Very large inverse problems in atmosphere and ocean modelling

For the very large nonlinear dynamical systems that arise in a wide range of physical, biological and environmental problems, the data needed to initialize a numerical forecasting model are seldom available. To generate accurate estimates of the expected states of the system, both current and future, the technique of ‘data assimilation’ is used to combine the numerical model predictions with observations of the system measured over time. Assimilation of data is an inverse problem that for very large-scale systems is generally ill-posed. In four-dimensional variational assimilation schemes, the dynamical model equations provide constraints that act to spread information into data sparse regions, enabling the state of the system to be reconstructed accurately. The mechanism for this is not well understood. Singular value decomposition techniques are applied here to the observability matrix of the system in order to analyse the critical features in this process. Simplified models are used to demonstrate how information is propagated from observed regions into unobserved areas. The impact of the size of the observational noise and the temporal position of the observations is examined. The best signal-to-noise ratio needed to extract the most information from the observations is estimated using Tikhonov regularization theory. Copyright © 2005 John Wiley & Sons, Ltd.

Reduced-order modelling of a high-voltage oscillatory impulse generator

The design of high-voltage equipment encompasses the study of oscillatory surges caused by transients such as those produced by switching. By obtaining a model, the response of which reconstructs that observed in the actual system, simulation studies and critical tests can be carried out on the model rather than on the equipment itself. In this paper, methods for the construction of simplified models are described and it is shown how the use of a complex model does not necessarily result in superior response pattern reconstruction.

The impact of the state of the troposphere on the response to stratospheric heating in a simplified GCM

Previous studies have made use of simplified general circulation models (sGCMs) to investigate the atmospheric response to various forcings. In particular, several studies have investigated the tropospheric response to changes in stratospheric temperature. This is potentially relevant for many climate forcings. Here the impact of changing the tropospheric climatology on the modeled response to perturbations in stratospheric temperature is investigated by the introduction of topography into the model and altering the tropospheric jet structure. The results highlight the need for very long integrations so as to determine accurately the magnitude of response. It is found that introducing topography into the model and thus removing the zonally symmetric nature of the model’s boundary conditions reduces the magnitude of response to stratospheric heating. However, this reduction is of comparable size to the variability in the magnitude of response between different ensemble members of the same 5000-day experiment. Investigations into the impact of varying tropospheric jet structure reveal a trend with lower-latitude/narrower jets having a much larger magnitude response to stratospheric heating than higher-latitude/wider jets. The jet structures that respond more strongly to stratospheric heating also exhibit longer time scale variability in their control run simulations, consistent with the idea that a feedback between the eddies and the mean flow is both responsible for the persistence of the control run variability and important in producing the tropospheric response to stratospheric temperature perturbations.

A simplified pricing model for volatility futures

We develop a general model to price VIX futures contracts. The model is adapted to test both the constant elasticity of variance (CEV) and the Cox–Ingersoll–Ross formulations, with and without jumps. Empirical tests on VIX futures prices provide out-of-sample estimates within 2% of the actual futures price for almost all futures maturities. We show that although jumps are present in the data, the models with jumps do not typically outperform the others; in particular, we demonstrate the important benefits of the CEV feature in pricing futures contracts. We conclude by examining errors in the model relative to the VIX characteristics

Calibration and evaluation of individual-based models using Approximate Bayesian Computation

This paper investigates the feasibility of using approximate Bayesian computation (ABC) to calibrate and evaluate complex individual-based models (IBMs). As ABC evolves, various versions are emerging, but here we only explore the most accessible version, rejection-ABC. Rejection-ABC involves running models a large number of times, with parameters drawn randomly from their prior distributions, and then retaining the simulations closest to the observations. Although well-established in some fields, whether ABC will work with ecological IBMs is still uncertain. Rejection-ABC was applied to an existing 14-parameter earthworm energy budget IBM for which the available data consist of body mass growth and cocoon production in four experiments. ABC was able to narrow the posterior distributions of seven parameters, estimating credible intervals for each. ABC’s accepted values produced slightly better fits than literature values do. The accuracy of the analysis was assessed using cross-validation and coverage, currently the best available tests. Of the seven unnarrowed parameters, ABC revealed that three were correlated with other parameters, while the remaining four were found to be not estimable given the data available. It is often desirable to compare models to see whether all component modules are necessary. Here we used ABC model selection to compare the full model with a simplified version which removed the earthworm’s movement and much of the energy budget. We are able to show that inclusion of the energy budget is necessary for a good fit to the data. We show how our methodology can inform future modelling cycles, and briefly discuss how more advanced versions of ABC may be applicable to IBMs. We conclude that ABC has the potential to represent uncertainty in model structure, parameters and predictions, and to embed the often complex process of optimizing an IBM’s structure and parameters within an established statistical framework, thereby making the process more transparent and objective.