On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

On the solvability of second kind integral equations on the real line

This paper considers general second kind integral equations of the form(in operator form φ − kφ = ψ), where the functions k and ψ are assumed known, with ψ ∈ Y, the space of bounded continuous functions on R, and k such that the mapping s → k(s, · ), from R to L1(R), is bounded and continuous. The function φ ∈ Y is the solution to be determined. Conditions on a set W ⊂ BC(R, L1(R)) are obtained such that a generalised Fredholm alternative holds: If W satisfies these conditions and I − k is injective for all k ∈ W then I − k is also surjective for all k ∈ W and, moreover, the inverse operators (I − k) − 1 on Y are uniformly bounded for k ∈ W. The approximation of the kernel in the integral equation by a sequence (kn) converging in a weak sense to k is also considered and results on stability and convergence are obtained. These general theorems are used to establish results for two special classes of kernels: k(s, t) = κ(s − t)z(t) and k(s, t) = κ(s − t)λ(s − t, t), where κ ∈ L1(R), z ∈ L∞(R), and λ ∈ BC((R\{0}) × R). Kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces. The general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations.

A uniqueness result for scattering by infinite rough surfaces

Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Effect of Rht alleles on the tolerance of wheat grain set to high temperature and drought stress during booting and anthesis

Factorial pot experiments were conducted to compare the responses of GA-sensitive and GA-insensitive reduced height (Rht) alleles in wheat for susceptibility to heat and drought stress during booting and anthesis. Grain set (grains/spikelet) of near isogenic lines (NILs) was assessed following three day transfers to controlled environments imposing day temperatures (t) from 20 to 40°C. Transfers were during booting and/or anthesis and pots maintained at field capacity (FC) or had water withheld. Logistic responses (y = c/1+e-b(t -m)) described declining grain set with increasing t, and t5 was that fitted to give a 5% reduction in grain set. Averaged over NIL, t5 for anthesis at FC was 31.7±0.47°C (S.E.M, 26 d.f.). Drought at anthesis reduced t5 by <2°C. Maintaining FC at booting conferred considerable resistance to high temperatures (t5=33.9°C) but booting was particularly heat susceptible without water (t5 =26.5°C). In one background (cv. Mercia), for NILs varying at the Rht-D1 locus, there was progressive reduction in t5 with dwarfing and reduced gibberellic acid (GA) sensitivity (Rht-D1a, tall, 32.7±0.72; Rht-D1b, semi-dwarf, 29.5±0.85; Rht-D1c, severe dwarf, 24.2±0.72). This trend was not evident for the Rht-B1 locus, or for Rht-D1b in an alternative background (Maris Widgeon). The GA-sensitive severe dwarf Rht12 was more heat tolerant (t5=29.4±0.72) than the similarly statured GA-insensitive Rht-D1c. The GA-sensitive, semi-dwarfing Rht8 conferred greater drought tolerance in one experiment. Despite the effects of Rht-D1 alleles in Mercia on stress tolerance, the inconsistency of the effects over background and locus led to the conclusion that semi-dwarfing with GA-insensitivity did not necessarily increase sensitivity to stress at booting and flowering. In comparison to effects of semi-dwarfing alleles, responses to heat stress are much more dramatically affected by water availability and the precise growth stage at which the stress is experienced by the plants.

Mid-Holocene land-surface conditions in northern Africa and the Arabian peninsula: a data set for the analysis of biogeophysical feedbacks in the climate system

Large changes in the extent of northern subtropical arid regions during the Holocene are attributed to orbitally forced variations in monsoon strength and have been implicated in the regulation of atmospheric trace gas concentrations on millenial timescales. Models that omit biogeophysical feedback, however, are unable to account for the full magnitude of African monsoon amplification and extension during the early to middle Holocene (˜9500–5000 years B.P.). A data set describing land-surface conditions 6000 years B.P. on a 1° × 1° grid across northern Africa and the Arabian Peninsula has been prepared from published maps and other sources of palaeoenvironmental data, with the primary aim of providing a realistic lower boundary condition for atmospheric general circulation model experiments similar to those performed in the Palaeoclimate Modelling Intercomparison Project. The data set includes information on the percentage of each grid cell occupied by specific vegetation types (steppe, savanna, xerophytic woods/scrub, tropical deciduous forest, and tropical montane evergreen forest), open water (lakes), and wetlands, plus information on the flow direction of major drainage channels for use in large-scale palaeohydrological modeling.

Software-based study of a non-sorting method for median calculation on a set of integer numbers

This paper presents a software-based study of a hardware-based non-sorting median calculation method on a set of integer numbers. The method divides the binary representation of each integer element in the set into bit slices in order to find the element located in the middle position. The method exhibits a linear complexity order and our analysis shows that the best performance in execution time is obtained when slices of 4-bit in size are used for 8-bit and 16-bit integers, in mostly any data set size. Results suggest that software implementation of bit slice method for median calculation outperforms sorting-based methods with increasing improvement for larger data set size. For data set sizes of N > 5, our simulations show an improvement of at least 40%.

A review of existing building benchmarks and the development of a set of reference office buildings for England and Wales

The modern built environment has become more complex in terms of building types, environmental systems and use profiles. This complexity causes difficulties in terms of optimising buildings energy design. In this circumstance, introducing a set of prototype reference buildings, or so called benchmark buildings, that are able to represent all or majority parts of the UK building stock may be useful for the examination of the impact of national energy policies on building energy consumption. This study proposes a set of reference office buildings for England and Wales based on the information collected from the Non-Domestic Building Stock (NDBS) project and an intensive review of the existing building benchmarks. The proposed building benchmark comprises 10 prototypical reference buildings, which in relation to built form and size, represent 95% of office buildings in England and Wales. This building benchmark provides a platform for those involved in building energy simulations to evaluate energy-efficiency measures and for policy-makers to assess the influence of different building energy policies.

Investigating hydrological regimes and processes in a set of catchments with temporary waters in Mediterranean Europe

Seven catchments of diverse size in Mediterranean Europe were investigated in order to understand the main aspects of their hydrological functioning. The methods included the analysis of daily and monthly precipitation, monthly potential evapotranspiration rates, flow duration curves, rainfall runoff relationships and catchment internal data for the smaller and more instrumented catchments. The results showed that the catchments were less dry than initially considered. Only one of them was really semi-arid throughout the year. All the remaining catchments showed wet seasons when precipitation exceeded potential evapotrans-piration, allowing aquifer recharge, wet runoff generation mechanisms and relevant baseflow contribution. Nevertheless, local infiltration excess (Hortonian) overland flow was inferred during summer storms in some catchments and urban overland flow in some others. The roles of karstic groundwater, human disturbance and low winter temperatures were identified as having an important impact on the hydrological regime in some of the catchments.

The rough guide to the Moon and Mars

Space is a dangerous place for humans, once we step beyond the rotection of Earth’s atmosphere and magnetic field. Galactic cosmic rays and bursts of charged particles from the Sun damaging to health happen with alarming frequency – the Apollo astronauts were very lucky. Understanding the physics of radiation from distinct sources in space will be useful to help future space voyagers plan journeys in greater safety, and produce effective shields for these unavoidable events on journeys to Mars or beyond.

The WFDEI meteorological forcing data set: WATCH Forcing Data methodology applied to ERA-Interim reanalysis data

The WFDEI meteorological forcing data set has been generated using the same methodology as the widely used WATCH Forcing Data (WFD) by making use of the ERA-Interim reanalysis data. We discuss the specifics of how changes in the reanalysis and processing have led to improvement over the WFD. We attribute improvements in precipitation and wind speed to the latest reanalysis basis data and improved downward shortwave fluxes to the changes in the aerosol corrections. Covering 1979–2012, the WFDEI will allow more thorough comparisons of hydrological and Earth System model outputs with hydrologically and phenologically relevant satellite products than using the WFD.

ERA-Interim/Land: a global land surface reanalysis data set

ERA-Interim/Land is a global land surface reanalysis data set covering the period 1979–2010. It describes the evolution of soil moisture, soil temperature and snowpack. ERA-Interim/Land is the result of a single 32-year simulation with the latest ECMWF (European Centre for Medium-Range Weather Forecasts) land surface model driven by meteorological forcing from the ERA-Interim atmospheric reanalysis and precipitation adjustments based on monthly GPCP v2.1 (Global Precipitation Climatology Project). The horizontal resolution is about 80 km and the time frequency is 3-hourly. ERA-Interim/Land includes a number of parameterization improvements in the land surface scheme with respect to the original ERA-Interim data set, which makes it more suitable for climate studies involving land water resources. The quality of ERA-Interim/Land is assessed by comparing with ground-based and remote sensing observations. In particular, estimates of soil moisture, snow depth, surface albedo, turbulent latent and sensible fluxes, and river discharges are verified against a large number of site measurements. ERA-Interim/Land provides a global integrated and coherent estimate of soil moisture and snow water equivalent, which can also be used for the initialization of numerical weather prediction and climate models.