Adjoint methods in data assimilation for estimating model error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Bayesian estimation of the spatial Durbin error model with an application to voter turnout in the 2004 presidential election

The potential for spatial dependence in models of voter turnout, although plausible from a theoretical perspective, has not been adequately addressed in the literature. Using recent advances in Bayesian computation, we formulate and estimate the previously unutilized spatial Durbin error model and apply this model to the question of whether spillovers and unobserved spatial dependence in voter turnout matters from an empirical perspective. Formal Bayesian model comparison techniques are employed to compare the normal linear model, the spatially lagged X model (SLX), the spatial Durbin model, and the spatial Durbin error model. The results overwhelmingly support the spatial Durbin error model as the appropriate empirical model.

Trends in Austral jet position in ensembles of high- and low-top CMIP5 models

Trends in the position of the DJF Austral jet have been analysed for multi-model ensemble simulations of a subset of high- and low-top models for the periods 1960-2000, 2000-2050, and 2050-2098 under the CMIP5 historical, RCP4.5, and RCP8.5 scenarios. Comparison with ERA-Interim, CFSR and the NCEP/NCAR reanalysis shows that the DJF and annual mean jet positions in CMIP5 models are equatorward of reanalyses for the 1979-2006 mean. Under the RCP8.5 scenario, the mean jet position in the high-top models moves 3 degrees poleward of its 1860-1900 position by 2098, compared to just over 2 degrees for the low-top models. Changes in jet position are linked to changes in the meridional temperature gradient. Compared to low-top models, the high-top models predict greater warming in the tropical upper troposphere due to increased greenhouse gases for all periods considered: up to 0.28 K/decade more in the period 2050-2098 under the RCP8.5 scenario. Larger polar lower-stratospheric cooling is seen in high-top models: -1.64 K/decade compared to -1.40 K/decade in the period 1960-2000, mainly in response to ozone depletion, and -0.41 K/decade compared to -0.12 K/decade in the period 2050-2098, mainly in response to increases in greenhouse gases. Analysis suggests that there may be a linear relationship between the trend in jet position and meridional temperature gradient, even under strong forcing. There were no clear indications of an approach to a geometric limit on the absolute magnitude of the poleward shift by 2100.

Identification of components of the Sigma B Regulon in Listeria monocytogenes that contribute to acid and salt tolerance

Sigma B (σB) is an alternative sigma factor that controls the transcriptional response to stress in Listeria monocytogenes and is also known to play a role in the virulence of this human pathogen. In the present study we investigated the impact of a sigB deletion on the proteome of L. monocytogenes grown in a chemically defined medium both in the presence and in the absence of osmotic stress (0.5 M NaCl). Two new phenotypes associated with the sigB deletion were identified using this medium. (i) Unexpectedly, the strain with the ΔsigB deletion was found to grow faster than the parent strain in the growth medium, but only when 0.5 M NaCl was present. This phenomenon was independent of the carbon source provided in the medium. (ii) The ΔsigB mutant was found to have unusual Gram staining properties compared to the parent, suggesting that σB contributes to the maintenance of an intact cell wall. A proteomic analysis was performed by two-dimensional gel electrophoresis, using cells growing in the exponential and stationary phases. Overall, 11 proteins were found to be differentially expressed in the wild type and the ΔsigB mutant; 10 of these proteins were expressed at lower levels in the mutant, and 1 was overexpressed in the mutant. All 11 proteins were identified by tandem mass spectrometry, and putative functions were assigned based on homology to proteins from other bacteria. Five proteins had putative functions related to carbon utilization (Lmo0539, Lmo0783, Lmo0913, Lmo1830, and Lmo2696), while three proteins were similar to proteins whose functions are unknown but that are known to be stress inducible (Lmo0796, Lmo2391, and Lmo2748). To gain further insight into the role of σB in L. monocytogenes, we deleted the genes encoding four of the proteins, lmo0796, lmo0913, lmo2391, and lmo2748. Phenotypic characterization of the mutants revealed that Lmo2748 plays a role in osmotolerance, while Lmo0796, Lmo0913, and Lmo2391 were all implicated in acid stress tolerance to various degrees. Invasion assays performed with Caco-2 cells indicated that none of the four genes was required for mammalian cell invasion. Microscopic analysis suggested that loss of Lmo2748 might contribute to the cell wall defect observed in the ΔsigB mutant. Overall, this study highlighted two new phenotypes associated with the loss of σB. It also demonstrated clear roles for σB in both osmotic and low-pH stress tolerance and identified specific components of the σB regulon that contribute to the responses observed.

Nonlinear error dynamics for cycled data assimilation methods

We investigate the error dynamics for cycled data assimilation systems, such that the inverse problem of state determination is solved at tk, k = 1, 2, 3, ..., with a first guess given by the state propagated via a dynamical system model from time tk − 1 to time tk. In particular, for nonlinear dynamical systems that are Lipschitz continuous with respect to their initial states, we provide deterministic estimates for the development of the error ||ek|| := ||x(a)k − x(t)k|| between the estimated state x(a) and the true state x(t) over time. Clearly, observation error of size δ > 0 leads to an estimation error in every assimilation step. These errors can accumulate, if they are not (a) controlled in the reconstruction and (b) damped by the dynamical system under consideration. A data assimilation method is called stable, if the error in the estimate is bounded in time by some constant C. The key task of this work is to provide estimates for the error ||ek||, depending on the size δ of the observation error, the reconstruction operator Rα, the observation operator H and the Lipschitz constants K(1) and K(2) on the lower and higher modes of controlling the damping behaviour of the dynamics. We show that systems can be stabilized by choosing α sufficiently small, but the bound C will then depend on the data error δ in the form c||Rα||δ with some constant c. Since ||Rα|| → ∞ for α → 0, the constant might be large. Numerical examples for this behaviour in the nonlinear case are provided using a (low-dimensional) Lorenz '63 system.

Are clinical documents optimised for patient safety? A critical analysis of patient safety outcomes using the EDA error model

Iatrogenic errors and patient safety in clinical processes are an increasing concern. The quality of process information in hardcopy or electronic form can heavily influence clinical behaviour and decision making errors. Little work has been undertaken to assess the safety impact of clinical process planning documents guiding the clinical actions and decisions. This paper investigates the clinical process documents used in elective surgery and their impact on latent and active clinical errors. Eight clinicians from a large health trust underwent extensive semi- structured interviews to understand their use of clinical documents, and their perceived impact on errors and patient safety. Samples of the key types of document used were analysed. Theories of latent organisational and active errors from the literature were combined with the EDA semiotics model of behaviour and decision making to propose the EDA Error Model. This model enabled us to identify perceptual, evaluation, knowledge and action error types and approaches to reducing their causes. The EDA error model was then used to analyse sample documents and identify error sources and controls. Types of knowledge artefact structures used in the documents were identified and assessed in terms of safety impact. This approach was combined with analysis of the questionnaire findings using existing error knowledge from the literature. The results identified a number of document and knowledge artefact issues that give rise to latent and active errors and also issues concerning medical culture and teamwork together with recommendations for further work.

Transition from an anti-phase error-correction-mode to a synchronization mode in the mutual hand tracking

Proactive motion in hand tracking and in finger bending, in which the body motion occurs prior to the reference signal, was reported by the preceding researchers when the target signals were shown to the subjects at relatively high speed or high frequencies. These phenomena indicate that the human sensory-motor system tends to choose an anticipatory mode rather than a reactive mode, when the target motion is relatively fast. The present research was undertaken to study what kind of mode appears in the sensory-motor system when two persons were asked to track the hand position of the partner with each other at various mean tracking frequency. The experimental results showed a transition from a mutual error-correction mode to a synchronization mode occurred in the same region of the tracking frequency with that of the transition from a reactive error-correction mode to a proactive anticipatory mode in the mechanical target tracking experiments. Present research indicated that synchronization of body motion occurred only when both of the pair subjects operated in a proactive anticipatory mode. We also presented mathematical models to explain the behavior of the error-correction mode and the synchronization mode.

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Remote sensing observations often have correlated errors, but the correlations are typically ignored in data assimilation for numerical weather prediction. The assumption of zero correlations is often used with data thinning methods, resulting in a loss of information. As operational centres move towards higher-resolution forecasting, there is a requirement to retain data providing detail on appropriate scales. Thus an alternative approach to dealing with observation error correlations is needed. In this article, we consider several approaches to approximating observation error correlation matrices: diagonal approximations, eigendecomposition approximations and Markov matrices. These approximations are applied in incremental variational assimilation experiments with a 1-D shallow water model using synthetic observations. Our experiments quantify analysis accuracy in comparison with a reference or ‘truth’ trajectory, as well as with analyses using the ‘true’ observation error covariance matrix. We show that it is often better to include an approximate correlation structure in the observation error covariance matrix than to incorrectly assume error independence. Furthermore, by choosing a suitable matrix approximation, it is feasible and computationally cheap to include error correlation structure in a variational data assimilation algorithm.

A new error measure for forecasts of household-level, high resolution electrical energy consumption

As low carbon technologies become more pervasive, distribution network operators are looking to support the expected changes in the demands on the low voltage networks through the smarter control of storage devices. Accurate forecasts of demand at the single household-level, or of small aggregations of households, can improve the peak demand reduction brought about through such devices by helping to plan the appropriate charging and discharging cycles. However, before such methods can be developed, validation measures are required which can assess the accuracy and usefulness of forecasts of volatile and noisy household-level demand. In this paper we introduce a new forecast verification error measure that reduces the so called “double penalty” effect, incurred by forecasts whose features are displaced in space or time, compared to traditional point-wise metrics, such as Mean Absolute Error and p-norms in general. The measure that we propose is based on finding a restricted permutation of the original forecast that minimises the point wise error, according to a given metric. We illustrate the advantages of our error measure using half-hourly domestic household electrical energy usage data recorded by smart meters and discuss the effect of the permutation restriction.

Estimating interchannel observation-error correlations for IASI radiance data in the Met Office system

The optimal utilisation of hyper-spectral satellite observations in numerical weather prediction is often inhibited by incorrectly assuming independent interchannel observation errors. However, in order to represent these observation-error covariance structures, an accurate knowledge of the true variances and correlations is needed. This structure is likely to vary with observation type and assimilation system. The work in this article presents the initial results for the estimation of IASI interchannel observation-error correlations when the data are processed in the Met Office one-dimensional (1D-Var) and four-dimensional (4D-Var) variational assimilation systems. The method used to calculate the observation errors is a post-analysis diagnostic which utilises the background and analysis departures from the two systems. The results show significant differences in the source and structure of the observation errors when processed in the two different assimilation systems, but also highlight some common features. When the observations are processed in 1D-Var, the diagnosed error variances are approximately half the size of the error variances used in the current operational system and are very close in size to the instrument noise, suggesting that this is the main source of error. The errors contain no consistent correlations, with the exception of a handful of spectrally close channels. When the observations are processed in 4D-Var, we again find that the observation errors are being overestimated operationally, but the overestimation is significantly larger for many channels. In contrast to 1D-Var, the diagnosed error variances are often larger than the instrument noise in 4D-Var. It is postulated that horizontal errors of representation, not seen in 1D-Var, are a significant contributor to the overall error here. Finally, observation errors diagnosed from 4D-Var are found to contain strong, consistent correlation structures for channels sensitive to water vapour and surface properties.

Representativity error for temperature and humidity using the Met Office high-resolution model

The observation-error covariance matrix used in data assimilation contains contributions from instrument errors, representativity errors and errors introduced by the approximated observation operator. Forward model errors arise when the observation operator does not correctly model the observations or when observations can resolve spatial scales that the model cannot. Previous work to estimate the observation-error covariance matrix for particular observing instruments has shown that it contains signifcant correlations. In particular, correlations for humidity data are more significant than those for temperature. However it is not known what proportion of these correlations can be attributed to the representativity errors. In this article we apply an existing method for calculating representativity error, previously applied to an idealised system, to NWP data. We calculate horizontal errors of representativity for temperature and humidity using data from the Met Office high-resolution UK variable resolution model. Our results show that errors of representativity are correlated and more significant for specific humidity than temperature. We also find that representativity error varies with height. This suggests that the assimilation scheme may be improved if these errors are explicitly included in a data assimilation scheme. This article is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.

A multiple error-correction model of housing supply

This paper considers supply dynamics in the context of the Irish residential market. The analysis, in a multiple error-correction framework, reveals that although developers did respond to disequilibrium in supply, the rate of adjustment was relatively slow. In contrast, however, disequilibrium in demand did not impact upon supply, suggesting that inelastic supply conditions could explain the prolonged nature of the boom in the Irish market. Increased elasticity in the later stages of the boom may have been a contributory factor in the extent of the house price falls observed in recent years.