Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Evaluation of suitable spectral intervals for near-IR laboratory detection of water vapour continuum absorption

The water vapour continuum absorption is an important component of molecular absorption of radiation in atmosphere. However, uncertainty in knowledge of the value of the continuum absorption at present can achieve 100% in different spectral regions leading to an error in flux calculation up to 3-5 W/m2 global mean. This work uses line-by-line calculations to reveal the best spectral intervals for experimental verification of the CKD water vapour continuum models in the currently least studied near-infrared spectral region. Possible sources of errors in continuum retrieval taken into account in the simulation include the sensitivity of laboratory spectrometers and uncertainties in the spectral line parameters in HITRAN-2004 and Schwenke-Partridge database. It is shown that a number of micro-windows in near-IR can be used at present for laboratory detection of the water vapour continuum with estimated accuracy from 30 to 5%.

The relationship between diffuse spectral reflectance of the soil and its cation exchange capacity is scale-dependent

Diffuse reflectance spectroscopy (DRS) is increasingly being used to predict numerous soil physical, chemical and biochemical properties. However, soil properties and processes vary at different scales and, as a result, relationships between soil properties often depend on scale. In this paper we report on how the relationship between one such property, cation exchange capacity (CEC), and the DRS of the soil depends on spatial scale. We show this by means of a nested analysis of covariance of soils sampled on a balanced nested design in a 16 km × 16 km area in eastern England. We used principal components analysis on the DRS to obtain a reduced number of variables while retaining key variation. The first principal component accounted for 99.8% of the total variance, the second for 0.14%. Nested analysis of the variation in the CEC and the two principal components showed that the substantial variance components are at the > 2000-m scale. This is probably the result of differences in soil composition due to parent material. We then developed a model to predict CEC from the DRS and used partial least squares (PLS) regression do to so. Leave-one-out cross-validation results suggested a reasonable predictive capability (R2 = 0.71 and RMSE = 0.048 molc kg− 1). However, the results from the independent validation were not as good, with R2 = 0.27, RMSE = 0.056 molc kg− 1 and an overall correlation of 0.52. This would indicate that DRS may not be useful for predictions of CEC. When we applied the analysis of covariance between predicted and observed we found significant scale-dependent correlations at scales of 50 and 500 m (0.82 and 0.73 respectively). DRS measurements can therefore be useful to predict CEC if predictions are required, for example, at the field scale (50 m). This study illustrates that the relationship between DRS and soil properties is scale-dependent and that this scale dependency has important consequences for prediction of soil properties from DRS data

An integrated model of soil-canopy spectral radiances, photosynthesis, fluorescence, temperature and energy balance

This paper presents the model SCOPE (Soil Canopy Observation, Photochemistry and Energy fluxes), which is a vertical (1-D) integrated radiative transfer and energy balance model. The model links visible to thermal infrared radiance spectra (0.4 to 50 μm) as observed above the canopy to the fluxes of water, heat and carbon dioxide, as a function of vegetation structure, and the vertical profiles of temperature. Output of the model is the spectrum of outgoing radiation in the viewing direction and the turbulent heat fluxes, photosynthesis and chlorophyll fluorescence. A special routine is dedicated to the calculation of photosynthesis rate and chlorophyll fluorescence at the leaf level as a function of net radiation and leaf temperature. The fluorescence contributions from individual leaves are integrated over the canopy layer to calculate top-of-canopy fluorescence. The calculation of radiative transfer and the energy balance is fully integrated, allowing for feedback between leaf temperatures, leaf chlorophyll fluorescence and radiative fluxes. Leaf temperatures are calculated on the basis of energy balance closure. Model simulations were evaluated against observations reported in the literature and against data collected during field campaigns. These evaluations showed that SCOPE is able to reproduce realistic radiance spectra, directional radiance and energy balance fluxes. The model may be applied for the design of algorithms for the retrieval of evapotranspiration from optical and thermal earth observation data, for validation of existing methods to monitor vegetation functioning, to help interpret canopy fluorescence measurements, and to study the relationships between synoptic observations with diurnally integrated quantities. The model has been implemented in Matlab and has a modular design, thus allowing for great flexibility and scalability.

Spectral properties, iron oxide content and provenance of Namib dune sands

This paper applies multispectral remote sensing techniques to map the Fe-oxide content over the entire Namib sand sea. Spectrometric analysis is applied to field samples to identify the reflectance properties of the dune sands which enable remotely sensed Fe-oxide mapping. The results indicate that the pattern of dune colour in the Namib sand sea arises from the mixing of at least two distinct sources of sand; a red component of high Fe-oxide content (present as a coating on the sand grains) which derives from the inland regions, particularly from major embayments into the Southern African escarpment; and a yellow coastal component of low Fe-oxide content which is brought into the area by northward-moving aeolian transport processes. These major provenances are separated by a mixing zone between 20 kin and 90 kin from the coast throughout the entire length of the sand sea. Previous workers have also recognised a third, fluvial, provenance, but the methodology applied here is not able to map this source as a distinct spectral component. (c) 2006 Elsevier B.V. All rights reserved.

Spectral properties, iron oxide content and provenance of Namib dune sands

This paper applies multispectral remote sensing techniques to map the Fe-oxide content over the entire Namib sand sea. Spectrometric analysis is applied to field samples to identify the reflectance properties of the dune sands which enable remotely sensed Fe-oxide mapping. The results indicate that the pattern of dune colour in the Namib sand sea arises from the mixing of at least two distinct sources of sand; a red component of high Fe-oxide content (present as a coating on the sand grains) which derives from the inland regions, particularly from major embayments into the Southern African escarpment; and a yellow coastal component of low Fe-oxide content which is brought into the area by northward-moving aeolian transport processes. These major provenances are separated by a mixing zone between 20 kin and 90 kin from the coast throughout the entire length of the sand sea. Previous workers have also recognised a third, fluvial, provenance, but the methodology applied here is not able to map this source as a distinct spectral component. (c) 2006 Elsevier B.V. All rights reserved.

Assessment of the consistency of near-infrared water vapor line intensities using high-spectral-resolution ground-based Fourier transform measurements of solar radiation

Calculations of the absorption of solar radiation by atmospheric gases, and water vapor in particular, are dependent on the quality of databases of spectral line parameters. There has been increasing scrutiny of databases such as HITRAN in recent years, but this has mostly been performed on a band-by-band basis. We report nine high-spectral-resolution (0.03 cm(-1)) measurements of the solar radiation reaching the surface in southern England over the wave number range 2000 to 12,500 cm(-1) (0.8 to 5 mm) that allow a unique assessment of the consistency of the spectral line databases over this entire spectral region. The data are assessed in terms of the modeled water vapor column that is required to bring calculations and observations into agreement; for an entirely consistent database, this water vapor column should be constant with frequency. For the HITRAN01 database, the spread in water vapor column is about 11%, with distinct shifts between different spectral regions. The HITRAN04 database is in significantly better agreement (about 5% spread) in the completely updated 3000 to 8000 cm(-1) spectral region, but inconsistencies between individual spectral regions remain: for example, in the 8000 to 9500 cm(-1) spectral region, the results indicate an 18% (+/- 1%) underestimate in line intensities with respect to the 3000 to 8000 cm(-1) region. These measurements also indicate the impact of isotopic fractionation of water vapor in the 2500 to 2900 cm(-1) range, where HDO lines dominate over the lines of the most abundant isotope of H2O.