorbed by the canopy, Itot, per unit leaf area: Alight=ϕItot-qLAI. here ϕ and q are model parameters representing photosynthetic efficiency (µmol s-1 W-1) and canopy level light compensation point (µmol m-2 s-1). Due to the lack of data constraints for carbon assimilation in our modelling framework, we normalise assimilation values and associated parameters. As a result, parameter values for ϕ and q in the above equation do not represent measurable values in the field, but instead scale between potential minimum and maximum photosynthetic rates within the model. While the absolute values of these two parameters have been scaled, the relative distributions across the globe can still be interpreted as having physical meaning. It is worth noting that the canopy level compensation point q and the direct and diffuse compensation points used in the calculation of the target LAI have different units (µmol m-2 s-1 and Wm-2, respectively), due to the units that the original PAR data are in as well as the structure of the model.

To account for water limitation to assimilation and, implicitly, phenological processes, we introduce a factor fW calculated as fw=s1(Ws)s2ϵLAI-uϵ, where Ws is volumetric soil moisture (unit less) obtained from the NCAR/NCEP data set, s1 and s2 are parameters associated with water extraction capacity from the soil, ϵ represent potential evapotranspiration and u is plant water use.

Similarly, we define an age factor fage to describe the declining carbon assimilation of leaves as they age: fa=min⁡(1,exp⁡μ(acrit-a)), where µ is the rate of decrease with age (yr-1) after a limit age acrit (years). Using both these factors, the overall assimilation is calculated as Atot=Alightfwfa. Overall, the leaf loss at any point in time t and all locations x for any group of leaves of the same age a (cohort) is L(x,t,a)=LAI(x,t,a),Atot(t,a)<Amin0,Atot(t,a)≥Amin.

To calculate the overall canopy LAI loss, we can then sum over all age groups.

We use five different model parametrisation to explore the extent to which the PFT approach is applicable to a data-constrained phenology model. The first such model setup, previously used in , is to fit a unique parameter set to each grid cell. We will term this the local model. This approach involves a very large number of parameters (14 parameters at each grid cell, for 2041 vegetated grid cell results in a total of 28 574 parameters). It is important to note, however, that the total amount of data available from sources such as MODIS is also very large, making it possible to parametrise extremely parameter-rich models, depending on the exact nature of the data.

The second model setup is using one set of parameters for each PFT, resulting in only 182 parameters for the entire globe. We term this the PFT model. To investigate the potential effects of geographical separation, we further separate each PFT into geographical regions (e.g. North American temperate deciduous broadleaf and European temperate deciduous broadleaf), resulting in 44 regions. This was done to test the assumption that species evolving in different geographical locations have different physiological parameters even when belonging to the same PFT. This setup is referred to as the region model and has 616 parameters. As a point of reference, we also introduce a global model where parameters are common for all grid cells, under the assumption that there is no difference in phenological behaviour between vegetation types or geographical regions.

To test the extent to which each parameter represents local characteristics, in the final model setup one or more parameters are location specific while the rest have PFT-wide values. We then term a parameter local if it has a specific value at each grid cell. This setup is the combination model. As there are a very large number of possible combinations of local parameters, we perform an initial analysis to determine which parameters would most improve the model performance, if local. We performed a principal components analysis (PCA) of the spatial variation in parameter values fitted for the local model. This highlighted that the principal axis of variation in all parameter values was strongly correlated with variation in Cdirect, while the second axis was dominated by variation in acrit (Appendix Table ). We also fit 14 different model parametrisations, allowing each parameter in turn to be local, while the other parameters are fitted at the PFT level. The two parameters identified by the PCA, the light compensation point Cdirect and the leaf age limit acrit, also show an increase in model performance, especially in terms of spatial variation explained (Table ). As a consequence of these two analyses, we focussed in detail on only one model that combined local and PFT parameters in which the Cdirect and acrit parameters are local. This model has 4238 parameters for the whole globe, compared to 28 574 for the local model and 182 for the PFT.

We fitted all models to the data using a custom Markov chain Monte Carlo (MCMC) algorithm, known as the Filzbach algorithm (http://research.microsoft.com/en-us/um/cambridge/groups/science/tools/filzbach/filzbach.htm), which has been described in detail in . Filzbach utilises MCMC with the Metropolis–Hastings algorithm to estimate the joint distribution of the parameter set θ. In our study we assume no prior information about θ and therefore our implementation is reduced to estimating the θ associated with the highest probability of the observations given by the model. To do this we need to define a likelihood function that gives the probability of the data for any set of predictions from the model with a given parametrisation. For the local model this likelihood function is maximised independently at each location x and is calculated as l(Zx|θx)=∑t(x)ln⁡nLAIobs(x,t),LAIpred(x,t,θx),σx, where LAIpred(x,t,θx) is the predicted LAI at location x at time t (this depends on the model parameters θx); LAIobs(x,t) is the observed MODIS LAI at location x at time t; and nLAIobs(x,t),LAIpred(x,t,θx),σx denotes the probability density for observing LAIobs(x,t) given a normal distribution with mean LAIpred(x,t,θx) and standard deviation σx, which expresses the magnitude of unexplained variation in LAI. The likelihood is calculated as a sum over all time steps at location x, expressed as t(x).

For the global, regional and PFT models, the likelihood estimation is carried out at the global, regional or PFT level, the likelihood being calculated as the sum at all locations x within a group G, x(G): lZG|θG=∑x(G)∑t(x)ln⁡nLAIobs(x,t),LAIpred(x,t,θG),σG, where ZG and θG denote observed LAI and model parameters for a given group of grid cells G. Within the combination model, the likelihood is again minimised for a whole PFT, but in addition to the PFT level parameters θG, the predicted LAI is also a function of local parameters θB,x. For all model parametrisations, we use years 2001–2005 as training data and 2006 for evaluation purposes. The model was run at a daily time step, but the likelihood was only computed when MODIS data were available, with a time step of 8 days.

Without separating training and test data in this way, the more parameter-rich models would be guaranteed to give a better fit to the data. Separating the training and test improves our ability to assess model performance, although, given that the training test data are separated by a relatively short time and not separated in space, we expect a tendency for the more parameter-rich models to provide superior performance against the test data.

To compare the different types of models described above, we define several model performance metrics against the test data. The best model should be able to capture both the timing and magnitude of the seasonal cycle at each location and the spatial variability in seasonal cycles across the globe. As an overall measure of fit we use the root mean squared error (RMSE) normalised by the mean LAI, which is a measure of the fit at each particular location. The mean LAI and LAI amplitude describe the magnitude of the seasonal cycle and we use the percent of variation explained to capture the extent to which the model describes their spatial distribution. Similarly, we use the start and end of the growing season to describe the timing. We define the start of the growing season as the first date of the year when the LAI reaches 0.2 of the maximum LAI, while the end of the growing season is the equivalent last date. To capture the timing in tropical areas with a less pronounced seasonal cycle, we also use the timing of maximum LAI. All metrics are reported for the model evaluation period (2006).

We choose not to use statistical information criteria (e.g. Bayesian information criteria) because our model fitting methodology does not easily allow for the computation of a single likelihood metric. The model structure is the same for all parametrisations, with the main model differences being the number of parameters at each grid cell. However, this means that different quantities of data are also used to fit different models. For example, since the local model is fitted separately at each location, it effectively consists of 2041 separate models, each with 14 parameters, while the PFT model contains 13 models each with 14 parameters. Rather than work out a global information criterion-based metric for the models, we instead opt to use the more meaningful metrics of the relationships between the model predictions and the data described

The most straightforward biological explanation for the observed results of the combined model is that the two local parameters – the light compensation point and leaf age limit – are location-specific plant traits that vary within a PFT sufficiently to affect model performance. Previous studies, which have included traits as a replacement for the PFT concept, have done so starting from biological principles, either based on trait databases or by evolving traits through plant competition . In contrast, in the current study we include no prior knowledge of which parameters correspond to known plant traits and the local parameters in the combination model are inferred from the fitted model. The question that arises is if the resulting parameters and parameter values provide any biological insights or if this is just a mathematical artefact, resulting either from the data used or the model structure.

The light compensation point is not a trait commonly used in models or included in trait data, but it is closely related to leaf photosynthetic parameters, such as Vcmax and Jmax, and could easily be derived in terms of these if our model included a biochemical description of photosynthesis. There is one other parameter in our model, the photosynthetic efficiency, ϕ, that is perhaps closer to the commonly used traits but did not emerge as the most important parameter in the PCA (Table ) or was able to explain the spatial variability in LAI (Table ). In contrast to the compensation point parameter, which drives leaf gain across the globe, ϕ mainly determines leaf loss in temperate and boreal regions, which are light and temperature limited . This result shows that leaf loss within a given PFT across temperate and boreal forests can be predicted well from environmental factors alone, without any inherent trait variation within a PFT. This could result from the real trait variation being low or the real trait variation having such a strong correlation with environmental factors that the effects of the trait variation cannot be separated from the effects of the environment. More ground measurements are needed to help confirm this. The model also contains a parameter for the diffuse light compensation point, Cdiffuse, introduced as a simple way of representing light in the canopy. This parameter is expressed relative to the Cdirect value at each location even though the parameter acts as a PFT level parameter, the absolute value for diffuse compensation point will vary spatially.

While the light compensation point is not a common parameter in vegetation models, measured values can be obtained from light response curves measured for individual plant species. Reported values range from 0.5 to 16.2 Wm-2 for tree species, varying with species and light environment . The compensation point values for the combined model agree broadly with these values (Fig. ), with the exception of values for boreal forests, where values can be much higher (up to 60 Wm-2), an error that we attribute to the poor quality of MODIS data in this region.

Leaf longevity is one of the main parameters used in vegetation models, which employ plant traits e.g. as well as in the analysis of the leaf trait spectrum . The second local parameter used in the combination model, the leaf age limit agecrit, does not have the same meaning as leaf lifespan, mainly because leaf ageing is only one of the three leaf loss mechanisms in our model, alongside temperature and light and water limitation. Thus, in regions where leaf loss is not age limited, for example in temperate areas, the parameter is poorly constrained and its age value is never reached, as leaf loss occurs much sooner. In wet tropical systems where leaf ageing is the main driver of leaf loss, this age parameter is the critical age where leaves start ageing; therefore, the effective lifespan can be much larger. However, according to our model, it is the main driver of leaf loss in tropical systems and thus a proxy for determining leaf lifespan seefor a detailed discussion of the physical interpretation of this parameter.

Our results show that allowing two critical traits to vary within a PFT among locations provides a superior model performance. It is likely that such traits vary due to underlying factors that are not explicitly included in our model. Two likely candidates for such hidden factors are nutrient availability and canopy structure. If the effects of these factors on traits could be understood and modelled explicitly, this could dramatically reduce the number of parameters required by the model, without making the assumption that the traits are constant within any PFT.

Leaf photosynthetic capacity is a function of leaf nitrogen content , a factor that has not been included in our model. According to current photosynthetic models, a higher leaf nitrogen content would lead to a higher light limited photosynthetic rate and hence a lower compensation point. Figure  shows the spatial distribution of the compensation point parameter as fitted in the combination model. The highest values are observed in grasslands, especially in the tropical region. In forest PFTs, the highest compensation point occurs over tropical forests, followed by temperate deciduous regions. This is supported by field studies, as higher latitude forests are generally more nitrogen limited while tropical and temperate grasslands are one of the most nutrient poor systems in terms of phosphorus . To explore the intra-PFT distribution of nitrogen availability and fully explain the locality of our compensation point parameter, we would need either a global data set of nitrogen availability such as the nutrient limitation index derived as a function of evapotranspiration and ecosystem production or coupling the phenology model with a full-scale vegetation model with an explicit representation of the nitrogen cycle e.g..

Canopy structure determines the light environment in the canopy and controls the actual amount of light that reaches the leaves for a given light intensity above the canopy. This means it can be of important value in determining the compensation point, both through model structure and long-term impact on plant behaviour. Within the model used in this study, we assume a homogeneous canopy, with a random distribution of leaf angles and no clumping, assumptions that can be considered valid at very large scales, but can potentially introduce errors for certain forest structure types. It has been shown that including leaf clumping in a carbon assimilation model has a major impact on resulting global gross primary productivity values. A leaf clumping factor would be used to adjust the attenuation coefficient α (Eq. ) to improve the description of light transmission through the canopy. It is possible that the compensation point parameter Cdirect artificially accounts for this variation in canopy structure, which explains its observed spatial variability. Further information such as the leaf clumping index map developed by would be needed to distinguish between the actual compensation point and canopy structure. This relationship is further complicated by the fact that plants adapt to their light environment; therefore, leaves in closed canopies will be better adapted to shaded conditions and will have lower compensation points and thus tropical forests, which are highly stratified, will have a much lower compensation point than other systems. The question is further complicated as canopy structure itself can be an adaptation to the available resources, such as light, water, or nitrogen, making it difficult to distinguish between all possible factors in the absence of further data.

One of the main possible sources of error in our conclusion is the way we have parametrised the PFT model. In most models, which use the PFT concept, grid cells are represented as a mix of PFTs, with PFT-specific parameters assigned to each fraction e.g., while we have chosen, in order to reduce the computational effort necessary for a global data-constrained model, to only use the dominant PFT in each grid cell. This approach, together with the low resolution that the model is run at could mean that the poor fit shown by the PFT model is due to a poor representation of PFTs rather than the unsuitability of the concept in vegetation models. If this was the case, we would expect high model errors in grid cells with a larger mix of vegetation types. However, the high errors in the PFT model are consistent throughout and do not show a significant correlation with the grid cell PFT heterogeneity (Fig. ), indicating that the mix of vegetation types within grid cells cannot be the only explanatory factor. For a more robust conclusion, we would need to re-run the analysis with either a higher spatial resolution or with a PFT mix in each grid cell. Given the high computational demand of the fitting procedure, this would ideally be done for a smaller number of PFTs rather than at global scales.

We use space borne vegetation data from the MODIS Terra sensor, as satellite measurements are one of the only sources of data for constraining global level vegetation models, although they suffer from instrument error and atmospheric contamination. We have attempted to filter the data robustly using data quality flags, as discussed in Sect.  and previous studies and the fitting procedure contains a parameter σx, which accounts for the error in the observations (Sect. ). The largest possible source of error is the seasonality shown by the MODIS data in the Amazon basin and other tropical regions, which is most likely to determine the spatial distribution of the agecrit parameter. Initial studies have shown that there is an increase in satellite observed LAI during the dry season over the Amazon , but more recent studies have argued that the observed change in LAI is due to sun-sensor geometry . This finding has been contradicted by subsequent papers and we do not attempt to present an answer to this debate. For the purpose of this study, we assume that this observed change in LAI is a reflection of actual changes in leaf cover, an assumption backed by observed changes in gross primary productivity and litterfall .

As more studies begin to acknowledge that the PFT concept is not necessarily the best approach to vegetation modelling, we need to quantify the extent to which the inclusion of spatially distributed parameters or plant traits improve our predictive capability and to identify the optimal number of parameters that both give a good model fit and minimise computational cost. In this study we have attempted not only to build a model with locally distributed parameters but also to quantify the extent to which a model with local parameters and one with PFT level parameters can capture the spatial variability in global LAI observations. Furthermore, we quantitatively identified which parameters need to be local to improve model performance with a view to reduce data and computational needs. We believe that the method used here for investigating the use of PFT level parameters has a high degree of generality and can be applied to a large variety of models and input data sets.

One of the advantages of the PFT concept is its capacity to represent future changes in vegetation distribution within DGVMs. Given a predicted change in climate, models using PFTs can then predict a change in PFT distribution, using either predefined climate envelopes or predefined plant competition rules . Models that instead use plant traits do not offer such a straightforward solution, but have a number of advantages. The PFT approach only allows abrupt changes in vegetation and cannot capture any plant adaptation to climate, whereas a trait approach can represent gradual changes; however, representing changing traits in response to climate is more difficult, both conceptually and computationally. Recent studies have proposed the use of plant competition and emergent traits to predict vegetation distribution . Therefore, a logical next step to our analysis would be to identify the environmental responses of the two combined parameters and their relationships with plant physiology responses, so parameter values can be estimated independent of data constraints for the purpose of model predictions under future climate change.