DALECv2 depicts a terrestrial ecosystem as a set of six carbon pools (labile Clab, foliar Cf, wood Cw, root Cr, litterfall Cl and soil organic matter Cs) linked via allocation fluxes. At a monthly time step the gross primary production (GPP) is calculated using the Aggregated Canopy Model as a non-linear function of meteorological drivers (temperature, radiation, atmospheric CO2 concentration), foliar carbon and foliar nitrogen. Following the mass conservation principle, GPP is then allocated to the different carbon pools or released in the atmosphere via respiration. The schematic for DALECv2 is represented in Fig.  and a complete description of the model can be found in . DALECv2 combines the two previous DALEC-evergreen and DALEC-deciduous models into a single model where the non-differentiable phenology process of DALEC-deciduous has been replaced with a differentiable process. DALECv2 is a non-linear dynamical system and the carbon pools are dynamical variables parametrised by their initial values C0 and by 17 parameters p, whose range and description can be found in Table . The magnitudes and ranges of the parameters and the initial values vary drastically; therefore, to avoid the computational problems caused by these different scales the variational methods will be formulated and implemented in terms of the log transformed variable x = log⁡([p, C0])T. However, in order to limit unnecessary notation and definition, in the remainder of this paper p and C0 will stand for their log⁡ transform.

The meteorological drivers are extracted from 0.125∘ × 0.125∘ ERA-Interim reanalysis data sets. For the purpose of our inverse modelling experiments we use four different observation streams: LAI, NEE, GPP and RESP (total respiration). LAI monthly mean observations for AmeriFlux sites are extracted from MOD15A2 LAI 8-day version 005 1 km resolution product. These observations together with the meteorological drivers are provided by A. Bloom and J. Exbrayat. Details about their construction can be found in . At AmeriFlux sites we use the level 4 data product (available at http://cdiac.ornl.gov/ftp/ameriflux/data/Level4/), which provides monthly means for NEE and GPP. NEE and GPP are then used to define RESP as RESP = NEE + GPP. The meteorological drivers span a period of 12 years from 2001 to 2013. LAI observations are available during the full period but for NEE and GPP, and thus RESP, shorter records are available depending on the AmeriFlux site. In this study we consider the Morgan Monroe State Forest located in Indiana, USA (39.3–86.4). This AmeriFlux site is composed in majority of mixed hardwood broadleaf deciduous trees and classifies as a humid subtropical climate.

In the remainder of the paper the main focus is on the vector x = log⁡([p, C0])T: in Sect.  first where we investigate the sensitivity of different outputs with respect to x and its components, and then in subsequent sections where x is estimated using inverse methods. The vector x, denoting fixed quantities as initial conditions and parameters for the dynamical system DALECv2, is seen as the variable from the point of view of sensitivity analysis and inverse modelling and therefore its components will be referred to as state variables, input variables or parameters interchangeably throughout the manuscript.

Over the last decade many inverse modelling studies have used NEE measurements from the FLUXNET network, together with other types of observations when available, to provide information about processes controlled by parameters with respect to which NEE is weakly sensitive. Though it contains an ever-increasing amount of information, the flux tower network only provides sparse coverage of terrestrial ecosystems. On the other hand, despite good spatial and temporal coverage, MODIS LAI monthly mean observations only constrain a limited set of DALECv2 state variables, and additional information is required in order to regularise the ill-posed problem and obtain a meaningful solution.

Additional information can be obtained by imposing priors on the variables or by adding other observation streams (biomass, soil organic matter, etc.). As an alternative, introduced a set of constraints, referred to as ecological and dynamical constraints (EDCs). These constraints, detailed in , can be divided into two groups: static and dynamic constraints. The static constraints which directly impose conditions on the parameters are as follows:

Turnover rate constraints which ensure that turnover rates ratios are consistent with knowledge of the carbon pools residence times.EDC1:p9<p8,EDC2:p9<p1,EDC3:p6<1/p5×365.25,EDC4:p7>p9exp⁡p10T‾,EDC5:p12+45<p15,

where T‾ denotes the mean temperature within the drivers time window. EDC4 is a modification to the constraint proposed in . It is currently used in the CARDAMON framework (http://www.geos.ed.ac.uk/homes/mwilliam/CARDAMOM.html).

demonstrated the efficiency of incorporating EDCs using a Monte Carlo method to improve parameter estimates and NEE predictions. We propose an approach to apply these extra constraints within a variational framework.

Sensitivity analysis studies how the variations of the output h of a model can be attributed to variations of the input variables xi. Such information is crucial for model design, inverse modelling and reduction of complex non-linear models. A global sensitivity analysis for DALEC was recently performed in . Here we consider a local approach where first-order derivatives are used to build sensitivity indices that help us understand the influence of input variables on the output.

We denote ht as the function that maps x = log⁡(p, C0) to the value of an output of the model (here LAI, NEE, GPP and RESP) at time t, and we denote the time series of the model output as h = (ht1, …, htN). Following , we consider the mean normalised sensitivity (MNS) defined as si=E∂h∂xi|σiσh|/∑j|∂h∂xjσjσh|, where E(⋅) denotes the average of the time series. The scalars σi and σh denote the parameter variance, set as 40 % of the parameter range, and the variance of the output respectively. The partial derivatives are computed using the adjoint derived using the method described in . The MNS si is a dimensionless number that allows us to compare among parameters.

We consider the Morgan Monroe State Forest over a 12-year period. We sample 100 parameter sets satisfying the ecological constraints. For each parameter set, we compute the MNS for DALEC simulated mean fluxes LAI and NEE. In Fig.  parameters are ranked with respect to their mean MNS. We see that for LAI only 12 out of the 23 variables are sensitive, namely p5, p17, p2, p13, p11, p15, p16, Cf, p12, p3, Clab and p14. Therefore, using LAI only in an inverse modelling experiment provides, at best, information about those twelve sensitive variables. For NEE we see that all variables are sensitive. Sensitivity analysis for GPP shows similar characteristics with LAI and so does RESP with NEE. For the four outputs under consideration (LAI, NEE, GPP and RESP) the most sensitive variables are the autotrophic respiration, p2, the annual leaf loss fraction, p5, the leaf mass per area, p17, the fraction of GPP allocated to labile pool, p13, the nitrogen use efficiency, p11, and the leaf fall day p15.

Here our focus is on the mean of the time series of DALEC fluxes (LAI, NEE) over a 12-year period. Finer analysis could be carried out by looking at seasonal aspects of the carbon cycle, identifying what variables are the most sensitive at certain times of the year, for example as studied in .

A generic inverse problem consists of finding a n dimensional state vector x such that h(x)=y, for a given N-dimensional observation vector y, including random noise, and a given model h. In the remainder of the paper the terms state vector, state variable, input variable and parameters will be used interchangeably to denote the vector x to be estimated using inverse methods and defined in the previous section as x = log⁡([p, C])T. The problem is well posed in the sense of if the three following conditions hold: (1) there exists a solution, (2) the solution is unique and (3) the solution depends continuously on the input data. If at least one of these conditions is violated the problem is said to be ill-posed. The inverse problem (Eq. ) is often ill-posed, and a regularisation method is required to replace the original problem with a well-posed problem. Solving Eq. () amounts to (1) constructing a solution x, (2) assessing the validity of the solution and (3) characterising its uncertainty. Each inverse problem has its own features which need to be understood in order to characterise properly the solution and its uncertainty.

Inverse problems are generally presented in a probabilistic framework where most methods can be expressed through a Bayesian formulation. The Bayesian approach provides a full characterisation of all possible solutions, their relative probabilities and uncertainties.

From Bayes' theorem, the probability density function (PDF) of the model state x given the set of observations y, p(x|y), is given by p(x|y)∝p(y|x)p(x), where p(y|x) is the PDF of the observations given x and p(x) is the prior PDF of x. A special case is given when p(y|x) and p(x) are Gaussian PDF given by p(x)=exp⁡-12x-x0TB-1x-x0, and p(y|x)=exp⁡-12(h(x)-y)TR-1(h(x)-y), where B is the covariance matrix of the prior term x0, and R is the covariance matrix of the observation error. When the operator h is linear then the posterior PDF p(x|y) is Gaussian and thus fully characterised by its mean and covariance matrix. The mean is obtained by minimising the modulus of the log of the joint probability distribution, which is the cost function J given by J(x)=J0(x)+Jy(x)=12‖x-x0‖B2+12‖h(x)-y‖R2. Many methods can be considered to minimise this cost function. A Monte Carlo method is employed in . Here we use a variational approach which applies a gradient-based method where the gradient is given by ∇J=B-1x-x0+HTR-1(h(x)-y), with HT denoting the adjoint operator. The covariance matrix of the solution, C, is given by the inverse of the Hessian of the cost function C=[Hess(J)]-1=B-1+HTR-1H-1. When the observation operator h is non-linear, the cost function J can have multiple local minima and the posterior PDF may no longer be a Gaussian PDF. However, locally, the PDF N(x̃, C), where C is given by Eq. () evaluated at a minimum x̃, provides a Gaussian approximation of the posterior PDF p(x|y).

The first term in the cost function (Eq. ) is a regularisation term encoding the Gaussian prior p(x). As we will show in the next sections the problem of assimilating Earth observations (LAI, GPP, NEE, RESP) into DALEC is a highly ill-posed problem and regularisation is required. The sensitivity analysis of Sect.  showed that LAI and GPP are not sensitive to all variables. Moreover, all observations streams show very low sensitivities to some variables. Therefore, as will be illustrated in Sect. , the solution (if any) is likely to be subject to large uncertainties. Apart from a couple of extensively studied sites, our prior knowledge about the variables is so far limited to their upper and lower bounds given in Table . As performed in , it is a common practice to use this information to define a Gaussian prior p(x) ∼ N(x0, B), where x0 is given by the centre of the variables ranges and B is the diagonal matrix whose diagonal elements are the squares of 40 % of the variables ranges. While using this kind of regularisation is necessary to ensure any solution at all when no better source of information is available, this introduces some biases in the solution. The EDCs introduced by provide new prior information about the variables. One of the purposes of this paper is to incorporate the EDCs as a regularisation term within 4DVAR. In the next section we propose a strategy to achieve this goal.

Incorporating the EDCs from an optimisation point of view can be easily performed by considering an inequality constraint optimisation problem where we aim at solving minxJy(x)subjecttol<x<uandg(x)<0, where g is the non-linear operator defining the EDCs described in Sect. , and l and u are the lower and upper bounds defined in Table . This approach provides an efficient, robust and quick strategy to find an acceptable solution; however, stability properties are not easily determined (see ).

We are seeking a multivariate Gaussian distribution that would encode the EDCs. At a forest site, we start by sampling the parameter space to obtain an ensemble of 1000 parameter sets satisfying the EDCs; each parameter set x is randomly created and required to satisfy g(x) < 0. We denote this ensemble by XEDCs. For most parameters, the sampling gives rise to undetermined PDFs which can certainly not be represented by Gaussian PDFs. However, upon inspecting the distribution g(x), for all x in XEDCs, we see that the distribution log⁡(g(x)) can be fairly accurately approximated by multivariate Gaussian PDFs N(c, Σ), where c denotes the mean of the distribution log⁡(g(x)) and Σ denotes its covariance matrix. As an example, Fig.  shows the marginals log⁡(g4(x)) and log⁡(g6(x)), corresponding to the EDCs 4 and 6 respectively, together with a Gaussian fit.

Using Bayes' theorem we can then write p(x|y,c)∝p(y|x)p(c|x)p(x). Finding a Gaussian approximation for p(x|y, c) amounts then to minimising the cost function J(x)=12‖h(x)-y‖R2+12‖log⁡(g(x))-c‖Σ2+12‖x-x0‖B2. The gradient of J is given by ∇J(x)=HTR-1(h(x)-y)+1g(x)GTΣ-1(log⁡(g(x))-c)+B-1x-x0, and the Hessian of the cost function can be approximated by Θ=HTR-1H-1(g(x))2GGTΣ-1(log⁡(g(x))-c)+1(g(x))2GTΣ-1G+B-1, evaluated at the minimiser x̃. The operator GT denotes the adjoint of the tangent linear model G whose key ingredient is given by the adjoint of DALECv2. The approximation of p(x|y, c) is then given by the Gaussian distribution N(x̃, Θ-1). In Sect.  we will perform experiments using real data to validate this approach.

We consider a singular value decomposition of H of the form H=USVT, where U is a N × N unitary matrix, V is a n × n unitary matrix and S is the N × n diagonal matrix whose diagonal elements are the singular values s1 ≥ … ≥ sn ≥ 0. Using this decomposition, the solution zLS to Eq. () can be written as zLS=VS†UTy=H†y. The matrix H† = VS†UT is the pseudo-inverse of H where S† is the diagonal matrix obtained by transposing S and replacing the non-zero elements with their inverse si-1. The covariance of the solution is given by CovzLS=H†TH†. Much can be learned about the stability of the solution (Eq. ) by inspecting the singular values of H. Assuming that H is full rank, it can be shown (see ) that the relative error in the solution, defined as the left-hand side of the above inequality, is bounded by ‖zLS-z0‖‖z0‖≤κ(H)‖ϵ‖‖d‖, where κ(H) is the condition number of H defined as κ(H) = s1/sn, z0 denotes the truth (possibly unknown) and ϵ represents observational noise. When the condition number is large the matrix is said to be ill-conditioned, the problem is ill-posed and the solution (Eq. ) is unstable: small perturbations to the system can lead to very large perturbations in the solution.

As an example we consider the problem of assimilating NEE observations into DALECv2 to estimate model parameters and initial conditions at Morgan Monroe State Forest. We linearise Eq. () about a point x* satisfying the EDCs, form the observability matrix H and compute its singular value decomposition. The singular values, shown in Fig. , reveal a condition number of the order of 105.

For a signal-to-noise ratio, namely ‖ϵ‖/‖d‖, of magnitude 0.1, inequality (Eq. ) gives an upper bound for the relative error in the solution of the order of 104, which does not give much credit to the least squares solution. How sharp is this bound? Are we overestimating the error? To answer these questions we create a set of noisy observations with noise variance σ = 0.1 and we compute the solution (Eq. ). The relative error for each component of the solution, ηi, and the variance νi, are given in Table . Despite a relatively good match between the modelled NEE perturbations and the observations, as shown in Fig. , the results of Table  show very large relative errors and variances for most variables. Moreover, these results are in agreement with the results of REFLEX: parameters directly linked to foliage and GPP are better estimated than parameters related to allocation to and turnover of fine root/wood. The results of Table  reflect the sensitivity analysis shown in Fig. . The variables with respect to which NEE is the most (least) sensitive are the less (more) affected by the noise.

To reduce the impact of observational noise on the solution, regularisation is required. The truncated singular value decomposition (TSVD) is a simple and popular method for regularisation. TSVD consists of truncating the pseudo-inverse in Eq. () in order to remove the smallest singular values, the most affected by the noise. The solution z(k) is then given by z(k)=VkSk†UkTy=Hk†y, where k is the truncation rank and where Sk, Uk and Vk are the rectangular matrices formed by the first k columns of S, U and V. The covariance of the solution is given by Covz(k)=VkSk†-2VkT. The truncation rank k can be chosen using the L-curve method. The L curve is a log–log plot of the norm of the solution ‖z(k)‖ against the norm of the residual ‖Hz(k) - d‖ parametrised by the regularisation parameter k. The optimal parameter corresponds to the point of maximum curvature of the L curve. Further details on the L-curve method can be found in .

In our example with NEE we use Hansen's regularisation tools (see ) to perform the TSVD method. The truncation rank obtained using the L-curve method is k = 7. The last three columns of Table , presenting the TSVD solution, the relative error of each components and the variances, can be compared with the unstable solution results. Whereas the relative errors in the unstable solution range from 5.3 × 10-2 to 3.8 × 104, the relative errors in the regularised solution range from 5.3 × 10-2 to 5.1. We see that TSVD has the effect of keeping small the variables that cannot be estimated correctly. As previously stated the results of the regularisation can be related to the sensitivity analysis depicted in Fig. : TSVD prevents the variables with respect to which NEE is the least sensitive from growing unbounded.

In the next section we consider the concept of a resolution matrix, which allows for finer analysis of the solution of the linear problem.

As suggested by Eqs. () and (), finding a solution z amounts to constructing a generalised inverse Hg such that formally z=Hgd. The generalised inverse is the operator representing any method, direct or iterative, used to solve the linear inverse problem, with or without any kind of regularisation. In the previous section we considered two examples of generalised inverse, the pseudo-inverse and the truncated inverse obtained using TSVD. The generalised inverse can be used to define operators which directly address the conditions for well-posedness for the linearised problem. Assuming a true state z* exists, possibly unknown, using Eqs. () and () we can then define an operator N called the model resolution matrix which relates the solution z to the true state z=HgHz*=Nz*. This matrix provides a practical tool to analyse the resolution power of an inverse method, that is, its ability to retrieve the true state, with or without using any regularisation method: the closer N is to the identity, the better the resolution. Moreover, the trace of the matrix defines a natural notion of information content (IC). Similarly a data resolution matrix can be defined to study how well data can be reconstructed and its diagonal elements naturally define a notion of data importance. For the two examples of generalised inverse presented in the previous section we obtain the following resolution matrices: N=H†H, for the pseudo-inverse and N=VkVkT, for the truncated pseudo-inverse. In the first case the IC equals the number of non-zero singular values, in the second case the IC equals the truncation rank k. An in-depth theoretical and practical analysis of these concepts and those introduced in the remainder of this section can be found in .

While the model resolution matrix allows us to see how a solution strategy maps the true state variables to the solution of the inverse problem, and to see how well and how independently the state variables can be recovered, one also needs to assess the uncertainty of the solution. This can be studied using the so-called unit covariance matrix, C, defined using the generalised inverse as C=HgTHg. By characterising the degree of error amplification that occurs in the mapping from the true state to the solution of the inverse problem, the unit covariance matrix is a crucial object for studying the stability of the solution with respect to observational noise. The unit covariance matrix defined by Eq. () agrees with the covariance matrices given in the previous section by Eq. () for the pseudo-inverse, and by Eq. () when TSVD is applied.

We now study the model resolution matrix for the LAI observation operator at Morgan Monroe State Forest. In the first instance we will demonstrate the resolution power of the LAI signal without regularisation using the pseudo-inverse as generalised inverse first, and then apply TSVD to show how using the truncated pseudo-inverse affects resolution. In a second case we will study the added value of the EDCs in terms of resolution.

As previously, we linearise Eq. () about the point x* given in Table . The trace of the resolution matrix obtained using the pseudo-inverse as generalised inverse is 10, and this means that 10 independent variables can be estimated using LAI. These independent variables are not the variables in which the system is expressed, but a linear transformation can be found to express the system in terms of the independent variables. Figure  shows the model resolution matrix for LAI. As shown in Sect.  with the sensitivity analysis, 11 out of the 23 variables are not sensitive to LAI, and this can be seen in the resolution matrix by the diagonal terms which are zero, represented in blue. In contrast the diagonal elements corresponding to sensitive variables have positive values, represented by colours ranging from light blue to red. Figure  also shows that whereas p5, p11, p12, p14, p15 and p16 are perfectly resolved (the corresponding elements are coloured brown or dark red), there exist linear combinations between the remaining sensitive variables, which explains why only 10 independent variables can be estimated from the 12 sensitive variables.

For the study of the unit covariance matrix we restrict ourselves to the sensitive variables. This amounts to removing the columns corresponding to the non-sensitive variables, containing only null elements, from the observability matrix. The dependency of the solution on observational noise can be studied by looking at Fig. , where the diagonal elements of the unit covariance matrix, corresponding to the variance of each element of the solution obtained using the pseudo-inverse, are represented in log scale. Except for p5, p12, p15 and p16, all variances are shown to be large.

As previously, we illustrate a simple regularisation strategy, TSVD, and show its effects on both resolution and stability. Figure  shows the resolution matrix for LAI with optimal truncation rank k = 6. The IC decreases to 6. We see that whereas p5, p12, p15 and p16 remain almost perfectly resolved, p13, p17 and Clab are only partially resolved and the remaining variables are not resolved properly. Figure  shows the corresponding diagonal elements of the unit covariance matrix, from which we see that the variances have been drastically reduced. This example shows how regularisation ensures stability at the price of losing resolution.

We now consider the effect of incorporating the static EDCs into the variational framework in terms of resolution. The static EDCs are given by the first seven EDCs, the linear problem is then given by HG̃z=df, where G̃=gx*-1Σ-1/2G, with Σ-1/2 the inverse of the symmetric square root of the covariance matrix Σ, defined in Sect. , restricted to the first seven components. The static EDCs depend only on 13 out of the 23 variables, namely p1 to p10, p12, p13 and p15. This can be seen on the matrix G where the columns corresponding to the remaining variables are null. Together with LAI observations, whose sensitive variables are represented in Fig. , we therefore have 19 sensitive variables. The model resolution matrix corresponding to the operator on the left-hand side of Eq. (), obtained using the pseudo-inverse, is depicted in Fig. . The trace of the model resolution matrix gives an IC of 16, 13 variables are perfectly resolved and 4 variables show linear dependencies (p2, p3, p4 and p13). However, although p9 and p10 are sensitive variables, they do not appear to be resolved at all: inspecting the linear operator G shows that the non-zero components corresponding to p9 and p10 are several order of magnitude smaller than the other components.

This example shows clearly the benefit of introducing the static EDCs to help estimate poorly constrained or otherwise undetermined components.