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Carrying out a Probabilistic Tsunami Hazard Assessment (PTHA) requires a large number of simulations done at a high resolution. Statistical emulation builds a surrogate to replace the simulator and thus reduces computational costs when propagating uncertainties from the earthquake sources to the tsunami inundations. To reduce further these costs, we propose here to build emulators that exploit multiple levels of resolution and a sequential design of computer experiments. By running a few tsunami simulations at high resolution and many more simulations at lower resolutions we are able to provide realistic assessments whereas, for the same budget, using only the high resolution tsunami simulations do not provide a satisfactory outcome. As a result, PTHA can be considered with higher precision using the highest spatial resolutions, and for impacts over larger regions. We provide an illustration to the city of Cilacap in Indonesia that demonstrates the benefit of our approach.

Deep Gaussian processes (DGPs) provide a rich class of models that can better represent functions with varying regimes or sharp changes, compared to conventional GPs. In this work, we propose a novel inference method for DGPs for computer model emulation. By stochastically imputing the latent layers, our approach transforms a DGP into a linked GP: a novel emulator developed for systems of linked computer models. This transformation permits an efficient DGP training procedure that only involves optimizations of conventional GPs. In addition, predictions from DGP emulators can be made in a fast and analytically tractable manner by naturally utilizing the closed form predictive means and variances of linked GP emulators. We demonstrate the method in a series of synthetic examples and empirical applications, and show that it is a competitive candidate for DGP surrogate inference, combining efficiency that is comparable to doubly stochastic variational inference and uncertainty quantification that is comparable to the fully-Bayesian approach. A $\texttt{Python}$ package $\texttt{dgpsi}$ implementing the method is also produced and available at https://github.com/mingdeyu/DGP., This article has been accepted for publication in Technometrics, published by Taylor & Francis

We develop a new method to analyze the total electron content (TEC) depression in the ionosphere after a tsunami occurrence. We employ Gaussian process regression to accurately estimate the TEC disturbance every 30 s using satellite observations from the global navigation satellite system (GNSS) network, even over regions without measurements. We face multiple challenges. First, the impact of the acoustic wave generated by a tsunami onto TEC levels is nonlinear and anisotropic. Second, observation points are moving. Third, the measured data are not uniformly distributed in the targeting range. Nevertheless, our method always computes the electron density depression volumes, along with estimated uncertainties, when applied to the 2011 Tohoku-Oki earthquake, even with random selections of only 5 % of the 1000 GPS Earth Observation Network System receivers considered here over Japan. Also, the statistically estimated TEC depression area mostly overlaps the range of the initial tsunami, which indicates that our method can potentially be used to estimate the initial tsunami. The method can warn of a tsunami event within 15 min of the earthquake, at high levels of confidence, even with a sparse receiver network. Hence, it is potentially applicable worldwide using the existing GNSS network.

Digital twins of physical and human systems informed by real-time data, are becoming ubiquitous across weather forecasting, disaster preparedness, and urban planning, but researchers lack the tools to run these models effectively and efficiently, limiting progress. One of the current challenges is to assimilate observations in highly nonlinear dynamical systems, as the practical need is often to detect abrupt changes. We developed a software platform to improve the use of real-time data in highly nonlinear system representations where non-Gaussianity prevents the use of more standard Data Assimilation. Optimal Particle filtering data assimilation (DA) techniques have been implemented within an user-friendly open source software platform in Julia – ParticleDA.jl. To ensure the applicability of the developed platform in realistic scenarios, emphasis has been placed on numerical efficiency, scalability and optimisation for high performance computing frameworks. Furthermore, the platform has been developed to be forward model agnostic, ensuring that it is applicable to a wide range of modelling settings, for instance unstructured and non-uniform meshes in the spatial domain or even state spaces that are not spatially organised. Applications to tsunami and numerical weather prediction demonstrate the computational benefits in terms of lower errors, lower computational costs (due to ensemble size and the algorithm's overheads being minimised) and versatility thanks to flexible I/O in a high level language Julia.

High resolution global climate simulations are computationally intractable due to the integration over long time scales. Therefore coarse spatial resolution models are used. One of the known failings of current climate simulations is the underrepresentation of cloud formation, which is largely due to the disparity in the spatial scale of the model and the underlying cloud generating physical processes. In this work emphasis has been placed on fusing an emulator with a climate model so as to embed high resolution variability into a coarse resolution climate model - with the aim of improving the representation of convective cloud formation. A multi-output Gaussian Process (MOGP) is trained on high resolution Unified Model (UM) runs to predict the variability on the temperature and specific humidity fields. A proof of concept study has been carried out where the trained MOGP model is then coupled in-situ with a simplified Atmospheric General Circulation Model (AGCM) named SPEEDY. The mean profiles of the SPEEDY model are perturbed at each timestep according to the predicted high resolution informed variability. The climate statistics from the fused model are then compared to the pure model run. Improvements in the precipitation, outgoing longwave and shortwave radiation patterns are observed.

This paper presents the first end-to-end example of a risk model for loss of assets in households due to possible future tsunamis. There is a significant need for Government to assess the generic risk to buildings, and the concrete impact on the full range of assets of households, including the ones that are key to livelihoods such as agricultural land, fishing boats, livestock and equipment. Our approach relies on the Oasis Loss Modelling Framework to integrate hazard and risk. We first generate 25 representative events of tsunamigenic earthquakes off the Southern coast of Java, Indonesia. We then create a new vulnerability function based upon the Indonesian household survey STAR1 of how much assets have been reduced in each household after the 2004 tsunami. We run a multinomial logit regression to precisely allocate the probabilistic impacts to bins that correspond with levels of financial reduction in assets. We focus on the town of Cilacap for which we build loss exceedance curves, which represent the financial losses that may be exceeded at a range of future timelines, using future tsunami inundations over a surveyed layout and value of assets over the city. Our loss calculations show that losses increase sharply, especially for events with return periods beyond 250 years. These series of computations will allow more accurate investigations of impacts on livelihoods and thus will help design mitigation strategies as well as policies to minimize suffering from tsunamis. Lloyd's Tercentenary Research Foundation; Lighthill Risk Network; Alan Turing Institute project "Uncertainty Quantification of multi-scale and multiphysics computer models: applications to hazard and climate models", EPSRC EP/N510129/1; Royal Society, the United Kingdom CHL/R1/180173.

We develop a new method to analyze the total electron content (TEC) depression in the ionosphere after a tsunami occurrence. We employ Gaussian process regression to accurately estimate the TEC disturbance every 30 s using satellite observations from the GNSS network, even over regions without measurements. We face multiple challenges. First, the impact of the acoustic wave generated by a tsunami onto TEC levels is non-linear and anisotropic. Second, observation points are moving. Third, the measured data is not uniformly distributed in the targeting range. Nevertheless, our method always computes the electron density depression volumes, along with estimated uncertainties, when applied to the 2011 Tohoku-Oki Earthquake, even with random selections of only 5 % of the 1,000 GPS Earth Observation Network System receivers considered here over Japan. Also, the statistically estimated TEC depression area mostly overlaps the range of the initial tsunami, which indicates that our method can potentially be used to estimate the initial tsunami. The method can warn of a tsunami event within 15 minutes of the earthquake, at high levels of confidence, even with a sparse receiver network. Hence, it is potentially applicable worldwide using the existing GNSS network.

The potential of a full-margin rupture along the Cascadia subduction zone poses a significant threat over a populous region of North America. Traditional probabilistic tsunami hazard assessments produce hazard maps based on simulated prediction of tsunami waves either under limited ranges of scenarios or at low resolution, due to cost. We use the GPU-accelerated tsunami simulator VOLNA-OP2 with a detailed representation of topographic and bathymetric features. We replace the simulator by a Gaussian Process emulator at each output location to overcome the large computational burden. The emulators are statistical approximations of the simulator behaviour. We train the emulators on a set of input-output pairs and use them to generate approximate output values over a six-dimensional scenario parameter space, e.g., uplift/subsidence ratio, maximum uplift, that represent the seabed deformation. We implement an advanced sequential design algorithm for the optimal selection of only sixty simulations. This approach allows for a first emulation-accelerated computation of probabilistic tsunami hazard in the region of the city of Victoria, British Columbia. The low cost of emulation provides for additional flexibility in the shape of the deformation, which we illustrate here, considering two families, buried rupture and splay-faulting, of 2,000 potential scenarios.

Tsunamis are unpredictable events and catastrophic in their potential for destruction of human lives and economy. The unpredictability of their occurrence poses a challenge to the tsunami community, as it is difficult to obtain from the tsunamigenic records estimates of recurrence rates and severity. Accurate and efficient mathematical/computational modeling is thus called upon to provide tsunami forecasts and hazard assessments. Compounding this challenge for warning centres is the physical nature of tsunamis, which can travel at extremely high speeds in the open ocean or be generated close to the shoreline. Thus, tsunami forecasts must be not only accurate but also delivered under severe time constraints. In the immediate aftermath of a tsunamigenic earthquake event, there are uncertainties in the source such as location, rupture geometry, depth, magnitude. Ideally, these uncertainties should be represented in a tsunami warning. However in practice, quantifying the uncertainties in the hazard intensity (i.e.,maximum tsunami amplitude) due to the uncertainties in the source is not feasible, since it requires a large number of high resolution simulations. We approximate the functionally complex and computationally expensive high resolution tsunami simulations with a simple and cheap statistical emulator. A workflow integrating the entire chain of components from the tsunami source to quantification of hazard uncertainties is developed here - quantification of uncertainties in tsunamigenic earthquake sources, high resolution simulation of tsunami scenarios using the GPU version of Volna-OP2 on a non-uniform mesh for an ensemble of sources, construction of an emulator using the simulations as training data, and prediction of hazard intensities with associated uncertainties using the emulator. Thus, using the massively parallelized finite volume tsunami code Volna-OP2 as the heart of the workflow, we use statistical emulation to compute uncertainties in hazard intensity at locations of interest. Such an integration also balances the trade-off between computationally expensive simulations and desired accuracy of uncertainties, within given time constraints. The developed workflow is fully generic and independent of the source (1945 Makran earthquake) studied here.

Instrumental global temperature records are derived from the network of in situ measurements of land and sea surface temperatures. This observational evidence is seen as being fundamental to climate science. Therefore, the accuracy of these measurements is of prime importance for the analysis of temperature variability. There are spatial gaps in the distribution of instrumental temperature measurements across the globe. This lack of spatial coverage introduces coverage error. An approximate Bayesian computation based multi-resolution lattice kriging is developed and used to quantify the coverage errors through the variance of the spatial process at multiple spatial scales. It critically accounts for the uncertainties in the parameters of this advanced spatial statistics model itself, thereby providing, for the first time, a full description of both the spatial coverage uncertainties along with the uncertainties in the modeling of these spatial gaps. These coverage errors are combined with the existing estimates of uncertainties due to observational issues at each station location. It results in an ensemble of 100 000 monthly temperatures fields over the entire globe that samples the combination of coverage, parametric and observational uncertainties from 1850 to 2018 over a 5∘×5∘ grid.

Historical earthquakes in the Java subduction zone have given genesis to tsunami affecting the southwest coasts of the island of Java, in Indonesia. The most recent earthquake on the 17th of July 2006, has given rise to a tsunami that killed more than 600 people. The tsunami was difficult to escape due to the small amount of ground shaking, which could have acted as an early warning, and due to the epicentre being very close to the shorelines, giving insufficient time for response. Historical data and scientific studies give little evidence for mega-thrust events in the Java trench, however such possibilities are not excluded and could have a devastating impact in the region. This work aims to assess the tsunami hazard occurring from a range of earthquake scenarios in the subduction zone. Taking as a benchmark the 2006 event, we initially validate our modelling approach against the wave observations recorded at three tide gauges. We then expand our work to account for future earthquake scenarios and their tsunamigenic consequences in the southern coasts of Java island. Bathymetry displacement is computed using the Okada elastic dislocation model. The nonlinear shallow water equation solver JAGURS is employed for the modelling of wave propagation. Our objective is to quantify the uncertainty of such events by using statistical surrogates: fast stochastic approximations of the model that can explore the likelihood of thousands of tsunami scenarios in a few moments of time. Gaussian process emulators are utilised to predict maximum wave amplification occurring from varying parameter distributions such as the moment magnitude of an earthquake. The resulting tsunami hazard footprints can be used in conjunction with existing socio-demographic information to assess tsunami risk in vulnerable areas. The end-data can eventually be used to inform policy making for better disaster mitigation planning.

In this paper, we model the full range of possible local impacts of future tsunamis in the Makran subduction zone (MSZ) at Karachi port, Pakistan. For the first time, the 3-D subduction geometry Sl...

The software package Volna-OP2 is a robust and efficient code capable of simulating the complete life cycle of a tsunami whilst harnessing the latest High Performance Computing (HPC) architectures. In this paper, a comprehensive error analysis and scalability study of the GPU version of the code is presented. A novel decomposition of the numerical errors into the dispersion and dissipation components is explored. Most tsunami codes exhibit amplitude smearing and/or phase lagging/leading, so the decomposition shown here is a new approach and novel tool for explaining these occurrences. It is the first time that the errors of a tsunami code have been assessed in this manner. To date, Volna-OP2 has been widely used by the tsunami modelling community. In particular its computational efficiency has allowed various sensitivity analyses and uncertainty quantification studies. Due to the number of simulations required, there is always a trade-off between accuracy and runtime when carrying out these statistical studies. The analysis presented in this paper will guide the user towards an acceptable level of accuracy within a given runtime.

A natural, if idealised, picture of the role of risk assessments in planning sees decision-makers drawing on the risk projections provided by natural and social scientific models and fashioning policies or plans that maximise expected benefit relative to this information. In this paper we draw on our study of the use tsunami science in development planning in Western India to identify ways in which this idealised picture fails to reflect important difficulties encountered by both the science and policy domains, including the representation and communication of scientific uncertainty and the management of this uncertainty within the planning system. We highlight aspects of the management of these uncertainties pose pressing problems and make some suggestions as to how they might be resolved.

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficie...

The state-of-the-art linked Gaussian process offers a way to build analytical emulators for systems of computer models. We generalize the closed form expressions for the linked Gaussian process under the squared exponential kernel to a class of Mat\'ern kernels, that are essential in advanced applications. An iterative procedure to construct linked Gaussian processes as surrogate models for any feed-forward systems of computer models is presented and illustrated on a feed-back coupled satellite system. We also introduce an adaptive design algorithm that could increase the approximation accuracy of linked Gaussian process surrogates with reduced computational costs on running expensive computer systems, by allocating runs and refining emulators of individual sub-models based on their heterogeneous functional complexity.

Copyright © The Author(s) 2019. The Indus Canyon in the northwestern Indian Ocean has been reported to be the site of numerous submarine mass failures in the past. This study is the first to investigate potential tsunami hazards associated with such mass failures in this region. We employed statistical emulation, i.e. surrogate modelling, to efficiently quantify uncertainties associated with slump-generated tsunamis at the slopes of the canyon. We simulated 60 slump scenarios with thickness of 100–300 m, width of 6–10.5 km, travel distances of 500–2000 m and submergence depth of 250–450 m. These scenarios were then used to train the emulator and predict 500,000 trial scenarios in order to study probabilistically the tsunami hazard over the near field. Due to narrow–deep canyon walls and the shallow continental shelf in the adjacent regions (

In this paper, we present the VOLNA-OP2 tsunami model and implementation; a finite-volume non-linear shallow-water equation (NSWE) solver built on the OP2 domain-specific language (DSL) for unstructured mesh computations. VOLNA-OP2 is unique among tsunami solvers in its support for several high-performance computing platforms: central processing units (CPUs), the Intel Xeon Phi, and graphics processing units (GPUs). This is achieved in a way that the scientific code is kept separate from various parallel implementations, enabling easy maintainability. It has already been used in production for several years; here we discuss how it can be integrated into various workflows, such as a statistical emulator. The scalability of the code is demonstrated on three supercomputers, built with classical Xeon CPUs, the Intel Xeon Phi, and NVIDIA P100 GPUs. VOLNA-OP2 shows an ability to deliver productivity as well as performance and portability to its users across a number of platforms.

Total column ozone variations estimated using ground-based stations provide important independent source of information in addition to satellite-based estimates. This estimation has been vigorously challenged by data inhomogeneity in time and by the irregularity of the spatial distribution of stations, as well as by interruptions in observation records. Furthermore, some stations have calibration issues and thus observations may drift. In this paper we compare the spatial interpolation of ozone levels using the novel stochastic partial differential equation (SPDE) approach with the covariance-based kriging. We show how these new spatial predictions are more accurate, less uncertain and more robust. We construct long-term zonal means to investigate the robustness against the absence of measurements at some stations as well as instruments drifts. We conclude that time series analyzes can benefit from the SPDE approach compared to the covariance-based kriging when stations are missing, but the positive impact of the technique is less pronounced in the case of drifts.

Numerical inversions for earthquake source parameters from tsunami wave data usually incorporate subjective elements to stabilize the search. In addition, noisy and possibly insufficient data result in instability and non-uniqueness in most deterministic inversions, which are barely acknowledged. Here, we employ the satellite altimetry data for the 2004 Sumatra–Andaman tsunami event to invert the source parameters. We also include kinematic parameters that improve the description of tsunami generation and propagation, especially near the source. Using a finite fault model that represents the extent of rupture and the geometry of the trench, we perform a new type of nonlinear joint inversion of the slips, rupture velocities and rise times with minimala prioriconstraints. Despite persistently good waveform fits, large uncertainties in the joint parameter distribution constitute a remarkable feature of the inversion. These uncertainties suggest that objective inversion strategies should incorporate more sophisticated physical models of seabed deformation in order to significantly improve the performance of early warning systems.

Downscaling is a technique that is used to extract high-resolution information from regional scale variables produced by coarse resolution models such as Chemical Transport Models (CTMs). Statistical downscaling methods in geophysics often rely on Empirical Orthogonal Functions (EOFs). EOFs are spatial Principal Components (PCs) that display space-time modes of variability of a quantity over a region. Here we present a novel statistical downscaling method that employs Fitted Empirical Orthogonal Functions (F-EOFs) to provide local forecasts. F-EOFs differ from EOFs in that they represent space-time variations associated with a particular location through the use of inverse regression. We illustrate our downscaling method, for ozone levels over the US, with the Regional chEmical trAnsport Model (REAM) whose outputs are over 70 by 70 km grid cells. We use ground level ozone observations from monitoring stations within the south-eastern US region to downscale REAM. We select the first leading F-EOFs and regress our observations on the corresponding F-EOF loadings. We also compare our results to linear regression and PC regression. The regression on F-EOFs shows the best predictive ability. To examine the consistency of our results we repeat the analysis for different fitting and validation periods. Furthermore, in our application, PFC regression also outperforms PC regression as a dimension reduction technique.

Statistical methods constitute a useful approach to understand and quantify the uncertainty that governs complex tsunami mechanisms. Numerical experiments may often have a high computational cost. This forms a limiting factor for performing uncertainty and sensitivity analyses, where numerous simulations are required. Statistical emulators, as surrogates of these simulators, can provide predictions of the physical process in a much faster and computationally inexpensive way. They can form a prominent solution to explore thousands of scenarios that would be otherwise numerically expensive and difficult to achieve. In this work, we build a statistical emulator of the deterministic codes used to simulate submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean, in two stages. First we calibrate, against observations of the landslide deposits, the parameters used in the landslide simulations. This calibration is performed under a Bayesian framework using Gaussian Process (GP) emulators to approximate the landslide model, and the discrepancy function between model and observations. Distributions of the calibrated input parameters are obtained as a result of the calibration. In a second step, a GP emulator is built to mimic the coupled landslide-tsunami numerical process. The emulator propagates the uncertainties in the distributions of the calibrated input parameters inferred from the first step to the outputs. As a result, a quantification of the uncertainty of the maximum free surface elevation at specified locations is obtained.

In this paper, we forecast ground level ozone concentrations over the USA, using past spatially distributed measurements and the functional linear regression model. We employ bivariate splines defined over triangulations of the relevant region of the USA to implement this functional data approach in which random surfaces represent ozone concentrations. We compare the least squares method with penalty to the principal components regression approach. Moderate sample sizes provide good quality forecasts in both cases with little computational effort. We also illustrate the variability of forecasts owing to the choice of smoothing penalty. Finally, we compare our predictions with the ones obtained using thin-plate splines. Predictions based on bivariate splines require less computational time than the ones based on thin-plate splines and are more accurate. We also quantify the variability in the predictions arising from the variability in the sample using the jackknife, and report that predictions based on bivariate splines are more robust than the ones based on thin-plate splines. © 2012 John Wiley & Sons, Ltd.

In earthquake occurrence studies, the so-called q value can be considered both as one of the parameters describing the distribution of interevent times and as an index of non-extensivity. Using simulated datasets, we compare four kinds of estimators, based on principle of maximum entropy (POME), method of moments (MOM), maximum likelihood (MLE), and probability weighted moments (PWM) of the parameters (q and τ0) of the distribution of inter-events times, assumed to be a generalized Pareto distribution (GPD), as defined by Tsallis (1988) in the frame of non-extensive statistical physics. We then propose to use the unbiased version of PWM estimators to compute the q value for the distribution of inter-event times in a realistic earthquake catalogue simulated according to the epidemic type aftershock sequence (ETAS) model. Finally, we use these findings to build a statistical emulator of the q values of ETAS model. We employ treed Gaussian processes to obtain partitions of the parameter space so that the resulting model respects sharp changes in physical behaviour. The emulator is used to understand the joint effects of input parameters on the q value, exploring the relationship between ETAS model formulation and distribution of inter-event times.

We present statistical evidence for a temporal link between variations in the El Niño–Southern Oscillation (ENSO) and the occurrence of earthquakes on the East Pacific Rise (EPR). We adopt a zero-inflated Poisson regression model to represent the relationship between the number of earthquakes in the Easter microplate on the EPR and ENSO (expressed using the southern oscillation index (SOI) for east Pacific sea-level pressure anomalies) from February 1973 to February 2009. We also examine the relationship between the numbers of earthquakes and sea levels, as retrieved by Topex/Poseidon from October 1992 to July 2002. We observe a significant (95% confidence level) positive influence of SOI on seismicity: positive SOI values trigger more earthquakes over the following 2 to 6 months than negative SOI values. There is a significant negative influence of absolute sea levels on seismicity (at 6 months lag). We propose that increased seismicity is associated with ENSO-driven sea-surface gradients (rising from east to west) in the equatorial Pacific, leading to a reduction in ocean-bottom pressure over the EPR by a few kilopascal. This relationship is opposite to reservoir-triggered seismicity and suggests that EPR fault activity may be triggered by plate flexure associated with the reduced pressure.

We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, in various situations where both the explanatory variable and the noise can be unbounded and dependent. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with a penalisation term. Under the assumptions that the regressors in the sample span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an application to ozone concentration forecasting over the USA that illustrates the predictive skills of the method.

International audience; In this paper, we introduce a functional semiparametric model, where a real-valued random variable is explained by the sum of a unknown linear combination of the components of a multivariate random variable and an unknown transformation of a functional random variable. The errors can be autocorrelated. We focus here on the parametric estimation of the coefficients in the linear combination. First, we use a nonparametric kernel method to remove the effect of the functional explanatory variable. Then, we use generalized least squares approach to obtain an estimator of these coefficients. Under some technical assumptions, we prove consistency and asymptotic normality of our estimator. Finally, we present Monte Carlo simulations that illustrate these characteristics.

There is a need for decadal predictions of the seabed evolution, for example to inform resurvey strategies when maintaining navigation channels. The understanding of the physical processes involved in morphological evolution, and the viability of process models to accurately model evolution over these time scales, are currently limited. As a result, statistical approaches are used to supply long-term forecasts. In this paper, we introduce a novel statistical approach for this problem: the autoregressive Hilbertian model (ARH). This model naturally assesses the time evolution of spatially-distributed measurements. We apply the technique to a coastal area in the East Anglian coast over the period 1846 to 2002, and compare with two other statistical methods used recently for seabed prediction: the autoregressive model and the EOF model. We evaluate the performance of the three methods by comparing observations and predictions for 2002. The ARH model enables a reduction of 10% of the root mean squared errors. Finally, we compute the variability in the predictions related to time sampling using the jackknife, a method that uses subsamples to quantify uncertainties.

The thermosphere–ionosphere electrodynamics general circulation model (TIE-GCM) of the upper atmosphere has a number of features that are a challenge to standard approaches to emulation, including a long run time, multivariate output, periodicity, and strong constraints on the interrelationship between inputs and outputs. These kinds of features are not unusual in models of complex systems. We show how they can be handled in an emulator and demonstrate the use of the outer product emulator for efficient calculation, with an emphasis on predictive diagnostics for model choice and model validation. We use our emulator to “verify” the underlying computer code and to quantify our qualitative physical understanding.

In this paper, we demonstrate a procedure for calibrating a complex computer simulation model having uncertain inputs and internal parameters, with application to the NCAR Thermosphere-Ionosphere-Electrodynamics General Circulation Model (TIE-GCM). We compare simulated magnetic perturbations with observations at two ground locations for various combinations of calibration parameters. These calibration parameters are: the amplitude of the semidiurnal tidal perturbation in the height of a constant-pressure surface at the TIE-GCM lower boundary, the local time at which this maximises and the minimum night-time electron density. A fully Bayesian approach, that describes correlations in time and in the calibration input space is implemented. A Markov Chain Monte Carlo (MCMC) approach leads to potential optimal values for the amplitude and phase (within the limitations of the selected data and calibration parameters) but not for the minimum night-time electron density. The procedure can be extended to include additional data types and calibration parameters.