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# Testing variational estimation of process parameters and initial conditions of an earth system model

Deceased

## Abstract

We present a variational assimilation system around a coarse resolution Earth System Model (ESM) and apply it for estimating initial conditions and parameters of the model. The system is based on derivative information that is efficiently provided by the ESM’s adjoint, which has been generated through automatic differentiation of the model’s source code. In our variational approach, the length of the feasible assimilation window is limited by the size of the domain in control space over which the approximation by the derivative is valid. This validity domain is reduced by non-smooth process representations. We show that in this respect the ocean component is less critical than the atmospheric component. We demonstrate how the feasible assimilation window can be extended to several weeks by modifying the implementation of specific process representations and by switching off processes such as precipitation.

Keywords:
How to Cite: Blessing, S., Kaminski, T., Lunkeit, F., Matei, I., Giering, R., Köhl, A., Scholze, M., Herrmann, P., Fraedrich, K. and Stammer, D., 2014. Testing variational estimation of process parameters and initial conditions of an earth system model. Tellus A: Dynamic Meteorology and Oceanography, 66(1), p.22606. DOI: http://doi.org/10.3402/tellusa.v66.22606
Published on 01 Dec 2014
Submitted on 14 Aug 2013

## 1. Introduction

State-of-the-art climate predictions rely on numerical models of the earth system. One of the major sources of uncertainty in these predictions is the correct representation and parameterisation of the processes underlying the climate system (see e.g. Cubasch et al., 2001). A further source is the uncertainty in the initial state, that is, the state of the climate system at the beginning of the integration. Systematic use of observational information has the potential to reduce both types of uncertainty. Due to their high complexity, state-of-the-art earth system models (ESMs) are extremely demanding in terms of computer time. This complicates the systematic estimation of process parameters (calibration) and of the initial state (initialisation) from observations.

These systematic approaches can, thus, typically only be pursued for models with reduced spatio-temporal resolution, simplified process representations, and/or reduced sets of uncertain (tunable) parameters. For example, Jones et al. (2005) employ FAMOUS, a reduced resolution version of its parent general circulation model HadCM3 to demonstrate the systematic tuning of eight process parameters. This subset of the full parameter space, which for the atmosphere component alone has about 100 dimensions (Murphy et al., 2004), had to be kept small for computational reasons. This is because even a parameter space of as few as eight dimensions can only be efficiently searched for an optimal parameter set by a gradient algorithm. Such gradient algorithms minimise the model–data misfit quantified by a cost function through the use of the cost function's gradient. Jones et al. (2005) had to restrict the dimension of the control space because they approximated the gradient (i.e. sensitivity) information in the optimisation procedure by inaccurate finite difference calculations of multiple model runs (depending on the chosen perturbation size), at a computational cost proportional to the number of tunable process parameters.

Computing parameter sensitivities with the adjoint avoids any restriction on the dimension of the parameter and initial state space. This is because the associated computational cost is independent of this dimension, as will be explained in Section 2.2 below. This concept has been demonstrated for several components of the earth system. To the atmospheric component, the adjoint approach is being routinely applied at operational centres for numerical weather prediction (NWP) for forecast initialisation (see e.g. Rabier et al., 2000). Adjoint-based calibration has been demonstrated (e.g. Blessing et al., 2004; Kaminski et al., 2007) for the Portable University Model of the Atmosphere (PUMA, Fraedrich et al., 2005c).

For the ocean component of the earth system, an adjoint-based assimilation system has been operated for more than a decade (Stammer et al., 2002, 2003). It is built around the MITgcm (Marshall et al., 1997a, 1997b) and infers a combination of initial and boundary conditions of the ocean circulation. Meanwhile, multiple versions of the system are being applied by several research groups around the world in different setups (e.g. Hoteit et al., 2005; Köhl and Stammer, 2008). As another example, the adjoint (Kauker et al., 2009) of the Arctic coupled sea-ice ocean model NAOSIM is employed to initialise seasonal predictions of the Arctic ice conditions (Kauker et al., 2010).

For the terrestrial biosphere component, this approach is demonstrated by the Carbon Cycle Data Assimilation System (CCDAS, http://ccdas.org, Rayner et al., 2005; Scholze et al., 2007; Kaminski et al., 2012, 2013), which performs a combined parameter and initial state estimation in the terrestrial biosphere model BETHY (Knorr and Heimann, 2001). CCDAS also features uncertainty propagation, based on second derivative information. The CCDAS concept is being transferred (see e.g. Luke, 2011; Kuppel et al., 2012; Kaminski et al., 2013; Schürmann et al., 2013) to several further terrestrial biosphere models (JSBACH, JULES, ORCHIDEE), all of which are component models in ESMs that contribute climate projections to the IPCC's 5th assessment report.

The construction of an analogous assimilation system around an entire ESM is clearly desirable. Such a system could allow, for example, the initialisation of climate model predictions in a way consistent with model dynamics. Another application could be the use of paleo records as constraints on the process parameters of the underlying ESM. Furthermore, the impact of all process parameters and the initial state on the model's climate sensitivity could be rigorously assessed in a single adjoint run. First steps into this direction were taken by Lee et al. (2000) and Galanti et al. (2003) who used ocean models (in the first case a beta plane model, and in the latter the MOM3 general circulation model) coupled to a simple statistical atmospheric component, derived through a singular value decomposition.

One of the challenges associated with the set-up of an assimilation system around an entire ESM is of a technical nature, imposed by the model's code size and complexity. For many of the above-listed component models (PUMA, MITgcm, NAOSIM, BETHY, JSBACH, JULES, ORCHIDEE), the derivative code has been generated by an automatic differentiation tool (Transformation of Algorithms in Fortran (TAF), Giering and Kaminski, 1998). Sugiura et al. (2008) pioneered assimilation into an ESM by coupling the adjoints of their component models. This approach is tedious, error prone, and inflexible as it requires hand coding the coupling on the derivative code level. The alternative approach, which consists of automatic differentiation of the entire ESM, has not been pursued yet. The present study demonstrates, for the first time, the feasibility of this coupled model differentiation, using an ESM consisting of the Planet Simulator (PlaSim, Fraedrich et al., 2005a, 2005b; Fraedrich, 2012) coupled to MITgcm (Marshall et al., 1997a, 1997b).

A more fundamental challenge results from the non-linearity of the climate system: The usefulness of derivative code depends on the capability of the linearisation around a point to represent the model in the point's neighbourhood. This capability is closely connected with the concept of predictability, which Lorenz (1963) analysed for a non-linear three-dimensional system that possesses a strange attractor. Lea et al. (2000) use this system to demonstrate that the usefulness of the linearisation of the long-term mean of the state variables around the system's parameters decreases with increasing integration period. Köhl and Willebrand (2002) analyse how this affects the parameter estimation from the long-term mean state via a gradient method for the same model as well as for a high-resolution quasi-geostrophic model of the oceanic circulation. In this estimation context, the poor linearisability of the long-term mean shows up in the form of multiple local minima in the model–data misfit. Köhl and Willebrand (2002) as well as Thuburn (2005) extend the adjoint approach by a statistical concept to enhance the usefulness of the gradient information. Pires et al. (1996) using the Lorenz model and Tanguay et al. (1995) using a β-plane model address the linearisation problem in the context of four-dimensional variational data assimilation, estimating initial conditions that minimise the model–data misfit. Pires et al. (1996) and Swanson et al. (1998) present a quasi-static variational assimilation approach that tracks the absolute cost function minimum through successive increments of the assimilation window. In the adjoint-based assimilation system around their ESM, Sugiura et al. (2008) are improving linearisability through the simulation of time-averaged fields and an approximate adjoint with an artificial damping term following an initial calibration of seven parameters through a Green's functions approach. Abarbanel et al. (2010) also suggest a variable damping term, and present an analysis of its effect on their cost function. A summary of the linearisation topic is provided by Lorenc and Payne (2007), together with a sketch of a seamless four-dimensional variational assimilation approach, which models probability density functions for the uncertain, small-scale processes.

In NWP, it is common to run the assimilation with ‘simplified physics’, that is, to remove a set of particularly non-linear processes or replace them by less complex and smoother formulations (Rabier et al., 2000). What is feasible for the short assimilation windows typical for NWP can be problematic on longer time scales, where it may result in considerable biases in the simulated state of the system. For the terrestrial biosphere component, it has been shown (Knorr et al., 2010; Kaminski et al., 2012) that the performance in the above-mentioned CCDAS is considerably improved by reformulation of some crucial process formulations (e.g. of leaf phenology). Formulations that rely on step functions or non-differentiabilities were replaced by formulations that resulted in smooth dependency of the simulation on initial conditions and process parameters. For the phenology this was achieved by adopting a statistical concept (Knorr et al., 2010) as opposed to a concept that, simply speaking, simultaneously removes all leaves within a given grid cell. The current study transfers this reformulation concept to our ESM.

A useful diagnostic for the performance of an adjoint-based assimilation system is the length of the feasible assimilation window, that is, the assimilation window over which the system can successfully operate. For a coarse resolution version of PUMA, the atmospheric component of our ESM, Kaminski et al. (2007) demonstrated feasible assimilation windows of up to 100 d for parameter estimation. Here we use the same diagnostic to study the performance of the assimilation system around our ESM.

The layout of the remainder of this paper is as follows. Section 2 will present the components of the assimilation system, Section 3 will describe the experimental setup, and Section 4 will present the results. Section 5 will provide a discussion and Section 6 a summary and conclusions.

## 2. The data assimilation system

### 2.1. The model

The ESM introduced here is the CESAM1 (CEN Earth System Assimilation Model). It consists of the PlaSim (Fraedrich et al., 2005a, 2005b; Fraedrich, 2012) coupled to the MITgcm (Marshall et al., 1997a, 1997b). The relevant components of the PlaSim include the spectral PUMA (Fraedrich et al., 2005c), including schemes for radiation, cloud cover, precipitation, runoff, soil temperature and wetness, surface fluxes, a thermodynamic sea-ice model, and a terrestrial biosphere component (SIMBA). The MITgcm is a state-of-the-art finite volume model of the general oceanic circulation, including a model of sea-ice dynamics and rheology (Zhang et al., 1998).

In the coupling, sea surface temperature and salinity are computed by the ocean model and used by the atmospheric model. In turn, the atmospheric model passes back heat flux, precipitation minus evaporation, runoff, wind stress, and, optionally, short wave radiative heat flux, atmospheric surface pressure, and snow and ice mass. Of the optional quantities, we use only short wave radiative heat flux since the sea-ice component of the MITgcm is deactivated in the present study. Instead, the thermodynamic ice model of PlaSim is used. In the current setup, the models run in turns, and the exchanged quantities are interpolated between the grids.

For all experiments, a resolution of 4° in the ocean and 5.6° (T21) in the atmosphere and land surface components is used. A time-step of 8 hours is used in the ocean and 48 minutes in the atmosphere. Configurations marked ‘slow’ use a 20-minute time-step in the atmosphere, and in one case (Exp. 4 of Table 3 described in Section 3), even a 10-minute time-step.

A number of modifications were made to PlaSim in order to enhance its performance in a variational assimilation system (see Appendix). Two configurations, called ‘standard’ and ‘minimal’ are used. ‘Standard’ uses most of PlaSim's components except for the terrestrial biosphere model, while in ‘minimal’ also the hydrological cycle is excluded, that is, evaporation, precipitation, and runoff. Moreover, in the moisture-free ‘minimal’ atmosphere there is no cloud-radiative feedback and the soil moisture is set to climatology. Configurations marked ‘w/o ocean’ replace the ocean with climatological sea surface temperature (SST). Table 1 gives an overview for quick reference. We further use the tags soft to mark experiments which do use smooth replacements for some occurrences of the if, where, min, max, abs, etc. statements, which proved problematic in initial tests, and hard for those which do not (see Appendix for details).

### 2.2. Assimilation

We use the observational information to constrain a vector of control variables, which can be a combination of initial and boundary conditions as well as parameters in the process formulations of the model. Our experiments will investigate several choices of control vectors summarised in Table 2. P10 is a control vector of process parameters from PlaSim, controlling the time scale for Rayleigh friction in the uppermost two atmospheric layers, the diffusion time scales for divergence, vorticity, and temperature, the point of mean long wave radiation transmissivity in a layer, and four degrees of freedom of diffusion and surface fluxes. I2 controls a globally uniform perturbation of initial conditions of atmospheric surface pressure and temperature at all levels. I4 is as I2, but additionally includes global-scale perturbations of salinity and temperature of the ocean. Finally, I3D controls a gridpoint-wise perturbation of atmospheric vorticity, divergence, surface pressure, and temperature, as well as of oceanic salinity and temperature.

Our assimilation system implements a probabilistic inversion concept (see Tarantola, 2005) that describes the state of information on a specific physical quantity by a probability density function (PDF). The prior information on the control variables is quantified by a PDF in control space and the observational information by a PDF in the space of observations, at all sampling times and locations. Their respective means are denoted by xprior and d and their respective covariance matrices by Cprior and Cobs, where Cobs accounts for uncertainties in the observations as well as uncertainties from errors in simulating their counterpart (model error). If the prior and observational PDFs were Gaussian and the model linear, the posterior PDF would be Gaussian, too, and completely characterised by its mean xpost and its covariance matrix Cpost. Further, xpost would minimise the following cost function:

where M(x) denotes the model operated as a mapping of the parameters onto simulated counterparts of the observations. In the non-linear case, we approximate the posterior PDF by a Gaussian with mean value xpost, which is also termed maximum a posteriori probability (MAP) estimate. Without the prior term, it is termed maximum likelihood estimate (MLE). The first term of eq. ((1)) quantifies the model–data misfit (observational term) and the second term the prior information. In NWP the latter term is called background term.

Figure 1 shows a schematic illustration of a cost function that includes a step function. The red curve displays the observational term (for perfect model and observations) with the step function while the green curve displays the effect of smoothing the step. In this case, the smoothing is not strong enough to avoid a secondary minimum. Adding a strong enough prior term (dark blue) removes the secondary minimum (magenta).

Fig. 1

Schematic illustration of a cost function that includes a step function, including the effects of smoothing and prior (background) term.

All of our experiments (see Section 3), with the exception of one, use pseudo observations produced from known true values of the control variables without added noise. In this context there are three options for the prior term:

Options (i) and (ii) allow us to assess the progress of the iterative minimisation of J(x) through the difference between the current and the true values of the control variables. For a successful minimisation, this difference, for example, expressed as a Euclidean norm, should converge to zero. By contrast, for Option (iii) we would expect the prior term to shift the minimum from the true value towards the prior. This is why we discard this option. We note, however, that Option (iii) is the usual choice for assimilation of real data, and is particularly important in underdetermined setups. The effect of Options (ii) and (iii) is to smooth the cost function, and thus mask potential problems in the observational term. Figure 1 schematically illustrates the smoothing effect for Option (ii). For Option (iii), the effect of the prior term will depend on its location relative to the two minima in the observational term. For our experiments, we choose Option (i) in order to make a clear assessment of the properties of the observational term. We demonstrate, however, the effect of including a prior term (Option (ii)) for two of our experiments, which use the P10 control vector (described in Section 3 and Table 2) with standard deviations set to 100% of the respective parameter values and zero off-diagonal elements in the uncertainty covariance matrix Cprior. We also use the prior uncertainty to scale the control vector in all our experiments, that is, the control vector is quantified in multiples of the prior uncertainty.

The assimilation consists of an iterative minimisation of J through variation of x by a quasi-Newton algorithm (Fletcher and Powell, 1963). This procedure determines the search direction through the gradient of J with respect to x. This gradient of the cost function with respect to the control variables is provided by automatic differentiation (Griewank, 1989) of the source code through TAF (Giering and Kaminski, 1998).

The automatic differentiation procedure decomposes the code that evaluates the entire function J into simple elementary functions such as ‘+' or ‘sin’ for which the derivative (local Jacobian matrix) is known. By applying the chain rule of calculus to the sequence of local Jacobians, the derivative of the composite function can then be evaluated accurately up to rounding error. This multiple matrix product can be evaluated in arbitrary order. The tangent linear model (TLM) uses the same order as the evaluation of the function, while the adjoint model (ADM) uses the reverse order. Both yield (up to rounding error) identical results for the gradient of J(x) of eq. ((1)), but the memory and CPU time requirements differ. While the requirements for the gradient calculations using one TLM run per control variable are proportional to the number of control variables, the requirements using the ADM are proportional to the number of dependent variables but virtually independent of the number of control variables. An efficient TLM and ADM pair (comprising 174 000 and 387 000 lines of Fortran code excluding comments, respectively) was generated through TAF. The TLM requires the CPU time of about 2.3 model runs to provide a single gradient component and the function value, while the ADM requires the CPU time of about four model runs to provide the entire gradient and the function value. This includes an efficient two-level-check-pointing scheme (Griewank, 1992) to allow long integrations. The model's MPI parallelisation capabilities were preserved in the derivative code without degradation of the above performance ratios. We note that repeated invocation of TAF can be used to generate code for evaluation of higher-order derivatives. For example, Kaminski et al. (2003) describe the generation of code for evaluation of the Hessian.

We note that TAF relies on a number of global analyses, that is, analyses of the entire function code. For example, an activity analysis traces all variables on the path from control variables to the cost function value. This analysis also covers the interfaces of the component models with the coupler. Treating the differentiation of the model components and the coupler separately, as demonstrated by Sugiura et al. (2008) for their ESM, would have required the user to perform this activity analysis and assure the correct coupling on the level of the generated component derivative codes. Even though there was pre-existing derivative code for some of the model components (Marotzke et al., 1999; Blessing, 2000; Blessing et al., 2008; Rivière et al., 2009), we apply our coupled model differentiation approach. This means we apply TAF to generate the derivative of J with respect to the parameters x to the entire coupled model at once. Thus, TAF automatically generates a single derivative code for all coupled model components, including the coupler. No further hand-coding is required, that is, for the reasons given, safer, more flexible and sustainable. In the context of this study, it allowed us to generate derivative code versions for a variety of combinations of model configurations, control vectors, and observational data sets.

### 2.3. Data sets for assimilation

‘Identical twin’ experiments use pseudo-data in the assimilation, which were generated with the model itself from a prescribed control vector, thus guaranteeing full consistency of data and model, and allowing us to know the ‘true’ control vector. For one other experiment, data interpolated from ERA-40 simulations (Uppala et al., 2005) are used in the atmosphere. In either case, data are provided at all grid points and levels of the respective subsystem, with the exception of atmospheric temperature at the uppermost level. For the atmosphere we are using vorticity, temperature, and surface pressure, while in the ocean these are temperature and salinity. As in Köhl and Willebrand (2002), all data are time-averaged over the assimilation window and no noise is added. Consequently, the cost function is evaluated at the end of the assimilation window. Given the spatial resolution of T21 and 10 levels in the atmosphere and 4° and 15 levels in the ocean, this amounts to 40 960 time-averaged observations from the atmosphere and (restricting to wet points) 61 942 from the ocean, totalling 102 902.

For Cobs of eq. ((1)), off-diagonal elements where set to zero, that is, we assume uncorrelated uncertainty. The diagonal elements are the squares of the following standard deviations: In the atmosphere we use 3 K for temperature, 5 hPa for surface pressure, and 2×105 1/s for vorticity. Our ocean uncertainties vary in space with standard deviations of 0.4–3.1 K for temperature and 0.15–0.8 PSU for salinity with the higher values towards the surface. These numbers are a rough guess of the actual uncertainties and effectively determine the relative weight given to the individual observation.

## 3. Experimental setup

Our experiments are designed to verify the correctness of the derivatives and to identify potential problems under a variety of situations. They present the first steps towards an assimilation system in a coupled model environment. We will examine several combinations of model configurations and control vectors.

For each of the model configurations, we generate a consistent snapshot of the model state (restart file) recorded at the end of a 10-yr integration. The restart file for the experiment with the ERA-40 data is derived by optimising the atmospheric initial conditions in a pre-assimilation over 200 iterations. This procedure uses as additional term in eq. ((1)), a surface pressure tendency penalty to suppress gravity waves as given in eq. 2.4 of Zou et al. (1993), summed over all time steps during the first 6 hours of a 1-d assimilation window, while the observational constraint was constructed as the time-averaged data over the full assimilation window. All but the aforementioned experiment will be conducted as identical twin experiments that assimilate pseudo data. This means we use default values of the control vector to generate pseudo data. Next, we start the iterative assimilation procedure from a perturbed control vector. For the identical twin experiments with short control vectors, that is, P10, I2, I4 as defined in Table 2, we call an experiment successful if we can accurately (Euclidean distance to default reduced by at least five orders of magnitude) recover the default parameter values through assimilation of the data, with a strongly (by more than five orders of magnitude) reduced gradient of the cost function.

Now, in the most favourable case of a linear model, we can expect a solution of this type of inverse problem to take as many iterations as there are components in the control vector (Powell, 1964). Since for an ESM one can usually only afford an iteration number of a few tens or hundreds, we can only expect setups with low dimensional control vectors to converge. For the setups with high-dimensional control vector, that is, I3D, we will only perform 20 iterations. In this context, we call an experiment successful, if reductions of the cost function and of the Euclidean distance of the control vector to the default values are achieved at the same time. In an apparently underdetermined setup such as I3D (with 125 430 control variables constrained by 102 902 observations) the improvement of the control vector is of particular importance.

Within the iterative minimisation, the trajectory of the control vector through the control space is highly dependent on the initial parameter vector. This means that two minimisations starting from neighbouring control vectors typically explore quite different regions in control space even if they converge to the same minimum. For an example, see Fig. 4 in Clerici et al. (2010). To assess the robustness of our experimental results, we carry out each experiment as a small ensemble with four members, each of which starts from a different point in control space. For the identical twin experiments with the I4 control vector the first member starts with the following perturbations of the control vector: in atmosphere and ocean 0.1 K for temperature, 1 per mil of the atmospheric surface pressure, 0.1 PSU for salinity. For the other members, the same magnitudes are used, but with varied signs. For example in the ‘min w ocean’ configuration this uniform initial state perturbation yields, after a 26 d integration, a perturbation of about 1.5 K in the lowest atmospheric layer (standard deviation), with maximum values of 18 K and 10 hPa. For the P10 control vector, a 10% perturbation of each component is used. The I3D control vector uses a globally uniform perturbation of the same magnitude as in the I4 case, but sign and magnitude are varied to generate the other ensemble members. An ensemble size of four is low, but appears to be sufficient for a first assessment.

## 4. Results

First we address model parameter estimation, that is, the control vector P10, in a set of identical twin experiments. Table 3 summarises our experimental results. Exp. 1 shows that in the most complex configuration ‘std w ocean’, we cannot even reliably recover our parameter vector over a 1-d assimilation window. Three out of four ensemble members fail to find a minimum. The minimisations get stuck at edges in the cost function. This is because, with a given minimal step size along a local downhill direction that is pointing towards an upward jump, the optimisation algorithm cannot achieve any further decrease of the cost function. Figure 2 illustrates this situation, where the stopping point is on the right hand side of the jump, and except for the jump point the cost function has ascending slope, that is, the downhill direction points towards the left. In our model such jumps can typically be traced back to if-statements in the convective precipitation. Removing the ocean does not help (Exp. 2). What helps is the simplification of the atmosphere (configuration ‘min w ocean’, Exp. 3). The simplification of the atmosphere can be regarded as a step towards the setup of the Kaminski et al. (2007) study, where the atmosphere is even reduced to its dynamical core. To approach the 100-d assimilation window of the Kaminski et al. study, we test a configuration ‘slow w/o ocean’ which removes the ocean, simplifies the atmosphere, and reduces its time step. The reduced time step is motivated by the study of Zhu and Kamachi (2000), who report stability problems for the linearisation of certain numerical time integration schemes. The reduced time step can thus be regarded as a way to render the cost function more regular. The configuration ‘slow w/o ocean’ works robustly for an assimilation window of 56 d (Exp. 4, Fig. 3).

Fig. 2

Cost function over a section of control space at the stopping point of one of the unsuccessful members of Exp. 1. Except for the x-value of the jump, the curve has a very small positive derivative (about 0.041 left, and 0.033 right of the jump), that is, an ascending slope. The units of the x-axis are relative to the stopping point.

Fig. 3

Convergence of the minimisation for control vector P10, configuration ‘slow w/o ocean’, assimilation of pseudo observations, and a 56-d assimilation window (Exp. 4): Cost function (solid red, ‘+’), norm of its gradient (green dashed, ‘×’), and absolute difference of components of control vector to true value over iteration number (par 1–10, see legend).

We note that repeating, as a test, Exp. 1 with a prior contribution still yields one successful member while repeating Exp. 3 with a prior term does increase the number of successful members from two to four. This behaviour confirms our expectation from Fig. 1: While a prior term cannot avoid step functions (probably induced by atmospheric processes in the ‘std’ configuration) it can help to avoid secondary minima in the smoother atmospheric ‘min’ configuration.

Exp. 5 tests the Exp. 4 setup with ERA data instead of pseudo data. Over an assimilation window of 1 d all four members converge to very proximate minima. Figure 4 shows the convergence for one of the members, which reduces the gradient norm by more than five orders of magnitude. The value of the cost function is reduced by 1%, and the parameter vector is considerably changed. The procedure has apparently found a minimum which is not necessarily a global one, but it is also possible that the model in the minimal configuration just cannot match the ERA-data any better. We note that the value of χ2, that is, twice that of the cost function at the minimum (Tarantola, 2005) of about 7414 is about a factor of 5.5 smaller than expected when assimilating 40 960 (≈5.5·7417) observations (see Section 2.3) to estimate 10 unknown parameters (see Table 2) without the use of prior information. One could fix this by scaling down our data uncertainties by a factor of $\sqrt{5.5}$ (see e.g. Ménard and Chang, 2000). We do not do this here because in the absence of a prior term a uniform scalar has no impact on the minimisation. Figure 5 shows that the estimated parameter vector achieves a slight improvement of the predictive skill beyond the assimilation window.

Fig. 4

Convergence of the minimisation for control vector P10, configuration ‘slow w/o ocean’, assimilation of ERA observations, and a 1-d assimilation window (Exp. 5): Norm of its gradient (top), cost function (centre), and absolute difference of the components of the control vector to the default value (labelled ‘true’ value in ID-twin experiments, bottom) over iteration number.

Fig. 5

RMS of temperature difference during and after assimilation window for Exp. 5.

Next, we present experiments where we estimate initial conditions (control vectors I2, I4, and I3D). We start with the most complex configuration, ‘std w ocean’, and the I4 control vector, which works for assimilation windows of 1 d with (Exp. 6) and mostly without (Exp. 7) soft switches. For a 3-d assimilation window, the assimilation does not work anymore (Exp. 8). Removing the ocean (Exp. 9) does not help. We can, however, achieve considerable extensions of the assimilation window if we simplify the atmospheric component. Configuration ‘min w ocean’ is mostly successful for an assimilation window of 26 d with (Exp. 10) and without (Exp. 11, Fig. 6) soft switches. Interestingly, reducing the time step deteriorates the estimation of initial conditions (Exp. 12), even though it improved the estimation of parameters. Fig. 7 shows the cost function over a section of the control space from the true value to the first guess of the first member of Exp. 12. At this large scale the cost function looks smooth. Note also the high curvature (expressed as the second derivative) of the cost function at the minimum of about 100 000, compared to a curvature of the prior term (not used in the experiment) of 1. This indicates a strong constraint by the observations. The discontinuities which hamper the minimisation are of the type shown in Fig. 2 and only visible at much finer scales.

Fig. 6

As Fig. 4 but for convergence of the minimisation for control vector I4, configuration ‘min w ocean’, assimilation of pseudo observations, and a 26-d assimilation window (Exp. 11).

Fig. 7

Cost function over a section of control space from the true value (origin) to the first guess (marked with vertical line) of the first of four (unsuccessful) members of Exp. 12 (control vector I4, ‘slow w ocean’; solid line, ‘+’) and a parabola fitted at the known minimum (dashed line; second deriv. is about 100 000).

As mentioned, we use the low dimensional control vectors because they have the potential to converge within an affordable number of iterations. For initialisation of climate predictions, however, we want to correct the 3D structure of the initial field. Our final set of experiments tests this type of control vector (I3D) and limits the number of iterations of our gradient algorithm to 20. Recall that in this context we call an experiment successful, if reductions of the cost function and of the Euclidean distance of the control vector to the default values are achieved at the same time. Over 1 d, our most complex configuration ‘std w ocean’ (Exp. 13) has three successful ensemble members (9, 13, and 14% reduction of norm of parameter difference to truth and cost function reductions of 32, 20, and 19%), while one member got stuck without reduction in the norm of the parameter difference to truth nor in the cost function. By contrast, over the same assimilation window in configuration ‘min w ocean’ (Exp. 14) all four ensemble members are successful, with 11, 12, 15, and 15% reduction of norm of parameter difference to truth and cost function reductions of 34, 20, 30, and 20%. In this latter configuration, three ensemble members were also successful for an assimilation window of 26 d (Exp. 15, Fig. 8) with 2, 5, and 10% reduction of norm of parameter difference to truth and cost function reductions of 10, 17, and 20%, while the same assimilation fails for the configuration ‘std w ocean’ (Exp. 16). Running the configuration ‘min w ocean’ with soft switches (Exp. 17) again yields three successful ensemble members with comparable results (parameter vector: 5, 7, and 10% reduction; cost function: 25, 14, and 28% reduction).

Fig. 8

As Fig. 4 but for convergence of the minimisation for control vector I3D, configuration ‘min w ocean’, assimilation of pseudo observations, and a 26-d assimilation window (Exp. 15).

## 5. Discussion

Running the uncoupled atmospheric model does not yield results superior to the coupled one, as we see in Exp. 2/1 and Exp. 9/8. This reflects the fact that the processes in the ocean model operate on longer time scales than those in the atmospheric model. In particular, switching off fast atmospheric processes such as precipitation increases the feasible assimilation window. The non-linear behaviour of these processes is aggravated by their numerical implementation, which often incorporates non-differentiable statements. A step function, for example, produces discontinuities in the cost function, which may provide an obstacle for gradient-based minimisation. In this study, the replacement of some of these formulations by differentiable approximations in the soft experiments has been limited to just a few parts in PlaSim. Hence, we expect future studies to reveal the full potential that lies in the reformulation of such processes in a differentiable way, possibly using statistical concepts as demonstrated by (Knorr et al., 2010) and (Kaminski et al., 2012).

The size of the time-step in dynamical models is typically a trade-off between performance and simulation quality. A long time-step results in a fast model integration, while a short time step typically reduces discretisation error and thus enhances the quality of the simulation. Exceeding a certain threshold (imposed by the CFL criterion) even results in an unstable integration. In our experiments, a reduction of the atmospheric time step improves the performance for parameter estimation (Exp. 4) but deteriorates it for the estimation of the initial state (Exp. 12). An obvious qualitative difference between parameter and initial state estimation is that model parameters directly influence the cost function at each time-step throughout the integration, while the influence of the initial state is indirect, because it has to be propagated from time step to time step through the integration of the dynamical system. This propagation has the potential to dampen or complicate the structure of the sensitivity of the cost function to an initial state change. Our experiments may show the balance of two mechanisms with opposite effect. On one hand a reduced time step enhances stability of the linearised model (Zhu and Kamachi, 2000), but on the other hand it increases the number of time steps required to cover a given assimilation window. It may be that the first mechanism is dominant for parameter estimation while the second mechanism is dominant for the estimation of initial conditions. To confirm these findings further research is required, including theoretical studies with simple models.

Assimilation of ERA-data in Exp. 5 certainly is more challenging than the identical twin experiments. Even though we used an assimilation procedure to prepare the initial state for the experiment, it is obvious from the RMS error growth in Fig. 5 that the model trajectory quickly diverges from the data. Still it is possible to improve this situation slightly by the parameter estimation. The difficulty lies in a combination of four factors: First, the atmospheric configuration is simplified. Second, the model resolution is coarse compared to the data source. Third, despite the pre-assimilation procedure the initial state is still sub-optimal. Fourth, the P10 control space is small. We can eliminate the first two of these factors by repeating the same procedure with the ERA data replaced by observations generated by the model from initial conditions of a different year. In this case the cost function reduction is stronger by a factor of more than 10. This indicates the potential for a more realistic model and higher resolution.

## 6. Summary and conclusions

We demonstrated a coupled model differentiation approach that applies, for the first time, automatic differentiation to an entire ESM at once. The generated derivative code is efficient and easy to maintain and to adapt to changes in the ESM code. No hand-coding is required at the derivative level. We further constructed a variational assimilation system around the ESM and demonstrated the assimilation of pseudo observations as well as of an atmospheric data set based on ERA, and addressed estimation of process parameters and initial conditions. For both applications, using pseudo and ERA data, we quantify the performance of the assimilation system by the length of the feasible assimilation window.

The focus of this study lies on the behaviour of the adjoint ESM in a standard assimilation environment, rather than on the construction of a sophisticated assimilation system, for example, with split in inner and outer optimisation loops (see e.g. Rabier et al., 2000). We also refrained from using a prior (or background) term in the cost function (eq. 1) in our experiments (except for a demonstration) in order to make a clearer assessment of the constraints on the coupled model provided by the observational term. Through its parabolic contribution to the cost function, a prior term would have stabilised the inverse problem. This would have clearly facilitated the minimisation and possibly would have masked convergence problems imposed by the observational term. In that respect we can regard our assessment as conservative.

We find that the performance of the coupled model in the assimilation system is highly dependent on the selection of atmospheric processes and their implementation. A reduced atmospheric configuration with a number of processes deactivated shows significantly better performance than the standard configuration, while inclusion or exclusion of the dynamical ocean component has only a minor effect. Reducing the atmospheric time-step helps the estimation of process parameters but complicates the estimation of initial conditions. We note that the absolute performance of the system is likely to change with the resolution of the model. For example, we would expect a degraded performance for enhanced resolution of the ocean or atmosphere component or both. Nevertheless, the above findings should hold over a range of resolutions, because the responsible mechanisms are resolution-independent.

The performance in the reduced configuration is much better when estimating parameters by assimilating pseudo data generated by the model itself instead of by assimilating ERA data, in spite of careful preparation of the initial conditions. Part of this difference may be attributed to the coarse resolution and a too large degree of simplification in the reduced configuration with a limited number of control variables. More work is required to improve the balance between realistic process representations and good performance in the assimilation system. The efficient handling of longer control vectors was demonstrated in the present study. Another perspective to further extend the feasible assimilation window is the combined use of a reduced and full configuration in a variational assimilation system, where the reduced configuration is used to provide an approximate gradient.

The study presented a first step towards a flexible variational assimilation system for initialisation of predictions and calibration of the ESM against observations. The system was demonstrated in an idealised set up. Obvious next steps are an increase of spatio-temporal resolution, extension/improvement of the process representations in a differentiable form, and the simultaneous use of observations of the entire climate system. A further obvious application is sensitivity studies based on the tangent and adjoint ESM. The TAF compliance of the system assures fast updates after any modification of the ESM code.

## 7. Acknowledgements

This work was supported by the European Community within the 6th Framework Programme for Research and Technological Development under contract no. 212643 (THOR) to Hamburg University and supported in part also through the DFG funded CLISAP excellence initiative. The presented research is building on work on an assimilation system developed around only PlaSim that was funded by NERC through its QUEST programme. M. Scholze contributed through that phase and acknowledges support through a CLISAP fellowship. P. Herrmann was supported through a Max-Planck Fellow Grant to D. Stammer. The authors wish to thank E. Kirk for his advice throughout this study and valuable comments on the manuscript.

## 8. Appendix

Based on a set of initial tests with PlaSim, we identified several spots in the process implementations that produced a non-smooth shape in the cost function. For these spots we modified the implementation. Even though these modifications are specific to the model at hand, it is instructive to give a few examples.

1. if- and where- statements which depend on the model state implement piecewise-defined functions. Especially in the boundary-layer parameterisation of the PlaSim it was possible to make these functions smooth and differentiable at the switching point by readjusting some of their coefficients.
2. In some places of the PlaSim, the above procedure was infeasible and one of the branches was selected and the other removed. Alternatively, an approach was used which gives a weighted combination of the results of both branches, using a sigmoid function depending on the if-condition.
3. The PlaSim does its time stepping in spectral space and has to deal with spurious negative moisture stemming from the Fourier-transform. The original model uses a redistribution algorithm which fills up negative moisture at affected grid cells, taking it from a certain domain. Out of vertical column containing the affected grid cell, latitude band, and global domain, it chooses the smallest domain that contains enough moisture. Switching this off had a positive effect on the smoothness of the cost function at the expense of formal moisture conservation. Given the limited assimilation window, we do not expect a strong detrimental effect for an assimilation. However, simulations including atmospheric moisture require a closed hydrological cycle, for example, rain, which is currently implemented in a form far from smooth.
4. In the PlaSim some of the min, max, and abs statements were replaced by smooth approximations.

We note that the smoothing effect of the above modifications is limited, because, unlike Knorr et al. (2010), in this initial study we refrained from redevelopment of entire process representations (such as convection) in smooth form.

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