Recent studies demonstrate that weather and climate predictions potentially improve by dynamically combining different models into a so-called "supermodel". Here, we focus on the weighted supermodel -the supermodel's time derivative is a weighted superposition of the time derivatives of the imperfect models, referred to as weighted supermodeling. A crucial step is to train the weights of the supermodel on the basis of historical observations. Here, we apply two different training methods to a supermodel of up to four different versions of the global atmosphere-ocean-land model SPEEDO. The standard version is regarded as truth. The first training method is based on an idea called cross pollination in time (CPT), where models exchange states during the training. The second method is a synchronization-based learning rule, originally developed for parameter estimation. We demonstrate that both training methods yield climate simulations and weather predictions of superior quality as compared to the individual model versions. Supermodel predictions also outperform predictions based on the commonly used multi-model ensemble (MME) mean. Furthermore, we find evidence that negative weights can improve predictions in cases where model errors do not cancel (for instance, all models are warm with respect to the truth). In principle, the proposed training schemes are applicable to state-of-the-art models and historical observations. A prime advantage of the proposed training schemes is that in the present context relatively short training periods suffice to find good solutions. Additional work needs to be done to assess the limitations due to incomplete and noisy data, to combine models that are structurally different (different resolution and state representation, for instance) and to evaluate cases for which the truth falls outside of the model class.tation of the physical processes the model is intended to describe. Nevertheless, with the best possible models in hand, more accurate predictions can be obtained by making good use of all of them, thus exploiting multi-model information.In order to reduce the impact of model errors on predictions, it is common practice to combine the predictions of a collection of different models in a statistical fashion. This is referred to as the multi-model ensemble (MME) approach: the MME mean prediction is often more skillful as model errors tend to average out (Weigel et al., 2008), whereas the spread between the model predictions is naturally interpreted as a Published by Copernicus Publications on behalf of the European Geosciences Union.

Abstract.Weather and climate models have improved steadily over time as witnessed by objective skill scores, although significant model errors remain. Given these imperfect models, predictions might be improved by combining them dynamically into a so-called "supermodel". In this paper a new training scheme to construct such a supermodel is explored using a technique called cross pollination in time (CPT). In the CPT approach the models exchange states during the prediction. The number of possible predictions grows quickly with time, and a strategy to retain only a small number of predictions, called pruning, needs to be developed. The method is explored using low-order dynamical systems and applied to a global atmospheric model. The results indicate that the CPT training is efficient and leads to a supermodel with improved forecast quality as compared to the individual models. Due to its computational efficiency, the technique is suited for application to state-of-the art high-dimensional weather and climate models.

The SPEEDO global climate model (an atmosphere model coupled to a land and an ocean/sea-ice model with about 250.000 degrees of freedom) is used to investigate the merits of a new multi-model ensemble approach to the climate prediction problem in a perfect model setting. Two imperfect models are generated by perturbing parameters. Connection terms are introduced that synchronize the two models on a common solution, referred to as the supermodel solution. A synchronization-based learning algorithm is applied to the supermodel through the introduction of an update rule for the connection coefficients. Connection coefficients cease updating when synchronization errors between the supermodel and solutions of the "true" equations vanish. These final connection coefficients define the supermodel. Different supermodel solutions, but with equivalent performance, are found depending on the initial values of the connection coefficients during learning. The supermodels have a climatology and a climate response to a CO increase in the atmosphere that is closer to the truth as compared to the imperfect models and the standard multi-model ensemble average, showing the potential of the supermodel approach to improve climate predictions.

Abstract. As an alternative to using the standard multi-model ensemble (MME) approach to combine the output of different models to improve prediction skill, models can also be combined dynamically to form a so-called supermodel. The supermodel approach enables a quicker correction of the model errors. In this study we connect different versions of SPEEDO, a global atmosphere-ocean-land model of intermediate complexity, into a supermodel. We focus on a weighted supermodel, in which the supermodel state is a weighted superposition of different imperfect model states. The estimation, “the training”, of the optimal weights of this combination is a critical aspect in the construction of a supermodel. In our previous works two algorithms were developed: (i) cross pollination in time (CPT)-based technique and (ii) a synchronization-based learning rule (synch rule). Those algorithms have so far been applied under the assumption of complete and noise-free observations. Here we go beyond and consider the more realistic case of noisy data that do not cover the full system's state and are not taken at each model's computational time step. We revise the training methods to cope with this observational scenario, while still being able to estimate accurate weights. In the synch rule an additional term is introduced to maintain physical balances, while in CPT nudging terms are added to let the models stay closer to the observations during training. Furthermore, we propose a novel formulation of the CPT method allowing the weights to be negative. This makes it possible for CPT to deal with cases in which the individual model biases have the same sign, a situation that hampers constructing a skillfully weighted supermodel based on positive weights. With these developments, both CPT and the synch rule have been made suitable to train a supermodel consisting of state of the art weather and climate models.

Abstract. In alternative to using the standard multi-model ensemble (MME) approach to combine the output of different models to improve prediction skill, models can also be combined dynamically to form a so-called supermodel. The supermodel approach allows for a quicker correction of the model errors. In this study we focus on weighted supermodels, in which the supermodel state is a weighted superposition of different imperfect model states. The estimation, “the training”, of the optimal weights of this combination is a critical aspect in the construction of a supermodel. In our previous works two algorithms were developed: (i) cross pollination in time (CPT-based technique), and, (ii) a synchronization based learning rule (synch rule). Those algorithms have been so far applied under the assumption of complete and noise-free observations. Here we go beyond and consider the more realistic case of noisy data that do not cover the full system's state and are not taken at each model's computational time step. We revise the training methods to cope with this observational scenario, while still being able to estimate accurate weights. In the synch rule an additional term is introduced to maintain physical balances, while in CPT nudging terms are added to let the models stay closer to the observations during training. Furthermore, we propose a novel formulation of the CPT method allowing for the weights to be negative. This makes it possible for CPT to deal with cases in which the individual model biases have the same sign, a situation that hampers constructing a skilful weighted supermodel based on positive weights. With these developments, both CPT and the synch rule have been made suitable to train a supermodel consisting of state-of-the-art weather or climate models.

<p><span>Instead of combining data from an ensemble of different models after the simulations are already performed, as in a standard multi-model ensemble, we let the models interact with each other during their simulation. This ensemble of interacting models is called a supermodel. By exchanging information, models can compensate for each other's errors before the errors grow and spread to other regions or variables. Effectively, we create a new dynamical system. The exchange between the models is frequent enough such that the models synchronize, in order to prevent loss of variance when the models are combined. In previous work, we experimented successfully with combining atmospheric models of intermediate complexity in the context of parametric error. Here we will show results of combining two different AGCMs, NorESM1-ATM and CESM1-ATM. The models have different horizontal and vertical resolutions. To combine states from the different grids, we convert the individual model states to a ‘common state space’ with interpolation techniques. The weighted superposition of different model states is called a ‘pseudo-observation’. The pseudo-observations are assimilated back into the individual models, after which the models continue their run. We apply recently developed methods to train the weights determining the superposition of the model states, in order to obtain a supermodel that will outperform the individual models and any weighted average of their outputs.</span></p>

We would like to thank Reviewer 1 for his/her positive comments. We revised our manuscript accordingly. We decided to use 'sampling error' instead of 'natural variability' since the differences between the averages of the runs provide an indication of the sampling error as a result of natural variability.

<p>The established benefits of post-processing the results of multi-model ensembles, even by simple averaging, suggest a more radical approach: The models should be combined more frequently in run-time so as to form a single “supermodel”.  Simple nudging of models to one another, as frequently as the models might assimilate data from observations, combines model fusion with a reasonable degree of model autonomy.</p> <p>Key to the success of the supermodeling approach is the phenomenon of chaos synchronization, known in the field of nonlinear dynamics, wherein two chaotic systems synchronize when connected through only a few of their variables, despite sensitive dependence on initial conditions. Synchronization gives rise to consensus among models. The nudging coefficients can be trained so that that consensus agrees with observations, because the effective dynamics of the trained supermodel, regarded as a single dynamical system, matches the dynamics of nature. Yet the number of independent nudging coefficients that must be trained is far less than the number of trainable parameters in a typical climate model.</p> <p>It is expected that supermodeling will be especially useful for improving the representation of localized structures, such as blocking patterns, which will wash out if de-synchronized output fields of different models are combined by averaging.  We confirm a hypothesis that such coherent structures will synchronize even when the underlying fields do not, because the internal synchronization within each structure re-enforces synchronization between models: A configuration of CAM4 and CAM5 models, of different resolution, connected by nudging, exhibits correlated blocking activity even when the flows do not otherwise synchronize.  </p> <p>We further explore the basis for correlated blocking activity in a pair of coupled quasi-geostrophic channel models. The local synchronization error is lower in a region of the channels where blocks form than elsewhere in the channels. Blocking correlations emerge as a vestige of “chimera synchronization”, the phenomenon in which complete synchronization of two spatially extended systems is intermittent in space as well as time. Such partial synchronization of different models in the regions of blocks - and of other structures such as jets, fronts, and large-scale convection - would be particularly useful for projecting climate-change patterns in extreme events associated with those structures.</p>