Article

Emulation of vessel motion simulators for computationally efficient uncertainty quantification

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Abstract

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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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.
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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.
Conference Paper
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.
Article
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.
Article
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.
Article
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.
Article
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.
Conference Paper
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.
Article
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).
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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