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Global biogeochemical ocean models help to investigate the present and potential future state of the ocean biogeochemistry, its productivity and cascading effects on higher trophic levels such as fish. They are often subjectively tuned against data sets of inorganic tracers and surface chlorophyll and only very rarely against organic components such as particulate organic carbon or zooplankton. The resulting uncertainty in biogeochemical model parameters (and parameterisations) associated with these components can explain some of the large spread of global model solutions with regard to the cycling of organic matter and its impacts on biogeochemical tracer distributions, such as oxygen minimum zones (OMZs). A second source of uncertainty arises from differences in the model spin-up length, as, so far, there seems to be no agreement on the required simulation time that should elapse before a global model is assessed against observations. We investigated these two sources of uncertainty by optimising a global biogeochemical ocean model against the root-mean-squared error (RMSE) of six different combinations of data sets and different spin-up times. Besides nutrients and oxygen, the observational data sets also included phyto- and zooplankton, as well as dissolved and particulate organic phosphorus. We further analysed the optimised model performance with regard to global biogeochemical fluxes, oxygen inventory and OMZ volume. The optimisations resulted in optimal model solutions that yield similar values of the RMSE of tracers mainly located in surface layers, showing a range of between 14 % of the average RMSE after 10 years and 24 % after 3000 years of simulation. Global biogeochemical fluxes, global oxygen bias and OMZ volume showed a much stronger divergence among the models and over time than RMSE, indicating that even models that are similar with regard to local surface tracer concentrations can perform very differently when assessed against the global diagnosti

In geoscience and other fields, researchers use models as a simplified representation of reality. The models include processes that often rely on uncertain parameters that reduce model performance in reflecting real-world processes. The problem is commonly addressed by adapting parameter values to reach a good match between model simulations and corresponding observations. Different optimization tools have been successfully applied to address this task of model calibration. However, seeking one best value for every single model parameter might not always be optimal. For example, if model equations integrate over multiple real-world processes which cannot be fully resolved, it might be preferable to consider associated model parameters as random parameters. In this paper, a random parameter is drawn from a wide probability distribution for every singe model simulation. We developed an optimization approach that allows us to declare certain parameters random while optimizing those that are assumed to take fixed values. We designed a corresponding variant of the well known Covariance Matrix Adaption Evolution Strategy (CMA-ES). The new algorithm was applied to a global biogeochemical circulation model to quantify the impact of zooplankton mortality on the underlying biogeochemistry. Compared to the deterministic CMA-ES, our new method converges to a solution that better suits the credible range of the corresponding random parameter with less computational effort.

Global biogeochemical ocean models are often tuned to match the observed distributions and fluxes of inorganic and organic quantities. This tuning is typically carried out “by hand”. However, this rather subjective approach might not yield the best fit to observations, is closely linked to the circulation employed and is thus influenced by its specific features and even its faults. We here investigate the effect of model tuning, via objective optimisation, of one biogeochemical model of intermediate complexity when simulated in five different offline circulations. For each circulation, three of six model parameters have been adjusted to characteristic features of the respective circulation. The values of these three parameters – namely, the oxygen utilisation of remineralisation, the particle flux parameter and potential nitrogen fixation rate – correlate significantly with deep mixing and ideal age of North Atlantic Deep Water (NADW) and the outcrop area of Antarctic Intermediate Waters (AAIW) and Subantarctic Mode Water (SAMW) in the Southern Ocean. The clear relationship between these parameters and circulation characteristics, which can be easily diagnosed from global models, can provide guidance when tuning global biogeochemistry within any new circulation model. The results from 20 global cross-validation experiments show that parameter sets optimised for a specific circulation can be transferred between similar circulations without losing too much of the model's fit to observed quantities. When compared to model intercomparisons of subjectively tuned, global coupled biogeochemistry–circulation models, each with different circulation and/or biogeochemistry, our results show a much lower range of oxygen inventory, oxygen minimum zone (OMZ) volume and global biogeochemical fluxes. Export production depends to a large extent on the circulation applied, while deep particle flux is mostly determined by the particle flux parameter. Oxygen inventory, OMZ volume, prim

Global biogeochemical ocean models rely on many parameters, which govern the interaction between individual components, and their response to the physical environment. They are often assessed/calibrated against quasi-synoptic data sets of dissolved inorganic tracers. However, a good fit to one observation might not necessarily imply a good match to another. We investigate whether two different metrics—the root-mean-square error to nutrients and oxygen and a metric measuring the overlap between simulated and observed oxygen minimum zones (OMZs)—help to constrain a global biogeochemical model in different aspects of performance. Three global model optimizations are carried out. Two single-objective optimizations target the root-mean-square metric and a sum of both metrics, respectively. We then present and explore multiobjective optimization, which results in a set of compromise solutions. Our results suggest that optimal parameters for denitrification and nitrogen fixation differ when applying different metrics. Optimization against observed OMZs leads to parameters that enhance fixed nitrogen cycling; this causes too low nitrate concentrations and a too high global pelagic denitrification rate. Optimization against nutrient and oxygen concentrations leads to different parameters and a lower global fixed nitrogen turnover; this results in a worse fit to OMZs. Multiobjective optimization resolves this antagonistic effect and provides an ensemble of parameter sets, which help to address different research questions. We finally discuss how systematic model calibration can help to improve models used for projecting climate change and its effect on fisheries and climate gas emissions.

Biogeochemical models, capturing the major feedbacks of the pelagic ecosystem of the world ocean, are today often embedded into Earth system models which are increasingly used for decision making regarding climate policies. These models contain poorly constrained parameters (e.g., maximum phytoplankton growth rate), which are typically adjusted until the model shows reasonable behavior. Systematic approaches determine these parameters by minimizing the misfit between the model and observational data. In most common model approaches, however, the underlying functions mimicking the biogeochemical processes are nonlinear and non-convex. Thus, systematic optimization algorithms are likely to get trapped in local minima and might lead to non-optimal results. To judge the quality of an obtained parameter estimate, we propose determining a preferably large lower bound for the global optimum that is relatively easy to obtain and that will help to assess the quality of an optimum, generated by an optimization algorithm. Due to the unavoidable noise component in all observations, such a lower bound is typically larger than zero. We suggest deriving such lower bounds based on typical properties of biogeochemical models (e.g., a limited number of extremes and a bounded time derivative). We illustrate the applicability of the method with two real-world examples. The first example uses real-world observations of the Baltic Sea in a box model setup. The second example considers a three-dimensional coupled ocean circulation model in combination with satellite chlorophyll a.

Global biogeochemical ocean models contain a variety of different biogeochemical components and often much simplified representations of complex dynamical interactions, which are described by many (≈10–≈100) parameters. The values of many of these parameters are empirically difficult to constrain, due to the fact that in the models they represent processes for a range of different groups of organisms at the same time, while even for single species parameter values are often difficult to determine in situ. Therefore, these models are subject to a high level of parametric uncertainty. This may be of consequence for their skill with respect to accurately describing the relevant features of the present ocean, as well as their sensitivity to possible environmental changes. We here present a framework for the calibration of global biogeochemical ocean models on short and long time scales. The framework combines an offline approach for transport of biogeochemical tracers with an Estimation of Distribution Algorithm (Covariance Matrix Adaption Evolution Strategy, CMAES). We explore the performance and capability of this framework by five different optimizations of six biogeochemical parameters of a global biogeochemical model. First, a twin experiment explores the feasibility of this approach. Four optimizations against a climatology of observations of annual mean dissolved nutrients and oxygen determine the extent, to which different setups of the optimization influence model's fit and parameter estimates. Because the misfit function applied focuses on the large-scale distribution of inorganic biogeochemical tracers, parameters that act on large spatial and temporal scales are determined earliest, and with the least spread. Parameters more closely tied to surface biology, which act on shorter time scales, are more difficult to determine. In particular the search for optimum zooplankton parameters can benefit from a sound knowledge of maximum and minimum parameter values, l

We present a new quantum-inspired evolutionary algorithm, the attractor population QEA (apQEA). Our benchmark problem is a classical and difficult problem from Combinatorics, namely finding low-discrepancy colorings in the hypergraph of arithmetic progressions on the first n integers, which is a massive hypergraph (e.g., with approx. 3.88 ×1011 hyperedges for n = 250 000). Its optimal low-discrepancy coloring bound is known and it has been a long-standing open problem to give practically and/or theoretically efficient algorithms. We show that apQEA outperforms known QEA approaches and the classical combinatorial algorithm (Sárközy 1974) by a large margin. Regarding practicability, it is also far superior to the SDP-based polynomial-time algorithm of Bansal (2010), the latter being a breakthrough work from a theoretical point of view. Thus we give the first practical algorithm to construct optimal colorings in this hypergraph, up to a constant factor. We hope that our work will spur further applications of Algorithm Engineering to Combinatorics.

This work deals with the design of exact algorithms and heuristics for complex optimization problems that origin from three practical applications and one classical combinatorial task. We obtain exact algorithms by modeling our problems in terms of mathematical optimization problems and applying suitable software tools to solve this models. Because of the complexity of our problems, exact algorithms cannot solve large instances within adequate time. Therefore, on the one hand, we derive efficient heuristics by adapting algorithms for similar mathematical problems in a suitable manner. On the other hand, evolutionary algorithms have shown to be successfully applicable to many hard mathematical problems. For that reason, we experimentally determine appropriate EA frameworks, hybridizing the evolutionary operators with our problem specific heuristics.

Sensitivity computations, parameter identification and optimiza-tion for an 1-D marine biogeochemical model ofN P ZDtype are pre-sented. For the optimization a hybrid algorithm combining quantum-evolutionary and local gradient-based search methods is used. It turnsout to be an efficient and flexible tool for optimization and can beeasily adopted for other simulation models. For the model under in-vestigation attainable data could be exactly identified. For realisticmeasurement data we argue that a certain parameter set leading toa non-optimal fit cannot be improved. Moreover we show that datauncertainty leads to a significant parameter spread. Thus we concludethat theN P ZDmodel needs to be modified or extended, maybe in-cluding a modification of external forcings and/or initial conditions.