Gaussian anamorphosis extension of the DEnKF for combined state and parameter estimation: application to a 1D ocean ecosystem model.

Transcription

1 Gaussian anamorphosis extension of the DEnKF for combined state and parameter estimation: application to a D ocean ecosystem model. Ehouarn Simon a,, Laurent Bertino a a Nansen Environmental and Remote Sensing Center, Thormøhlensgate 47, 56 Bergen, Norway Abstract We consider the problem of combined state-parameter estimation in biased nonlinear models with non-gaussian extensions of the Deterministic Ensemble Kalman Filter (DEnKF). We focus on the particular framework of ocean ecosystem models. Such models present important obstacles to the use of data assimilation methods based on Kalman filtering due to the non-linearity of the models, the constraints of positiveness that apply to the variables and parameters, and the non-gaussian distribution of the variables in which they result. We present extensions of the DEnKF dealing with these difficulties by introducing a nonlinear change of variables (anamorphosis function) in order to execute the analysis step with Gaussian transformed variables and parameters. Several strategies to build the anamorphosis functions are investigated and compared within the framework of twin experiments realized in a simple D ocean ecosystem model. A solution to the problem of the specification of the observation error for transformed observations is suggested. The study highlights the inability of the plain DEnKF with a simple post-processing of the negative values to properly estimate parameters when constraints of positiveness apply to the variables. It goes on to show that the introduction of the Gaussian anamorphosis can remedy these assimilation biases. Keywords: Data assimilation, Ensemble Kalman filter, Gaussian anamorphosis, Combined state-parameter estimation, Ecosystem modelling Preprint submitted to Journal of Marine Systems June 4,

2 . Introduction The knowledge of the status of marine resources should be closely monitored in a changing climate, so analysis and short term forecasts of the primary production are needed by environmental agencies for monitoring algal blooms and possible movement of the fish populations (Johannessen et al., 7; Allen et al., 8). To that end, within the framework of the MyOcean project, research activities attempt the reanalysis of the primary production and the biological components of the oceans, notably for the Arctic through the Arctic Marine Forecasting Center. The numerical ecosystem models developed during the last decades, as well as their coupling with existing physical ocean models, are a necessary step, together with the direct observation of the ocean biology, to meet these goals. Nevertheless these models present numerous uncertainties linked to the complexities of the processes that they attempt to represent and the parameterizations that they introduce. Even though many improvements have been made in the modelling of ocean ecosystems, the models are still too simple in comparison to the complexity of the ocean biology. Parameters remain poorly known, cannot be observed (lack of direct measure) and may vary in space and in time (Losa et al.,, 4). Furthermore, due to the key role played by the parameters or the mathematical model chosen for the parameterizations, wrong specifications of parameters can lead to large model error. The data assimilation methods, thanks to their ability to combine in an optimal way the heterogeneous and uncertain information provided by the models and the observations, are relevant tools to tackle the problem of parameter calibration. The feasibility and the potentialities of simultaneous state and parameter estimation with ensemble-based Kalman filters have been demonstrated by Anderson (), who augmented the state vector with the parameters to estimate. In the same way, experiments of combined state-parameter estimation conducted in a simple linear scalar model (Evensen, 6) highlighted the abilities of the Ensemble Kalman Filter (EnKF; Evensen, 994) to calibrate a poorly known parameter. This approach has also proved to be efficient even for large scale applications, as Corresponding author. Tel: ; Fax: addresses: ehouarn.simon@nersc.no (Ehouarn Simon ), laurent.bertino@nersc.no (Laurent Bertino)

3 highlighted by the twin experiments in an earth system model of intermediate complexity (Annan et al., 5). Furthermore, the authors introduce logarithm transformations to guarantee the positiveness of several diffusion parameters during the estimation. The performance of the EnKF in combined state-parameter estimation have been carefully evaluated in a -D sea-breeze model by Aksoy et al. (6). In the framework of ocean biogeochemistry, Losa et al. (4) successfully applied weak constraint variational data assimilation to estimate parameters in a -D ecosystem model. Nevertheless application of data assimilation methods to ecosystem models in an efficient way is a theoretically and practically challenging issue. On the one hand, the strongly nonlinear behavior of ecosystem models (especially during the period of the spring bloom) raises the question of which stochastic model to use (Bertino et al., ). On the other hand, one is also confronted with the model constraints. Most variables of ecosystem models are concentrations of a biological tracer, and cannot be negative. In the same way, parameters are positive. These non-gaussian distributions of most biogeochemical variables and parameters break an important assumption of the linear analysis, leading to a loss of optimality of the EnKF (and other linear filters). The optimality of the linear statistical analysis is proved under some assumptions, notably an assumption of Gaussianity made on the distribution of the variables (of the model and the observations) and the errors. Twin experiments of combined state-parameter estimation with the Lorenz model done by Kivman () highlighted the difficulties indeed the inability of the EnKF to recover the true value of positive parameters in nonlinear models. This study confirmed also the benefits of using nonlinear methods like particle filters in such non-gaussian frameworks. Unlike the EnKF, the application of an extension of a Sequential Importance Resampling filter (SIR; see Doucet et al., ) led to successful estimations of the true value of parameters. In the same way, Losa et al. () successfully applied a SIR filter for a combined state-parameter estimation in a -D ecosystem model. Indeed, particle filters seem attractive for such models as they are variance minimizing schemes for any probability density function. However, the size of the ensemble required for an efficient application of such a filter is too large to be considered for realistic configurations (Snyder et al., 8). We refer to the review of van Leeuwen (9) for more details about particle filters. An easy method of performing Kalman filter estimation in an extended framework of variables with non-gaussian distributions involves the in-

4 troduction of the Gaussian anamorphosis into the filter, as suggested by Bertino et al. (). The idea is to introduce non-linear changes of variables (anamorphosis functions) in order to realize the analysis step with Gaussian distributed transformed variables. The numerical experiments of model state estimation that they conducted in a -D ocean ecosystem model highlighted the potential of this approach. In a previous study (Simon and Bertino, 9), we demonstrated that this non-gaussian extension of the EnKF could be easily applied to perform model state estimations in realistic configurations. Twin experiments done in a -D configuration of the North Atlantic and Arctic oceans highlighted a slight advantage in effectiveness compared to the plain EnKF with post-processing of the negative values. This advantage has been recently confirmed by Doron et al. () in the framework of twin experiments of combined state-parameter estimation in a -D ocean-coupled physical-biogeochemical model. The unique analysis performed during the spring bloom highlighted the ability of this non-gaussian extension of the Kalman filter to retrieve consistently the maps of parameters and thus reduce their prior uncertainty, as stated by the authors. Nevertheless, this study does not provide information about potential assimilation bias that might occur after several cycles of analysis. The present study extends our previous research to the problem of combined state-parameter estimation in inevitably biased ecosystem models, and we focus on Kalman filtering. More information concerning the more general problem of Gaussian statistical modeling in data assimilation can be found in Bocquet et al. (). The aim of this study is to demonstrate that the Deterministic Ensemble Kalman filter (DEnKF; Sakov and Oke, 8) and more generally ensemble-based Kalman filters remains a high-performance tool for the estimation of biased parameters in such non-gaussian frameworks involving positive state variables and parameters, provided that the variables and parameters are appropriately transformed before and after the analysis. In that way, we focus on the strategies to empirically design the anamorphosis functions. Attention is also given to the problem of the specification of the observation error for the transformed variables. The outline of this paper is as follows. We present a non-gaussian extension of the DEnKF and different strategies to build the anamorphosis function in section. We describe our experimental framework in section, present and discuss our results in section 4 and make our conclusions in section 5. 4

5 . Non-Gaussian extensions of the Deterministic Ensemble Kalman Filter We describe in this section a way to design a non-gaussian extension of the DEnKF. Essentially we introduce nonlinear changes of variables in order to realize the analysis step with Gaussian distributed transformed variables, while the forecast step is done in the physical or biological space... The Deterministic Ensemble Kalman Filter with Gaussian anamorphosis As suggested by Bertino et al. () for the EnKF, the algorithm is based on the skeleton of the DEnKF and divides into two steps: Forecast : the forecast step is a propagation step that uses a Monte- Carlo sampling to approximate the forecast density by N realizations: i = : N, x f,i n = f n (x a,i n, ɛ m,i n ) () with x n the state vector at time t n, f n the nonlinear model and ɛ m n the model error. The superscripts f and a stand for forecast and analysis. Analysis: the analysis step conditions each forecast member to the new observation y n by a linear update. The anamorphosis functions are introduced in this step. For each variable of the model, at time t n, we apply a function ψ n which is a nonlinear bijective function from the physical space to a Gaussian space. We transform each variable separately. In order to simplify the notations, we consider the monovariate case (i.e., there is only one function ψ n ). It reads: i = : N, x f,i n = ψ n (x f,i n ) () In practice, this means that we apply a transformation to each variable in every point of the discretized domain. In the same way, we introduce an anamorphosis function χ n for the observations y n at time t n : ỹ n = χ n (y n ). () The observation operator H links the physical variables and the observations. We define the observation operator H n linking the transformed variables and observations by the formula H n = χ n H ψ n (4) 5

6 where defines the function composition. The linear analysis is done with the transformed variables and observations according to the equations for the updates of the mean and the ensemble anomalies of the DEnKF described in Sakov and Oke (8). The transformed Kalman gain matrix is built on the forecast error covariance matrix C f n approximated by the covariance of ( x f,i n ) i=:n. The inverse transformation to the physical space is done by using the inverse of the anamorphosis function: i = : N, x a,i n = ψn ( x a,i n ) (5) The analyzed mean x a n and the covariance matrix C a n are approximated by the ensemble average and covariance of (x a,i n ) i=:n... Strategies to design a monovariate anamorphosis function The performances of the non-gaussian extensions of the DEnKF described above are strongly dependent on the choice of the anamorphosis functions ψ n and χ n. One solution is to use analytic functions like the logarithm. However, this requires prior knowledge of the distribution of variables. Another solution is to construct the anamorphosis functions directly from a sample of variables. The idea is to build the anamorphosis functions from the empirical marginal distributions of the variables. The algorithm of the construction of a monovariate anamorphosis function (one function per variable) divides into three parts: the construction of the empirical anamorphosis function based on the marginal distribution of the variable, the interpolation of this empirical step function in order to get a bijective function, and the definition of the tails that are necessary to process the most unlikely values. A more complete description of the construction of a monovariate anamorphosis function into the framework of Kalman filtering can be found in Simon and Bertino (9). We briefly recall the ideas behind the construction of the monovariate empirical anamorphosis.... The empirical anamorphosis function Let Z(x) be a random variable that is function of space. We assume that we do not know the marginal distribution of Z(x), but we have access to an approximation via a sample (z i ) i=:n of this variable. The aim is to build a step function ψ such that Z(x) = ψ(y (x)) (6) 6

7 with Y (x) following a predefined marginal distribution. In our assimilation framework, Y (x) is assumed to have a normal distribution N (, ). Computing a sample (y i ) i=:n of the Gaussian variable Y (x). i = : N, y i = G ( i N ) (7) where G is the cumulative distribution function of Y (x), the empirical anamorphosis function reads: ψ(y) = N z i [yi,y i [(y) (8) i= with [yi,y i [ the support function on the interval [y i, y i [, for which y i is included and y i is excluded. This function is equal to one on the interval [y i, y i [ and equal to zero everywhere else. In the present study, we focus on the strategy to choose a relevant sample (z i ) i=:n for the variables to transform. It raises the issue of the specification of likely and unlikely values of the variables. Furthermore, important issues come from the problem of model bias and the dependance of the transformations on a reference model run. This problem leads to observations that may be out of the range of the sample obtained from model realizations (ensemble, multi-year simulation). It raises the question of how to transform such observations that appear unlikely from the model. However, this problem is not necessarily related to model bias and can occur with a perfect model during extreme events. We thus consider three approaches.... Static approach In this approach, the sample is made of restarts gathered from a preintegration of the model. It has been used in Simon and Bertino (9). These restarts can come from a climatology, or at least a multiyear simulation, or several members of an annual ensemble simulation. Furthermore, assuming that seasonality is stronger than differences in locations, all variables and observations are assumed to be identically distributed in space, and identically distributed in time conditionally on a period centered on the datum of the analysis step. This allows the handling of slight differences in distribution between the ensemble and the climatology. For the special case of ocean ecosystem model, we decided to use a time period of three months centered on the datum of the analysis step. This choice is motivated by the time scale of bloom phenomena which is about 4 7

8 months. Such a moving window allows for a representation of the differences of distribution at the beginning and the end of the bloom in the construction of the anamorphosis functions. On one hand, such an approach is expected to provide samples with a large spread of values, or at least samples that do not depend on the evolution of the ensemble during the assimilation and its potential collapse. It leads to a stable assimilation system. On the other hand, it raises problems of inefficiency in the anamorphosis functions when the ensemble drifts away from the climatology. This might occur with the presence of model bias or the use of an irrelevant original climatology, leading to non-gaussian transformed variables. Due to the significant risk of using the tails during the transformation of the forecast ensemble (drift), we propose to use linear tails as done in Simon and Bertino (9), because arbitrary nonlinear transformations might damage the analyzed solution. When using linear interpolation to build the anamorphosis function, it consists simply of extending to infinity the first and last segments of the interpolation with the same slope.... Dynamic approach In this approach, the sample is populated by the members of the forecast ensemble at the time of the analysis. Both the model variables and the observations are assumed to be identically distributed in space so that the sample fed into the Gaussian anamorphosis is larger than the sole ensemble size. This was found useful for the quality of the fit. Nevertheless, the dynamic approach is sensitive to model biases that can make the observations fall out of the forecast distribution. A solution would be to merge the two samples (model and observation ensembles) while computing the anamorphosis. As we noted in previous studies (not shown), this remedies the above problem and reduces a shift of the mean to high values. However, it does not guarantee that the merged distribution is continuous, and may result in an awkward anamorphosis of disjoint model and observed modes. We propose to use nonlinear tails (see Equations (4) and (5) for example). The aim is to obtain an infinite Gaussian domain in order to reduce the risk of collapse of the ensemble after analysis, notably into the framework of parameter estimation for which there is no evolution during the forecast steps. To the best of our knowledge, the impact of anamorphosis functions built on the forecast ensemble on the assimilation system has not been properly studied. The twin experiments realized by Doron et al. () highlighted 8

9 the potential of this approach, but on a single shot of assimilation...4. A hybridization A hybridization of the two previous approaches is realized by combining the static approach for the observed variables and the observations and the dynamic approach for the non-observed variables. With its large range of values the static approach allows for handling differences of distribution between the observations and the observed variables while the dynamic approach leads to Gaussian transformed non-observed variables. We expect a more stable approach than the pure dynamic one and more efficient transformations than the static approach... Observation error and transformed variables According to the Kalman filter assumptions, we force the observation error in the transformed space to have a normal distribution: ɛ o,i n N (, Σ o ). An important issue is the specification of the variance Σ o of this error. In Simon and Bertino (9), we proposed to specify the variance of the relative error for each observation p: p = : N obs, ɛ o (p) N (, σ o (p)), where N obs is the number of assimilated observations. Due to the risks of overestimation of the transformed forecast error compared with the transformed observation error that is intrinsic to this strategy, we suggest a new way to define the variance of the observation error in the transformed space. The algorithm is based on the transformation of perturbed observations with the anamorphosis function. For each observation p = : N obs, we do the following:. Generation of perturbed observations (d i (p)) i=:n in the biological space. It is necessary to specify a distribution for the observation error in the biological space (Gaussian distribution, log-normal distribution, etc). This depends of the nature of the problem and the prior information that we have concerning the observation error. In ocean ecosystem models, the observation error is usually assumed to be log-normally distributed. The perturbed observations are defined by: i = : N, d i (p) = y(p) e (E i σ (p)/) (9) with (E i ) i=:n a sample of the random variable E N (, σ (p)) and σ(p) is the relative observation error. σ (p) is a bias reduction term. 9

10 . Transformation of (d i (p)) i=:n. Each perturbed observation is transformed with the anamorphosis function dedicated to observations (see Equation ). It reads: i = : N, di (p) = χ n (d i (p)) (). Definition of the variance of the observation error σ o (p). σ o (p) is defined as the sample variance of ( d i (p)) i=:n. The transformed observation error for observation p is defined according to ɛ o (p) N (, σ o (p)) This approach might exaggerate the spread of the transformed perturbed observations for those observations that are close to the tails of the anamorphosis function. Observations that appear to be unlikely in the biological space will have a large error variance in the transformed space where the analysis is performed. It can be seen as a conservative quality control step: the confidence in observations increases as their probability expressed in the empirical anamorphosis function increases.. Description of the experimental framework.. The D ecosystem model The experiments are realized in a simple D vertical ecosystem model that represents the yearly cycles of phytoplankton blooms. The original nondimensional model has been introduced by Evans and Parslow (985). Eknes and Evensen () extended this model to a vertical water column for their data assimilation experiments. This D version is the one used by Bertino et al. () to highlight the benefits of the introduction of anamorphosis functions. The model includes three components: nutrients (N), phytoplankton (P) and herbivores (H). The equations describing their evolution are given by: [ ] N α(t, z, P ) t = j + N r P + z (K z(z, M(t)) N z ) [ ] P α(t, z, P ) t = j + N r P c(p P )H + K + P P z (K z(z, M(t)) P z ) H t = fc(p P )H K + P P gh + z (K z(z, M(t)) H z ) ()

11 Table : Biological parameters. In bold, parameters that are estimated during the data assimilation experiments and differ from the true value. Parameter Nominal value True value c Maximum grazing rate.day.day f Grazing efficiency.5.7 g Loss to carnivores.7day.day j Uptake half saturation.5mmol N m.5mmol N m r Plant metabolic loss.7day.day K Grazing half saturation.mmol N m.mmol N m P Grazing threshold.mmol N m.mmol N m where M(t) is the mixed layer depth and is defined as an input for the ecosystem model, α is the photosynthetic light rate and K z is the depth dependent diffusion parameter. We refer to Eknes and Evensen () for more details concerning the computation of these quantities. The description of the biological parameters is given in Table. The values correspond to the ones used in Evans and Parslow (985) and Eknes and Evensen () and are appropriate to the Flemish Cape. The depth is meters and the model has uniform vertical layers. We increased the nutrient concentration specified in Eknes and Evensen () to 5mmol N m in the bottom. This guarantees a constant contribution of nutrients in the bottom of the ocean that is brought to the mixed layer via the vertical mixing. The phytoplankton and herbivore concentrations have been also increased to. 8 mmol N m in the bottom. There is no flux at the surface... Data assimilation experiments We focus on the impact of the strategy to build the anamorphosis functions on the performances of the combined state-parameter estimation. In that way, twin experiments have been conducted: the true state and the observations are produced by a deterministic simulation of the model involving parameters that can be different from the nominal ones (Table ). Our aim is to investigate the ability of the different methods to estimate the true value of the parameters used to build the reference.

12 The state and parameter estimations are conducted jointly. As suggested in Anderson () and Evensen (9), the assimilation is done by augmenting the usual state vector with the parameters that we plan to estimate. In this present study, the state vector is made up of all the vertical components of the three state variables (P, H and N) and three parameters (the grazing efficiency (f), the loss to carnivores (g), and the plant metabolic loss (r)). These parameters have been chosen due to the prospect of future experiments in a D configuration of the ecosystem model NOR- WECOM (Skogen and Søiland, 998). First, the ability of the methods to estimate uncertain parameters involved in the closure terms of the model is evaluated through the estimation of the plant metabolic loss (r) and the zooplankton loss to carnivores (g). Preliminary experiments realized in a D configuration of NORWECOM (Simon and Bertino, ) suggested that reductions of model bias could be obtained by controlling the phytoplankton death rates (parameters equivalent to r). Finally, the ability of the methods to optimize the uncertain grazing rate ( fc(p P )H K+P P ) is investigated through the estimation of the parameter f (grazing efficiency) rather than the estimation of c (maximum grazing rate) and/or K (grazing half saturation). Due to the simplicity of the model, we focus on a parameter that is not directly involved in the dynamics of the observed variables (phytoplankton). We referred to Buitenhuis et al. (6, ) for more details about observation-based parameterizations of zooplankton compartments. Several assimilation systems have been implemented. They are built over the DEnKF. The first one does not include anamorphosis functions. It corresponds to the direct application of the filter. It includes a post-processing step that removes negative values by increasing them to zero. This postprocessing is done at the end of the analysis step and makes sure that the forecast step runs from an ensemble that satisfies to the constraint of positiveness of the model. The four other systems include anamorphosis functions and differ only in the strategy used to build these ones: the three empirical approaches described in. (static, dynamic and the hybridization) and the usual analytic approach based on the assumption of log-normally distributed variables and parameters. For that particular case, equations () and () read i = : N, x f,i n = log(x f,i n ) and ỹ n = log(y n ). No post-processing steps are included, as the methods do not require any. The true state x t is generated as follows. A ten years simulation is conducted starting from rest. The value of the parameters representing the grazing efficiency (f), the loss to carnivores (g), and the plant metabolic

13 loss (r) are chosen different from the nominal ones (see Table ). They will be called the true values of the parameter in the following. The solution obtained during the last five years represents the true state. The observations are the phytoplankton (P) in the two first layers of the model and are defined as follows y n = H n x t n G, with G Γ(, σ σ o), σ o o =. () We construct the observations by multiplying the true surface phytoplankton with a Gamma distributed observation error with a standard deviation around % (average should be ). The Gamma distribution has a thinner tail than the log-normal one and would be less prone to favor large observation error. However, the observation error may reach high values as noted for the case of real data. Experiments realized with normally distributed observation error led to similar results (not shown). The strategy for estimating the observation error ɛ o changes with the assimilation systems. Without anamorphosis functions, the observation error for each observation p is assumed to have a Gaussian distribution with a mean of zero and a standard deviation of % of the value of the observation: ɛ o (p) N (, σ o =. y n (p)). For all the systems with anamorphosis functions (static, dynamic, hybrid and logarithm), the observation error in the transformed space has a Gaussian distribution with a mean of zero and a standard deviation computed following the algorithm described in.: ɛ o N (, σ o ). For the generation of the perturbed observations, we assume that the observation error in the biological space is log-normally distributed, with a variance of σ o =. as highlighted in the equation (9). This leads to an observation error with a constant variance of. for the system with the logarithm anamorphosis. The ensemble is made of members and is generated as follows. First, a 5 year spin-up simulation is integrated with the nominal values of the parameters (see Table ). The differences between the true and nominal values for the parameters f, g, and r lead to model biases. The spring blooms (P) start too early, are too short and too weak compared to the true state. In the same way, the production of herbivores (H) starts too early and is too weak. At the end of the spin-up, the background state ensemble is generated by adding a truncated-gaussian perturbation to the solution x(t = 5years). i = : N, x i b = max(, x(t = 5years) ( + b i )), with b N (, σ b ) ()

14 σ b is chosen to be equal to. for the phytoplankton and the herbivores, and.5 for the nutrients. The parameter ensemble is initialized by assuming that the parameters are log-normally distributed around the nominal values with a 5% error. Starting from this background, a one-year ensemble simulation is performed. The model includes truncated-gaussian random perturbations on the state variables at each time step. The standard deviation is chosen to be equal to % of the value for the phytoplankton and the herbivores, and 4% of the value for the nutrients, see Equation (), in the first twelve upper layers. The standard deviation decreases linearly in the eight deepest layers in order to obtain a smooth transition between the deep layers and the bottom layer. No perturbations are added to the parameters and these ones remain constant during the integration of the model. Assimilation cycles are then performed over four years with a frequency of one analysis step every four days. It has to be noted that the introduction of perturbations on the ecosystem variables, as well as the use of perturbed parameters, leads to a slight increase in the concentration of phytoplankton for the mean of the ensemble during the cold period. Even without model bias, the primary production is larger in winter with ensemble simulations. In order to check the robustness of the estimation, we repeated the experiment twenty times. That is, twenty initial ensembles (combined stateparameter background), twenty sets of observations, and twenty data sets for the static anamorphosis functions (see.) were generated. Nevertheless, the different assimilation systems used the same background ensemble and observations for each of the twenty realizations. The diagnostics shown in section 4 are averaged over these twenty experiments... Construction of the monovariate anamorphosis function For all the experimental approaches (static, dynamic and hybrid), the anamorphosis functions are piecewise linear, using linear interpolation of the experimental anamorphosis function. The middles of step intervals are used to interpolate the empirical anamorphosis functions, with the exception of the leftmost and rightmost intervals for which the minimal and maximal values of the data set are used. Concerning the biological bounds, for all the approaches (even the logarithmic one), the minimum values are equal to zero (constraint of positiveness) and the maximum values are unlikely, high values that are summarized in Table. On the other hand, the definition of the Gaussian bounds depends on the approach used to build the anamorphosis functions. 4

15 Table : Anamorphosis functions: upper biological bounds. Note that given the simplicity of the model, we permit grazing efficiencies superior to, although this should not be done in a model respecting the mass balance. Variables and parameters Upper bound P Phytoplankton mmol N m N Nutrients mmol N m H Herbivores 5mmol N m f Grazing efficiency. g Loss to carnivores.5day r Plant metabolic loss.5day Practical details for the static and dynamic approaches are given in the next two sections.... Static approach The experimental anamorphosis functions for the state variables are computed from output sampled every 4 days from a one-year ensemble simulation. The use of a deterministic multi-year simulation in the special framework of combined state-parameter estimation could not be sufficient to provide a good representation of the error subspace. Because Monte Carlo simulations are more suitable to handle and represent the large uncertainties present in the system, particularly the ones related to poorly known parameters, we preferred to run an ensemble simulation with nine members that were arbitrary chosen. The small size of the ensemble is motivated by practical issues related to the storage capacity of future realistic systems. The ensemble simulation has been performed with the biased model. Only outputs present in a time period of three months centered on the datum of the analysis step are used. The repetition of the experiments leads to the creation of twenty different data sets (one per realization) obtained from twenty ensemble simulations. The experimental anamorphosis functions of the parameters are computed from the parameter ensemble generated during the initialization. For the particular case of the phytoplankton, data from two years of observations are included in the model data set to build the empirical anamorphosis function. These data include the observations that are assimilated. 5

16 Nevertheless, only observations anterior to the analysis datum were selected (we do not use unknown future observations). Such inclusion of observational data in the model data set is expected to remedy the problem of observations out of range of pure model data set that might occur in presence of model bias (see..). The tails of the anamorphoses are linear and consist of prolonging the last segment towards the specified biological minimum and maximum values. The Gaussian bounds are defined as the abscissa of the intersection between this straight line and the biological bounds.... Dynamic approach The experimental anamorphosis functions are computed from the one hundred members of the forecast ensemble. Again, for the particular case of the phytoplankton, data from two years of observations are included in the model data set to build the empirical anamorphosis function. The same observations were used as in the static approach. As explained in.., the tails are nonlinear. These functions are chosen to converge towards the biological bounds at infinity and to guarantee a differentiable transition towards the piecewise linear function issuing from the interpolation of the empirical anamorphosis. The left tail is defined as follows: y ], y ], f l (y) = z exp( φ (y y ) z ) (4) where the Gaussian variable y and the biological variable z are described in.. and φ is the slope of the straight line interpolating the first step of the empirical anamorphosis function. This function converges as expected towards zero at. The right tail is defined as follows: y [y N, + [, f r (y) = Z max (Z max z N ) exp( φ N(y N y) Z max z N ) (5) where the Gaussian variable y N and the biological variable z N are described in.., Z max is the upper bound and φ N is the slope of the straight line interpolating the last step of the empirical anamorphosis function. This function converges as expected towards Z max at +. 6

17 4. Data assimilation results 4.. Observation error We are interested in the time evolution of the error variance for the transformed observations. Since the observation error in equation () is timevarying, a first verification of its values is necessary. Our aim is to highlight potential bias in the estimation of the observation error that might result from the anamorphosis functions. Figure represents the evolution in time of σ, the standard deviation of the transformed observation error, and the standard deviation of the surface phytoplankton component of the transformed forecast error. First, as expected, we note that the transformed observation error is constant with time for the logarithmic case, except for the period of the spinup. Because the perturbed observations are log-normally distributed around the observations, the use of the logarithm leads to an observation error with a constant standard deviation of. in the transformed space. Furthermore, except for peaks during the bloom periods, we note that σ is twice as large as the standard deviation of the transformed forecast error. Observations are less reliable than the model and this results in low corrections of variables and parameters and a low decrease of the spread of the ensemble during the whole year, except for spring blooms. For such events, the forecast error becomes much larger than the observation error and large corrections are expected. The dynamic anamorphosis leads to a specification of the observation error that is always larger than the forecast error, even during spring blooms. σ is much larger than the forecast error in autumn and during the bloom, leading to a loss of performance for the assimilation during these periods. This is caused by some of the observations being out of the ensemble range (see..). On the contrary, both errors are equivalent in winter. During that period, the innovations are low due to low concentrations of phytoplankton. It results in low impact on the variables but potentially large reductions of the spread of the ensemble. In the same way, the static anamorphosis leads to a specified observation error that is larger than the forecast error, except for the first spring bloom. Nevertheless, the discrepancies between these two errors are lower, especially during the blooms, than the one observed in the dynamic case. Compared to the logarithmic case, the corrections during the bloom are weaker and a larger reduction of the spread of the ensemble is expected during the cold period. The hybrid and static anamorphoses lead to the 7

18 .5 Logarithm Observation Forecast.5 Dynamic Observation Forecast Standard deviation.5 Standard deviation Static Observation Forecast.5 Hybrid Observation Forecast Standard deviation.5 Standard deviation Figure : Observation error in the transformed space: time evolution of the averaged standard deviation of the surface observation error σ (grey line) and the surface phytoplankton component of the forecast error (black line). 8

19 same results, as the hybrid uses the static approach for the observed variables. 4.. Error evolution for the state variables We are interested in the evolution in time of the true Root Mean Square error (RMS) and the ensemble standard deviations (STD) of the solution of the different systems. These diagnostics are averaged over the number of experiments that were done. The expression at time t n of these two quantities is as follows: RMS(t n ) = N exp STD(t n ) = N exp N exp i= N exp i= #Ω (x t (t n, k) x(t n, k, i)) k Ω N #Ω k Ω m= N (x m (t n, k, i) x(t n, k, i)) (6) where Ω is the domain of computation, #Ω is the number of grid points of the domain used for the computation of the RMS and STD, N is the number of members, x m is the forecast member m, N exp is the number of experiments, x t is the true state, and x is the mean of the forecast ensemble. Figure represents the evolution of the RMS error and the standard deviations over five years for the phytoplankton, the nutrients and the herbivores. These diagnostics are computed over the entire water column. In that case, Ω represents the twenty vertical layers. We recall that no assimilation is performed during the first year. Furthermore, in order to highlight the temporal evolution of the relative RMS error and standard deviation, the evolution of the spatial mean of the true state has been plotted (green dashed line) for the phytoplankton and the herbivores components. The spatial mean of the nutrient component of the true state is evolving with time around 7 mmol N m, a value too large to be plotted. First we note that the assimilation in all the systems leads to a significant reduction of the RMS error and STD for the phytoplankton, the variable that is observed in the surface. The errors are almost null except for the spring bloom periods. We note for these periods a continuous decrease of the error during the four years of assimilation. The solution resulting from the assimilation with the logarithm anamorphosis produces the lowest RMS error. Furthermore, the introduction of the different anamorphosis functions 9

20 prevents the occurrence of large peaks in the errors that are observed at the end of the bloom for the plain DEnKF when only a simple post-processing of the negative values is used. Nevertheless, the occurrence of peaks in the error for the static and hybrid anamorphoses may point out a minor inadequacy in winter of the static sample used to build the phytoplankton anamorphosis function. The problem of discrepancy between the forecast ensemble and the data set used to build the static anamorphosis functions occurs also for the herbivores. The solutions with the static anamorphosis have their highest errors in winter during the first year of assimilation. The irrelevance of the static anamorphosis functions during the cold period results in the appearance of outliers in the transformed forecast ensemble, which leads to analysis steps that damage the solution. This problem can be remedied by the introduction of dynamic and hybrid anamorphoses, that reduce significantly the error of the solutions. Again, we note that the solutions with the lowest errors are produced by the assimilation with the logarithm anamorphosis, while the assimilation without anamorphosis functions cannot significantly reduce the error for the herbivores. In the same way, we note that the assimilation without anamorphosis functions cannot reduce the RMS error for the nutrients (indeed, it actually increases it). Furthermore, we note that the time evolution of the RMS error and the standard deviation are out of phase, revealing strong assimilation biases. On the contrary, the introduction of the anamorphosis function significantly reduces the RMS error of the solutions. The logarithm and hybrid anamorphoses have the lowest errors. We note also that the assimilation with the static anamorphosis is slower to reduce the errors for the nutrients, so the solutions contain large errors during the first two years. Finally, we note that the standard deviation is higher than the RMS error for the different systems, expressing an over-estimation of the error for the nutrients by the filters, which is uncommon in realistic settings. 4.. Evolution of the parameters We are interested in the benefits of the introduction of anamorphosis functions on parameter estimation. Figure represents the time evolution of the mean and the mean plus/minus the standard deviation of the ensemble for the three estimated parameters. Again these quantities are averaged over the twenty experiments. The true value of the parameters has also been plotted with a dark grey dash-dot line. First, we note that the estimation of the plant metabolic loss (r) quickly converges towards the true value for all

21 Without anamorphoses Phytoplankton Nutrients Herbivores Phytoplankton P RMS STD mean of the true Nutrients N Herbivores H 5 5 Logarithm anamorphosis Phytoplankton P Nutrients N Herbivores H 5 5 Dynamic anamorphosis Phytoplankton P Nutrients N Herbivores H 5 5 Static anamorphosis Phytoplankton P Nutrients N Herbivores H 5 5 Hybrid anamorphosis Phytoplankton P Nutrients N Herbivores H 5 5 Figure : Observation error of %: time evolution of the averaged RMS error (blue line) and standard deviation (red line) computed over the water column and averaged over the twenty experiments. The mean of the true state is plotted for the phytoplankton and herbivores (green dashed curve)

22 Table : Parameters: mean and standard deviation (computed over the twenty experiments) of the means of parameters obtained at the final time. The true values are: r =.day, f =.7 and g =.day Without ana. Logarithm ana. Dynamic ana. Static ana. Hybrid ana. r (day ).98 ±.4.98 ±..98 ±..99 ±..98 ±. f.88 ± ±.4.6 ± ±.5.65 ±.69 g (day ).54 ±.4.94 ±.7.79 ±..88 ±.5.85 ±. the systems of assimilation. The mean and standard deviation of the twenty means of the ensemble obtained at the end of the experiments are summarized in Table. Estimates produced by systems including anamorphosis functions are slightly better than those without anamorphosis functions. As expected, coefficients of linear parameterization involving the observed variables can be efficiently estimated with basic ensemble methods. On the other hand, the estimation of coefficients of nonlinear parameterizations or parameterizations that do not affect the observed variables directly is a challenging issue. We note that the corrections on parameters are in the opposite direction during the first bloom for the assimilation system without anamorphosis functions. This leads to final averaged means that are lower than the nominal values. Furthermore, the estimation seems sensitive to the random contribution of the observation errors, as well as the perturbations, as highlighted by the large values reached by the standard deviation of the mean of the parameters obtained at the end of the twenty experiments (see Table ). These results are confirmed by the scatter plot, in Figure 4, of the final estimate (mean of the ensemble) of the grazing efficiency f against the final estimate of the loss to carnivores g for the twenty experiments. First we note that the assimilation degrades both parameters in most of the experiments: 9% of the points belong to the left bottom corner, indicating corrections in the wrong direction for both parameters. The dearth of points in the right top corner shows that the filter can not improve simultaneously the two parameters. This inability to estimate the true value of parameters is in agreement with the results of Kivman (). Furthermore, the DEnKF, and other ensemble based Kalman filters, are linear estimation methods whose direction of corrections is determined by the forecast covariance error matrix (P f H T ) and the amplitude by the innovations (weighted by the confidence in the observations and forecast). The parameters being constant during the forecast steps, only the successive lin-

23 Without anamorphoses Mean and standard deviations (d! ) Plant metabolic loss (r) Grazing efficiency (f) Loss to carnivores (g) Plant metabolic loss r 5 5 Mean STD True Mean and standard deviations (d! ) Grazing efficiency f 5 5 Mean and standard deviations (d! ) Loss to carnivores g 5 5 Logarithm anamorphosis Mean and standard deviations (d! ) Plant metabolic loss r 5 5 Mean and standard deviations (d! ) Grazing efficiency f 5 5 Mean and standard deviations (d! ) Loss to carnivores g 5 5 Dynamic anamorphosis Mean and standard deviations (d! ) Plant metabolic loss r 5 5 Mean and standard deviations (d! ) Grazing efficiency f 5 5 Mean and standard deviations (d! ) Loss to carnivores g 5 5 Static anamorphosis Mean and standard deviations (d! ) Plant metabolic loss r 5 5 Mean and standard deviations (d! ) Grazing efficiency f 5 5 Mean and standard deviations (d! ) Loss to carnivores g 5 5 Hybrid anamorphosis Mean and standard deviations (d! ) Plant metabolic loss r 5 5 Mean and standard deviations (d! ) Grazing efficiency f 5 5 Mean and standard deviations (d! ) Loss to carnivores g 5 5 Figure : Observation error of %: time evolution of the averaged mean (black line) and averaged mean plus/minus the standard deviation (shade area) of the parameters. The true value is highlighted with a dark dash-dote line.

24 ear corrections determine their evolution in time. The final estimates of f and g are obtained by adding to the background estimates (biased nominal N [ ] σ f,n N values) the vector d f,o n σ f,n := d n D n where d n are the weighted n= g,o n= innovations and σ.,o f,n the forecast error covariance between the parameters and the observed variable at analysis time t n. If we look at the scatter plot of the final estimates of f against g, a good representation of the model error all along the four-year estimation (or at least after the first year of assimilation) from the forecast covariance error matrix should lead to parameter estimates that are close to the straight line defined by the initial and true points. Thus, we expect to note a linear relation, parallel to the straight line defined by the initial and true points, between the final estimates of the twenty experiments. On the contrary, the presence of numerous outliers or a large spread of the cloud will highlight wrong representations, even selective (a few directions D n ), of the error subspace from the ensemble during the assimilation. This would lead to a strong sensitivity to the random processes that are involved in the filtering process, notably the observation error. This is what we observe for the DEnKF without anamorphoses, even if a linear pattern with a relevant slope emerges. The introduction of anamorphoses leads to a significant improvement of the parameter estimation. The averaged mean of both parameters has been improved suggesting that the assimilation provided corrections in the direction of the true value of the parameters. The linear relation, that emerges between the two optimized parameters, is parallel to, and indeed coincides with, the line defined by the initial and true points. It confirms the good representation of the errors from the ensemble during the greater part of the experiments. The introduction of the anamorphosis functions leads to a linear estimation which is more stable to the nonlinear consequences of the random process. Nevertheless, the width of the cloud suggests that the static approach is less precise, as noted in Table. Again, the system with the logarithm anamorphosis leads to the best estimates. At the end of the experiments, the true values of both parameters are within the ensemble spread on average and we note that the assimilation jointly improved both parameters for all the twenty experiments (see Figure 4). On average, the use of static and hybrid anamorphoses lead to similar estimates. Nevertheless, a look at Figure 4 confirms the large spread of the estimation with the static anamorphosis. So the number of wrong estimations the final points that are not in the top right corner is larger with 4