Stochastic modelling of rainfall from satellite data

Satellite-based rainfall monitoring is widely used for climatological studies because of its full global coverage but it is also of great importance for operational purposes especially in areas such as Africa where there is a lack of ground-based rainfall data. Satellite rainfall estimates have enormous potential benefits as input to hydrological and agricultural models because of their real time availability, low cost and full spatial coverage. One issue that needs to be addressed is the uncertainty on these estimates. This is particularly important in assessing the likely errors on the output from non-linear models (rainfall-runoff or crop yield) which make use of the rainfall estimates, aggregated over an area, as input. Correct assessment of the uncertainty on the rainfall is non-trivial as it must take account of • the difference in spatial support of the satellite information and independent data used for calibration • uncertainties on the independent calibration data • the non-Gaussian distribution of rainfall amount • the spatial intermittency of rainfall • the spatial correlation of the rainfall field This paper describes a method for estimating the uncertainty on satellite-based rainfall values taking account of these factors. The method involves firstly a stochastic calibration which completely describes the probability of rainfall occurrence and the pdf of rainfall amount for a given satellite value, and secondly the generation of ensemble of rainfall fields based on the stochastic calibration but with the correct spatial correlation structure within each ensemble member. This is achieved by the use of geostatistical sequential simulation. The ensemble generated in this way may be used to estimate uncertainty at larger spatial scales. A case study of daily rainfall monitoring in the Gambia, west Africa for the purpose of crop yield forecasting is presented to illustrate the method.

Advances in the stochastic modelling of satellite-derived rainfall estimates using a sparse calibration dataset

As satellite technology develops, satellite rainfall estimates are likely to become ever more important in the world of food security. It is therefore vital to be able to identify the uncertainty of such estimates and for end users to be able to use this information in a meaningful way. This paper presents new developments in the methodology of simulating satellite rainfall ensembles from thermal infrared satellite data. Although the basic sequential simulation methodology has been developed in previous studies, it was not suitable for use in regions with more complex terrain and limited calibration data. Developments in this work include the creation of a multithreshold, multizone calibration procedure, plus investigations into the causes of an overestimation of low rainfall amounts and the best way to take into account clustered calibration data. A case study of the Ethiopian highlands has been used as an illustration.

Impacts of climate change scenarios on dissolved oxygen in the River Thames, UK

A water quality model is used to assess the impact of possible climate change on dissolved oxygen (DO) in the Thames. The Thames catchment is densely populated and, typically, many pressures are anthropogenic. However, that same population also relies on the river for potable water supply and as a disposal route for treated wastewater. Thus, future water quality will be highly dependent on future activity. Dynamic and stochastic modelling has been used to assess the likely impacts on DO dynamics along the river system and the probability distributions associated with future variability. The modelling predictions indicate that warmer river temperatures and drought act to reduce dissolved oxygen concentrations in lowland river systems

Achieving quality through statistical prediction for building services systems

The application of prediction theories has been widely practised for many years in many industries such as manufacturing, defence and aerospace. Although these theories are not new, their application has not been widely used within the building services industry. Collectively, the building services industry should take a deeper look at these approaches in comparison with the traditional deterministic approaches currently being practised. By extending the application into this industry, this paper seeks to provide the industry with an overview of how simplified stochastic modelling coupled with availability and reliability predictions using historical data compiled from various sources could enhance the quality of building services systems.

Introduction. Stochastic physics and climate modelling

Finite computing resources limit the spatial resolution of state-of-the-art global climate simulations to hundreds of kilometres. In neither the atmosphere nor the ocean are small-scale processes such as convection, clouds and ocean eddies properly represented. Climate simulations are known to depend, sometimes quite strongly, on the resulting bulk-formula representation of unresolved processes. Stochastic physics schemes within weather and climate models have the potential to represent the dynamical effects of unresolved scales in ways which conventional bulk-formula representations are incapable of so doing. The application of stochastic physics to climate modelling is a rapidly advancing, important and innovative topic. The latest research findings are gathered together in the Theme Issue for which this paper serves as the introduction.

Comparison of stochastic parameterisation approaches in a single-column model

We discuss and test the potential usefulness of single-column models (SCMs) for the testing of stochastic physics schemes that have been proposed for use in general circulation models (GCMs). We argue that although single column tests cannot be definitive in exposing the full behaviour of a stochastic method in the full GCM, and although there are differences between SCM testing of deterministic and stochastic methods, SCM testing remains a useful tool. It is necessary to consider an ensemble of SCM runs produced by the stochastic method. These can be usefully compared to deterministic ensembles describing initial condition uncertainty and also to combinations of these (with structural model changes) into poor man's ensembles. The proposed methodology is demonstrated using an SCM experiment recently developed by the GCSS (GEWEX Cloud System Study) community, simulating transitions between active and suppressed periods of tropical convection.

Epidemiological risk assessment using linked network and grid based modelling: Phytophthora ramorum and Phytophthora kernoviae in the UK

We developed a stochastic simulation model incorporating most processes likely to be important in the spread of Phytophthora ramorum and similar diseases across the British landscape (covering Rhododendron ponticum in woodland and nurseries, and Vaccinium myrtillus in heathland). The simulation allows for movements of diseased plants within a realistically modelled trade network and long-distance natural dispersal. A series of simulation experiments were run with the model, representing an experiment varying the epidemic pressure and linkage between natural vegetation and horticultural trade, with or without disease spread in commercial trade, and with or without inspections-with-eradication, to give a 2 x 2 x 2 x 2 factorial started at 10 arbitrary locations spread across England. Fifty replicate simulations were made at each set of parameter values. Individual epidemics varied dramatically in size due to stochastic effects throughout the model. Across a range of epidemic pressures, the size of the epidemic was 5-13 times larger when commercial movement of plants was included. A key unknown factor in the system is the area of susceptible habitat outside the nursery system. Inspections, with a probability of detection and efficiency of infected-plant removal of 80% and made at 90-day intervals, reduced the size of epidemics by about 60% across the three sectors with a density of 1% susceptible plants in broadleaf woodland and heathland. Reducing this density to 0.1% largely isolated the trade network, so that inspections reduced the final epidemic size by over 90%, and most epidemics ended without escape into nature. Even in this case, however, major wild epidemics developed in a few percent of cases. Provided the number of new introductions remains low, the current inspection policy will control most epidemics. However, as the rate of introduction increases, it can overwhelm any reasonable inspection regime, largely due to spread prior to detection. (C) 2009 Elsevier B.V. All rights reserved.

Modelling the propagation of terahertz radiation through a tissue simulating phantom

Terahertz (THz) frequency radiation, 0.1 THz to 20 THz, is being investigated for biomedical imaging applications following the introduction of pulsed THz sources that produce picosecond pulses and function at room temperature. Owing to the broadband nature of the radiation, spectral and temporal information is available from radiation that has interacted with a sample; this information is exploited in the development of biomedical imaging tools and sensors. In this work, models to aid interpretation of broadband THz spectra were developed and evaluated. THz radiation lies on the boundary between regions best considered using a deterministic electromagnetic approach and those better analysed using a stochastic approach incorporating quantum mechanical effects, so two computational models to simulate the propagation of THz radiation in an absorbing medium were compared. The first was a thin film analysis and the second a stochastic Monte Carlo model. The Cole–Cole model was used to predict the variation with frequency of the physical properties of the sample and scattering was neglected. The two models were compared with measurements from a highly absorbing water-based phantom. The Monte Carlo model gave a prediction closer to experiment over 0.1 to 3 THz. Knowledge of the frequency-dependent physical properties, including the scattering characteristics, of the absorbing media is necessary. The thin film model is computationally simple to implement but is restricted by the geometry of the sample it can describe. The Monte Carlo framework, despite being initially more complex, provides greater flexibility to investigate more complicated sample geometries.

A new modelling framework for statistical cumulus dynamics

We propose a new modelling framework suitable for the description of atmospheric convective systems as a collection of distinct plumes. The literature contains many examples of models for collections of plumes in which strong simplifying assumptions are made, a diagnostic dependence of convection on the large-scale environment and the limit of many plumes often being imposed from the outset. Some recent studies have sought to remove one or the other of those assumptions. The proposed framework removes both, and is explicitly time-dependent and stochastic in its basic character. The statistical dynamics of the plume collection are defined through simple probabilistic rules applied at the level of individual plumes, and van Kampen's system size expansion is then used to construct the macroscopic limit of the microscopic model. Through suitable choices of the microscopic rules, the model is shown to encompass previous studies in the appropriate limits, and to allow their natural extensions beyond those limits.

Multiscale evolving complex network model of functional connectivity in neuronal cultures

Cultures of cortical neurons grown on multielectrode arrays exhibit spontaneous, robust and recurrent patterns of highly synchronous activity called bursts. These bursts play a crucial role in the development and topological selforganization of neuronal networks. Thus, understanding the evolution of synchrony within these bursts could give insight into network growth and the functional processes involved in learning and memory. Functional connectivity networks can be constructed by observing patterns of synchrony that evolve during bursts. To capture this evolution, a modelling approach is adopted using a framework of emergent evolving complex networks and, through taking advantage of the multiple time scales of the system, aims to show the importance of sequential and ordered synchronization in network function.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Asset liability modelling and pension schemes: the application of robust optimization to USS

This paper uses a novel numerical optimization technique - robust optimization - that is well suited to solving the asset-liability management (ALM) problem for pension schemes. It requires the estimation of fewer stochastic parameters, reduces estimation risk and adopts a prudent approach to asset allocation. This study is the first to apply it to a real-world pension scheme, and the first ALM model of a pension scheme to maximise the Sharpe ratio. We disaggregate pension liabilities into three components - active members, deferred members and pensioners, and transform the optimal asset allocation into the scheme’s projected contribution rate. The robust optimization model is extended to include liabilities and used to derive optimal investment policies for the Universities Superannuation Scheme (USS), benchmarked against the Sharpe and Tint, Bayes-Stein, and Black-Litterman models as well as the actual USS investment decisions. Over a 144 month out-of-sample period robust optimization is superior to the four benchmarks across 20 performance criteria, and has a remarkably stable asset allocation – essentially fix-mix. These conclusions are supported by six robustness checks.