The SEIB-DGVM simulates establishment, growth, and decay of the individuals of prescribed PFTs within a spatially explicit virtual forest (Sato et al., 2007), forced by climate conditions such as air temperature, soil temperature, cloudiness, precipitation, humidity, and winds. We used version 2.71 (Sato and Ise, 2012) but with minimal modifications for DA. The model simulates daily states, but the original model outputs were only once per year. Outputs are needed for DA once every 4 days; thus, we modified the model code to output the model states every 4 days. In addition, the original model code assumed running for many years continuously, and the initial seed for the random number generator was fixed. As a result, in this study, we stop the model every 4 days, and the same seed is repeated every time when we start the model. Therefore, we modified the model code to randomly generate the seed for the random number generator every time when we initiate the model. Other modifications are summarized in Appendix A.

The size of the model state space is determined by the prognostic variables for tree, grass, forest as a whole, and soil. Each individual tree has 13 prognostic variables such as biomass of root, leaf and trunk, and we assume that up to 300 trees can exist in the forest area. Therefore, the number of tree variables is less than or equal to 3900 (i.e., 300×13). As for grass, the forest area is divided into 30 by 30 grid cells, and each grid cell has four variables such as biomass of root and leaf. Hence, the number of grass variables is fixed at 3600 (i.e., 30×30×4). In addition, forest as a whole has eight prognostic variables such as snow and soil carbon mass, and finally, soil moisture (one variable) is defined for 30 soil layers. Therefore, the number of state variables is between 3638 (no tree, i.e., 0 + 3600 + 8 + 30) and 7538 (300 trees, i.e., 3900 + 3600 + 8 + 30).

Among the various model outputs ranging from individual tree height to soil water content (Sato et al., 2007, with updated information available from the package of version 2.71), we focus on LAI because it is the key to the vegetation model, and because previous studies show a promise in assimilating satellite-based LAI data with a static vegetation model (Stöckli et al., 2011) and a “non-individual-based” DGVM (Demarty et al., 2007). We extend the previous studies to assimilate the LAI data with the individual-based DGVM.

Individual-based DGVMs include highly non-linear processes such as occasional establishment and death of individual plants. These processes produce and eliminate state variables, and the phase space changes time to time. DA methods that have been used in geophysical applications usually assume that the state variables are defined uniquely for the given dynamical system and that the phase space dimension stays the same. The widely used ensemble Kalman filter, for example, finds the best linear combination of the ensemble with optimal fit to the observations, but it is not trivial to define a linear combination or even the ensemble mean for the variables missing in some ensemble members. Therefore, it would not be trivial to apply the widely used DA methods to individual-based DGVMs.

Alternatively, particle filters run independent parallel simulations or particles and represent the probability density function (PDF) explicitly by assigning probability to each particle. Therefore, particle filters can handle non-Gaussianity and non-linearity explicitly, and can be applied to the individual-based DGVMs in a straightforward manner (e.g., Garreta et al., 2010) even though the phase space dimension is different for each particle.

Here, we adopt a particle filter approach known as sequential importance resampling (SIR; Fig. 1) (Gordon et al., 1993). Although the method is not efficient for large dimensional systems (e.g., Bickel et al., 2008; Snyder et al., 2008, 2015; Snyder, 2012), we tested this well-known method as the first attempt to construct the DA system with SEIB-DGVM. First, n parallel simulations are performed, and each simulation is considered as a particle representing the true state of the system with equal probability. Next, likelihood lt(i) is calculated for each particle using the Gaussian likelihood function:

lt(i)=pyt|xt|t-1(i)=12π⋅σ2exp-(yt-xt|t-1(i))22⋅σ2fori=1,…,n. Here, xt|t-1(i) denotes the simulated LAI of the ith particle at time t from the previous time step t-1, yt the observed LAI at time t, and σ the observation error standard deviation. Since the prior probability is uniform, Bayes' rule gives that the posterior probability of the ith particle is proportional to lt(i), i.e., the particles closer to the observation have more probability. Next, we resample the particles, so that each particle has equal probability. The particles with more probability (larger lt(i)) are duplicated, and the particles with less probability (smaller lt(i)) are removed. If n is sufficiently large, we can evaluate the posterior PDF accurately. Each resampled particle represents the true state of the system with equal probability and acts as the initial particle for the next time step. This Bayesian framework is repeated.

We first perform a series of idealized observing system simulation experiments (OSSEs). The OSSE (e.g., Atlas, 1997) is a widely used approach in meteorological DA to test the general performance of a DA system and to evaluate the impact of specific observing systems. OSSE has the nature run, which is usually generated by running a simulation for a certain period. Observation data are simulated from the nature run by applying the observation operator, i.e., converting the model variables to the observed variables. Here, we add artificial random noise to simulate the observation error. DA experiments are initiated from the state independent of the nature run, and the simulated observations are assimilated. The resulting analyses and subsequent forecasts are compared with the nature run to evaluate the performance of DA. Once an OSSE is done, it is straightforward to extend the OSSE to the real world by simply replacing the simulated observations with the real-world observations.

To generate the nature run, the SEIB-DGVM was initialized with the bare ground (i.e., no plant at the beginning) and was run for 107 years using the climate forcing data from 2001 to 2010 available at the SEIB-DGVM web page (http://seib-dgvm.com/). Here, the 10-year forcing data are repeated for the 107-year simulation, and the last 7 years from years 101 to 107 use the actual climate forcing of 2001 to 2007; thus, we refer to years 101 to 107 as 2001 to 2007. The daily climate data were generated by the procedure of Sato and Ise (2012) with updated information available at the SEIB-DGVM web page, based on the monthly Climate Research Unit observation-based data (CRU-TS3.22 0.5∘ monthly climate time series) (Harris et al., 2014) and the daily data from the National Centers for Environmental Prediction (NCEP)/National Center for Atmospheric Research (NCAR) reanalysis (Kalnay et al., 1996). We chose the study area at one of the AsiaFlux sites, the Siberia Yakutsk larch forest site at Spasskaya Pad, the middle basin of the Lena River (62∘15′18′′ N, 129∘14′29′′ E). The observed climate data at this site were not directly used in this study, but these data may have been included in the NCEP/NCAR reanalysis. Field-observed carbon flux data are available as the ground truth to verify the DA results at this site. Forced by the climate data, the SEIB-DGVM simulates the vegetation shifts from the bare ground to a grassland, and then to a forest. The two PFTs, the boreal deciduous needleleaf trees and C3 grass, are the dominant PFTs in this study area. Therefore, we do not consider the other PFTs in this study following Sato et al. (2010). We call these two PFTs simply “tree” and “grass”.

The nature run (Fig. 2a) was performed with the “true” parameter values Pmax = 15 µmolCO2 m-2 s-1 and Dor = 230 DOY (day of year) for tree and Pmax = 9 µmolCO2 m-2 s-1 and Dor = 270 DOY for grass, where Pmax and Dor stand for the maximum photosynthesis rate and the start date of the dormancy, respectively (Fig. 2b). Hereafter, we omit the units for Pmax (µmolCO2 m-2 s-1) and Dor (DOY) for simplicity. The LAI observations for the last 4 years from 2004 to 2007 were created by adding independent Gaussian random noise to the LAI values from the nature run (Fig. 2a) every 4 days, simulating the MODIS LAI product. Here, the observation error standard deviation was given by 10 % of the nature run LAI value. The observed LAI values of less than 0.5 were not used for DA because the MODIS data for the real-world experiment did not include LAI values of less than 0.5. There are too few data with real MODIS LAI values of less than 0.5, and we assign the missing value in preprocessing. Since the LAI is observed only for values of 0.5 or larger, the LAI observation exists only in the summer season.

Next, 8000 particles (parallel simulations) were generated with uniformly perturbed parameters: Pmax = [0, 60] for tree, Pmax = [0, 15] for grass, and Dor = [200, 300] for both. Here, [a, b] denotes random draws from the uniform distribution between a and b. These initial perturbation sizes are based on the previous studies (Kolari et., 2006; Zeng et al., 2011; Zhao et al., 2015; Takagi et al., 2015). We ran 8000 parallel simulations for 103 years for spin-up from the bare ground using the same climate forcing data as the nature run. In the course of the vegetation succession, these randomly perturbed parameter sets result in a variety of LAI simulations (Fig. 2b).

The 8000 particles at the end of the 103-year spin-up runs are used as the initial conditions for DA. The simulated LAI observations are assimilated every 4 days. The nature run and particle filter use the same climate forcing data, so that the difference comes from the model parameter values. The particles continue to be the free runs until the first LAI observation is assimilated in the summer season. The state variables and model parameters are estimated together at DA, and the model systematic errors associated with the model parameters are corrected by DA with parameter estimation. No explicit bias correction is applied. To avoid the exact duplications after resampling, the model parameters Pmax and Dor are randomly perturbed for the duplicated particles. The random perturbations avoid particle degeneracy, which usually causes filter divergence. After some tuning, we found proper perturbation sizes that work for stable filtering without causing particle degeneracy, especially for biomass which is found to be the most sensitive to the perturbation sizes. Here, random draws [-4, 4] are added to Pmax for tree and to Dor for both tree and grass, and [-1, 1] are added to Pmax for grass because the initial Pmax perturbation size for grass is a quarter of that of tree. The sensitivity to the resampling perturbation sizes will be discussed in the next section. In the case that these perturbed parameters exceed the corresponding initial parameter range, the excess value was bounced back from the limits. To assess the impact of DA, we also perform an experiment without DA (“NODA” hereafter) and compare it to the experiment with DA (“TEST” hereafter).

Figure 3 shows the time series of LAI for NODA (left) and TEST (right). The observations (Fig. 3a, blue dots with error bars) cannot distinguish the tree and grass, but the model simulates LAIs for tree and grass separately (Fig. 3b, c). Although the particles without DA are widely spread (left, gray areas), DA makes the particles much narrower (right) and consistent with the nature run (red curves). With DA, the median of the particles for tree is almost identical to the nature run for the entire 4 years (Fig. 3b, right). As for grass, the median of the particles is also very close to the nature run with DA, but in the first 3 years the dormancy period is delayed (Fig. 3c, right).

The model parameters are estimated accurately (Fig. 4). There is no direct observation of these parameters, so that the estimations are purely due to DA of the LAI observations. Although the particles of the NODA experiment are uniformly distributed (Fig. 4, left), DA makes the particles close to the true parameters (Fig. 4, right). Since we assimilated the LAI only when it was 0.5 or larger, DA has an impact only in the summer season when the leaves grow. It takes 1–4 years until the true values fall within the quartiles of the particles. The Pmax estimates for both tree and grass show occasional jumps but tend to stay around the true values (Fig. 4a, b). Dor for tree seems the most accurate and stable after the dormancy period of the first year (Fig. 4c). Dor for grass takes the longest; the estimation is not accurate until the dormancy of the fourth year (Fig. 4d). This may be related to the previous results showing the erroneous estimates of the grass LAI near the dormancy period in the first 3 years (Fig. 3c). The systematic errors in NODA come from the uncertain parameter settings. TEST can estimate the parameters through DA and can reduce the systematic errors. This is different from the bias-correction strategy of the first guess.

Other model variables such as GPP, RE, NEE, and biomass show large improvements (Fig. 5). Although the particles of the NODA experiment are widely spread, DA with only LAI observations greatly reduces the uncertainties for the four variables, and the estimations are generally reasonable.

To investigate the sensitivity to the choice of the nature run, we performed two additional OSSEs, which we call “OSSE2” and “OSSE3”, by generating different nature runs with different parameter sets (Table 1). The random numbers for the observation errors are also different. The other settings follow the TEST experiment.

The results show that both OSSE2 and OSSE3 perform well in general. Namely, the LAI and parameters are estimated generally well (Fig. 6). We find the main difference between OSSE2 and OSSE3 in the parameters for grass (Fig. 6c, e). OSSE3 shows significantly larger uncertainties for the parameters for grass. In OSSE2, the Pmax value for grass is larger and produces more grass LAI. Since grass starts to grow earlier and stays longer than tree, it is critical to have LAI observations near the emerging and falling periods for estimating the grass parameters. Due to the larger Pmax value for grass in OSSE2, LAI can be observed with the observing threshold of LAI of 0.5 near the emerging and falling periods. By contrast, in OSSE3, the Pmax value for grass is smaller, and the small grass LAI of less than 0.5 cannot be observed. We can see this in the LAI time series (Fig. 6a, right) near the tails in the spring and fall seasons every year. The uncertainties of LAI are not reduced year by year, corresponding to the large uncertainties of the grass parameters. In the summer, LAI becomes larger mostly due to trees, so that the tree parameters can be estimated well.

Here, we investigate the sensitivity to the initial perturbation sizes with particle sizes ranging from 1000 to 16 000. Table 2 shows the three initial perturbation settings: small, moderate, and large. For the TEST experiment, the moderate initial perturbation sizes were used. We perform additional sensitivity experiments with the small and large initial perturbation sizes. Except for the initial perturbation sizes and the particle size, the experiments follow the TEST experiment.

Table 3 shows the mean absolute errors (MAEs) and the widths of the 1–99 % quantiles, respectively, averaged over a year in 2007. We consider that the filter diverges when the MAE is larger than the half width of the 1–99 % quantiles, as shown by the italic font in the tables. The results show that the filter diverges for biomass in 10 out of 15 experiments. The five experiments that do not diverge are (4000; small), (8000; small), (16 000; small), (8000; moderate) = TEST, and (16 000; moderate), where ( ; ) denotes (particle size; initial perturbation sizes). The (1000; large) experiment causes filter divergence for most variables and parameters. The (2000; large) experiment shows filter divergence for Dor for grass in addition to biomass. Sampling a wider interval with a smaller particle size generally reduces the particle density, or the effective number of the particles, so that the results seem to be reasonable.

Here, we investigate the sensitivity to the resampling perturbation sizes with particle sizes ranging from 500 to 16 000, in a similar way as the previous subsection. Resampling perturbations add random perturbations to Pmax and Dor when resampling and avoid particle degeneracy. Table 4 shows the three resampling perturbation settings: small, moderate, and large. For the TEST experiment, the moderate resampling perturbation sizes were used.

Table 5 shows similar tables as Table 3 but for the sensitivity to the resampling perturbation sizes. We use the similar notation of ( ; ) denoting (particle size; resampling perturbation setting). The results show that the filter diverges for biomass in 13 out of 18 experiments. The five experiments that do not diverge are (4000; large), (8000; moderate) = TEST, (8000; large), (16 000; moderate), and (16 000; large). The (500; small) experiment is the most unstable, with more variables and parameters showing filter divergence. Resampling perturbations act as variance inflation in the ensemble filters (e.g., Anderson and Anderson, 1999). It is known that variance inflation generally stabilizes the filter, and the results obtained here seem to be consistent. With 4000 particles or more, the parameters and state variables except for biomass were estimated accurately, although the filter collapsed for biomass with smaller perturbations even with large particle sizes.

Here, the OSSE is extended to the real world by replacing the simulated observations with the real observations. The sensitivity results in the previous section showed that the settings used for the TEST experiment provided stable filter performance; therefore, we follow the TEST experiment here with the moderate initial and resampling perturbation sizes and with 8000 particles.

Since the OSSE used the actual climate forcing in 2004 to 2007, we used the quality-controlled MODIS LAI product of MCD15A3 for those years with flagged as “good quality”, “Terra or Aqua”, “detectors apparently fine for up to 50 %”, “significant clouds not present”, and “main method used with or without saturation”. We took the median of the LAI observations in the 10 km radius from the study site (62∘15′18′′ N, 129∘14′29′′ E). There are a number of missing data in the quality-controlled MODIS data. Therefore, if the number of the data in the 10 km radius is less than 300, we set these data as the missing data for DA. Since the MODIS data resolution is 1 km, the 10 km radius area contains about 314 data. The observation error standard deviations are assigned to each LAI datum in the original MODIS product (Knyazikhin et al., 1999). We rely on the estimate of the observation error standard deviations and take the median of the error standard deviations in the same way as getting the LAI data. The observation error standard deviation is used in the particle filter when computing the likelihood function (Eq. 1).

The model-simulated NEE was validated with the field observation data at this AsiaFlux site (Ohta et al., 2001, 2008, 2014). The data were quality controlled by the steady-state test as indicated by the quality flag 0. Although the model simulates daily-average NEE, the field observation data represent instantaneous NEE every 30 min. The observation data are missing frequently, and it is not trivial to derive daily averages. Therefore, the raw data are compared with the DA results directly. This allows only a rough verification about whether or not the simulated NEE is in a reasonable range, but this is the only possible verification with an independent source.

Figures 7, 8, and 9 show similar time series to Figs. 3, 4, and 5, respectively, but with the real MODIS LAI observations. Although the particles of the NODA experiments are widely spread, DA makes the particles much narrower (right) for all variables and parameters. With DA, the median of LAI is very close to the observations, within the range of the observation error standard deviations (Fig. 7a). The grass and tree LAIs are estimated separately (Fig. 7b, c), but there is no direct observation or other verification truth to compare with. This is similar to the model parameters (Fig. 8) and other model variables (Fig. 9) except for NEE, for which direct field observation data are available. As in the OSSE results, the range of uncertainties for NEE is reduced significantly by DA (Fig. 9c). Since the field observations are made instantaneously every 30 min, the observation values (red) appear to have a wider range. However, the SEIB-DGVM simulates only daily-average NEE, and it is not straightforward to compare the outputs from SEIB-DGVM with the field observations. We still find that the median of NEE becomes closer to the observations, particularly near the dormancy period. The simulated NEE generally stays within the reasonable range compared with the field observations. In general, the particle filter shows promising results with the real MODIS LAI data.