Continuous Spatial Variation Research Articles

Measurement of dye diffusion in scattering tissue phantoms using dual-wavelength low-coherence interferometry

We demonstrate low-coherence interferometry (LCI) for dye diffusion measurements in scattering tissue phantoms. The diffusion coefficient of a phthalocyanine dye in 1.5% agar gel containing scattering Intralipid was measured using a dual-wavelength interfero-meter. One wavelength was matched to the absorption peak of the dye at 675 nm. The other, 805 nm, was not affected by the dye, and was used to correct for varying sample scattering as a function of depth, assuming a constant ratio between scattering at the two wavelengths. The same wavelength dependence of scattering is assumed for the entire sample, but no a priori knowledge about the amount of scattering is needed. The dye diffusion coefficient was estimated by fitting a mathematical model of the interferometer signal to the measured LCI envelope. We compare results obtained using both a constant-scattering and a depth-resolved-scattering approach to determine the sample scattering. The presented method provides robust estimation of the diffusion coefficient when spatial resolution in determining the depth-resolved scattering is varied. Results indicate that the method is valid for samples having continuous spatial variations in the scattering coefficient over lengths as short as the coherence length of the probing light. The method allows in situ characterization of diffusion in scattering media.

Multifaceted nature of hydrogeologic scaling and its interpretation

Geology is ubiquitously heterogeneous, exhibiting both discrete and continuous spatial variations on a multiplicity of scales. It is therefore natural to expect that hydrogeologic and other geophysical variables would do likewise. We present evidence that hydrogeologic variables exhibit isotropic and directional dependencies on scales of measurement (data support), observation (extent of phenomena such as a dispersing plume), sampling window (domain of investigation), spatial correlation (structural coherence), and spatial resolution (descriptive detail). We then show that it is possible to interpret these multiple scale dependencies within a unified theoretical framework. This and similar theoretical frameworks may be applicable to a wider range of geophysical scale issues.

A Non‐Gaussian Spatial Process Model for Opacity of Flocculated Paper

ABSTRACT. Product quality in the paper‐making industry can be assessed by opacity of a linear trace through continuous production sheets, summarized in spectral form. We adopt a class of non‐Gaussian stochastic models for continuous spatial variation to describe data of this type. The model has flexible covariance structure, physically interpretable parameters and allows several scales of variation for the underlying process. We derive the spectral properties of the model, and develop methods of parameter estimation based on maximum likelihood in the frequency domain. The methods are illustrated using sample data from a UK mill.

Spatial scales of 137Cs-derived soil flux by wind in a 25 km 2 arable area of eastern England

The area effected by wind erosion in England is estimated to be small, but the magnitude of the problem within this area is unknown. Direct measurement of the process is difficult because of very high spatial and temporal variability, selectivity and its slow, insidious nature. The artificial radionuclide caesium-137 ( 137Cs), offers an alternative method. It was used here to estimate the net (ca. 35 year) soil flux (erosion and deposition) in a 5×5 km area (ca. 19 km 2 area sampled) of East Anglia. It is the first study in the UK to investigate the continuous spatial variation of 137Cs over an area of such a size and one of only a few in the world to focus specifically on the redistribution of soil by wind. A two-stage, nested sampling frame captured approximately 50% of the variation that occurred between and within fields. A total of 148 samples were taken and their analysis was used to produce a variogram of the spatial variation. A spherical model was fitted to the experimental variogram using a weighted least-squares procedure. Simulations of sampling configurations on a regular grid did not provide a practical improvement to the nested sampling frame within the specified tolerance. Instead, the parameters were used in ordinary kriging to map 137Cs every 50 m at unsampled locations across the area. A tentative value for the newly established 137Cs reference inventory for the region was 2068±130 Bq m −2. Owens' mass–balance model of the relationship between 137Cs movement and soil redistribution was modified to the spatially distributed situation and also so that it could account better for the factors that control wind erosion and deposition (Towards improved interpretation of caesium-137 measurements in soil erosion studies. Unpublished PhD thesis, Exter University). A calibration relationship for each field was used to calculate the net soil flux at every 50 m. The sample region was found to approximately balance with a net soil loss of 0.6 t ha −1 year −1; the range was −32.6 to +37.5 t ha −1 year −1. Soil from the high-loss fields was accumulating in field boundaries. Despite little of the material leaving the region, the effect of soil nutrient loss on the fields may be considerable. It appears that wind erosion may have as great or greater impact in the areas where it is active than does water erosion. Net soil flux was inferred to be the result of wind erosion, but soil loss on harvested crops (primarily sugar beet), and perhaps losses during tillage (pulverizing erosion) may be other contributors, and this requires future study. Moreover, the quantity of soil lost may not be as important as its quality. Future work is therefore also required to investigate the mass–balance of soil nutrient enrichment and depletion as a consequence of soil erosion and fertiliser application.

Numerical analysis of uniform rectangular waveguides filled by inhomogeneous dielectrics

AbstractA numerical analysis of a uniform rectangular waveguide filled by an inhomogeneous dielectric with a continuous spatial variation of the relative permittivity is presented in this Letter. The proposed method, based on a method‐of‐moments (MoM) solution of the differential equations for TE, LSE, and LSM modes, is applied to find out the cutoff frequencies of the waveguide. Some numerical examples are also presented, considering both continuous and stepped variations of the permittivity along a transverse direction. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 34: 313–316, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10446

Design of Inhomogeneous Slabs for Filtering Applications Via Closed Form Solutions of the Reflection Coefficient

—A novel efficient design technique for microwave inhomogeneous media components (filters and matching lines) is developed in this paper. The inhomogeneity here considered can be modeled via a continuous spatial variation of the permittivity along the power flow direction and can be taken into account in the transmission line equivalent circuit representation in terms of a longitudinal line impedance and a transverse line admittance varying along the same direction. In order to design a filter or a matching line layout with such a material, new analytical expressions of the reflection and transmission coefficients for nonuniform transmission lines are found. Some examples are also presented in order to illustrate the validity of this technique for an efficient realization of proper layouts involving inhomogeneous layers which satisfy prescribed filtering characteristics in the frequency domain.

Tapered stripline embedded in inhomogeneous media as microwave matching line

A novel design method for a stripline microwave matching line is developed in this paper. Striplines considered have a tapered inner conductor embedded in an inhomogeneous dielectric material with continuous spatial variation of the relative permittivity. The employment of this kind of waveguide, as it is shown in this paper, ensures good matching properties in a wide frequency range. These matching properties can be controlled by means of two different factors: the taper of the stripline inner conductor and the relative permittivity spatial variation of the dielectric material filling the stripline. Starting from the nonuniform transmission-line theory, a novel closed analytical form for the input reflection coefficient of such lines is derived, and design formulas for the matching line are carried out. Finally, several applications that show the capability, flexibility, and fastness of the developed synthesis method are presented.

A new efficient method of analysis for inhomogeneous media shields and filters

In this paper, we discuss the shielding characteristics of an inhomogeneous shield for oblique incidence. The inhomogeneity considered can be modeled via a continuous spatial variation of the permittivity along the power-flow direction and can be taken into account in the mathematical model involving the transmission line equivalent-circuit representation in terms of a longitudinal-line impedance and a transverse-line admittance varying along the same direction. In order to design a filter or a matching line layout with such a material, new analytical expressions of the reflection coefficient and of the shielding factor for nonuniform dielectric slabs are found. Some numerical examples are also presented for both the TE and TM polarization cases.

A New Stripline High Pass Filter Layout

A new kind of stripline microwave high pass filter is proposed in this paper. Striplines here used have a tapered inner conductor and are filled by inhomogeneous dielectric materials with continuous spatial variation of the refraction index. Thus, filtering properties of such components can be controlled by means of two different keys: the inner conductor taper and the refraction index spatial variation of the dielectric material filling the stripline. A design method for such a filter is carried out starting from the generalized nonuniform transmission line theory by which a novel closed analytical form for the input reflection coefficient is derived. This closed form allows to obtain straightforwardly the design formulas for the high pass filters. Finally, some applications that show the capability of the new filter here proposed and the fastness of the developed synthesis method are presented.

Diffraction topography using white X-ray beams with low effective divergence

The divergence of the incident X–ray beam as seen from a point in the specimen, the effective divergence α, is of the order of a microradian at a third generation synchrotron radiation source. This entails two effects on white– beam X–ray diffraction topography. 1. The specimen–detector distance can be varied at will in the metre range without appreciable blurring of the image. Thus the discontinuous change in distortion associated with magnetic domains, or implanted layers in a piezoelectric material, can often be directly measured. It was also possible to observe focusing effect due to continuous spatial variations of lattice plane orientation. This effect was visualized in the cases of the images of single dislocations, from elastic resonance patterns, and on wafer–bonded samples. 2. The small value of α can also be described as yielding appreciable lateral coherence of the beam. Propagation of the diffracted beam, i.e. Fresnel diffraction, can turn variations of the phase of the diffracted beam into changes in intensity, hence contrast. In the case of a periodic spatial variation of the phase due to periodic poling, the Talbot effect in the diffracted beam provides the possibility of measuring the difference in phase of a structure factor in inversion–related domains, i.e. for Friedel pairs.

3-D reconstruction and volume modelling of the grain fabric of geological materials

This paper describes three-dimensional reconstruction and volume modelling of geological materials at the macroscopic and the microscopic scale. By means of a precision lapping device, samples are eroded in plan-parallel steps, erosion intervals being adapted to the size of minerals and fabric features of interest. For reconstruction of macroscopic features, each newly eroded sample surface is recorded by a high-resolution color flatbed scanner. Supervised image classification applied to the derived images yields color based mineral recognition. At the microscopic scale, scanning is done with a microanalyzer to portray mineral phases and microchemical variation. Minerals and fabric features are three dimensionally reconstructed by stacking and interpolating the acquired raster images of successive erosion levels in a voxel array. The voxel data structure features analyical flexibility, integration of both discrete and continuous spatial variation and versatile 3-D volume model visualization.

Elastoplastic analysis of thermal cycling: layered materials with compositional gradients

Elastopllastic analyses are presented for the cyclic thermal response in multi-layered materials which comprise layers of fixed compositions of a metal and a ceramic, and a compositionally graded interface. Analytical solutions for the characteristic temperature at which the onset of thermally induced plastic deformation occurs are derived for the layered composite. Solutions for the evolution of curvature and thermal strains, and for the initiation of plastic yielding are also obtained for different combinations of the geometry, physical properties and compositional gradation for both thermoelastic and thermoplastic deformation. Finite-element formulations incorporating continuous and smooth spatial variations in the composition and properties of the graded layer are used to simulate the evolution of thermal stresses, the accumulation of plastic strains, and the development of monotonic and cyclic plastic zones at the interfaces, edges and free surfaces of different layers during thermal cycling. Engineering diagrams detailing the effects of compositional gradients are also presented for optimizing thermal residual stresses, layer geometry, and plastic strain accumulation.

Spatial Variations in UV-Optical Lines Across the Ring Nebula

IUE spectra of the Ring Nebula (M57=NGC 6720) were taken 16–18 May 1991 using the large aperture (10 × 20 arc sec oval) at low dispersion. SWP and LWP spectra of seven locations were acquired at 10 arc sec intervals extending along a PA = 124° line and passing through the central star of the nebula. This direction was such that the long axis of the large aperture was aligned and overlapped, thus enabling the line profiles to be “spliced” together. This provided continuous spatial variation curves at approximately 1 arc sec spatial resolution over the entire diameter (80 arc sec) of the main body of the Ring Nebula.

REFLECTIONS ON REALISTIC INTERFACES

Discrete echoes in non-destructive testing are usually associated with a step change in impedance between two different lossless media. While maintaining the problem as a one-dimensional one, different refinements to this model are briefly summarised. These include a continuous spatial impedance variation, interfaces comprising discrete layers of finite thickness, the inclusion of losses in the media on the two sides of a discontinuous boundary, and lossy media in which the losses also vary continuously in space. Some of these models have found application in medical ultrasonics and underwater acoustics and may prove useful also in NDE.

An Adaptive Filter for Estimating Spatially-Varying Parameters: Application to Modeling Police Hours Spent in Response to Calls for Service

The Spatial Adaptive Filter (SAF), introduced in this paper, uses generalized damped negative feedback to estimate spatially-varying parameters for multivariate models. Previous adaptive filters have been designed to estimate time-varying parameters and process data recursively in time sequence. SAF processes all data simultaneously in an iterative algorithm. Monte Carlo studies show that SAF is successful in automatically identifying and estimating step-jump and continuous spatial variation in the parameters of causal variables. A case study on census-tract data from Columbus, Ohio, relating police-vehicle hours spent in responding to calls to socio-economic indicators, has systematic spatial variation in estimated parameters. Independent variables that are significant in inner-city areas of Columbus become progressively less significant in moving to outlying areas.

Photoeffects on the Cu/H3PO4 interface Part II. Band model of the interface in darkness

We describe a continuous spatial variation of copper concentration, from metal towards the solution, inside the oxide anodic film. This corresponds to an evolution of the nature of the oxide, from Cu 2 O on the metal side to CuO on that of the solution. In this paper we give an interface model in the band diagram. This model accounts for the electrochemical phenomenon observed.

Photoeffects on the Cu/H 3PO 4 interface : Part II. band model of the interface in darkness

We describe a continuous spatial variation of copper concentration, from metal towards the solution, inside the oxide anodic film. This corresponds to an evolution of the nature of the oxide, from Cu 2O on the metal side to CuO on that of the solution. In this paper we give an interface model in the band diagram. This model accounts for the electrochemical phenomenon observed.

Elementary solutions of the linear transport equation for continuously varying spatial media

We demonstrate that it is possible to construct elementary solutions (eigenfunctions) of the linear transport equation for certain types of continuously varying spatial media. In general, both discrete and continuum modes result, which appear to be complete on the half-range. A detailed analysis is given for an ’’exponential’’ medium, including numerical results and a half-range completeness proof. A ’’linear’’ medium is also considered. A general method is presented for constructing, jointly, the spatial variation of a medium and the corresponding functional forms of the eigenfunctions. Our results represent a partial generalization of the singular eigenfunction technique to media with continuous spatial variation.