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.

Landscape, regional and global estimates of nitrogen flux from land to sea: errors and uncertainties

Regional to global scale modelling of N flux from land to ocean has progressed to date through the development of simple empirical models representing bulk N flux rates from large watersheds, regions, or continents on the basis of a limited selection of model parameters. Watershed scale N flux modelling has developed a range of physically-based approaches ranging from models where N flux rates are predicted through a physical representation of the processes involved, through to catchment scale models which provide a simplified representation of true systems behaviour. Generally, these watershed scale models describe within their structure the dominant process controls on N flux at the catchment or watershed scale, and take into account variations in the extent to which these processes control N flux rates as a function of landscape sensitivity to N cycling and export. This paper addresses the nature of the errors and uncertainties inherent in existing regional to global scale models, and the nature of error propagation associated with upscaling from small catchment to regional scale through a suite of spatial aggregation and conceptual lumping experiments conducted on a validated watershed scale model, the export coefficient model. Results from the analysis support the findings of other researchers developing macroscale models in allied research fields. Conclusions from the study confirm that reliable and accurate regional scale N flux modelling needs to take account of the heterogeneity of landscapes and the impact that this has on N cycling processes within homogenous landscape units.

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.

Error, reproducibility and sensitivity: a pipeline for data processing of Agilent oligonucleotide expression arrays

Background: Expression microarrays are increasingly used to obtain large scale transcriptomic information on a wide range of biological samples. Nevertheless, there is still much debate on the best ways to process data, to design experiments and analyse the output. Furthermore, many of the more sophisticated mathematical approaches to data analysis in the literature remain inaccessible to much of the biological research community. In this study we examine ways of extracting and analysing a large data set obtained using the Agilent long oligonucleotide transcriptomics platform, applied to a set of human macrophage and dendritic cell samples. Results: We describe and validate a series of data extraction, transformation and normalisation steps which are implemented via a new R function. Analysis of replicate normalised reference data demonstrate that intrarray variability is small (only around 2 of the mean log signal), while interarray variability from replicate array measurements has a standard deviation (SD) of around 0.5 log(2) units (6 of mean). The common practise of working with ratios of Cy5/Cy3 signal offers little further improvement in terms of reducing error. Comparison to expression data obtained using Arabidopsis samples demonstrates that the large number of genes in each sample showing a low level of transcription reflect the real complexity of the cellular transcriptome. Multidimensional scaling is used to show that the processed data identifies an underlying structure which reflect some of the key biological variables which define the data set. This structure is robust, allowing reliable comparison of samples collected over a number of years and collected by a variety of operators. Conclusions: This study outlines a robust and easily implemented pipeline for extracting, transforming normalising and visualising transcriptomic array data from Agilent expression platform. The analysis is used to obtain quantitative estimates of the SD arising from experimental (non biological) intra- and interarray variability, and for a lower threshold for determining whether an individual gene is expressed. The study provides a reliable basis for further more extensive studies of the systems biology of eukaryotic cells.

Comparison of TAMSAT and CPC rainfall estimates with raingauges, for southern Africa

Two different TAMSAT (Tropical Applications of Meteorological Satellites) methods of rainfall estimation were developed for northern and southern Africa, based on Meteosat images. These two methods were used to make rainfall estimates for the southern rainy season from October 1995 to April 1996. Estimates produced by both TAMSAT methods and estimates produced by the CPC (Climate Prediction Center) method were then compared with kriged data from over 800 raingauges in southern Africa. This shows that operational TAMSAT estimates are better over plateau regions, with 59% of estimates within one standard error (s.e.) of the kriged rainfall. Over mountainous regions the CPC approach is generally better, although all methods underestimate and give only 40% of estimates within 1 s.e. The two TAMSAT methods show little difference across a whole season, but when looked at in detail the northern method gives unsatisfactory calibrations. The CPC method does have significant overall improvements by building in real-time raingauge data, but only where sufficient raingauges are available.

A comparison of GOCE and drifter-based estimates of the North Atlantic steady-state surface circulation

Over the last decade, due to the Gravity Recovery And Climate Experiment (GRACE) mission and, more recently, the Gravity and steady state Ocean Circulation Explorer (GOCE) mission, our ability to measure the ocean’s mean dynamic topography (MDT) from space has improved dramatically. Here we use GOCE to measure surface current speeds in the North Atlantic and compare our results with a range of independent estimates that use drifter data to improve small scales. We find that, with filtering, GOCE can recover 70% of the Gulf Steam strength relative to the best drifter-based estimates. In the subpolar gyre the boundary currents obtained from GOCE are close to the drifter-based estimates. Crucial to this result is careful filtering which is required to remove small-scale errors, or noise, in the computed surface. We show that our heuristic noise metric, used to determine the degree of filtering, compares well with the quadratic sum of mean sea surface and formal geoid errors obtained from the error variance–covariance matrix associated with the GOCE gravity model. At a resolution of 100 km the North Atlantic mean GOCE MDT error before filtering is 5 cm with almost all of this coming from the GOCE gravity model.

Observation impact in data assimilation: the effect of non-Gaussian observation error.

Data assimilation methods which avoid the assumption of Gaussian error statistics are being developed for geoscience applications. We investigate how the relaxation of the Gaussian assumption affects the impact observations have within the assimilation process. The effect of non-Gaussian observation error (described by the likelihood) is compared to previously published work studying the effect of a non-Gaussian prior. The observation impact is measured in three ways: the sensitivity of the analysis to the observations, the mutual information, and the relative entropy. These three measures have all been studied in the case of Gaussian data assimilation and, in this case, have a known analytical form. It is shown that the analysis sensitivity can also be derived analytically when at least one of the prior or likelihood is Gaussian. This derivation shows an interesting asymmetry in the relationship between analysis sensitivity and analysis error covariance when the two different sources of non-Gaussian structure are considered (likelihood vs. prior). This is illustrated for a simple scalar case and used to infer the effect of the non-Gaussian structure on mutual information and relative entropy, which are more natural choices of metric in non-Gaussian data assimilation. It is concluded that approximating non-Gaussian error distributions as Gaussian can give significantly erroneous estimates of observation impact. The degree of the error depends not only on the nature of the non-Gaussian structure, but also on the metric used to measure the observation impact and the source of the non-Gaussian structure.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Assessing the performance of dynamical trajectory estimates

Estimating trajectories and parameters of dynamical systems from observations is a problem frequently encountered in various branches of science; geophysicists for example refer to this problem as data assimilation. Unlike as in estimation problems with exchangeable observations, in data assimilation the observations cannot easily be divided into separate sets for estimation and validation; this creates serious problems, since simply using the same observations for estimation and validation might result in overly optimistic performance assessments. To circumvent this problem, a result is presented which allows us to estimate this optimism, thus allowing for a more realistic performance assessment in data assimilation. The presented approach becomes particularly simple for data assimilation methods employing a linear error feedback (such as synchronization schemes, nudging, incremental 3DVAR and 4DVar, and various Kalman filter approaches). Numerical examples considering a high gain observer confirm the theory.

Uncertainties in tidally adjusted estimates of sea level rise flooding (bathtub model) for the Greater London

Sea-level rise (SLR) from global warming may have severe consequences for coastal cities, particularly when combined with predicted increases in the strength of tidal surges. Predicting the regional impact of SLR flooding is strongly dependent on the modelling approach and accuracy of topographic data. Here, the areas under risk of sea water flooding for London boroughs were quantified based on the projected SLR scenarios reported in Intergovernmental Panel on Climate Change (IPCC) fifth assessment report (AR5) and UK climatic projections 2009 (UKCP09) using a tidally-adjusted bathtub modelling approach. Medium- to very high-resolution digital elevation models (DEMs) are used to evaluate inundation extents as well as uncertainties. Depending on the SLR scenario and DEMs used, it is estimated that 3%–8% of the area of Greater London could be inundated by 2100. The boroughs with the largest areas at risk of flooding are Newham, Southwark, and Greenwich. The differences in inundation areas estimated from a digital terrain model and a digital surface model are much greater than the root mean square error differences observed between the two data types, which may be attributed to processing levels. Flood models from SRTM data underestimate the inundation extent, so their results may not be reliable for constructing flood risk maps. This analysis provides a broad-scale estimate of the potential consequences of SLR and uncertainties in the DEM-based bathtub type flood inundation modelling for London boroughs.

Robust inference in an heteroscedastic measurement error model

In this paper we deal with robust inference in heteroscedastic measurement error models Rather than the normal distribution we postulate a Student t distribution for the observed variables Maximum likelihood estimates are computed numerically Consistent estimation of the asymptotic covariance matrices of the maximum likelihood and generalized least squares estimators is also discussed Three test statistics are proposed for testing hypotheses of interest with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels Results of simulations and an application to a real data set are also reported (C) 2009 The Korean Statistical Society Published by Elsevier B V All rights reserved

Inference and local influence assessment in skew-normal null intercept measurement error model

In this article, we discuss inferential aspects of the measurement error regression models with null intercepts when the unknown quantity x (latent variable) follows a skew normal distribution. We examine first the maximum-likelihood approach to estimation via the EM algorithm by exploring statistical properties of the model considered. Then, the marginal likelihood, the score function and the observed information matrix of the observed quantities are presented allowing direct inference implementation. In order to discuss some diagnostics techniques in this type of models, we derive the appropriate matrices to assessing the local influence on the parameter estimates under different perturbation schemes. The results and methods developed in this paper are illustrated considering part of a real data set used by Hadgu and Koch [1999, Application of generalized estimating equations to a dental randomized clinical trial. Journal of Biopharmaceutical Statistics, 9, 161-178].

Convex combinations of long memory estimates from different sampling rates

Convex combinations of long memory estimates using the same data observed at different sampling rates can decrease the standard deviation of the estimates, at the cost of inducing a slight bias. The convex combination of such estimates requires a preliminary correction for the bias observed at lower sampling rates, reported by Souza and Smith (2002). Through Monte Carlo simulations, we investigate the bias and the standard deviation of the combined estimates, as well as the root mean squared error (RMSE), which takes both into account. While comparing the results of standard methods and their combined versions, the latter achieve lower RMSE, for the two semi-parametric estimators under study (by about 30% on average for ARFIMA(0,d,0) series).

Intercomparison of integrated water vapor estimates from multisensors in the amazonian region

Water vapor is an atmospheric component of major interest in atmospheric science because it affects the energy budget and plays a key role in several atmospheric processes. The Amazonian region is one of the most humid on the planet, and land use change is able to affect the hydrologic cycle in several areas and consequently to generate severe modifications in the global climate. Within this context, accessing the error associated with atmospheric humidity measurement and the validation of the integrated water vapor (IWV) quantification from different techniques is very important in this region. Using data collected during the Radiation, Cloud, and Climate Interactions in Amazonia during the Dry-to-Wet Transition Season (RACCI/DRY-TO-WET), an experiment carried out in southwestern Amazonia in 2002, this paper presents quality analysis of IWV measurements from RS80 radiosondes, a suite of GPS receivers, an Aerosol Robotic Network (AERONET) solar radiometer, and humidity sounding from the Humidity Sounder for Brazil (HSB) aboard the Aqua satellite. When compared to RS80 IWV values, the root-mean-square (RMS) from the AERONET and GPS results are of the order of 2.7 and 3.8 kg m(-2), respectively. The difference generated between IWV from the GPS receiver and RS80 during the daytime was larger than that of the nighttime period because of the combination of the influence of high ionospheric activity during the RACCI experiment and a daytime drier bias from the RS80.

Estimates of covariance functions for growth from birth to 630 days of age in Nelore cattle

Weight records of Brazilian Nelore cattle, from birth to 630 d of age, recorded every 3 mo, were analyzed using random regression models. Independent variables were Legendre polynomials of age at recording. The model of analysis included contemporary groups as fixed effects and age of dam as a linear and quadratic covariable. Mean trends were modeled through a cubic regression on orthogonal polynomials of age. Up to four sets of random regression coefficients were fitted for animals' direct and maternal, additive genetic, and permanent environmental effects. Changes in measurement error variances with age were modeled through a variance function. Orders of polyno-mial fit from three to six were considered, resulting in up to 77 parameters to be estimated. Models fitting random regressions modeled the pattern of variances in the data adequately, with estimates similar to those from corresponding univariate analysis. Direct heritability estimates decreased after birth and tended to be lowest at ages at which maternal effect estimates tended to be highest. Maternal heritability estimates increased after birth to a peak around 110 to 120 d of age and decreased thereafter. Additive genetic direct correlation estimates between weights at standard ages (birth, weaning, yearling, and final weight) were moderate to high and maternal genetic and environmental correlations were consistently high. © 2001 American Society of Animal Science. All rights reserved.