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The development and use of numerical simulators to predict vessel motions is essential to design and operational decision making in offshore engineering. Increasingly, probabilistic analyses of these simulators are being used to quantify prediction uncertainty. In practice, obtaining the required number of model evaluations may be prohibited by time and computational constraints. Emulation reduces the computational burden by forming a statistical surrogate of the model. The method is Bayesian and treats the numerical simulator as an unknown function modelled by a Gaussian process prior, with covariances of the model outputs constructed as a function of the covariances of the inputs. In offshore engineering, simulator inputs include directional quantities and we describe a way to build this information into the covariance structure. The methodology is discussed with reference to a numerical simulator which computes the mean turret offset amplitude of a FPSO in response to environmental forcing. It is demonstrated through statistical diagnostics that the emulator is well designed, with evaluations executed around 60,000 times faster than the numeric simulator. The method is generalisable to many offshore engineering numerical simulators that require directional inputs and is widely applicable to industry.

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... The computational efficiency of the GP surrogate model [8], coupled with accuracy comparable to the high-fidelity FEM simulations, allow the possibility of probabilistic prediction of fatigue damage in the bolted ring-flange. A Monte Carlo (MC) simulation [12] to propagate uncertainty from inputs to predicted fatigue damage may be feasible using a GP surrogate model, where it is prohibitive with the FEM model [8][9][10]. ...

... To perform SRA, where a large number of calls to the model are required, we are motivated to consider emulating the FEM simulation results through the application of a GP surrogate model. Emulation using a GP surrogate model leverages both computational efficiency and the accuracy of the FEM simulation [8,10]. The GP surrogate model is a statistical surrogate of the output of the FEM simulation, which we assume to be some unknown function f (·). ...

Achieving and maintaining a suitable level of bolt pre-load is critical to ensure structural reliability under the Fatigue Limit State for bolted ring-flanges in offshore wind turbine structures. Bolt pre-load is likely to vary over lifetime, with re-tensioning applied if relaxation exceeds design guideline allowance. An approach to assess the influence of varying bolt pre-load may be useful in the operational context. Recent work has demonstrated the suitability of a Gaussian Process surrogate model to emulate Finite Element Method structural simulations models of bolted ring-flanges, with computational efficiency gains. In this paper we predict cumulative fatigue damage in bolts over time, given uncertainty in bolt pre-load estimation, using a Gaussian Process surrogate model. We perform Structural Reliability Analysis to deliver approximations of annual Probability of Failure and the Reliability Index, under the Fatigue Limit State. Our approximations are compared to targets defined in relevant design standards. Furthermore, we incorporate observations, and maintenance actions, in updating the Structural Reliability Analysis during operation, and suggest practical applications of this method to inform inspection and maintenance practices.

... Forecasting products have become vital to offshore engineering operations. Forecasts have been produced for future meteorological and oceanographic states (Schiller et al., 2020), structural responses (Zhao et al., 2018a;Astfalck et al., 2019a), and energy demands (Foley et al., 2012). Broadly, engineers categorize forecasting models as either numerical, data-only, or hybrid (Sikorska et al., 2011). ...

Maritime engineering relies on model forecasts for many different processes, including meteorological and oceanographic forcings, structural responses, and energy demands. Understanding the performance and evaluation of such forecasting models is crucial in instilling reliability in maritime operations. Evaluation metrics that assess the point accuracy of the forecast (such as root-mean-squared error) are commonplace, but with the increased uptake of probabilistic forecasting methods such evaluation metrics may not consider the full forecasting distribution. The statistical theory of proper scoring rules provides a framework in which to score and compare competing probabilistic forecasts, but it is seldom appealed to in applications. This translational paper presents the underlying theory and principles of proper scoring rules, develops a simple panel of rules that may be used to robustly evaluate the performance of competing probabilistic forecasts, and demonstrates this with an application to forecasting surface winds at an asset on Australia’s North–West Shelf. Where appropriate, we relate the statistical theory to common requirements by maritime engineering industry. The case study is from a body of work that was undertaken to quantify the value resulting from an operational forecasting product and is a clear demonstration of the downstream impacts that statistical and data science methods can have in maritime engineering operations.

... Gaussian Process (GP) emulators are non-parametric statistical regression models that flexibly represent chosen model output or performance metrics as a function of a subset of input parameters, together with an uncertainty on that prediction (Rasmussen and Williams, 2006;Astfalck et al., 2019). We describe the model output y as a function of a vector of input 230 parameters expressed as: ...

The neodymium (Nd) isotope composition (εNd) of seawater can be used to trace large-scale ocean circulation features. Yet, due to the elusive nature of marine Nd cycling, particularly in discerning non-conservative particle-seawater interactions, there remains considerable uncertainty surrounding a complete description of marine Nd budgets. Here, we present an optimisation of the Nd isotope scheme within the fast coupled atmosphere-ocean general circulation model (FAMOUS), using a statistical emulator to explore the parametric uncertainty and optimal combinations of three key model inputs relating to: (1) the efficiency of reversible scavenging, (2) the magnitude of the seafloor benthic flux, and (3) a riverine source scaling, accounting for release of Nd from river sourced particulate material. Furthermore, a suite of sensitivity tests provide insight on the regional mobilisation and spatial extent (i.e., testing a margin-constrained versus a seafloor-wide benthic flux) of certain reactive sediment components. In the calibrated scheme, the global marine Nd inventory totals 4.27 × 1012 g and has a mean residence time of 727 years. Atlantic Nd isotope distributions are represented well, and the weak sensitivity of North Atlantic Deep Water to highly unradiogenic sedimentary sources implies an abyssal benthic flux is of secondary importance in determining the water mass εNd properties under the modern vigorous circulation condition. On the other hand, Nd isotope distributions in the North Pacific are 3 to 4 εNd-units too unradiogenic compared to water measurements, and our simulations indicate that a spatially uniform flux of bulk sediment εNd does not sufficiently capture the mobile sediment components interacting with seawater. Our results of sensitivity tests suggest that there are distinct regional differences in how modern seawater acquires its εNd signal, in part relating to the complex interplay of Nd addition and water advection.

... As a probabilistic encoder of expert knowledge, formal elicitation procedures (O'Hagan et al., 2006) have contributed greatly to physical domain sciences where complex models are essential to our understanding of the underlying processes. From climatology, meteorology and oceanography (Kennedy et al., 2008) to geology and geostatistics (Walker andCurtis, 2014, andLark et al., 2015) to hydrodynamics and engineering (Astfalck et al., 2018(Astfalck et al., , 2019, the central role of expert elicitation is being increasingly recognised. The complexity and parameterisations of geophysical models, as well as the expert knowledge that resides within the geophysical community, suggest this domain should be no different. ...

Unlike some other well-known challenges such as facial recognition, where
machine learning and inversion algorithms are widely developed, the
geosciences suffer from a lack of large, labelled data sets that can be used
to validate or train robust machine learning and inversion schemes. Publicly
available 3D geological models are far too restricted in both number and the
range of geological scenarios to serve these purposes. With reference to
inverting geophysical data this problem is further exacerbated as in most
cases real geophysical observations result from unknown 3D geology, and
synthetic test data sets are often not particularly geological or
geologically diverse. To overcome these limitations, we have used the Noddy
modelling platform to generate 1 million models, which represent the first
publicly accessible massive training set for 3D geology and resulting
gravity and magnetic data sets (https://doi.org/10.5281/zenodo.4589883, Jessell, 2021). This model suite
can be used to train machine learning systems and to provide comprehensive
test suites for geophysical inversion. We describe the methodology for
producing the model suite and discuss the opportunities such a model suite
affords, as well as its limitations, and how we can grow and access this
resource.

... where τ ≥ 6, c ∈ (0, π], (a) + = max(0, a), and d(i, j) is the geodesic distance between locations i and j. The C 4 -Wendland covariance function is commonly chosen so as to define a smooth process on the sphere; (see, for example Astfalck et al., 2019). Parameters are specified as κ 2 = 1.61, c = 0.92, and τ = 6; this represents the prior-beliefs of domain experts as to the behaviour of U X . ...

Any experiment with climate models relies on a potentially large set of spatio-temporal boundary conditions. These can represent both the initial state of the system and/or forcings driving the model output throughout the experiment. Whilst these boundary conditions are typically fixed using available reconstructions in climate modelling studies, they are highly uncertain, that uncertainty is unquantified, and the effect on the output of the experiment can be considerable. We develop efficient quantification of these uncertainties that combines relevant data from multiple models and observations. Starting from the coexchangeability model, we develop a coexchangable process model to capture multiple correlated spatio-temporal fields of variables. We demonstrate that further exchangeability judgements over the parameters within this representation lead to a Bayes linear analogy of a hierarchical model. We use the framework to provide a joint reconstruction of sea-surface temperature and sea-ice concentration boundary conditions at the last glacial maximum (19-23 ka) and use it to force an ensemble of ice-sheet simulations using the FAMOUS-Ice coupled atmosphere and ice-sheet model. We demonstrate that existing boundary conditions typically used in these experiments are implausible given our uncertainties and demonstrate the impact of using more plausible boundary conditions on ice-sheet simulation.

... While the abovementioned studies suggest deterministic surrogate models, recently developed stochastic surrogate models have shown promising results for using probabilistic analysis in offshore engineering problems. For instance, Astfalck et al. (2019) developed a statistical surrogate using the Gaussian Process (GP) concept (Rasmussen et al., 2006) for stochastic prediction of the offsets of a ship-shaped floating offshore facility subjected to ocean wave, wind and current loads. The GPs can rapidly conduct nonlinear regression, which is the process of estimating the relationship between dependent and independent variables or predictors (Gelman et al., 2013), in high-dimensional spaces. ...

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Quantifying the risk of system failure is a key input to a risk-based decision-making process for the in-service integrity management of aging steel catenary risers (SCRs), which are prone to fatigue failure within the touchdown zone (TDZ). However, in practice the computational cost of assessing the probability of failure (POF) for SCRs is prohibitive. In this paper, we propose an efficient framework for quantifying the fatigue POF within the SCR TDZ by using the Bayesian machine learning technique adopting Gaussian Processes (GP) for regression. GP-based predictive models perform stochastic response prediction rapidly and are therefore attractive for rapid evaluation of the SCR fatigue POF. In this paper, we introduce innovative techniques for predicting fatigue damage profiles within the SCR TDZ generated by an irregular sea-state applied to the host vessel at its mean offset position, and for dimensionality reduction in the regression analysis. We utilize a realistic case study to demonstrate the framework, which consists of a representative 20″ SCR connected to a semi-submersible host vessel located in 950 m water depth. In addition, we use a site-specific 32-year hindcast wave data and 1-year measured current data, with associated statistical distributions, for the long-term fatigue loading conditions. We also employ numerical simulation and a random sampling technique to create simulation-based input and output datasets for training and testing of the derived probabilistic simulation-based surrogate. The results show that the proposed method is able to predict the fatigue life of the representative riser efficiently and accurately. Finally, we demonstrate the usefulness of the proposed method by rapidly generating the fatigue POF for the 20″ SCR considering the uncertainties associated with input loads, material properties, geometric parameters, and soil stiffness.

... For example, to understand the uncertainty associated with models for bolt stress, a Monte Carlo (MC) approximation to the distribution of input parameters may be propagated through FEM solutions. However, if the FEM simulation is computationally expensive MC methods may be prohibitive, especially in real operations (Astfalck et al. 2019). ...

Fatigue assessment of bolted ring-flanges in offshore wind turbine structures is a critical step in the overall structural design process. Analytical and numerical methods are employed to predict bolt stress, offering the convenience of speed (analytical) or accuracy (numerical). In this paper , we investigate the application of a Gaussian Process (GP) surrogate model, leveraging the accuracy of numerical techniques with significant gains in computation times. We present the notion of predictive uncertainty available from the GP surrogate model. Finally, we demonstrate the capacity of the GP surrogate model to propagate input space uncertainty in a computationally efficient manner.

... More recently, as stochastic simulation has become more prominent, so has the interest in the emulation of stochastic computer simulators, for example, Astfalck et al. (2019), Rocchetta et al. (2018), Boys et al. (2018). There are a variety of approaches to (GP based) emulation of stochastic computer models; see Baker et al. (2020) for a recent overview. ...

Increasingly, stochastic computer models are being used in science and engineering to predict and understand complex phenomena. Despite the power of modern computing, these simulators are often too computationally costly to be of practical use due to their complexity. Hence the emulation of stochastic computer models is a problem of increasing interest. Many stochastic computer models can be run at different levels of complexity, which incurs a trade-off with simulation accuracy. More complex simulations are more expensive to run, but will often be correlated with less complex but cheaper to run versions. We present a heteroscedastic Gaussian process approach to emulation of stochastic simulators which utilises cheap approximations to a stochastic simulator, motivated by a stochastic reliability and maintenance model of a large offshore windfarm. The performance of our proposed methodology is demonstrated on two synthetic examples (a simple, tractable example and a predator-prey model) before being applied to the stochastic windfarm simulator.

The Digital Twin (DT) paradigm offers an extension of simulation model utility into the operational phase of an engineering asset. The goal is a simulation “twinned” with observed data that reflects the actual performance of the asset. However, exploring sources of uncertainty for both the physical asset and the simulation model are a challenge. For example, random metocean conditions, and uncertainty on model parameters and response behaviour of offshore wind turbine (OWT) structures, contribute to uncertainty for predicted life under fatigue. In-service assessment of OWT structures will benefit from twinning simulations and observed data, where a framework to treat this uncertainty is defined. The DT needs to capture state, condition, and behaviour to a level that allows quantification and propagation of uncertainty for reliability analysis. Using a DT, built for fatigue assessment of bolted ring-flanges on OWT support structures, this paper explores the challenges and opportunities in defining uncertainties of interest. We propagate these uncertainties through the DT in a coherent manner using a Gaussian Process (GP) surrogate modelling approach, efficiently emulating a computationally expensive numerical simulator. The GP is an attractive surrogate model method given this computational efficiency, in addition to providing an estimate of prediction uncertainty at unobserved points in the output space. The use of the GP surrogate model is included within a definition of the Surrogate DT, a framework including “fast” and “slow” twinning processes. We define six requirements to apply DTs to OWT structures which provide practical guidelines for modelling this complex asset under uncertainty.

The environment experienced by offshore industries is inherently volatile, providing a strong impetus for accurate modelling and prediction of meteorological and oceanographic conditions. Numerical forecasts, obtained from physics-based deterministic models, are informative yet imperfect in their prediction, motivating the quantification of their associated uncertainty. To achieve this, we present a Bayesian hierarchical model that updates numerical forecasts of metocean conditions and examines their associated error structure. This involves a linear regression to remove systematic forecast biases and the use of time series techniques, namely autoregressive and generalised autoregressive conditional heteroscedasticity models, to remove residual time-evolving error structure. We find that the hierarchical statistical model outperforms a numerical weather prediction model in terms of mean squared error and continuous ranked probability score, a metric which is able to account for uncertainty information contained in probabilistic forecasts. While the method is generalisable to a range of metocean conditions with an available set of past measurements and numerical forecasts, we present our findings using a case study of sea surface wind data. The data is collected from a location on Australia’s North-West Shelf, a region home to an extent of offshore operations and often characterised by extreme unexpected metocean conditions.

This paper describes the development of a data assimilation framework for deterministic vessel motion prediction in real-time, through the coupling of artificial neural network (ANN) and Ensemble Kalman Filter (EnKF). The ANN uses measurements of vessel motion and upstream wave elevation as inputs to predict future vessel motions. However, due to the uncertainties in the measurements and ANN, large errors may be built up over time. To improve the accuracy of the predictions, an EnKF has been used to combine measured data with the ANN predictions, to generate an updated prediction, which is subsequently used as input for the next prediction. Using a turret-moored Floating Production Storage Offloading (FPSO) vessel, the developed framework was first tested using synthesized measurements generated from numerical simulations and subsequently applied on experimental measurements of vessel motion. Sensitivity study was carried out to determine the data assimilation parameters, including the ensemble size and data assimilation interval. The data assimilation framework using ANN-EnKF is promising, in that it is able to improve the motion prediction accuracy for heave and pitch as compared to ANN, which do not incorporate measurements. The improvement is more obvious for the experimental measurements that have higher uncertainties.

A novel application of a convolutional neural network (CNN) for the identification of mooring line failure of a turret-moored FPSO is demonstrated. The CNN was trained on images of the turret horizontal displacement history, simulated for both an intact mooring and a system with one line that had failed. When tested on operational and extreme environments representative of the North West Shelf of Australia, the CNN successfully distinguished between the turret responses associated with the intact and broken mooring. Classification accuracy was found to be lower for relatively benign conditions when the turret offset response was minimal. This was significantly improved through the use of additional hidden layers and retraining. As the CNN does not explicitly utilise metocean data as input, apart from training, it is envisaged that it offers an effective and lower-cost alternative to existing mooring failure detection approaches for the offshore industry.

Prediction of the extremal responses of dynamic structures is a vital step in the risk management of offshore assets. Often when modelling structural response the outputs are dependent on covariates defined on a continuous input domain. We demonstrate a methodology to allow for continuous covariates in extremal modelling by building latent variable models, whereby output dependencies are incorporated by smooth processes in the latent parameters. This allows information from close-by input regions to be shared when forming inference at unseen inputs. We illustrate the methodology using a computational simulation of Floating Production Storage and Offloading (FPSO)vessel motions, modelled as functions of the peak wave period. We provide methodologies and diagnostics for the modelling of the time-domain maxima, quantiles, and threshold exceedance data. There are three contributions made by this research: a methodology to predict the extremal outputs from a time-domain simulator, with incorporation of continuous covariate knowledge; significant speed increase when using the developed methodology as a computational proxy to the simulator; and a framework for the probabilistic quantification of the output uncertainty of the extremal data.

Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulator's input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulator's behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was 10(11) times smaller than the original input space. Simulator evaluations made within this region were found to have a 65% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.

The microphysical properties of convective clouds determine their radiative effects on climate, the amount and intensity of precipitation as well as dynamical features. Realistic simulation of these cloud properties presents a major challenge. In particular, because models are complex and slow to run, we have little understanding of how the considerable uncertainties in parameterized processes feed through to uncertainty in the cloud responses. Here we use statistical emulation to enable a Monte Carlo sampling of a convective cloud model to quantify the sensitivity of twelve cloud properties to aerosol concentrations and nine model parameters representing the main microphysical processes. We examine the response of liquid and ice-phase hydrometeor concentrations, precipitation and cloud dynamics for a deep convective cloud in a continental environment. Across all cloud responses, the concentration of the Aitken and accumulation aerosol modes and the collection efficiency of droplets by graupel particles have the most influence on the uncertainty. However, except at very high aerosol concentrations, uncertainties in precipitation intensity and amount are affected more by interactions between drops and graupel than by large variations in aerosol. The uncertainties in ice crystal mass and number are controlled primarily by the shape of the crystals, ice nucleation rates and aerosol concentrations. Overall, although aerosol particle concentrations are an important factor in deep convective clouds, uncertainties in several processes significantly affect the reliability of complex microphysical models. The results suggest that our understanding of aerosol-cloud interaction could be greatly advanced by extending the emulator approach to models of cloud systems. This article is protected by copyright. All rights reserved.

Cosmologists at the Institute of Computational Cosmology, Durham University,
have developed a state of the art model of galaxy formation known as Galform,
intended to contribute to our understanding of the formation, growth and
subsequent evolution of galaxies in the presence of dark matter. Galform
requires the specification of many input parameters and takes a significant
time to complete one simulation, making comparison between the model's output
and real observations of the Universe extremely challenging. This paper
concerns the analysis of this problem using Bayesian emulation within an
iterative history matching strategy, and represents the most detailed
uncertainty analysis of a galaxy formation simulation yet performed.

Surrogate modeling, also called metamodeling, has evolved and been
extensively used over the past decades. A wide variety of methods and
tools have been introduced for surrogate modeling aiming to develop and
utilize computationally more efficient surrogates of high-fidelity
models mostly in optimization frameworks. This paper reviews, analyzes,
and categorizes research efforts on surrogate modeling and applications
with an emphasis on the research accomplished in the water resources
field. The review analyzes 48 references on surrogate modeling arising
from water resources and also screens out more than 100 references from
the broader research community. Two broad families of surrogates namely
response surface surrogates, which are statistical or empirical
data-driven models emulating the high-fidelity model responses, and
lower-fidelity physically based surrogates, which are simplified models
of the original system, are detailed in this paper. Taxonomies on
surrogate modeling frameworks, practical details, advances, challenges,
and limitations are outlined. Important observations and some guidance
for surrogate modeling decisions are provided along with a list of
important future research directions that would benefit the common
sampling and search (optimization) analyses found in water resources.

The reanalysis at National Centers for Environmental Prediction (NCEP) focuses on atmospheric states reports generated by a constant model and a constant data assimilation system. The datasets have been exchanged among national and international partners and used in several more reanalyses. The new data assimilation techniques have been introduced including three-dimensional variational data assimilation (3DVAR), 4DVAR, and ensembles of analyses such as ensemble Kalman filter (EnKF), which produce not only an ensemble mean analysis but also a measure of the uncertainty. The new climate forecast system reanalysis (CFSR) was executed to create initial states for the atmosphere, ocean, land, and sea ice that are consistent as possible with the next version of the climate forecast system (CFS) version 2, which is to be implemented operationally at NCEP in 2010. Several graphical plots were generated automatically at the end of each reanalyzed month and were displayed on the CFSR Web site in real time.

A climate model emulator is developed using neural network techniques and trained with the data from the multithousand-member climateprediction.net perturbed physics GCM ensemble. The method recreates nonlinear interactions between model parameters, allowing a simulation of a much larger ensemble that explores model parameter space more fully.
The emulated ensemble is used to search for models closest to observations over a wide range of equilibrium response to greenhouse gas forcing. The relative discrepancies of these models from observations could be used to provide a constraint on climate sensitivity. The use of annual mean or seasonal differences on top-of-atmosphere radiative fluxes as an observational error metric results in the most clearly defined minimum in error as a function of sensitivity, with consistent but less well-defined results when using the seasonal cycles of surface temperature or total precipitation.
The model parameter changes necessary to achieve different values of climate sensitivity while minimizing discrepancy from observation are also considered and compared with previous studies. This information is used to propose more efficient parameter sampling strategies for future ensembles.

The estimated range of climate sensitivity has remained unchanged for decades, resulting in large un- certainties in long-term projections of future climate under increased greenhouse gas concentrations. Here the multi-thousand-member ensemble of climate model simulations from the climateprediction.net project and a neural network are used to establish a relation between climate sensitivity and the amplitude of the seasonal cycle in regional temperature. Most models with high sensitivities are found to overestimate the seasonal cycle compared to observations. A probability density function for climate sensitivity is then calculated from the present-day seasonal cycle in reanalysis and instrumental datasets. Subject to a number of assumptions on the models and datasets used, it is found that climate sensitivity is very unlikely (5% probability) to be either below 1.5-2 K or above about 5-6.5 K, with the best agreement found for sensitivities between 3 and 3.5 K. This range is narrower than most probabilistic estimates derived from the observed twentieth-century warming. The current generation of general circulation models are within that range but do not sample the highest values.

The CRASH computer model simulates the effect of a vehicle colliding against different barrier types. If it accurately represents real vehicle crash-worthiness, the computer model can be of great value in various aspects of vehicle design, such as the setting of timing of air bag releases. The goal of this study is to address the problem of validating the computer model for such design goals, based on utilizing computer model runs and experimental data from real crashes. This task is complicated by the fact that (i) the output of this model consists of smooth functional data, and (ii) certain types of collision have very limited data. We address problem (i) by extending existing Gaus-sian process-based methodology developed for models that produce real-valued output, and resort to Bayesian hierarchical modeling to attack problem (ii).

Mathematical models, usually implemented in computer programs known as simulators, are widely used in all areas of science and technology to represent complex real-world phenomena. Simulators are often suciently complex that they take appreciable amounts of computer time or other resources to run. In this context, a methodology has been developed based on building a statistical representation of the simulator, known as an emulator. The principal approach to building emulators uses Gaussian processes. This work presents some diagnostics to validate and assess the adequacy of a Gaussian process emulator as surrogate for the simulator. These diagnostics are based on comparisons between simulator outputs and Gaussian process emulator outputs for some test data, known as validation data, dened by a sample of simulator runs not used to build the emulator. Our diagnostics take care to account for correlation between the validation data. In order to illustrate a validation procedure, these diagnostics are applied to two dierent data sets.

Mathematical modelers from different disciplines and regulatory agencies worldwide agree on the importance of a careful sensitivity analysis (SA) of model-based inference. The most popular SA practice seen in the literature is that of ’one-factor-at-a-time’ (OAT). This consists of analyzing the effect of varying one model input factor at a time while keeping all other fixed. While the shortcomings of OAT are known from the statistical literature, its widespread use among modelers raises concern on the quality of the associated sensitivity analyses. The present paper introduces a novel geometric proof of the inefficiency of OAT, with the purpose of providing the modeling community with a convincing and possibly definitive argument against OAT. Alternatives to OAT are indicated which are based on statistical theory, drawing from experimental design, regression analysis and sensitivity analysis proper.

This paper is intended to review a number of variance-based methods used in Sensitivity Analysis (SA) to ascertain how much a model (numerical or otherwise) depends on each or some of its input parameters. A class of variance-based methods (correlation ratio or importance measure) that is capable of measuring only the main effect contribution of each input parameter on the output variance are described briefly. In addition, two methods (Sobol' and FAST) that are capable of computing the so-called "Total Sensitivity Indices" (TSI), which measures a parameter's main effect and all the interactions (bf any order) involving that parameter, are described in details. An illustrated example demonstrates that the incorporation of total effect indices is the only way to perform a rigorous quantitative sensitivity analysis.

In many scientific disciplines complex computer models are used to understand the behaviour of large scale physical systems. An uncertainty anal- ysis of such a computer model known as Galform is presented. Galform models the creation and evolution of approximately one million galaxies from the begin- ning of the Universe until the current day, and is regarded as a state-of-the-art model within the cosmology community. It requires the specification of many in- put parameters in order to run the simulation, takes significant time to run, and provides various outputs that can be compared with real world data. A Bayes Linear approach is presented in order to identify the subset of the input space that could give rise to acceptable matches between model output and measured data. This approach takes account of the major sources of uncertainty in a consistent and unified manner, including input parameter uncertainty, function uncertainty, observational error, forcing function uncertainty and structural uncertainty. The approach is known as History Matching, and involves the use of an iterative suc- cession of emulators (stochastic belief specifications detailing beliefs about the Galform function), which are used to cut down the input parameter space. The analysis was successful in producing a large collection of model evaluations that exhibit good fits to the observed data.

New measurements from a drillship operating in a swell dominated region are used to demonstrate that generally reliable estimates of the hydrodynamic motions can be obtained from numerically hindcast directional wave spectra. Independent estimates of the directional wave spectra derived from the ship motions using a parametric approach are shown to be in close alignment with the hindcast spectra. Deviations between predictions and observations in this methodology are largely explained through the idealised nature of the hindcast spectra and limitations in the ability of the numerical model to capture swell arrival over the long fetch on approach to the project site, which is demonstrated through comparison of observed and modelled spectra. In general, the study provides encouraging support for the application of numerical hindcast techniques for vessel motion assessments, particularly for sites for which wave buoy measurements are not available.

Probability distributions that describe metocean conditions are essential for design and operational decision making in offshore engineering. When data are insufficient to estimate these distributions an alternative is expert elicitation – a collection of techniques that translate personal qualitative knowledge into subjective probability distributions. We discuss elicitation of surface currents on the Exmouth Plateau, North-Western Australia, a region of intense oil and gas drilling and exploration. Metocean and offshore engineering experts agree that surface currents on the plateau exhibit large spatio-temporal variation, and that recorded observations do not fully capture this variability. Combining such experts’ knowledge, we elicit the joint distribution of magnitude and direction by first focusing on the marginal distribution of direction, followed by the conditional distribution of magnitude given direction. Although we focus on surface currents, the direction/magnitude components are common to many metocean processes. The directional component complicates the problem by introducing circular probability distributions. The subjectivity of elicitation demands caution and transparency, and this is addressed by embedding our method into the established elicitation protocol, the Sheffield Elicitation Framework. The result is a general framework for eliciting metocean conditions when data are insufficient to estimate probabilistic summaries.

The present work is concerned with a dynamic modeling and optimization method to minimize the vibration transmission through the marine propulsion shafting system by the FRF-based substructuring method and the sensitivity analysis. The dynamic model of the coupled propeller - propulsion shafting - hull system under propeller exciting forces and excitation forces from the propulsion electric machine is established. Vibration response characteristics for the coupled system are analyzed. Mean-square forces and power flow transmitted to the foundation is employed as objective functions to evaluate the optimal design of propulsion shafting system. The proposed method can take into consideration of real propellers as well as the fluid-structure interaction between the propeller, hull and the surrounding water. It is shown that propeller flexibility contributes obvious peaks in the vibration responses. For the studied model, the stiffness of the bearings and the isolators should be lowered; the stiffness of the coupling should be increased to suppress vibration transmission. The mean-square force and the power flow transmitted to the foundation in the concerned frequency range is decreased to 0.712 and 0.525 of the initial value, respectively. The proposed dynamic modeling and optimization method is capable of performing optimization with greatly improved efficiency.

Maintaining the expected position is critical to the overall safe operation of a floating oil platform. Mooring systems are critical to the integrity of the platform. Relying on instrumentation for monitoring the mooring line tensions represents multi-faceted challenges. Therefore, alternative methods have been introduced across the industry to reduce the costs and complexities of maintaining these systems.
The paper discusses implementation of the Position Response Learning System (PRLS), a novel concept for addressing the integrity of mooring systems. PRLS is based on emerging data technologies and particularly machine learning that bridges the gap between a variety of global position measurements of the oil-platform and the mooring integrity paradigm. Machine learning enables learning from large amount of data without explicit programming. The PRLS concept does not require expensive line-tension measurement systems, but rather the global motion systems enhanced with the metocean monitoring. The global motion systems that include DGPS and MRU are typically already installed on oil platforms. In addition to measured data, PRLS can utilize a plethora of other data sources, including numerical simulations, model test data, and most importantly, the real-time and archived field data other than the line tensions. When the data are coupled with machine learning methods, they provide reliable, robust, and cost-effective solutions to address the integrity of the mooring system in real time of the oil platform.
The article illustrates how the PRLS can identify a mooring line failure and even indicate which of the mooring line fails. Preliminary results based on the simulated data show that the accuracy of such predictions is better than 98%. The PRLS runs in the background independent of other integrity monitoring systems. It requires retraining periodically with new field data to improve the prediction robustness and accuracy. PRLS may be deployed on all types of floating platforms under a relatively moderate capital expense, and with very low operational costs when compared to high capital and operational expenses of a subsea mooring line monitoring system.

This paper presents the validation of tool which combines a model of the steady wind, current and wave loads with numerical estimates of the metocean conditions to predict the heading of a turret-moored vessel such as an FPSO. Estimates of the wave-induced mean drift contribution were obtained using an open-source three-dimensional boundary element code. Empirical data from reduced-scale tests were used to obtain the wind and current loads, which were shown to enable more reliable estimates than would be possible using data for very large crude carriers. The environmental conditions were predicted numerically using multi-resolution nested models which assimilate observed data. Using measurements from an operating FPSO for full-scale validation, the model was shown to be able to predict the heading for a range of environmental conditions including non-collinear seas, swells and winds to an accuracy typically within 5%. Preliminary estimates of the vessel roll amplitudes also agreed favourably with measured values. It is envisaged that the model will be attractive for operators of FPSOs in better facilitating the identification of dangerous conditions. This could ultimately lead to safer and more efficient operations.

This paper details a case study, where Bayesian methods are used to estimate the model parameters of an offshore platform. This first involves running a series of Finite Element simulations using the Ramboll Offshore Structural Analysis Programs (ROSAP)—developed by Ramboll Oil & Gas—thus establishing how the modal characteristics of an offshore structure model vary as a function of its material properties. Data based modelling techniques are then used to emulate the Finite Element model, as well as estimates of model error. The uncertainties associated with estimating the hyperparameters of the data based modelling techniques are then analysed utilising Markov chain Monte Carlo (MCMC) methods. The resulting analysis takes account of the uncertainties which arise from measurement noise, model error, model emulation and parameter estimation.

In this study sensitivity analyses were conducted to investigate the relative importance of different uncertain variables on the life-cycle cost (LCC) estimation of a steel jacket offshore platform subjected to seismic loads. The sensitivity analysis was conducted using different methods such as tornado diagram analysis (TDA), first-order second-moment (FOSM) and Latin hypercube sampling (LHS). The analysis results showed that the uncertain variables related to loss estimation and seismic hazard had a more dominant influence on the LCC variability compared to the other variables. Among the structural uncertain parameters, the variability in plastic hinge strength and modal damping ratio had the most significant impact on the LCC. Variability in the initial cost showed higher impact on LCC estimations compared to other cost component variables. It was also observed that the application of members with energy dissipation capability resulted in more economical design compared to use of conventional members.

We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work.

Dynamic positioning capability/dynamic station-keeping capability (DPCap/DynCap) analysis determines the maximum environmental forces that the dynamic positioning (DP) thrust system can counteract for a vessel on any given heading. These forces and moment are statically/dynamically balanced by the thrust forces and moment provided by the thrust system comprising several types and configurations of thrusters. The thrust provided by each thruster is vital for the positioning capability of the vessel; however, to date no method has been reported in the literature to quantitatively identify the influence that each individual thruster exerts on the combined positioning capability of a vessel. This study proposes a novel thrust sensitivity analysis method based on a newly defined synthesized positioning capability criterion for DPCap/DynCap analysis, which allows the sensitivity of the thrusters to be determined. Once the most sensitive thruster is determined, it should then be given the most attention since, if it fails, the remaining thrusters cannot supply sufficient forces and moment to hold the vessel on course or in position. It is demonstrated that the analysis is helpful for determining the most sensitive thruster and may provide reliable guidance in the design of the thrust system.

This work focuses on substituting a computationally expensive simulator by a cheap emulator to enable studying applications where running the simulator is prohibitively expensive. The procedure consists of two steps. In a first step, the emulator is calibrated to closely mimic the simulator response for a number of pre-defined cases. In a second step the calibrated emulator is used as surrogate for the simulator in the otherwise prohibitively expensive application. An appealing feature of the proposed framework contrary to other approaches is that the uncertainty on the emulator prediction can be determined. While the proposed framework is applicable in virtually all areas of natural sciences, we discuss the approach and evaluate its performance based on a typical example in the realm of computational wind engineering, namely the determination of the wind field in an urban area.

This paper presents a new methodology to calibrate structural reliability models output. The methodology combines data from experience and prediction models to correct the structural reliability models. The paper gives an overview of the techniques used for model calibrations, summarises the methodology in an overall discussion, and proposes data processing to sanitise sensitive information. The proposed methodology is inherently adaptable and can be applied to many other fields that require cost effective maintenance, as well as providing data for calibrating methods and codes.

S ummary
The optimal design problem is tackled in the framework of a new model and new objectives. A regression model is proposed in which the regression function is permitted to take any form over the space of independent variables. The design objective is based on fitting a simplified function for prediction. The approach is Bayesian throughout.
The new designs are more robust than conventional ones. They also avoid the need to limit artificially design points to a predetermined subset of . New solutions are also offered for the problems of smoothing, curve fitting and the selection of regressor variables.

Squalls have been present in the environmental specifications for floating units in West Africa for the last couple of years. However it appears that such phenomena tend to be the designing factor for mooring systems of deepwater FPSO’s (in spread or turret configuration) and offloading buoys. At that stage, due to the lack of proper modelling/characterisation, squalls tend to be represented for design purposes by on-site recorded time series of varying wind velocity and associated relative headings applied from any direction. This leads to rapid changes in offsets and loads in the mooring lines induced by the transient response of the vessel to sudden load increase generated by such squall signal. Through diverse exemplary simplified calculations, this paper illustrates the influence of the consideration of squalls in the design process, together with the present shortcomings in the modelling process, either in terms of extreme conditions, or in terms of operating conditions, knowing that such events are difficult to forecast. In addition the effect of tugs, and associated operating limitations are also discussed. Areas needing further investigation are finally identified.

A thirty one year wave hindcast (1979-2009) using NCEP's latest
high resolution Climate Forecast System Reanalysis (CFSR) wind and ice
database has been developed and is presented here. The hindcast has been
generated using the third generation wind wave model WAVEWATCH III®
with a mosaic of 16 two-way nested grids. The resolution of the grids
ranged from 1/2° to 1/15°. Validation results for bulk
significant wave height Hs and 10 m (above Mean Sea Level)
wind speeds U10 have been presented using both altimeter
records and NDBC buoys. In general the database does a good job of
representing the wave climate. At most buoys there is excellent
agreement between model and data out to the 99.9th percentile. The
agreement at coastal buoys is not as good as the offshore buoys due to
unresolved coastal features (topographic/bathymetric) as well as issues
related to interpolating wind fields at the land-sea margins. There are
some concerns about the wave climate in the Southern Hemisphere due to
the over prediction of winds (early part of the database) as well as the
lack of wave blocking due to icebergs (in 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.

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.

Global climate models (GCMs) contain imprecisely defined parameters that account, approximately, for subgrid-scale physical processes. The response of a GCM to perturbations in its parameters, which is crucial for quantifying uncertainties in simulations of climate change, can - in principle - be assessed by simulating the GCM many times. In practice, however, such "perturbed physics" ensembles are small because GCMs are so expensive to simulate. Statistical tools can help in two ways. First, they can be used to combine ensembles from different but related experiments, increasing the effective number of simulations. Second, they can be used to describe the GCM's response in ways that cannot be extracted directly from the ensemble(s). The authors combine two experiments to learn about the response of the Hadley Centre Slab Climate Model version 3 (HadSM3) climate sensitivity to 31 model parameters. A Bayesian statistical framework is used in which expert judgments are required to quantify the relationship between the two experiments; these judgments are validated by detailed diagnostics. The authors identify the entrainment rate coefficient of the convection scheme as the most important single parameter and find that this interacts strongly with three of the large-scale-cloud parameters.

We consider the use of emulator technology as an alternative method to second-order Monte Carlo (2DMC) in the uncertainty analysis for a percentile from the output of a stochastic model. 2DMC is a technique that uses repeated sampling in order to make inferences on the uncertainty and variability in a model output. The conventional 2DMC approach can often be highly computational, making methods for uncertainty and sensitivity analysis unfeasible. We explore the adequacy and efficiency of the emulation approach, and we find that emulation provides a viable alternative in this situation. We demonstrate these methods using two different examples of different input dimensions, including an application that considers contamination in pre-pasteurised milk.

The author shows how geostatistical data that contain measurement errors can be analyzed objectively by a Bayesian approach using Gaussian random fields. He proposes a reference prior and two versions of Jeffreys' prior for the model parameters. He studies the propriety and the existence of moments for the resulting posteriors. He also establishes the existence of the mean and variance of the predictive distributions based on these default priors. His reference prior derives from a representation of the integrated likelihood that is particularly convenient for computation and analysis. He further shows that these default priors are not very sensitive to some aspects of the design and model, and that they have good frequentist properties. Finally, he uses a data set of carbon/nitrogen ratios from an agricultural field to illustrate his approach.Analyse bayésienne objective de données spatiaies entachées d'erreurs de mesuresL'auteur montre comment des données géostatistiques entachées d'erreurs de mesure peuvent ětre analysées objectivement par une approche bayésienne à l'aide de champs aléatoires gaussiens. II propose une loi a priori de référence et deux versions de la loi de Jeffreys pour les paramètres du modèle. II étudie l'intégrabilité et l'existence des moments des lois a posteriori correspondantes. II démontre aussi l'existen-ce de l'espérance et de la variance des lois prévisionnelles déduites de ces lois a priori objectives. Sa loi a priori de référence découle d'une représentation de la vraisemblance intégrée qui est très commode aux fins de calcul et d'analyse. II montre par ailleurs que ces lois a priori sont plutǒt insensibles à certaines caracté-ristiques du plan d'expérience et du modèle, en plus de bien se comporter au plan fréquentiste. Enfin, il se sert de données sur le rapport carbone/azote d'une terre agricole pour illustrer son propos.

Atmospheric data assimilation techniques rely on parametric models for spatial correlation functions. This article proposes and discusses various families of homogeneous and isotropic correlation models on Euclidean spaces and on the sphere. In particular, three simply parametrized classes of compactly supported, smooth, and analytically simple correlation functions are proposed. the first two classes approximate standard second- and third-order autoregressive functions, and a member of the third family approximates the Gaussian function within a maximal error of 0.0056. Furthermore, correlation models suggested previously for meteorological applications are checked for permissibility, with both positive and negative results.

In many areas of science and technology, mathematical models are built to simulate complex real world phenomena. Such models are typically implemented in large computer programs and are also very complex, such that the way that the model responds to changes in its inputs is not transparent. Sensitivity analysis is concerned with understanding how changes in the model inputs influence the outputs. This may be motivated simply by a wish to understand the implications of a complex model but often arises because there is uncertainty about the true values of the inputs that should be used for a particular application. A broad range of measures have been advocated in the literature to quantify and describe the sensitivity of a model's output to variation in its inputs. In practice the most commonly used measures are those that are based on formulating uncertainty in the model inputs by a joint probability distribution and then analysing the induced uncertainty in outputs, an approach which is known as probabilistic sensitivity analysis. We present a Bayesian framework which unifies the various tools of prob- abilistic sensitivity analysis. The Bayesian approach is computationally highly efficient. It allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than standard Monte Carlo methods. Furthermore, all measures of interest may be computed from a single set of runs.

Computer models are widely used in scientific research to study and predict the behaviour of complex systems. The run times of computer-intensive simulators are often such that it is impractical to make the thousands of model runs that are conventionally required for sensitivity analysis, uncertainty analysis or calibration. In response to this problem, highly efficient techniques have recently been developed based on a statistical meta-model (the emulator) that is built to approximate the computer model. The approach, however, is less straightforward for dynamic simulators, designed to represent time-evolving systems. Generalisations of the established methodology to allow for dynamic emulation are here proposed and contrasted. Advantages and difficulties are discussed and illustrated with an application to the Sheffield Dynamic Global Vegetation Model, developed within the UK Centre for Terrestrial Carbon Dynamics.

The Bayesian approach to quantifying, analysing and reducing uncertainty in the application of complex process models is attracting increasing attention amongst users of such models. The range and power of the Bayesian methods is growing and there is already a sizeable literature on these methods. However, most of it is in specialist statistical journals. The purpose of this tutorial is to introduce the more general reader to the Bayesian approach.

The prediction of ship stability during the early stages of design is very important from the point of vessel’s safety. Out of the six motions of a ship, the critical motion leading to capsize of a vessel is the rolling motion. In the present study, particular attention is paid to the performance of a ship in beam sea. The linear ship response in waves is evaluated using strip theory. Critical condition in the rolling motion of a ship is when it is subjected to synchronous beam waves. In this paper, a nonlinear approach has been tried to predict the roll response of a vessel. Various representations of damping and restoring terms found in the literature are investigated. A parametric investigation is undertaken to identify the effect of a number of key parameters like wave amplitude, wave frequency, metacentric height, etc.

Variance based methods have assessed themselves as versatile and effective among the various available techniques for sensitivity analysis of model output. Practitioners can in principle describe the sensitivity pattern of a model Y=f(X1,X2,…,Xk) with k uncertain input factors via a full decomposition of the variance V of Y into terms depending on the factors and their interactions. More often practitioners are satisfied with computing just k first order effects and k total effects, the latter describing synthetically interactions among input factors. In sensitivity analysis a key concern is the computational cost of the analysis, defined in terms of number of evaluations of f(X1,X2,…,Xk) needed to complete the analysis, as f(X1,X2,…,Xk) is often in the form of a numerical model which may take long processing time. While the computational cost is relatively cheap and weakly dependent on k for estimating first order effects, it remains expensive and strictly k-dependent for total effect indices. In the present note we compare existing and new practices for this index and offer recommendations on which to use.

An algorithm was developed to derive the concentrations of phytoplankton pigment, suspended matter and gelbstoff, and the aerosol path radiance from 'Rayleigh corrected' top-of-atmosphere reflectances over turbid coastal waters. The procedure is designed for MERIS, the Medium Resolution Imaging Spectrometer, which will be flown onboard the Earth observation satellite Envisat of the European Space Agency (ESA). The algorithm is a neural network (NN) which is used to parameterize the inverse of a radiative transfer model. It is used in this study as a multiple nonlinear regression technique. The NN is a feedforward backpropagation model with two hidden layers. The NN was trained with computed reflectances covering the range of 0.5-50mugl-1 phytoplankton pigment, 1-100mgl-1 suspended matter, gelbstoff absorption at 420nm of 0.02-2m-1 and a horizontal visibility of 2-50km. Inputs to the NN are the reflectances of the 16 spectral channels which were under discussion for MERIS. The outputs are the three water constituent concentrations and the aerosol concentration, here expressed as the horizontal ground visibility. Tests with simulated reflectances show: (1) that concentations are correctly retrieved for a wide range covering oligotrophic Case I and turbid Case II water; (2) that the atmospheric correction can be performed even over very turbid water where the reflectance of the water cannot be neglected for the atmospheric correction channels in the near-infrared spectral range; and (3) that the algorithm is robust against errors in the input data. Although the training of the NN is time consuming, the utilization of the NN algorithm is extremely fast and can be applied routinely for satellite data mass production.

Isotropic positive definite functions on spheres play important roles in
spatial statistics, where they occur as the correlation functions of
homogeneous random fields and star-shaped random particles. In approximation
theory, strictly positive definite functions serve as radial basis functions
for interpolating scattered data on spherical domains. We review
characterizations of positive definite functions on spheres in terms of
Gegenbauer expansions and apply them to dimension walks, where monotonicity
properties of the Gegenbauer coefficients guarantee positive definiteness in
higher dimensions. Subject to a natural support condition, isotropic positive
definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and
Wendland's functions, allow for the direct substitution of the Euclidean
distance by the great circle distance on a one-, two- or three-dimensional
sphere, as opposed to the traditional approach, where the distances are
transformed into each other. Completely monotone functions are positive
definite on spheres of any dimension and provide rich parametric classes of
such functions, including members of the powered exponential, Mat\'{e}rn,
generalized Cauchy and Dagum families. The sine power family permits a
continuous parameterization of the roughness of the sample paths of a Gaussian
process. A collection of research problems provides challenges for future work
in mathematical analysis, probability theory and spatial statistics.

Computer codes are used in scientific research to study and predict the behaviour of complex systems. Their run times often
make uncertainty and sensitivity analyses impractical because of the thousands of runs that are conventionally required, so
efficient techniques have been developed based on a statistical representation of the code. The approach is less straightforward
for dynamic codes, which represent time-evolving systems. We develop a novel iterative system to build a statistical model
of dynamic computer codes, which is demonstrated on a rainfall-runoff simulator.

We consider a problem of inference for the output of a computationally expensive computer model. We suppose that the model is to be used in a context where the values of one or more inputs are uncertain, so that the input configuration is a random variable. We require to make inference about the induced distribution of the output. This distribution is called the uncertainty distribution, and the general problem is known to users of computer models as uncertainty analysis. To be specific, we develop Bayesian inference for the distribution and density functions of the model output. Modelling the output, as a function of its inputs, as a Gaussian process, we derive expressions for the posterior mean and variance of the distribution and density functions, based on data comprising observed outputs at a sample of input configurations. We show that direct computation of these expressions may encounter numerical difficulties. We develop an alternative approach based on simulating approximate realisations from the posterior distribution of the output function. Two examples are given to illustrate our methods. Copyright Biometrika Trust 2002, Oxford University Press.

We consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models. Such models, implemented as computer codes, are often generic in the sense that by a suitable choice of some of the model's input parameters the code can be used to predict the behaviour of the system in a variety of specific applications. However, in any specific application the values of necessary parameters may be unknown. In this case, physical observations of the system in the specific context are used to learn about the unknown parameters. The process of fitting the model to the observed data by adjusting the parameters is known as calibration. Calibration is typically effected by ad hoc fitting, and after calibration the model is used, with the fitted input values, to predict the future behaviour of the system. We present a Bayesian calibration technique which improves on this traditional approach in two respects. First, the predictions allow for all sources of uncertainty, including the remaining uncertainty over the fitted parameters. Second, they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best-fitting parameter values. The method is illustrated by using data from a nuclear radiation release at Tomsk, and from a more complex simulated nuclear accident exercise.

Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this inconvenience consists in replacing the complex computer code by a reduced model, called a metamodel, or a response surface that represents the computer code and requires acceptable calculation time. One particular class of metamodels is studied: the Gaussian process model that is characterized by its mean and covariance functions. A specific estimation procedure is developed to adjust a Gaussian process model in complex cases (non-linear relations, highly dispersed or discontinuous output, high-dimensional input, inadequate sampling designs, etc.). The efficiency of this algorithm is compared to the efficiency of other existing algorithms on an analytical test case. The proposed methodology is also illustrated for the case of a complex hydrogeological computer code, simulating radionuclide transport in groundwater.

Multivariate probabilistic projections using imperfect climate models part 1: outline of methodology

- Sexton

Bayesian forecasting for complex systems using computer simulators

- Craig