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Fires occurring over the peatlands in Indonesian Borneo accompanied by droughts have posed devastating impacts on human health, livelihoods, economy and the natural environment, and their prevention requires a comprehensive understanding of climate-associated risk. We want to strengthen the possibility of early warning triggers of drought, which is a strong predictor of the prevalence of fires, and evaluate the climate risk relevant to the formulation of long-term policies to eliminate fires. Although it is widely known that the droughts are often associated with El Niño events, the onset process of El Niño and thus the drought precursors and their possible changes under the future climate are not clearly understood. Here we use a causal network approach to quantify the strength of teleconnections to droughts at a seasonal timescale shown in (1) observational and reanalysis data (2) CMIP6 models and (3) seasonal hindcasts. We consider two drivers of JJA droughts identified through literature review and causal analysis, namely Niño 3.4 SST in JJA (abbreviated as ENSO) and SST anomaly over the eastern North Pacific to the east of the Hawaiian Islands (abbreviated as Pacific SST) in MAM. The observational and reanalysis data proves that the droughts are strongly linked to ENSO variability, with drier years corresponding to El Niño conditions, and droughts can be predicted with a lead time of three months based on their associations with Pacific SST, with higher SST preceding drier conditions. We find that some CMIP6 models are showing unrealistic amounts of JJA rainfall and underestimate drought risks in Indonesian Borneo and their teleconnections, owing to the underestimation of ENSO amplitude and overestimation of local convections. Under the SSP585 scenario, the CMIP6 multi-model ensembles show significant increase in both the maximum number of consecutive dry days in the Indonesian Borneo region in JJA (p = 0.006) and its linear association with Pacific SST in MAM (p = 0.001) from year 2061 – 2100 compared with the historical baseline. On the other hand, seasonal hindcast models are (1) overestimating the variability of maximum number of consecutive dry days, (2) showing varied skills in simulating the mean rainfall and drought indicators, and (3) underestimating the teleconnections to Borneo droughts, making it difficult to assess the likelihood of unprecedented drought and fire risk under El Niño conditions. Our study agrees with previous studies regarding the limited skill of fire risk prediction by state-of-the-art seasonal forecasting models, with their shortfalls caused by a lack of proper representation of relevant teleconnections.

Radar altimeters have so far had limited use in the coastal zone, the area with most societal impact. This is due to both lack of, or insufficient accuracy in the necessary corrections, and more complicated altimeter signals. This letter examines waveform data from the Envisat RA-2 as it passes regularly over Pianosa (a 10-km2 island in the northwestern Mediterranean). Forty-six repeat passes were analyzed, with most showing a reduction in signal upon passing over the island, with weak early returns corresponding to the reflections from land. Intriguingly, one third of cases showed an anomalously bright hyperbolic feature. This feature may be due to extremely calm waters in the Golfo della Botte (northern side of the island), but the cause of its intermittency is not clear. The modeling of waveforms in such a complex land/sea environment demonstrates the potential for sea surface height retrievals much closer to the coast than is achieved by routine processing. The long-term development of altimetric records in the coastal zone will not only improve the calibration of altimetric data with coastal tide gauges but also greatly enhance the study of storm surges and other coastal phenomena.

Predicting the trajectory of rare degenerative diseases can be extremely beneficial, especially when these predictions are personalised to be relevant for a specific patient. These predictions can help inform and advise patients, families, and clinicians about the next stages of treatment and care. Obtaining such predictions, however, can be challenging, especially when data is limited. In particular, it is important that these predictions do not rely too heavily on general trends from the wider afflicted population while not relying exclusively on the, potentially sparse, data from the patient in question. We present a case study, wherein a modelling framework is developed for predicting a patient’s long term trajectory, using a mix of data from the patient of concern and a database of previously observed patients. This framework directly accounts for the temporal structure of a patient’s trajectory, effortlessly handles a large amount of missing data, allows for a wide range of patient progression, and offers a robust quantification of the various uncertainties. We showcase this framework to an example involving Duchenne Muscular Dystrophy, where it provides promising results.

In this paper, a sliding mode-based guidance strategy is proposed for the control of an autonomous vehicle. The aim of the autonomous vehicle deployment is the study of unknown environmental spatial features. The proposed approach allows the solution of both boundary tracking and source-seeking problems with a single autonomous vehicle capable of sensing the value of the spatial field at its position. The movement of the vehicle is controlled through the proposed guidance strategy, which is designed on the basis of the collected measurements without the necessity of preplanning or human intervention. Moreover, no a priori knowledge about the field and its gradient is required. The proposed strategy is based on the so-called suboptimal sliding mode controller. The guidance strategy is demonstrated by computer-based simulations and a set of boundary tracking experimental sea trials. The efficacy of the algorithm to autonomously steer the C-Enduro surface vehicle to follow a fixed depth contour in a dynamic coastal region is demonstrated by the results from the trial described in this paper.

A key challenge in developing flagship climate model configurations is the process of setting uncertain input parameters at values that lead to credible climate simulations. Setting these parameters traditionally relies heavily on insights from those involved in parameterisation of the underlying climate processes. Given the many degrees of freedom and computational expense involved in evaluating such a selection, this can be imperfect leaving open questions about whether any subsequent simulated biases result from mis-set parameters or wider structural model errors (such as missing or partially parameterised processes). Here, we present a complementary approach to identifying plausible climate model parameters, with a method of bias correcting subcomponents of a climate model using a Gaussian process emulator that allows credible values of model input parameters to be found even in the presence of a significant model bias. A previous study (McNeall et al., 2016) found that a climate model had to be run using land surface input parameter values from very different, almost non-overlapping, parts of parameter space to satisfactorily simulate the Amazon and other forests respectively. As the forest fraction of modelled non-Amazon forests was broadly correct at the default parameter settings and the Amazon too low, that study suggested that the problem most likely lay in the model's treatment of non-plant processes in the Amazon region. This might be due to modelling errors such as missing deep rooting in the Amazon in the land surface component of the climate model, to a warm–dry bias in the Amazon climate of the model or a combination of both. In this study, we bias correct the climate of the Amazon in the climate model from McNeall et al. (2016) using an “augmented” Gaussian process emulator, where temperature and precipitation, variables usually regarded as model outputs, are treated as model inputs alongside land surface input parameters. A sensitivity analysis finds that the forest fraction is nearly as sensitive to climate variables as it is to changes in its land surface parameter values. Bias correcting the climate in the Amazon region using the emulator corrects the forest fraction to tolerable levels in the Amazon at many candidates for land surface input parameter values, including the default ones, and increases the valid input space shared with the other forests. We need not invoke a structural model error in the land surface model, beyond having too dry and hot a climate in the Amazon region. The augmented emulator allows bias correction of an ensemble of climate model runs and reduces the risk of choosing poor parameter values because of an error in a subcomponent of the model. We discuss the potential of the augmented emulator to act as a translational layer between model subcomponents, simplifying the process of model tuning when there are compensating errors and helping model developers discover and prioritise model errors to target.

The dynamic emulation of non-linear deterministic computer codes where the output is a time series, possibly multivariate, is examined. Such computer models simulate the evolution of some real-world phenomenon over time, for example models of the climate or the functioning of the human brain. The models we are interested in are highly non-linear and exhibit tipping points, bifurcations and chaotic behaviour. However, each simulation run could be too time-consuming to perform analyses that require many runs, including quantifying the variation in model output with respect to changes in the inputs. Therefore, Gaussian process emulators are used to approximate the output of the code. To do this, the flow map of the system under study is emulated over a short time period. Then, it is used in an iterative way to predict the whole time series. A number of ways are proposed to take into account the uncertainty of inputs to the emulators, after fixed initial conditions, and the correlation between them through the time series. The methodology is illustrated with two examples: the highly non-linear dynamical systems described by the Lorenz and van der Pol equations. In both cases, the predictive performance is relatively high and the measure of uncertainty provided by the method reflects the extent of predictability in each system.

Contributions to climate uncertainty in the modeling of the near surface temperature of the ocean come from several sources: the knowledge of the initial state of the atmosphere, the inadequacy of a model to include or correctly represent all aspects of the dynamics, the uncertain knowledge of the initial state of the ocean and small-scale (or internal, “weather”) noise. This paper concentrates on the latter uncertainty, the uncertainty in the knowledge of the initial field, resulting in a map of the uncertainty of the near surface temperature field. To make the full estimate of this uncertainty, the outcomes from a relatively small, 30 member, ensemble were expanded using statistical emulators. The uncertainty varies in space and decreases in time. There are broad areas of similar uncertainty, consistent with uncertainties in the initial conditions and the variability in local processes. After 10 years, the average spatial uncertainty calculated from this expanded data set gives a range of +/− 0.4 $$^{\circ }$$ C with 95% of the spatial areas having an uncertainty of less than +/− 0.9 $$^{\circ }$$ C. When compared to other estimates of similar metrics, the map represents a lower bound on the uncertainty because this first experiment examines the ocean/ice system in isolation from feedbacks that occur between the atmosphere and the ocean. This uncertainty also does not include any contribution from the lack of knowledge about how a future state of the air/ocean system will be forced by changes in greenhouse gases.

This paper describes results of an experiment that perturbed the initial conditions for the ocean's temperature field of the Community Earth System Model (CESM) with a well defined design. The resulting 30-member ensemble of CESM simulations, each of 10 years in length, is used to create an emulator (a nonlinear regression relating the initial conditions to various outcomes) from the simulators. Through the use of the emulator to expand the output distribution space, we estimate the spatial uncertainties at 10 years for surface air temperature, 25 m ocean temperature, precipitation, and rain. Outside the tropics, basin averages for the uncertainty in the ocean temperature field range between 0.48 ∘C (Indian Ocean) and 0.87 ∘C (North Pacific) (2 standard deviation). The tropical Pacific uncertainty is the largest due to different phasings of the ENSO signal. Over land areas, the regional temperature uncertainty varies from 1.03 ∘C (South America) to 10.82 ∘C (Europe) (2 standard deviation). Similarly, the regional average uncertainty in precipitation varies from 0.001 cm day−1 over Antarctica to 0.163 cm day−1 over Australia with a global average of 0.075 cm day−1. In general, both temperature and precipitation uncertainties are larger over land than over the ocean. A maximum covariance analysis is used to examine how ocean temperatures affect both surface air temperatures and precipitation over land. The analysis shows that the tropical Pacific influences the temperature over North America, but the North America surface temperature is also moderated by the state of the North Pacific outside the tropics. It also indicates which regions show a high degree of variance between the simulations in the ensemble and are, therefore, less predictable. The calculated uncertainties are also compared to an estimate of internal variability within CESM. Finally, the importance of feedback processes on the solution of the simulation over the 10 years of the experiment is quantified. These estimates of uncertainty do not take into consideration the anthropogenic effect on warming of the atmosphere and ocean.

The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support during the Infectious Dynamics of Pandemics programme where work on this paper was undertaken. This work was supported by EPSRC grant no. EP/R014604/1. L.H. is supported by the Wellcome Trust (block grant no. RG92770). C.D. acknowledges the University of Edinburgh’s Data Driven Innovation Programme under Scottish Funding Council support, the Alan Turing Institute sponsored ‘Managing Uncertainty in Government Modelling’ project, and EPSRC (grant no. EP/P001173/1). B.S. acknowledges the Scottish Covid-19 Response Consortium. The COVID-19 pandemic has seen infectious disease modelling at the forefront of government decision-making. Models have been widely used throughout the pandemic to estimate pathogen spread and explore the potential impact of different intervention strategies. Infectious disease modellers and policymakers have worked effectively together, but there are many avenues for progress on this interface. In this paper, we identify and discuss seven broad challenges on the interaction of models and policy for pandemic control. We then conclude with suggestions and recommendations for the future. Publisher PDF

Many computer models possess high-dimensional input spaces and substantial computational time to produce a single model evaluation. Although such models are often ‘deterministic’, these models suffer from a wide range of uncertainties. We argue that uncertainty quantification is crucial for computer model validation and reproducibility. We present a statistical framework, termed history matching, for performing global parameter search by comparing model output to the observed data. We employ Gaussian process (GP) emulators to produce fast predictions about model behaviour at the arbitrary input parameter settings allowing output uncertainty distributions to be calculated. History matching identifies sets of input parameters that give rise to acceptable matches between observed data and model output given our representation of uncertainties. Modellers could proceed by simulating computer models’ outputs of interest at these identified parameter settings and producing a range of predictions. The variability in model results is crucial for inter-model comparison as well as model development. We illustrate the performance of emulation and history matching on a simple one-dimensional toy model and in application to a climate model. This article is part of the theme issue ‘Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico ’.

Downscaling aims to link the behaviour of the atmosphere at fine scales to properties measurable at coarser scales, and has the potential to provide high resolution information at a lower computational and storage cost than numerical simulation alone. This is especially appealing for targeting convective scales, which are at the edge of what is possible to simulate operationally. Since convective scale weather has a high degree of independence from larger scales, a generative approach is essential. We here propose a statistical method for downscaling moist variables to convective scales using conditional Gaussian random fields, with an application to wet bulb potential temperature (WBPT) data over the UK. Our model uses an adaptive covariance estimation to capture the variable spatial properties at convective scales. We further propose a method for the validation, which has historically been a challenge for generative models., Comment: 32 pages, 19 figures

Continuous structural parameterization (CSP) is a proposed method for approximating different numerical model parameterizations of the same process as functions of the same grid‐scale variables. This allows systematic comparison of parameterizations with each other and observations or resolved simulations of the same process. Using the example of two convection schemes running in the Met Office Unified Model (UM), we show that a CSP is able to capture concisely the broad behavior of the two schemes, and differences between the parameterizations and resolved convection simulated by a high resolution simulation. When the original convection schemes are replaced with their CSP emulators within the UM, basic features of the original model climate and some features of climate change are reproduced, demonstrating that CSP can capture much of the important behavior of the schemes. Our results open the possibility that future work will estimate uncertainty in model projections of climate change from estimates of uncertainty in simulation of the relevant physical processes. Plain Language Summary Numerical models are used to provide estimates of future weather and climate change. The models contain “parameterizations,” which are algorithms that simulate the effect of processes too small or poorly understood to represent using physical equations. Although they are based as much as possible on physics, parameterizations are a large source of modeling uncertainty because there can be large disagreements on how to best represent a given process. The method and even the variables used by two different parameterizations may differ. It is therefore very difficult to know how different parameterizations cause numerical models to produce different results and whether the parameterizations we have are adequate and span the range of uncertainty concerning our knowledge of the processes they represent. Using the example of small‐scale atmospheric convection linked to rain and thunderstorms, this paper describes a mathematical method for expressing different parameterizations within the same framework. This allows easy but formal mathematical comparison of different parameterizations and gives future work the potential to understand whether our parameterizations perform as they should in conjunction with future observations.

History Matching is a technique used to calibrate complex computer models, that is, finding the input settings which lead to the simulated output matching up with real world observations. Key to this technique is the construction of emulators, which provide fast probabilistic predictions of future simulations. In this work, we adapt the History Matching framework to tackle the problem of level set estimation, that is, finding input settings where the output is below (or above) some threshold. The developed methodology is heavily motivated by a specific case study: how can one design a building that will be sufficiently protected against overheating and sufficiently energy efficient, whilst considering the expected increases in temperature due to climate change? We successfully manage to address this - greatly reducing a large initial set of candidate building designs down to a small set of acceptable potential buildings., Comment: 19 pages, 3 figures

In many real-world applications, we are interested in approximating black-box, costly functions as accurately as possible with the smallest number of function evaluations. A complex computer code is an example of such a function. In this work, a Gaussian process (GP) emulator is used to approximate the output of complex computer code. We consider the problem of extending an initial experiment (set of model runs) sequentially to improve the emulator. A sequential sampling approach based on leave-one-out (LOO) cross-validation is proposed that can be easily extended to a batch mode. This is a desirable property since it saves the user time when parallel computing is available. After fitting a GP to training data points, the expected squared LOO (ES-LOO) error is calculated at each design point. ES-LOO is used as a measure to identify important data points. More precisely, when this quantity is large at a point it means that the quality of prediction depends a great deal on that point and adding more samples nearby could improve the accuracy of the GP. As a result, it is reasonable to select the next sample where ES-LOO is maximised. However, ES-LOO is only known at the experimental design and needs to be estimated at unobserved points. To do this, a second GP is fitted to the ES-LOO errors and where the maximum of the modified expected improvement (EI) criterion occurs is chosen as the next sample. EI is a popular acquisition function in Bayesian optimisation and is used to trade-off between local/global search. However, it has a tendency towards exploitation, meaning that its maximum is close to the (current) "best" sample. To avoid clustering, a modified version of EI, called pseudo expected improvement, is employed which is more explorative than EI yet allows us to discover unexplored regions. Our results show that the proposed sampling method is promising.

Polynomial chaos and Gaussian process emulation are methods for surrogate-based uncertainty quantification and have been developed independently in their respective communities over the last 25 years. Despite tackling similar problems in the field, to our knowledge there has yet to be a critical comparison of the two approaches in the literature. We begin by providing a detailed description of polynomial chaos and Gaussian process approaches for building a surrogate model of a black-box function. The accuracy of each surrogate method is then tested and compared for two simulators used in industry: a land-surface model (adJULES) and a launch vehicle controller (VEGACONTROL). We analyze surrogates built on experimental designs of various size and type to investigate their performance in a range of modeling scenarios. Specifically, polynomial chaos and Gaussian process surrogates are built on Sobol sequence and tensor grid designs. Their accuracy is measured by their ability to estimate the mean, standard de...

We use observations of forest fraction to constrain carbon cycle and land surface input parameters of the reduced resolution global climate model, FAMOUS. Using a history matching approach along with a computationally cheap statistical proxy (emulator) of the climate model, we compare an ensemble of simulations of forest fraction with observations, and rule out parameter settings where the forests are poorly simulated. Regions of parameter space where FAMOUS best simulates the Amazon forest fraction are incompatible with the regions where FAMOUS best simulates other forests, indicating a structural error in the model. Using observations of the Amazon forest to constrain input parameters leads to very different conclusions about the acceptable values of input parameters than using the other forests. We characterise the structural model discrepancy, and explore the consequences of ignoring it in a history matching exercise. We use sensitivity analysis to find the parameters which have most impact on simulator error. We use the emulator to simulate the forest fraction at the best set of parameters implied by matching the model to the Amazon, and to other major forests in turn. We can find parameters that lead to a realistic forest fraction in the Amazon, but using the Amazon alone to tune the simulator would result in a significant overestimate of forest fraction in the other forests. Conversely, using the other forests to calibrate the model leads to a larger underestimate of the Amazon forest fraction. Finally, we perform a history matching exercise using credible estimates for simulator discrepancy and observational uncertainty terms. We are unable to constrain the parameters individually, but just under half of joint parameter space is ruled out as being incompatible with forest observations. We discuss the possible sources of the discrepancy in the simulated Amazon, including missing processes in the land surface component, and a bias in the climatology of the Amazon.

There is a growing practical interest in the ability to increase the sea states at which marine operations can be safely undertaken by exploiting the quiescent periods that are well known to exist under a wide range of sea conditions. While the actual prediction of quiescent periods at sea for the control of operations is a deterministic process, the long term planning of future maritime tasks that rely on these quiescent periods is a statistical process involving the anticipated quiescence properties of the forecasted sea conditions in the geographical region of interest. It is in principle possible to obtain such data in tabular form either large scale simulation or from field data. However, such simulations are computationally intensive and libraries of appropriate field data are not common. Thus, it is clearly attractive to develop techniques that exploit standard wave spectral models for describing the quiescence statistics directly from such spectra. The present study focuses upon such techniques and is a first step towards the production of a computationally low-cost quiescence prediction tool and compares its efficacy against simulations. Two significant properties emerge for a large class of wave spectral models that encompasses the ubiquitous Neumann and Pierson Moskowitz or Bretschneider forms. Firstly, the auto-correlation function of the wave profile that are required to produce the quiescence property can be obtained analytically in terms of standard special functions. This considerably reduces the computational cost making desktop computer-based planning tools a reality. Secondly, for each class of these parametric spectra, the probability of a given number of consecutive wave heights (normalised to the significant wave heights) less than some critical value is in fact independent of absolute wave height. Thus, for a broad class of practically interesting wave spectra all that is required to obtain the statistical distribution of the quiescent periods is simple rescaling.

The analysis of computer models can be aided by the construction of surrogate models, or emulators, that statistically model the numerical computer model. Increasingly, computer models are becoming stochastic, yielding different outputs each time they are run, even if the same input values are used. Stochastic computer models are more difficult to analyse and more difficult to emulate - often requiring substantially more computer model runs to fit. We present a method of using deterministic approximations of the computer model to better construct an emulator. The method is applied to numerous toy examples, as well as an idealistic epidemiology model, and a model from the building performance field.

The ocean warming trend is a well-known global phenomenon. As early as 2001, and then reiterated in 2007, the Intergovernmental Panel on Climate Change (IPCC) reported that the global average sea surface temperature (SST) will increase by about 0.2 °C per decade. To date, however, only a limited number of studies have been published reporting the spatio-temporal trends in SST in the Indo-Pacific region, one the richest marine ecosystems on Earth. In this research, the monthly 1° spatial resolution NOAA Optimum Interpolation (OI) sea surface temperature (SST) V2 dataset (OISSTv2) derived from measurements made by the Advanced Very High Resolution Radiometer (AVHRR) and in situ measurements, were used to examine the spatio-temporal trends in SST in the region. The multi-model mean SST from the Representative Concentration Pathways (RCP2.6) mitigation scenario of the Coupled Model Intercomparison Project Phase 5 (CMIP5) was also used to forecast future SST from 2020 to 2100, decadally. Three variables from the OISSTv2, namely maximum (MaxSST), mean (MeanSST) and minimum (MinSST) monthly mean SST, were regressed against time measured in months from 1982 to 2010 using linear regression. Results revealed warming trends detected for all three SST variables. In the Coral Triangle a warming trend with a rate of 0.013 °C year −1 , 0.017 °C year −1 , and 0.019 °C year −1 was detected over 29 years for MaxSST, MeanSST and MinSST, respectively. In the SCS, the warming rate was 0.011 °C year −1 , (MaxSST), 0.012 °C year −1 (MeanSST) and 0.015 °C year −1 (MinSST) over 29 years. The CMIP5 RCP2.6 forecast suggested a future warming rate to 2100 of 0.004 °C year −1 for both areas, and for all three SST variables. The warming trends reported in this study provide useful insights for improved marine-related management.

Biogeochemical ocean circulation models used to investigate the role of plankton ecosystems in global change rely on adjustable parameters to capture the dominant biogeochemical dynamics of a complex biological system. In principle, optimal parameter values can be estimated by fitting models to observational data, including satellite ocean colour products such as chlorophyll that achieve good spatial and temporal coverage of the surface ocean. However, comprehensive parametric analyses require large ensemble experiments that are computationally infeasible with global 3-D simulations. Site-based simulations provide an efficient alternative but can only be used to make reliable inferences about global model performance if robust quantitative descriptions of their relationships with the corresponding 3-D simulations can be established. The feasibility of establishing such a relationship is investigated for an intermediate complexity biogeochemistry model (MEDUSA) coupled with a widely used global ocean model (NEMO). A site-based mechanistic emulator is constructed for surface chlorophyll output from this target model as a function of model parameters. The emulator comprises an array of 1-D simulators and a statistical quantification of the uncertainty in their predictions. The unknown parameter-dependent biogeochemical environment, in terms of initial tracer concentrations and lateral flux information required by the simulators, is a significant source of uncertainty. It is approximated by a mean environment derived from a small ensemble of 3-D simulations representing variability of the target model behaviour over the parameter space of interest. The performance of two alternative uncertainty quantification schemes is examined: a direct method based on comparisons between simulator output and a sample of known target model "truths" and an indirect method that is only partially reliant on knowledge of the target model output. In general, chlorophyll records at a representative array of oceanic sites are well reproduced. The use of lateral flux information reduces the 1-D simulator error considerably, consistent with a major influence of advection at some sites. Emulator robustness is assessed by comparing actual error distributions with those predicted. With the direct uncertainty quantification scheme, the emulator is reasonably robust over all sites. The indirect uncertainty quantification scheme is less reliable at some sites but scope for improving its performance is identified. The results demonstrate the strong potential of the emulation approach to improve the effectiveness of site-based methods. This represents important progress towards establishing a robust site-based capability that will allow comprehensive parametric analyses to be achieved for improving global models and quantifying uncertainty in their predictions.

This article examines the dynamics of the US economy over the last five decades using Bayesian analysis of dynamic stochastic general equilibrium (DSGE) models. It highlights an example application in what is commonly referred to as the new macroeconometrics, which combines macroeconomics with econometrics. The article describes a benchmark New Keynesian DSGE model that incorporates four types of agents: households that consume, save, and supply labour to a labour ‘packer’; a labour ‘packer’ that puts together the labour supplied by different households into an homogeneous labour unit; intermediate good producers, who produce goods using capital and aggregated labour; and a final good producer that mixes all the intermediate goods. It also considers the application of the model in policy analysis for public institutions such as central banks, along with private organizations and businesses. Finally, it discusses three avenues for further research in the estimation of DSGE models.

In this paper, an unknown three-dimensional oceanic feature is analyzed using a single autonomous vehicle. The problem of mapping the three-dimensional underwater feature is posed from a boundary mapping perspective. An autonomous underwater vehicle, specifically a glider, is considered. A dynamic and a kinematic model of the glider is used in order to mimic its movement and to design the control actions to be applied. Differently from previous work, the glider is not requested to follow a predefined trajectory of waypoints, but is considered capable of ‘intelligently’ determining its own trajectory. In particular, the glider defines its direction of movement through a suboptimal sliding mode control method, using only the measurement of the analyzed feature at its current position, in order to achieve the boundary mapping objective. Knowledge of the vehicle position and of the gradient of the analyzed feature is not required. A set of simulations results are provided, based on a realistic dataset relating to a sea temperature front in the Iceland Faroes Front.

Submarine landslides can be far larger than those on land, and are one of the most important processes for moving sediment across our planet. Landslides that are fast enough to disintegrate can generate potentially very hazardous tsunamis, and produce long run-out turbidity currents that break strategically important cable networks. It is important to understand their frequency and triggers. We document the distribution of recurrence intervals for large landslide-triggered turbidity currents (>0.1 km3) in three basin-plains. A common distribution of recurrence intervals is observed, despite variable ages and disparate locations, suggesting similar underlying controls on slide triggers and frequency. This common distribution closely approximates a temporally-random Poisson distribution, such that the probability of a large disintegrating slide occurring along the basin margin is independent of the time since the last slide. This distribution suggests that non-random processes such as sea level are not a dominant control on frequency of these slides. Recurrence intervals of major (>M 7.3) earthquakes have an approximately Poissonian distribution, suggesting they could be implicated as triggers. However, not all major earthquakes appear to generate widespread turbidites, and other as yet unknown triggers or sequential combinations of processes could produce the same distribution. This is the first study to show that large slide-triggered turbidites have a common frequency distribution in distal basin plains, and that this distribution is temporally random. This result has important implications for assessing hazards from landslide-tsunamis and seafloor cable breaks, and the long-term tempo of global sediment fluxes.

In this paper, the experimental results obtained from the implementation of a novel boundary tracking algorithm based on the ‘suboptimal sliding mode’ concept are presented. In March 2016, the C-Enduro Autonomous Surface Vehicle (ASV) was deployed to validate the algorithm in a dynamic coastal region in Ardmucknish bay, Argyll and Bute, Scotland. The algorithm was tested in a bathymetric context. The capability of the algorithm to autonomously steer the surface vehicle to follow a fixed depth contour in the ocean, in the presence of wind and oceanic currents, is demonstrated by the results from the trial described in this paper.

In this paper, a circular formation of autonomous vehicles is deployed such that the centroid tracks a temperature contour in the region of a tidal mixing front. Only local temperature measurements at the vehicles positions are taken for this purpose. The European north-west continental shelf sea surface temperature forecast data, provided by the Met Office, UK, has been used for this study as ‘prior information’. A probabilistic ‘belief’ model of the temperature profile over the region of operation is developed using instantaneous measurements from the vehicles and the available forecast information. While tracking a contour, the ‘belief’ model is updated and used for predicting local temperature gradient information. A quasi-continuous sliding mode contour following algorithm, which relies on local temperature measurements and the predicted temperature gradient, is proposed to steer the centroid of the formation.

Metamodels (or emulators) are statistical tools for the analysis of large complex simulation models. They consist of a Gaussian (or second order) process (kriging) fitted to a designed set of simulator runs. Once an emulator has been built, it is important that it is validated against some independent runs of the simulator. This paper considers the design of experiments for the validation of the fitted metamodel. All the proposed designs are based on maximin Latin hypercubes and add an extra criterion to be optimised based on the distances between the points in the validation and original designs. Simulation experiments are carried out to determine how well each design performs against the alternative criteria.

We measure the potential of an observational data set to constrain a set of inputs to a complex and computationally expensive computer model. We use each member in turn of an ensemble of output from a computationally expensive model, corresponding to an observable part of a modelled system, as a proxy for an observational data set. We argue that, given some assumptions, our ability to constrain uncertain parameter inputs to a model using its own output as data, provides a maximum bound for our ability to constrain the model inputs using observations of the real system. The ensemble provides a set of known parameter input and model output pairs, which we use to build a computationally efficient statistical proxy for the full computer model, termed an emulator. We use the emulator to find and rule out "implausible" values for the inputs of held-out ensemble members, given the computer model output. As we know the true values of the inputs for the ensemble, we can compare our constraint of the model inputs with the true value of the input for any ensemble member. Measures of the quality of constraint have the potential to inform strategy for data collection campaigns, before any real-world data is collected, as well as acting as an effective sensitivity analysis. We use an ensemble of the ice sheet model Glimmer to demonstrate our measures of quality of constraint. The ensemble has 250 model runs with 5 uncertain input parameters, and an output variable representing the pattern of the thickness of ice over Greenland. We have an observation of historical ice sheet thickness that directly matches the output variable, and offers an opportunity to constrain the model. We show that different ways of summarising our output variable (ice volume, ice surface area and maximum ice thickness) offer different potential constraints on individual input parameters. We show that combining the observational data gives increased power to constrain the model. We investigate the impact of uncertainty in observations or in model biases on our measures, showing that even a modest uncertainty can seriously degrade the potential of the observational data to constrain the model.

We apply an established statistical methodology called history matching to constrain the parameter space of a coupled non-flux-adjusted climate model (the third Hadley Centre Climate Model; HadCM3) by using a 10,000-member perturbed physics ensemble and observational metrics. History matching uses emulators (fast statistical representations of climate models that include a measure of uncertainty in the prediction of climate model output) to rule out regions of the parameter space of the climate model that are inconsistent with physical observations given the relevant uncertainties. Our methods rule out about half of the parameter space of the climate model even though we only use a small number of historical observations. We explore 2 dimensional projections of the remaining space and observe a region whose shape mainly depends on parameters controlling cloud processes and one ocean mixing parameter. We find that global mean surface air temperature (SAT) is the dominant constraint of those used, and that the others provide little further constraint after matching to SAT. The Atlantic meridional overturning circulation (AMOC) has a non linear relationship with SAT and is not a good proxy for the meridional heat transport in the unconstrained parameter space, but these relationships are linear in our reduced space. We find that the transient response of the AMOC to idealised CO2 forcing at 1 and 2 % per year shows a greater average reduction in strength in the constrained parameter space than in the unconstrained space. We test extended ranges of a number of parameters of HadCM3 and discover that no part of the extended ranges can by ruled out using any of our constraints. Constraining parameter space using easy to emulate observational metrics prior to analysis of more complex processes is an important and powerful tool. It can remove complex and irrelevant behaviour in unrealistic parts of parameter space, allowing the processes in question to be more easily studied or emulated, perhaps as a precursor to the application of further relevant constraints.

This paper examines the uncertainty in the change in the heat content in the ocean component of a general circulation model. We describe the design and implementation of our statistical methodology. Using an ensemble of model runs and an emulator, we produce an estimate of the full probability distribution function (PDF) for the change in upper ocean heat in an Atmosphere/Ocean General Circulation Model, the Community Climate System Model v. 3, across a multi-dimensional input space. We show how the emulator of the GCM’s heat content change and hence, the PDF, can be validated and how implausible outcomes from the emulator can be identified when compared to observational estimates of the metric. In addition, the paper describes how the emulator outcomes and related uncertainty information might inform estimates of the same metric from a multi-model Coupled Model Intercomparison Project phase 3 ensemble. We illustrate how to (1) construct an ensemble based on experiment design methods, (2) construct and evaluate an emulator for a particular metric of a complex model, (3) validate the emulator using observational estimates and explore the input space with respect to implausible outcomes and (4) contribute to the understanding of uncertainties within a multi-model ensemble. Finally, we estimate the most likely value for heat content change and its uncertainty for the model, with respect to both observations and the uncertainty in the value for the input parameters.

The effect of a Gaussian process parameter known as the nugget, on the development of computer model emulators is investigated. The presence of the nugget results in an emulator that does not interpolate the data and attaches a non-zero uncertainty bound around them. The limits of this approximation are investigated theoretically, and it is shown that they can be as large as those of a least squares model with the same regression functions as the emulator, regardless of the nugget's value. The likelihood of the correlation function parameters is also studied and two mode types are identified. Type I modes are characterised by an approximation error that is a function of the nugget and can therefore become arbitrarily small, effectively yielding an interpolating emulator. Type II modes result in emulators with a constant approximation error. Apart from a theoretical investigation of the limits of the approximation error, a practical method for automatically imposing restrictions on its extent is introduced. This is achieved by means of a penalty term that is added to the likelihood function, and controls the amount of unexplainable variability in the computer model. The main findings are illustrated on data from an Energy Balance climate model.

The article of record as published may be located at http://dx.doi.org/10.1175/JTECH-D-11-00110.1 Gaussian process emulators are a powerful tool for understanding complex geophysical simulators, including oceanic and atmospheric general circulation models. Concern has been raised about their ability to emulate complex nonlinear systems. For the first time, using the simple Stommel model, the way in which emulators can reasonably represent the full sampling space of an extreme nonlinear, bimodal system is illustrated. This simple example also shows how an emulator can help to elucidate interactions between parameters. The ideas are further illustrated with a second, more realistic, intermediate complex climate simulator. The paper describes what is meant by an emulator, the methodology of emulators, how emulators can be assessed, and why they are useful. It is shown how simple emulators can be useful to explore the parameter space (initial conditions, process parameters, and boundary conditions) of complex computer simulators, such as ocean and climate general circulation models, even when simulator outcomes contain steps in the response.

In this paper, a source seeking problem is considered exploiting a single autonomous vehicle. The aim of the control scheme is to direct the vehicle to move towards the source of the considered phenomenon and to reach it, in order to determine its position. The vehicle's movement is controlled through a suboptimal sliding mode steering law. This is a gradient-free control law which exploits local measurements of the phenomenon at the vehicle's position only, in order to direct it towards the source. A numerical example is proposed and further results are obtained exploiting realistic data from the DIMES experiment.

To study climate change on multi-millennial timescales or to explore a model's parameter space, efficient models with simplified and parameterised processes are required. However, the reduction in explicitly modelled processes can lead to underestimation of some atmospheric responses that are essential to the understanding of the climate system. While more complex general circulations are available and capable of simulating a more realistic climate, they are too computationally intensive for these purposes. In this work, we propose a multi-level Gaussian emulation technique to efficiently estimate the outputs of steady-state simulations of an expensive atmospheric model in response to changes in boundary forcing. The link between a computationally expensive atmospheric model, PLASIM (Planet Simulator), and a cheaper model, EMBM (energy–moisture balance model), is established through the common boundary condition specified by an ocean model, allowing for information to be propagated from one to the other. This technique allows PLASIM emulators to be built at a low cost. The method is first demonstrated by emulating a scalar summary quantity, the global mean surface air temperature. It is then employed to emulate the dimensionally reduced 2-D surface air temperature field. Even though the two atmospheric models chosen are structurally unrelated, Gaussian process emulators of PLASIM atmospheric variables are successfully constructed using EMBM as a fast approximation. With the extra information gained from the cheap model, the multi-level emulator of PLASIM's 2-D surface air temperature field is built using only one-third the amount of expensive data required by the normal single-level technique. The constructed emulator is shown to capture 93.2 % of the variance across the validation ensemble, with the averaged RMSE of 1.33 °C. Using the method proposed, quantities from PLASIM can be constructed and used to study the effects introduced by PLASIM's atmosphere.

Due to their potential volume and speed, large submarine landslides can generate destructive tsunamis or damage expensive seafloor structures. Understanding their timing is therefore important for hazard assessments; however, dating large numbers of landslides close to their origin is logistically difficult. Previous landslide studies are typically limited to fewer than ten observations of ages. To address this we analyse extensive, continuous and long-term turbidite records from four deep-sea basins which are interpreted to be the distal deposits of large, disintegrative landslides. Our records include sufficient numbers of turbidites (N = 151–1571) for robust statistical analysis of long-term controls on event timing and testing for relationships with triggering mechanisms such as earthquakes, sea level, climate change and volcanic activity. We explore statistical methods developed by medical, economic and biological disciplines and show how they can be applied to analysis of submarine landslide frequency and triggering. Frequency analysis of field data reveals two different distribution forms for landslide recurrence – exponential and log-normal. We discuss possible individual and combined effects of controlling factors that result in these distributions. Rescaled range and Gaussian finite mixture models determine whether and how landslides are clustered in time. Parametric Generalised Linear Models and non-parametric Proportional Hazards Models are used to test for the significance and influence of different variables and their rate of change. We demonstrate the value of unusually detailed long-term landslide records, and show how statistical analysis provides quantitative inputs for future hazard assessments, landslide-climate studies and understanding the tempo of deep-sea sediment flux.

Large (> ~ 1 km3) submarine landslides can potentially generate very destructive tsunamis and damage expensive sea floor infrastructure. It is therefore important to understand their frequency and triggers, and whether their frequency is likely to change significantly due to future climatic and sea level change. It is expensive to both collect seafloor samples and to date landslides accurately; therefore we need to know how many landslides we need to date, and with what precision, to answer whether sea level is a statistically significant control. Previous non-statistical analyses have proposed that there is strong correlation between climate driven changes and landslide frequency. In contrast, a recent statistical analysis by Urlaub et al. (2013) of a global compilation of 41 large (> 1 km3) submarine landslide ages in the last 30 ka concluded that these ages have a temporally random distribution. This would suggest that landslide frequency is not strongly controlled by a single non-random global factor, such as eustatic sea level. However, there are considerable uncertainties surrounding the age of almost all large landslides, as noted by Urlaub et al. (2013). This contribution answers a key question that Urlaub et al. (2013) posed, but could not address — are large submarine landslides in this global record indeed temporally random, or are the uncertainties in landslide ages simply too great to tell? We use simulated age distributions in order to determine the significance of available age constraints from real submarine landslides. First, it is shown that realistic average uncertainties in landslide ages of ± 3 kyr may indeed result in a near-random distribution of ages, even where there are non-random triggers such as sea level. Second, we show how combination of non-random landslide ages from just 3 different settings, can easily produce an apparently random distribution if the landslides from different settings are out of phase. Third, if landslide frequency was directly proportional to sea level, we show that at least 10 to 53 landslides would need to be dated perfectly globally — to show this correlation. We conclude that it is prudent to focus on well-dated landslides from one setting with similar triggers, rather than having a poorly calibrated understanding of ages in multiple settings.

The generalised Pareto distribution (GPD) is often used to model the distribution of storm peak wave heights exceeding a high threshold, from which return values can be calculated. There are large differences in the performance of various parameter and quantile estimators for the GPD. Commonly used estimation methods such as maximum likelihood or probability weighted moments are not optimal, especially for smaller sample sizes. The performance of several estimators for the GPD is compared by the Monte Carlo simulation and the implications for estimating return values of significant wave height are discussed. Of the estimators compared, the likelihood-moment (LM) estimator has close to the lowest bias and variance over a wide range of sample sizes and GPD shape parameters. The LM estimator always exists, is simple to compute and has a low sensitivity to choice of threshold. It is recommended that the LM estimator is used for calculating return values of significant wave height when the sample size is less than 500. For sample sizes above 500 the NEW estimator of Zhang and Stephens (2009) can give accurate results for low computational cost.

The use of Gaussian process emulators is now widespread in the analysis of computer experiments. These methods generally assume that all the simulator inputs are continuous. In this paper we consider the design problem for the case where one or more simulator inputs is a switch, a factor that can take the values on or off. We propose two possible designs: One based on Sobol sequences and one on Latin Hypercubes. In both cases a small, but space filling, subset of simulator runs are carried out at both switch settings. This design is then nested within larger space filling designs one for each of the switch settings. If the switch is found to not affect the results these two designs can be combined into a much larger also space tilling design.

Autonomous underwater vehicles (AUVs) are used increasingly to explore hazardous marine environments. Risk assessment for such complex systems is based on subjective judgment and expert knowledge as much as on hard statistics. Here, we describe the use of a risk management process tailored to AUV operations, the implementation of which requires the elicitation of expert judgment. We conducted a formal judgment elicitation process where eight world experts in AUV design and operation were asked to assign a probability of AUV loss given the emergence of each fault or incident from the vehicle's life history of 63 faults and incidents. After discussing methods of aggregation and analysis, we show how the aggregated risk estimates obtained from the expert judgments were used to create a risk model. To estimate AUV survival with mission distance, we adopted a statistical survival function based on the nonparametric Kaplan-Meier estimator. We present theoretical formulations for the estimator, its variance, and confidence limits. We also present a numerical example where the approach is applied to estimate the probability that the Autosub3 AUV would survive a set of missions under Pine Island Glacier, Antarctica in January–March 2009.

The uncertainty in estimates of the energy yield from a wave energy converter (WEC) is considered. The study is presented in two articles. The first article considered the accuracy of the historic data and the second article, presented here, considers the uncertainty which arises from variability in the wave climate. Mean wave conditions exhibit high levels of interannual variability. Moreover, many previous studies have demonstrated longer-term decadal changes in wave climate. The effect of interannual and climatic changes in wave climate on the predictability of long-term mean WEC power is examined for an area off the north coast of Scotland. In this location anomalies in mean WEC power are strongly correlated with the North Atlantic Oscillation (NAO) index. This link enables the results of many previous studies on the variability of the NAO and its sensitivity to climate change to be applied to WEC power levels. It is shown that the variability in 5, 10 and 20 year mean power levels is greater than if annual power anomalies were uncorrelated noise. It is also shown that the change in wave climate from anthropogenic climate change over the life time of a wave farm is likely to be small in comparison to the natural level of variability. Finally, it is shown that despite the uncertainty related to variability in the wave climate, improvements in the accuracy of historic data will improve the accuracy of predictions of future WEC yield.