Exponential Correlation Function Research Articles

Beams of gravitationally bound ultracold neutrons in rough waveguides

We investigate the propagation of ultracold neutrons through a rough waveguide in conjunction with recent experiments in which the ultracold neutrons were beamed between a perfect mirror and a rough scatterer and absorber. The main goal is to find a way to resolve the lowest gravitationally quantized discrete states in the peV range. We compare the neutron count for various types of mirrors with Gaussian, power-law, and exponential correlation functions of surface inhomogeneities. The main conclusion is that all the information about inhomogeneities, including their amplitude, correlation radius, and the rate of decay of the correlation function, enter the exit neutron count via just a single constant $\ensuremath{\Phi}$, which effectively renormalizes the amplitude of roughness. To observe well-defined quantum steps, one should have an experimental setup with $\ensuremath{\Phi}g40$. For a wide variety of correlation functions, the constant $\ensuremath{\Phi}$ is proportional to the square of the amplitude of the surface roughness and is inversely proportional to the square root of the correlation radius. The strong dependence of $\ensuremath{\Phi}$ on roughness parameters and the shape of the correlation function opens a novel way for improving the resolution of gravitationally bound states by optimizing the roughness pattern without reverting to an undesirable strong roughness. We discuss how to optimize the scatterer and absorber by first generating numerically the desired roughness profile and then transferring it to the mirror. We also study the effect of beam preparation on the initial occupancies of gravitational states before the beam enters the waveguide. It turns out that there are simple ways to manipulate the beam in front of the waveguide that can help to resolve the gravitationally bound quantum states. Our results are in good agreement with available experimental data.

Statistical simulation of an exponentially correlated many-dimensional random field

A model of a many-dimensional homogeneous isotropic random field with an exponential correlation function and a given arbitrary one-dimensional distribution suitable for numerical implementation is constructed. The model is based on partition of the phase space by a field of random hyperplanes determined by some parametric Poisson point field. The possibility of application of this model to the solution of stochastic problems of radiation balance based on the combination of the methods of ‘double randomization’ and ‘maximal section’ is discussed.

The heterogeneity of segmental dynamics of filled EPDM by 1H transverse relaxation NMR

Residual second moment of dipolar interactions M∼2 and correlation time segmental dynamics distributions were measured by Hahn-echo decays in combination with inverse Laplace transform for a series of unfilled and filled EPDM samples as functions of carbon-black N683 filler content. The fillers–polymer chain interactions which dramatically restrict the mobility of bound rubber modify the dynamics of mobile chains. These changes depend on the filler content and can be evaluated from distributions of M∼2. A dipolar filter was applied to eliminate the contribution of bound rubber. In the first approach the Hahn-echo decays were fitted with a theoretical relationship to obtain the average values of the 1H residual second moment 〈M∼2〉 and correlation time 〈τc〉. For the mobile EPDM segments the power-law distribution of correlation function was compared to the exponential correlation function and found inadequate in the long-time regime. In the second approach a log-Gauss distribution for the correlation time was assumed. Furthermore, using an averaged value of the correlation time, the distributions of the residual second moment were determined using an inverse Laplace transform for the entire series of measured samples. The unfilled EPDM sample shows a bimodal distribution of residual second moments, which can be associated to the mobile polymer sub-chains (M∼2≅6.1rad2s-2) and the second one associated to the dangling chains M∼2≅5.4rad2s-2). By restraining the mobility of bound rubber, the carbon-black fillers induce diversity in the segmental dynamics like the apparition of a distinct mobile component and changes in the distribution of mobile and free-end polymer segments.

Backscattering Coefficients, Coherent Reflectivities, and Emissivities of Randomly Rough Soil Surfaces at L-Band for SMAP Applications Based on Numerical Solutions of Maxwell Equations in Three-Dimensional Simulations

We used Numerical Maxwell Model in 3-D Simulations (NMM3D) to study the backscattering coefficients, coherent reflectivities, and emissivities of soil surfaces using Gaussian random rough surfaces with exponential correlation functions. The surface area used is 8 by 8 square wavelengths. A total of close to 200 cases are computed by varying rms height, correlation length, and soil permittivity. We consider a 40° incidence angle. For each case, 15 realizations of rough surface profiles are generated, and 30 solutions of Maxwell equations are computed because of two polarizations. The method for solving the Maxwell equations is based on the Method of Moments (MoM) with Rao-Wilton-Glisson (RWG) basis functions. The solutions are accelerated by the sparse matrix canonical grid method implemented on parallel computing. The rms height varies up to 0.126 wavelength. The results are compared with the Dubois formulation, Small Perturbation Method (SPM), Kirchhoff Approximation (KA), and Advanced Integral Equation Model (AIEM). The NMM3D results are also compared with VV and HH backscatter data of soil surfaces where ground truth rms heights and correlation lengths were both measured. Good agreement is found between the NMM3D results and experimental measurement data. Based on the computed cases, interpolation tables are derived that can be directly applied to L-band active and passive microwave remote sensing of soil moisture, such as for the upcoming Soil Moisture Active and Passive (SMAP) mission.

Numerical Backscattering Analysis for Rough Surfaces Including a Cloddy Structure

In recent years, the presence of a new type of agricultural-surface tillage used for the sowing of wheat and corn has been observed with increasing frequency. It illustrates less roughly ploughed soils, with a greater quantity of small clods distributed over the soil surface. In this paper, a new description of such rough agricultural surfaces is proposed. It is based on a composite model, including a classical surface represented by an exponential correlation function, together with a random cloddy structure. This description enables volumetric structures to be introduced over the soil's surface. A numerical moment-modeling method, based on integral equations, is used to evaluate the contribution of clods to the radar backscattering behavior of agricultural surfaces. It is found that the presence of clods explains the very small correlation lengths which are often found in cloddy agricultural fields. The classical approach, in which the surface is described by a correlation function only based on two statistical parameters, rms height and correlation length, overestimates the backscattering coefficients when compared with an approach that includes the clods. This overestimation is often observed with real radar data for such fields.

Higher Moment Estimation for Shallow-Water Reverberation Prediction

The modeling of the higher moments of shallow-water reverberation is addressed for problems where non-Rayleigh reverberation is caused by scattering from rough boundaries with chi-squared height distributions. Expressions for the first and third spatial moments of chi-squared height-distributed roughness are derived for exponential correlation functions corresponding to von-Karman second-order spectra. The trispectra for these roughness distributions are also derived, and differences are highlighted between these results and the results that are obtained for the limiting case of Gaussian height-distributed higher order spectra. A normal mode model for propagation to and from these scatterers is combined with first-order perturbation theory to compute the higher moments of the reverberation pressure envelope in a 2-D shallow-water waveguide as a function of time, correlation length scale, and non-Gaussianity as parameterized by the degrees of freedom (dof) of the chi-squared distribution. Results show that the non-Rayleighness of shallow-water reverberation with non-Gaussian chi-squared roughness is controlled by both the degree of non-Gaussianity and the ratio of the ensonified scatterer patch size to the correlation length scale. In shallow-water waveguides, the size of the ensonified patch increases with time, due to multipath effects. It is shown that this effect causes a trend towards Rayleighness in the reverberation pressure envelope. It is also shown that when the correlation length scale of the scatterers is much greater than the size of the ensonified patch, the effects of multipath time spread on the reverberation statistics is much reduced, and non-Rayleighness can persist.

Statistical distribution of the field scattered by rough layered interfaces: formulae derived from the small perturbation method

We present a statistical study of electromagnetic wave scattering by a stack of two random rough interfaces that are characterised by Gaussian distributed heights and by exponential correlation functions. These interfaces can be correlated or not. The coherent and incoherent intensities and the statistical distribution of the scattered field in modulus and phase are obtained using the Rayleigh expansion and the small perturbation method. For a structure of finite extension, we show that the modulus follows a Hoyt law and the phase is not uniform. For a structure of infinite extension, whether the interfaces are correlated or not, the modulus of the field follows a Rayleigh law while the phase is uniform.

Generalized linear spatial models in epidemiology: A case study of zoonotic cutaneous leishmaniasis in Tunisia

Generalized linear spatial models (GLSM) are used here to study spatial characters of zoonotic cutaneous leishmaniasis (ZCL) in Tunisia. The response variable stands for the number of affected by district during the period 2001–2002. The model covariates are: climates (temperature and rainfall), humidity and surrounding vegetation status. As the environmental and weather data are not available for all the studied districts, Kriging based on linear interpolation was used to estimate the missing data. To account for unexplained spatial variation in the model, we include a stationary Gaussian process S with a powered exponential spatial correlation function. Moran coefficient, DIC criterion and residuals variograms are used to show the high goodness-of-fit of the GLSM. When compared with the statistical tools used in the previous ZCL studies, the optimal GLSM found here yields a better assessment of the impact of the risk factors, a better prediction of ZCL evolution and a better comprehension of the disease transmission. The statistical results show the progressive increase in the number of affected in zones with high temperature, low rainfall and high surrounding vegetation index. Relative humidity does not seem to affect the distribution of the disease in Tunisia. The results of the statistical analyses stress the important risk of misleading epidemiological conclusions when non-spatial models are used to analyse spatially structured data.

Non-Markovian dynamics for a free quantum particle subject to spontaneous collapse in space: General solution and main properties

We analyze the non-Markovian dynamics of a quantum system subject to spontaneous collapse in space. After having proved, under suitable conditions, the separation of the center-of-mass and relative motions, we focus our analysis on the time evolution of the center of mass of an isolated system (free-particle case). We compute the explicit expression of the Green's function, for a generic Gaussian noise, and analyze in detail the case of an exponential correlation function. We study the time evolution of average quantities, such as the mean position, momentum, and energy. We eventually specialize to the case of Gaussian wave functions and prove that all basic facts about collapse models, which are known to be true in the white-noise case, hold also in the more general case of non-Markovian dynamics.

Non-Markovian Quantum Trajectories: An Exact Result

We analyze the non-Markovian stochastic Schrödinger equation describing a particle subject to spontaneous collapses in space (in the language of collapse models), or subject to a continuous measurement of its position (in the language of continuous quantum measurement). For the first time, we give the explicit general solution for the free particle case (H=p(2)/2m) and discuss the main properties. We analyze the case of an exponential correlation function for the noise, giving a quantitative description of the dynamics and of its dependence on the correlation time.

Empirically Adopted IEM for Retrieval of Soil Moisture From Radar Backscattering Coefficients

The integral equation model (IEM) is considered as a promising algorithm for soil moisture retrieval from active microwave data over bare soil and sparsely vegetated conditions. However, the soil dielectric constant is implicitly embedded in the complicated IEM; inversion of soil moisture is often accomplished through iteration and is thus computationally expensive, particularly when it is applied to retrieve soil moisture from active microwave data on a large scale. To simplify the inversion process of soil moisture directly from the active microwave data, basic math functions were adopted to fit the simulation results of the original IEM so that the radar backscattering coefficient becomes an explicit function of soil dielectric constant or the soil dielectric constant is an explicit function of radar backscattering coefficient. Soil moisture is then calculated directly from radar backscattering coefficient without iteration. We called this model empirically adopted IEM (EA-IEM). The accuracy of the EA-IEM to the original IEM and its applicability are analyzed through three processes: model intercomparison, sensitivity analysis, and model comparison with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">situ</i> measurements. The average differences of backscattering coefficients between the EA-IEM and the original IEM are 0.14 dB for HH-polarization and 0.12 dB (Gaussian correlation function) and 0.2 dB (exponential correlation function) for VV-polarization. The sensitivity of soil moisture variation is examined under the consideration of absolute and relative calibration errors. A comparison between the soil moisture estimated and the measurements is performed, and the root-mean-square (rms) error is found to be 3.4%, suggesting that the EA-IEM performs well in these real cases. All these analyses indicate that the EA-IEM is a good representative of the original IEM and can be used to retrieve soil moisture under the tested range of model parameters: incidence angles between 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deg</sup> and 60 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deg</sup> , soil dielectric constants between 4 and 42, surface rms height from 4 to 31 mm, and correlation length from 50 to 250 mm.

Ozone correlation lengths and measurement uncertainties from analysis of historical ozonesonde data in North America and Europe

A spatial and temporal correlation analysis is performed on the WOUDC (World Ozone and Ultraviolet Radiation Data Centre) ozonesonde data from 13 midlatitude stations in North America and Europe. The data records span more than 40 years at some stations, and a total of more than 27,000 ozonesonde profiles are utilized. The spatial correlation coefficients between pairs of stations decrease with increasing station separation distance, following a power exponential correlation function. The horizontal distance for the correlation coefficient to decrease by a factor of e is about 1000–2000 km in the stratosphere with a peak at around 22‐km altitude, and is about 500–1000 km in the troposphere. The autocorrelation coefficient decreases rapidly with time lag, and the timescale of the autocorrelation varies between about 1.5 and 3.5 days in the troposphere but is generally longer in the stratosphere at 2–6 days. The extrapolation of the correlation functions to zero station distance or zero time lag yields estimates of the intrinsic uncertainty of ozonesonde measurements. The uncertainty is found to be less than 7% for 20–30‐km altitudes in the stratosphere, about 15% in the troposphere, and to have larger values near the tropopause and at the surface. The results are broadly consistent with those from the recent JOSIE and BESOS experiments, and other intercomparisons, with the additional measurement uncertainty probably reflecting changes in ozonesonde type, model, manufacture, and preparation procedure during the period of the record.

Fluorine dynamics in BaF 2 crystal lattice as an example of complex motion in a simple system

Fluorine relaxation profiles for a BaF 2 single crystal collected at several temperatures have been analyzed in terms of essentially different motional models: free rotational and free translational diffusion. The analysis has been performed to critically review the sensitivity of field dependent relaxation studies to mechanisms of molecular motions. The tested motional models do not realistically describe the fluorine dynamics within the crystal lattice. They have been chosen to attempt to answer quite fundamental questions regarding the feasibility of the field dependent nuclear spin relaxation studies to provide unique information on dynamic processes: 1. Is it possible to get information about the motional mechanisms by analyzing relaxation profiles collected in a broad frequency range? 2. To what extent is it possible to reasonably reproduce relaxation profiles in terms of unrealistic motional models? It has been concluded from the analysis that the rotational model leading to a single exponential correlation function explains the experimental data much better than the translational one. Validity regimes of the second order perturbation theory have been discussed in the context of the investigated system and the applied models.

Chromophoric dissolved organic matter optical characteristics and spatial distribution in the lakes of the middle reaches of the Yangtze River

根据2007年秋季洪湖、东湖和梁子湖的调查结果,分析了长江中游3个典型湖泊CDOM光学特性及其空间分布差异.结果表明:(1)CDOM吸收系数在洪湖最高,梁子湖最低;(2)在洪湖,CDOM吸收系数受陆源影响较大,与悬浮泥沙浓度呈现较好正相关关系,而在梁子湖,CDOM吸收系数主要受到浮游植物降解贡献,与叶绿素浓度具有显著的正相关关系;(3)洪湖与东湖指数函数斜率S值变化不大;梁子湖CDOM吸收系数空间分布差异较大,指数函数斜率S值与400nm波段CDOM吸收系数ag(400)显著负相关.在建立梁子湖CDOM遥感反演模型时,应引入上述指数函数斜率S值与400nm波段CDOM吸收系数之间的线性函数,对CDOM吸收系数的指数模型进行优化.;Chromophoric dissolved organic matter (CDOM) is the colored fraction of the dissolved organic carbon (DOC) pool. According to investigations and filed measurements in Lake Honghu, Lake Donghu, Lake Liangzi in September and October of 2007, optical characteristics and spatial distribution of CDOM in these lakes are analyzed. Results show that CDOM absorption coefficient in Lake Honghu is the highest, with the mean value of the CDOM absorption coefficient at 400nm (a_g(400)) is 1.45m^-¹, while the lowest is in Lake Liangzi, with the mean value of the CDOM absorption coefficient at 400nm is 0.68m^-¹. A significantly positive correlation between CDOM absorption coefficient and suspended sediment concentration is found in Lake Honghu. In Lake Liangzi, CDOM absorption coefficient is positive correlated with chlorophyll concentration, which means CDOM in the lake is mainly contributed by Phytoplankton degradation. The slope (S) in the exponential correlation function does not vary much in Lake Honghu and Lake Donghu. Fulivc acid is the dominated constituent of CDOM in Lake Liangzi, and the absorption coefficient of CDOM vary extremely. There is a negative correlation relationship (-0.9) between S and a_g(400) in Lake Liangzi.

Probabilistic quantitative precipitation field forecasting using a two-stage spatial model

Short-range forecasts of precipitation fields are needed in a wealth of agricultural, hydrological, ecological and other applications. Forecasts from numerical weather prediction models are often biased and do not provide uncertainty information. Here we present a postprocessing technique for such numerical forecasts that produces correlated probabilistic forecasts of precipitation accumulation at multiple sites simultaneously. The statistical model is a spatial version of a two-stage model that represents the distribution of precipitation by a mixture of a point mass at zero and a Gamma density for the continuous distribution of precipitation accumulation. Spatial correlation is captured by assuming that two Gaussian processes drive precipitation occurrence and precipitation amount, respectively. The first process is latent and drives precipitation occurrence via a threshold. The second process explains the spatial correlation in precipitation accumulation. It is related to precipitation via a site-specific transformation function, so as to retain the marginal right-skewed distribution of precipitation while modeling spatial dependence. Both processes take into account the information contained in the numerical weather forecast and are modeled as stationary isotropic spatial processes with an exponential correlation function. The two-stage spatial model was applied to 48-hour-ahead forecasts of daily precipitation accumulation over the Pacific Northwest in 2004. The predictive distributions from the two-stage spatial model were calibrated and sharp, and outperformed reference forecasts for spatially composite and areally averaged quantities.

Investigation on electromagnetic scattering from rough soil surface of layered medium using the small perturbation method

Electromagnetic scattering from a rough surface of layered medium is investigated, and the formulae of the scattering coefficients for different polarizations are derived using the small perturbation method. A rough surface with exponential correlation function is presented for describing a rough soil surface of layered medium, the formula of its scattering coefficient is derived by considering the spectrum of the rough surface with exponential correlation function; the curves of the bistatic scattering coefficient of HH polarization with variation of the scattering angle are obtained by numerical calculation. The influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the roughness surface parameters and the frequency of the incident wave on the bistatic scattering coefficient is discussed. Numerical results show that the influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the rms and the correlation length of the rough surface, and the frequency of the incident wave on the bistatic scattering coefficient is very complex.

Modeling higher order statistics of shallow water reverberation

The modeling of the higher moments of shallow water reverberation is addressed for problems where non-Rayleigh reverberation is caused by scattering from rough boundaries with non-Gaussian height distributions. Expressions for the first three spatial moments of exponentially height-distributed roughness are derived for both Gaussian and exponential correlation functions corresponding to Gaussian or von-Karman second order spectra. The bi and tri-spectra for these roughness distributions are also derived, and differences are highlighted between these results and the results that are obtained for Gaussian height-distributed higher order spectra. Finally results for the higher moments of reverberation pressure are derived for exponentially height-distributed bottom roughness with the Gaussian spatial correlation function in the presence of multipath as a function of time, bandwidth, and correlation length scale. [Work supported by ONR]

The range of influence between cattle herds is of importance for the local spread of Salmonella Dublin in Denmark

The objective of the study was to estimate the range of influence between cattle herds with positive Salmonella Dublin herd status. Herd status was a binary outcome of high/low antibody levels to Salmonella Dublin in bulk-tank milk and blood samples collected from all cattle herds in Denmark for surveillance purposes. Two methods were used. Initially, a spatial generalised linear mixed model was developed with an exponential correlation function to estimate the range of influence simultaneously with the effect of potential risk factors. An iteratively reweighted generalised least squares procedure was used as a second method for verifying the range of influence estimates. With this iterative procedure, deviance residuals were calculated based on a generalised linear model and the range of influence was estimated based on the residuals using an exponential semivariogram. The range of influence was estimated for six different regions in Denmark using both methods. The analyses were performed on data collected during 1 year after initiation of the Salmonella Dublin surveillance program providing herd classifications for the 4th year-quarter of 2003 and 2 years later for the 4th year-quarter of 2005. The prevalence of dairy herds with a positive Salmonella Dublin herd classification status in this period had decreased from 22.1 to 17.0%. In non-dairy herds, the prevalence was nearly unchanged during the same period (3.4 and 3.7% in 4th quarter of 2003 and 2005, respectively). For all cattle herds, the range of influence was 2.3–6.4 km in 2003 and 1.5–8.3 km in 2005. There seemed to be no association between the range of influence and the density of herds in the different regions.

Band-Limited Exponential Correlation Function for Rough-Surface Scattering

A band-limited exponential correlation function is presented for use in studies of rough-surface scattering. This model of surface statistics is developed by multiplying the power spectral density of a standard exponential correlation function surface with a Gaussian rolloff at high frequencies in order to limit the high-frequency spectral content of the surface. A parameter is introduced to describe the Gaussian rolloff, and the corresponding correlation function is determined. The primary proposed use of the model is in assessing the contributions of surface high-frequency content in the scattering process, particularly in approximate theories of surface scattering. Methods for incorporating this surface description into standard approximate scattering theories are described, and sample backscattering results provided to illustrate use of the model

Integrated Rate Equation Considering Product Inhibition and Its Application to Kinetic Assay of Serum Ethanol

Kinetic assay of serum ethanol was investigated by predicting maximal product absorbance at 340 nm (A(mk)) through fitting to the yeast alcohol dehydrogenase reaction curve with the integrated rate equation, taking into account product inhibition in the presence of semicarbazide. Predicted A(mk) linearly responded to the preset constant of steady-state concentration of acetaldehyde (C(ald)). An exponential correlation function was established between desired C(ald) and putative A(mk) for authentic ethanol. For unknown samples, iterative fitting to reaction curve till preset constant C(ald) and resultant A(mk) satisfied this exponential correlation function yielded A(mk) with variation coefficient <4.3%. Variations in enzyme activity, data range and kinetic parameters showed negligible effects. The recovery was consistent to 100% with resistance to methanol and isopropanol. The upper limit of linear response for A(mk) was about 40 times of the lower limit. These results indicated that this kinetic method was reliable for serum ethanol assays with obvious advantages.