On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Part load operation coefficient of air-conditioning system of public building

IPLV overall coefficient, presented by Air-Conditioning and Refrigeration Institute (ARI) of America, shows running/operation status of air-conditioning system host only. For overall operation coefficient, logical solution has not been developed, to reflect the whole air-conditioning system under part load. In this research undertaking, the running time proportions of air-conditioning systems under part load have been obtained through analysis on energy consumption data during practical operation in all public buildings in Chongqing. This was achieved by using analysis methods, based on the statistical energy consumption data distribution of public buildings month-by-month. Comparing with the weight number of IPLV, part load operation coefficient of air-conditioning system, based on this research, does not only show the status of system refrigerating host, but also reflects and calculate energy efficiency of the whole air-conditioning system. The coefficient results from the processing and analyzing of practical running data, shows the practical running status of area and building type (actual and objective) – not clear. The method is different from model analysis which gets IPLV weight number, in the sense that this method of coefficient results in both four equal proportions and also part load operation coefficient of air-conditioning system under any load rate as necessary.

Conditioning model output statistics of regional climate model precipitation on circulation patterns

Dynamical downscaling of Global Climate Models (GCMs) through regional climate models (RCMs) potentially improves the usability of the output for hydrological impact studies. However, a further downscaling or interpolation of precipitation from RCMs is often needed to match the precipitation characteristics at the local scale. This study analysed three Model Output Statistics (MOS) techniques to adjust RCM precipitation; (1) a simple direct method (DM), (2) quantile-quantile mapping (QM) and (3) a distribution-based scaling (DBS) approach. The modelled precipitation was daily means from 16 RCMs driven by ERA40 reanalysis data over the 1961–2000 provided by the ENSEMBLES (ENSEMBLE-based Predictions of Climate Changes and their Impacts) project over a small catchment located in the Midlands, UK. All methods were conditioned on the entire time series, separate months and using an objective classification of Lamb's weather types. The performance of the MOS techniques were assessed regarding temporal and spatial characteristics of the precipitation fields, as well as modelled runoff using the HBV rainfall-runoff model. The results indicate that the DBS conditioned on classification patterns performed better than the other methods, however an ensemble approach in terms of both climate models and downscaling methods is recommended to account for uncertainties in the MOS methods.

Soil conditioning and plant-soil feedbacks in a modified forest ecosystem are soil-context dependent

Aims There is potential for altered plant-soil feedback (PSF) to develop in human-modified ecosystems but empirical data to test this idea are limited. Here, we compared the PSF operating in jarrah forest soil restored after bauxite mining in Western Australia with that operating in unmined soil. Methods Native seedlings of jarrah (Eucalyptus marginata), acacia (Acacia pulchella), and bossiaea (Bossiaea ornata) were grown in unmined and restored soils to measure conditioning of chemical and biological properties as compared with unplanted control soils. Subsequently, acacia and bossiaea were grown in soils conditioned by their own or by jarrah seedlings to determine the net PSF. Results In unmined soil, the three plant species conditioned the chemical properties but had little effect on the biological properties. In comparison, jarrah and bossiaea conditioned different properties of restored soil while acacia did not condition this soil. In unmined soil, neutral PSF was observed, whereas in restored soil, negative PSF was associated with acacia and bossiaea. Conclusions Soil conditioning was influenced by soil context and plant species. The net PSF was influenced by soil context, not by plant species and it was different in restored and unmined soils. The results have practical implications for ecosystem restoration after human activities.

Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.