On variational data assimilation in continuous time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalization of what is known as weakly constrained four-dimensional variational assimilation (4D-Var) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two-point boundary value problem. For imperfect models, the trade-off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. This trade-off turns out to be trivial if the model is perfect. However, even in this situation, allowing for minute deviations from the perfect model is shown to have positive effects, namely to regularize the problem. The presented formalism is dynamical in character. No statistical assumptions on dynamical or observational noise are imposed.

Sensitivity and out-of-sample error in continuous time data assimilation

Data assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time–tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade-off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade-off. A relation between the sensitivity and the out-of-sample error is established, which allows the latter to be calculated under operational conditions. A minimum out-of-sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade-off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, also known as synchronization. Numerical examples demonstrate the feasibility of the approach.

Resolution of sharp fronts in the presence of model error in variational data assimilation

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L1-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L2-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L1-norm penalty approach and a mixed total variation (TV) L1–L2-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L1–L2-norm penalty performs better than either the L1-norm method or the strong constraint 4DVar (L2-norm)method. A strength of the mixed TV L1–L2-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices.

eRules: a modular adaptive classification rule learning algorithm for data streams

Advances in hardware and software in the past decade allow to capture, record and process fast data streams at a large scale. The research area of data stream mining has emerged as a consequence from these advances in order to cope with the real time analysis of potentially large and changing data streams. Examples of data streams include Google searches, credit card transactions, telemetric data and data of continuous chemical production processes. In some cases the data can be processed in batches by traditional data mining approaches. However, in some applications it is required to analyse the data in real time as soon as it is being captured. Such cases are for example if the data stream is infinite, fast changing, or simply too large in size to be stored. One of the most important data mining techniques on data streams is classification. This involves training the classifier on the data stream in real time and adapting it to concept drifts. Most data stream classifiers are based on decision trees. However, it is well known in the data mining community that there is no single optimal algorithm. An algorithm may work well on one or several datasets but badly on others. This paper introduces eRules, a new rule based adaptive classifier for data streams, based on an evolving set of Rules. eRules induces a set of rules that is constantly evaluated and adapted to changes in the data stream by adding new and removing old rules. It is different from the more popular decision tree based classifiers as it tends to leave data instances rather unclassified than forcing a classification that could be wrong. The ongoing development of eRules aims to improve its accuracy further through dynamic parameter setting which will also address the problem of changing feature domain values.

Crossbred cow adoption and milk market participation in a multivariate count data framework

Cross-bred cow adoption is an important and potent policy variable precipitating subsistence household entry into emerging milk markets. This paper focuses on the problem of designing policies that encourage and sustain milkmarket expansion among a sample of subsistence households in the Ethiopian highlands. In this context it is desirable to measure households’ ‘proximity’ to market in terms of the level of deficiency of essential inputs. This problem is compounded by four factors. One is the existence of cross-bred cow numbers (count data) as an important, endogenous decision by the household; second is the lack of a multivariate generalization of the Poisson regression model; third is the censored nature of the milk sales data (sales from non-participating households are, essentially, censored at zero); and fourth is an important simultaneity that exists between the decision to adopt a cross-bred cow, the decision about how much milk to produce, the decision about how much milk to consume and the decision to market that milk which is produced but not consumed internally by the household. Routine application of Gibbs sampling and data augmentation overcome these problems in a relatively straightforward manner. We model the count data from two sites close to Addis Ababa in a latent, categorical-variable setting with known bin boundaries. The single-equation model is then extended to a multivariate system that accommodates the covariance between crossbred-cow adoption, milk-output, and milk-sales equations. The latent-variable procedure proves tractable in extension to the multivariate setting and provides important information for policy formation in emerging-market settings

An integrated view of data quality in Earth observation

Data quality is a difficult notion to define precisely, and different communities have different views and understandings of the subject. This causes confusion, a lack of harmonization of data across communities and omission of vital quality information. For some existing data infrastructures, data quality standards cannot address the problem adequately and cannot fulfil all user needs or cover all concepts of data quality. In this study, we discuss some philosophical issues on data quality. We identify actual user needs on data quality, review existing standards and specifications on data quality, and propose an integrated model for data quality in the field of Earth observation (EO). We also propose a practical mechanism for applying the integrated quality information model to a large number of datasets through metadata inheritance. While our data quality management approach is in the domain of EO, we believe that the ideas and methodologies for data quality management can be applied to wider domains and disciplines to facilitate quality-enabled scientific research.

Growth rate and vertical propagation of the initial error in baroclinic models

The present study investigates the growth of error in baroclinic waves. It is found that stable or neutral waves are particularly sensitive to errors in the initial condition. Short stable waves are mainly sensitive to phase errors and the ultra long waves to amplitude errors. Analysis simulation experiments have indicated that the amplitudes of the very long waves become usually too small in the free atmosphere, due to the sparse and very irregular distribution of upper air observations. This also applies to the four-dimensional data assimilation experiments, since the amplitudes of the very long waves are usually underpredicted. The numerical experiments reported here show that if the very long waves have these kinds of amplitude errors in the upper troposphere or lower stratosphere the error is rapidly propagated (within a day or two) to the surface and to the lower troposphere.

An effective approach for land-cover classification from airborne lidar fused with co-registered data

Airborne lidar provides accurate height information of objects on the earth and has been recognized as a reliable and accurate surveying tool in many applications. In particular, lidar data offer vital and significant features for urban land-cover classification, which is an important task in urban land-use studies. In this article, we present an effective approach in which lidar data fused with its co-registered images (i.e. aerial colour images containing red, green and blue (RGB) bands and near-infrared (NIR) images) and other derived features are used effectively for accurate urban land-cover classification. The proposed approach begins with an initial classification performed by the Dempster–Shafer theory of evidence with a specifically designed basic probability assignment function. It outputs two results, i.e. the initial classification and pseudo-training samples, which are selected automatically according to the combined probability masses. Second, a support vector machine (SVM)-based probability estimator is adopted to compute the class conditional probability (CCP) for each pixel from the pseudo-training samples. Finally, a Markov random field (MRF) model is established to combine spatial contextual information into the classification. In this stage, the initial classification result and the CCP are exploited. An efficient belief propagation (EBP) algorithm is developed to search for the global minimum-energy solution for the maximum a posteriori (MAP)-MRF framework in which three techniques are developed to speed up the standard belief propagation (BP) algorithm. Lidar and its co-registered data acquired by Toposys Falcon II are used in performance tests. The experimental results prove that fusing the height data and optical images is particularly suited for urban land-cover classification. There is no training sample needed in the proposed approach, and the computational cost is relatively low. An average classification accuracy of 93.63% is achieved.

On the growth of errors in data assimilation systems

Considerable progress has taken place in numerical weather prediction over the last decade. It has been possible to extend predictive skills in the extra-tropics of the Northern Hemisphere during the winter from less than five days to seven days. Similar improvements, albeit on a lower level, have taken place in the Southern Hemisphere. Another example of improvement in the forecasts is the prediction of intense synoptic phenomena such as cyclogenesis which on the whole is quite successful with the most advanced operational models (Bengtsson (1989), Gadd and Kruze (1988)). A careful examination shows that there are no single causes for the improvements in predictive skill, but instead they are due to several different factors encompassing the forecasting system as a whole (Bengtsson, 1985). In this paper we will focus our attention on the role of data-assimilation and the effect it may have on reducing the initial error and hence improving the forecast. The first part of the paper contains a theoretical discussion on error growth in simple data assimilation systems, following Leith (1983). In the second part we will apply the result on actual forecast data from ECMWF. The potential for further forecast improvements within the framework of the present observing system in the two hemispheres will be discussed.

Problems in four-dimensional data assimilation

With the introduction of new observing systems based on asynoptic observations, the analysis problem has changed in character. In the near future we may expect that a considerable part of meteorological observations will be unevenly distributed in four dimensions, i.e. three dimensions in space and one in time. The term analysis, or objective analysis in meteorology, means the process of interpolating observed meteorological observations from unevenly distributed locations to a network of regularly spaced grid points. Necessitated by the requirement of numerical weather prediction models to solve the governing finite difference equations on such a grid lattice, the objective analysis is a three-dimensional (or mostly two-dimensional) interpolation technique. As a consequence of the structure of the conventional synoptic network with separated data-sparse and data-dense areas, four-dimensional analysis has in fact been intensively used for many years. Weather services have thus based their analysis not only on synoptic data at the time of the analysis and climatology, but also on the fields predicted from the previous observation hour and valid at the time of the analysis. The inclusion of the time dimension in objective analysis will be called four-dimensional data assimilation. From one point of view it seems possible to apply the conventional technique on the new data sources by simply reducing the time interval in the analysis-forecasting cycle. This could in fact be justified also for the conventional observations. We have a fairly good coverage of surface observations 8 times a day and several upper air stations are making radiosonde and radiowind observations 4 times a day. If we have a 3-hour step in the analysis-forecasting cycle instead of 12 hours, which is applied most often, we may without any difficulties treat all observations as synoptic. No observation would thus be more than 90 minutes off time and the observations even during strong transient motion would fall within a horizontal mesh of 500 km * 500 km.

Design of a forecasting system for ECMWF

As laid out in its convention there are 8 different objectives for ECMWF. One of the major objectives will consist of the preparation, on a regular basis, of the data necessary for the preparation of medium-range weather forecasts. The interpretation of this item is that the Centre will make forecasts once a day for a prediction period of up to 10 days. It is also evident that the Centre should not carry out any real weather forecasting but merely disseminate to the member countries the basic forecasting parameters with an appropriate resolution in space and time. It follows from this that the forecasting system at the Centre must from the operational point of view be functionally integrated with the Weather Services of the Member Countries. The operational interface between ECMWF and the Member Countries must be properly specified in order to get a reasonable flexibility for both systems. The problem of making numerical atmospheric predictions for periods beyond 4-5 days differs substantially from 2-3 days forecasting. From the physical point we can define a medium range forecast as a forecast where the initial disturbances have lost their individual structure. However we are still interested to predict the atmosphere in a similar way as in short range forecasting which means that the model must be able to predict the dissipation and decay of the initial phenomena and the creation of new ones. With this definition, medium range forecasting is indeed very difficult and generally regarded as more difficult than extended forecasts, where we usually only predict time and space mean values. The predictability of atmospheric flow has been extensively studied during the last years in theoretical investigations and by numerical experiments. As has been discussed elsewhere in this publication (see pp 338 and 431) a 10-day forecast is apparently on the fringe of predictability.

Estimation of optimal gravity wave parameters for climate models using data assimilation

There is a current need to constrain the parameters of gravity wave drag (GWD) schemes in climate models using observational information instead of tuning them subjectively. In this work, an inverse technique is developed using data assimilation principles to estimate gravity wave parameters. Because mostGWDschemes assume instantaneous vertical propagation of gravity waves within a column, observations in a single column can be used to formulate a one-dimensional assimilation problem to estimate the unknown parameters. We define a cost function that measures the differences between the unresolved drag inferred from observations (referred to here as the ‘observed’ GWD) and the GWD calculated with a parametrisation scheme. The geometry of the cost function presents some difficulties, including multiple minima and ill-conditioning because of the non-independence of the gravity wave parameters. To overcome these difficulties we propose a genetic algorithm to minimize the cost function, which provides a robust parameter estimation over a broad range of prescribed ‘true’ parameters. When real experiments using an independent estimate of the ‘observed’ GWD are performed, physically unrealistic values of the parameters can result due to the non-independence of the parameters. However, by constraining one of the parameters to lie within a physically realistic range, this degeneracy is broken and the other parameters are also found to lie within physically realistic ranges. This argues for the essential physical self-consistency of the gravity wave scheme. A much better fit to the observed GWD at high latitudes is obtained when the parameters are allowed to vary with latitude. However, a close fit can be obtained either in the upper or the lower part of the profiles, but not in both at the same time. This result is a consequence of assuming an isotropic launch spectrum. The changes of sign in theGWDfound in the tropical lower stratosphere, which are associated with part of the quasi-biennial oscillation forcing, cannot be captured by the parametrisation with optimal parameters.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

What variables condition syntactic transfer? A look at the L3 initial state

This study investigates transfer at the third-language (L3) initial state, testing between the following possibilities: (1) the first language (L1) transfer hypothesis (an L1 effect for all adult acquisition), (2) the second language (L2) transfer hypothesis, where the L2 blocks L1 transfer (often referred to in the recent literature as the ‘L2 status factor’; Williams and Hammarberg, 1998), and (3) the Cumulative Enhancement Model (Flynn et al., 2004), which proposes selective transfer from all previous linguistic knowledge. We provide data from successful English-speaking learners of L2 Spanish at the initial state of acquiring L3 French and L3 Italian relating to properties of the Null-Subject Parameter (e.g. Chomsky, 1981; Rizzi, 1982). We compare these groups to each other, as well as to groups of English learners of L2 French and L2 Italian at the initial state, and conclude that the data are consistent with the predictions of the ‘L2 status factor’. However, we discuss an alternative possible interpretation based on (psycho)typologically-motivated transfer (borrowing from Kellerman, 1983), providing a methodology for future research in this domain to meaningfully tease apart the ‘L2 status factor’ from this alternative account.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.