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.

Evaluating raw ensembles with the continuous ranked probability score

The continuous ranked probability score (CRPS) is a frequently used scoring rule. In contrast with many other scoring rules, the CRPS evaluates cumulative distribution functions. An ensemble of forecasts can easily be converted into a piecewise constant cumulative distribution function with steps at the ensemble members. This renders the CRPS a convenient scoring rule for the evaluation of ‘raw’ ensembles, obviating the need for sophisticated ensemble model output statistics or dressing methods prior to evaluation. In this article, a relation between the CRPS score and the quantile score is established. The evaluation of ‘raw’ ensembles using the CRPS is discussed in this light. It is shown that latent in this evaluation is an interpretation of the ensemble as quantiles but with non-uniform levels. This needs to be taken into account if the ensemble is evaluated further, for example with rank histograms.

Nitrogen speciation and phosphorus fractionation dynamics in a lowland chalk catchment

A detailed analysis of temporal and spatial trends in nitrogen (N) speciation and phosphorus (P) fractionation in the Wylye, a lowland Chalk sub-catchment of the Hampshire Avon, UK is presented, identifying the sources contributing to nutrient enrichment, and temporal variability in the fractionation of nutrients in transit from headwaters to lower reaches of the river. Samples were collected weekly from ten monitoring stations with daily sampling at three further sites over one year, and monthly inorganic N and total reactive P (TRP) concentrations at three of the ten weekly monitoring stations over a ten year period are also presented. The data indicate significant daily and seasonal variation in nutrient fractionation in the water column, resulting from plant uptake of dissolved organic and inorganic nutrient fractions in the summer months, increased delivery of both N and P from diffuse sources in the autumn to winter period and during high flow events, and lack of dilution of point source discharges to the Wylye from septic tank, small package Sewage Treatment Works (STW) and urban Waste Water Treatment Works (WwTW) during the summer low flow period. Weekly data show that contributing source areas vary along the river with headwater N and P strongly influenced by diffuse inorganic N and particulate P fluxes, and SRP and organic-rich point source contributions from STW and WwTW having a greater influence in the lower reaches. Long-term data show a decrease in TRP concentrations at all three monitoring stations, with the most pronounced decrease occurring downstream from Warminster WwTW, following the introduction of P stripping at the works in 2001. Inorganic N demonstrates no statistically significant change over the ten year period of record in the rural headwaters, but an increase in the lower reaches downstream from the WwTW which may be due to urban expansion in the lower catchment.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 01. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

Continuous sky view factor maps from high resolution urban digital elevation models

This paper presents a new method to calculate sky view factors (SVFs) from high resolution urban digital elevation models using a shadow casting algorithm. By utilizing weighted annuli to derive SVF from hemispherical images, the distance light source positions can be predefined and uniformly spread over the whole hemisphere, whereas another method applies a random set of light source positions with a cosine-weighted distribution of sun altitude angles. The 2 methods have similar results based on a large number of SVF images. However, when comparing variations at pixel level between an image generated using the new method presented in this paper with the image from the random method, anisotropic patterns occur. The absolute mean difference between the 2 methods is 0.002 ranging up to 0.040. The maximum difference can be as much as 0.122. Since SVF is a geometrically derived parameter, the anisotropic errors created by the random method must be considered as significant.

In Vitro Fermentation of NUTRIOSE® FB06, a wheat dextrin soluble fibre, in a continuous culture human colonic model system

Wheat dextrin soluble fibre may have metabolic and health benefits, potentially acting via mechanisms governed by the selective modulation of the human gut microbiota. Our aim was to examine the impact of wheat dextrin on the composition and metabolic activity of the gut microbiota. We used a validated in vitro three-stage continuous culture human colonic model (gut model) system comprised of vessels simulating anatomical regions of the human colon. To mimic human ingestion, 7 g of wheat dextrin (NUTRIOSE® FB06) was administered to three gut models, twice daily at 10.00 and 15.00, for a total of 18 days. Samples were collected and analysed for microbial composition and organic acid concentrations by 16S rRNA-based fluorescence in situ hybridisation and gas chromatography approaches, respectively. Wheat dextrin mediated a significant increase in total bacteria in vessels simulating the transverse and distal colon, and a significant increase in key butyrate-producing bacteria Clostridium cluster XIVa and Roseburia genus in all vessels of the gut model. The production of principal short-chain fatty acids, acetate, propionate and butyrate, which have been purported to have protective, trophic and metabolic host benefits, were increased. Specifically, wheat dextrin fermentation had a significant butyrogenic effect in all vessels of the gut model and significantly increased production of acetate (vessels 2 and 3) and propionate (vessel 3), simulating the transverse and distal regions of the human colon, respectively. In conclusion, wheat dextrin NUTRIOSE® FB06 is selectively fermented in vitro by Clostridium cluster XIVa and Roseburia genus and beneficially alters the metabolic profile of the human gut microbiota.