Homogeneous Random Process Research Articles

The emergence of stochastic user equilibria in day-to-day traffic models

Stochastic user equilibrium (SUE) is a commonly used deterministic model of traffic assignment. It is well known that SUE provides an approximation to the stationary mean of a wide class of Markov day-to-day traffic assignment models. However, as is shown in this article, it is generally impossible to find any time homogeneous random process models for discrete traffic in which SUE precisely matches the stationary mean. In response, we develop a time-inhomogeneous model with this property. That model relies on a rather unnatural description of route choice utility. We therefore propose a second time-inhomogeneous model with a more standard type of utility function, where the stationary mean equals a generalized version of SUE.While the time-inhomogeneous models are largely of theoretical interest, they also provide highly serviceable approximations to standard types of day-to-day traffic models in which travellers update their route choice utilities relatively slowly in response to recent events. For such models, the generalized form of SUE provides a closer approximation to the stationary mean than classical SUE, particularly when traffic demands are small.

Realizability of the rapid distortion theory spectrum: The mechanism behind the Kelvin–Townsend equations

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramér’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.

Near-inertial-wave scattering by random flows

The impact of a turbulent flow on wind-driven oceanic near-inertial waves is examined using a linearised shallow-water model of the mixed layer. Modelling the flow as a homogeneous and stationary random process with spatial scales comparable to the wavelengths, we derive a transport (or kinetic) equation governing wave-energy transfers in both physical and spectral spaces. This equation describes the scattering of the waves by the flow which results in a redistribution of energy between waves with the same frequency (or, equivalently, with the same wavenumber) and, for isotropic flows, in the isotropisation of the wave field. The time scales for the scattering and isotropisation are obtained explicitly and found to be of the order of tens of days for typical oceanic parameters. The predictions inferred from the transport equation are confirmed by a series of numerical simulations. Two situations in which near-inertial waves are strongly influenced by flow scattering are investigated through dedicated nonlinear shallow-water simulations. In the first, a wavepacket propagating equatorwards as a result from the $\beta$-effect is shown to be slowed down and dispersed both zonally and meridionally by scattering. In the second, waves generated by moving cyclones are shown to be strongly disturbed by scattering, leading again to an increased dispersion.

Identification of flexible vehicle parameters on bridge using particle filter method

A conditional probability based approach known as Particle Filter Method (PFM) is a powerful tool for system parameter identification. In this paper, PFM has been applied to identify the vehicle parameters based on response statistics of the bridge. The flexibility of vehicle model has been considered in the formulation of bridge-vehicle interaction dynamics. The random unevenness of bridge has been idealized as non homogeneous random process in space. The simulated response has been contaminated with artificial noise to reflect the field condition. The performance of the identification system has been examined for various measurement location, vehicle velocity, bridge surface roughness factor, noise level and assumption of prior probability density. Identified vehicle parameters are found reasonably accurate and reconstructed interactive force time history with identified parameters closely matches with the simulated results. The study also reveals that crude assumption of prior probability density function does not end up with an incorrect estimate of parameters except requiring longer time for the iterative process to converge.

An efficient space–time based simulation approach of wind velocity field with embedded conditional interpolation for unevenly spaced locations

The classical spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian random processes. Due to the application of fast Fourier transform (FFT), the simulation is usually efficient. However, for processes with a large number of simulation points, it becomes necessary to enhance the simulation efficiency. One example is the wind velocity field along a large-span bridge, where hundreds of wind velocity fluctuations are required. In the case of bridges built over a homogeneous terrain such as coastal area or flat plain, the wind velocity field can be modeled as a multivariate homogeneous random process i.e., the auto power spectral densities (PSDs) at evenly-spaced simulation points are same and the cross PSD is a function of separation distance between two simulation points. Furthermore, in some applications, additional simulation points need to be included to a set of uniformly distributed points in order to make the wind velocity field consistent to the structural dynamic analysis requirement.In this paper, a hybrid approach of space–time random-field based SRM and proper orthogonal decomposition (POD)-based interpolation is developed for simulating the above wind velocity process. In this approach, the random-field based SRM is used to simulate the multivariate homogeneous random process composed of a set of uniformly distributed simulation locations while POD-based interpolation is used to conditionally generate the wind velocities at a few unevenly distributed points using the previously simulated wind velocities. The idea of the former is based on transforming the simulation of the homogeneous random process into that of the corresponding space–time random field where the phase angle is assumed to be zero and the coherence function must be an even function in terms of separation distance. Through this procedure, customary requirement for spectral matrix decomposition is eliminated and application of two dimensional FFT can improve the simulation efficiency dramatically. The shortcomings of this method include a slight approximation regarding the simulated sample and the non-ergodicity for the correlation function. The numerical example of a homogeneous wind velocity field along a bridge deck shows that the proposed random field-based method is very efficient in terms of accuracy and efficiency when the number of simulation locations is large and the POD-based interpolation also has good performance.

Effect of vehicle flexibility on the vibratory response of bridge

In the recent times, dimensions of heavy load carrying vehicle have changed significantly incorporating structural flexibility in vehicle body. The present paper outlines a procedure for the estimation of bridge response statistics considering structural bending modes of the vehicle. Bridge deck roughness has been considered to be non homogeneous random process in space. Influence of pre cambering of bridge surface and settlement of approach slab on the dynamic behavior of the bridge has been studied. A parametric study considering vehicle axle spacing, mass, speed, vehicle flexibility, deck unevenness and eccentricity of vehicle path have been conducted. Dynamic amplification factor (DAF) of the bridge response has been obtained for several of combination of bridge-vehicle parameters. The present study reveals that flexible modes of vehicle can reduce dynamic response of the bridge to the extent of 30-37% of that caused by rigid vehicle model. However, sudden change in the bridge surface profile leads to significant amount of increment in the bridge dynamic response even if flexible bending modes remain active. The eccentricity of vehicle path and flexural/torsional rigidity ratios plays a significant role in dynamic amplification of bridge response.

The Navier Wall Law at a Boundary with Random Roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size $${\varepsilon \ll 1}$$ . In the parent paper [8], we derived a homogenized boundary condition of Navier type as $${\varepsilon \rightarrow 0}$$ . We show here that for a large class of boundaries, this Navier condition provides a $${O(\varepsilon^{3/2} |\ln \varepsilon|^{1/2})}$$ approximation in L 2, instead of $${O(\varepsilon^{3/2})}$$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

Apparent scale correlations in a random multifractal process

We discuss various properties of a homogeneous random multifractal process, which are related to the issue of scale correlations. By design, the process has no built-in scale correlations. However, when it comes to observables like breakdown coefficients, which are based on a coarse-graining of the multifractal field, scale correlations do appear. In the log-normal limit of the model process, the conditional distributions and moments of breakdown coefficients reproduce the observations made in fully developed small-scale turbulence. These findings help to understand several puzzling empirical details, which have been extracted from turbulent data already some time ago.

Fatigue life prediction of horizontally curved thin walled box girder steel bridges

The fatigue damage accumulation rates of horizontally curved thin walled box-girder bridge have been estimated from vehicle-induced dynamic stress history using rain flow cycle counting method in the time domain approach. The curved box-girder bridge has been numerically modeled using computationally efficient thin walled box-beam finite elements, which take into account the important structural actions like torsional warping, distortion and distortional warping in addition to the conventional displacement and rotational degrees of freedom. Vehicle model includes heave-pitch-roll degrees of freedom with longitudinal and transverse input to the wheels. The bridge deck unevenness, which is taken as inputs to the vehicle wheels, has been assumed to be a realization of homogeneous random process specified by a power spectral density (PSD) function. The linear damage accumulation theory has been applied to calculate fatigue life. The fatigue life estimated by cycle counting method in time domain has been compared with those found by estimating the PSD of response in frequency domain. The frequency domain method uses an analytical expression involving spectral moment characteristics of stress process. The effects of some of the important parameters on fatigue life of the curved box bridge have been studied.

Systematic corrections to the measured cosmological constant as a result of local inhomogeneity

We explicitly calculate the lowest order systematic inhomogeneity-induced corrections to the cosmological constant that one would infer from an analysis of the luminosities and redshifts of type Ia supernovae, assuming a homogeneous universe. The calculation entails a post-Newtonian expansion within the framework of second-order perturbation theory, wherein we consider the effects of subhorizon density perturbations in a flat, dust-dominated universe. Within this formalism, we calculate luminosity distances and redshifts along the past light cone of an observer. The luminosity distance-redshift relation is then averaged over viewing angles and ensemble averaged, assuming that density fluctuations at a given cosmic time are a homogeneous random process. The resulting relation is fit to that of a homogeneous model containing dust and a cosmological constant, in order to deduce the best-fit cosmological constant density ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$. We find that the luminosity distance-redshift relation is indeed modified, even for large sample sizes, but only by a very small fraction, of order ${10}^{\ensuremath{-}5}$ for $z\ensuremath{\sim}0.1$. This lowest order deviation depends on the peculiar velocities of the source and the observer. However, when fitting this perturbed relation to that of a homogeneous universe, via maximizing a likelihood function, we find that the inferred cosmological constant can be surprisingly large, depending on the range of redshifts sampled. For a sample of supernovae extending from ${z}_{\mathrm{min}}=0.02$ out to a limiting redshift ${z}_{\mathrm{max}}=0.15$, we find that ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.004$. The value of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ has a large variance, and its magnitude tends to get progressively larger as the limiting redshift ${z}_{\mathrm{max}}$ gets smaller, implying that precision measurements of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ from nearby supernova data will require taking this effect into account. This effect has been referred to in the past as the ``fitting problem,'' and more recently as subhorizon ``backreaction.'' We find that it is likely too small to explain the observed value ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.7$. There have been previous claims of much larger backreaction effects. In contrast to those calculations, our work is directly related to how observers deduce cosmological parameters from astronomical data.

Dynamic Analysis of Horizontally Curved Thin-Walled Box-Girder Bridge due to Moving Vehicle

The impact on curved box-girder bridges due to vehicle moving across rough bridge deck have been analyzed using bridge-vehicle coupled dynamics. The bridge deck unevenness has been assumed to be a homogeneous random process in space specified by a PSD function. The analysis incorporates the effect of centrifugal forces due to vehicle moving on curved bridge. The curved box-girder bridge has been numerically modeled using computationally efficient thin-walled box-beam finite elements which take into account the torsional warping, distortion and distortional warping, that are important features of thin-walled box girders. Rigid vehicle with longitudinal and transverse input to the wheels giving rise to heave-pitch-roll degrees of freedom has been considered. The theoretical bridge model used in simulation study has been validated by a free vibration experiment using impact excitation. The impact factors for several response parameters such as bending moment, shear force, torsional moment, torsional bi-moment, distortional moment, distortional bi-moment and vertical deflections have been obtained for various bridge-vehicle parameters. Both constant velocity and forward acceleration of the vehicle have been considered to examine impact factor. The results highlighted that the impact factors of a curved box girder bridge corresponding to torsion, distortion and their corresponding bimoments have been observed to be generally very high, while those of the other responses are also relatively higher than that of corresponding straight box girder bridge.

Computation of the exact cramer-rao lower bound for 2-D ARMA parameter Estimation-I: the quarter-plane case

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper

Diffusion of high-frequency energy in fluid-saturated porous media

The modern mathematical theory of microlocal analysis shows that the energy associated with the high-frequency solutions of hyperbolic partial differential equations (such as the wave or the Navier equations) satisfy Liouville-type transport equations, or radiative transfer equations for randomly heterogeneous media. For long propagation times the latter can be approached by diffusion equations. Some classical results of the structural acoustics literature about the heat conduction analogy and the statistical energy analysis of structural dynamics at higher frequencies are recovered in this process. The purpose of this communication is to focus on such a diffusive regime for isotropic, fluid-saturated porous media. More specifically, we have derived the diffusion parameters (transport mean-free path and diffusion constant) for such media. Our model considers Biot’s equations of poroelasticity, where thermal and viscous effects are modelized by dynamic tortuosity and compressibility with singular memory kernels. The macroscopic bulk modulus and density of the dry solid phase are assumed to be homogeneous random processes, while tortuosity and porosity remain constant.

A critical molar mass separating the ductile and brittle regimes as revealed by thermal oxidation in polypropylene

The thermal oxidation of unstabilized polypropylene films has been studied at 70, 90, 110 and 130 °C in air by IR, SEC, rheometry and tensile testing. Embrittlement is observed when the weight average molar mass reaches 210±10 kg mol −1. Polydispersity data are consistent with a homogeneous random chain scission process. This especially high sensitivity of fracture properties to chain scission is assumed to be specific of semi-crystalline polymers having their amorphous phase in rubbery state.

A new look at the effect of bandwidth and non‐normality on fatigue damage

ABSTRACTA new attempt is made in this paper to quantify the effect of bandwidth and non‐normality in fatigue damage analysis. For the lack of actual stress history, a series of non‐Gaussian and homogeneous random processes are generated with fast Fourier transform (FFT) acceleration. A factor is defined on the basis of rain‐flow counting and Palmgren–Miner rule to correct the narrow band and normality assumption. It is revealed that the fatigue damage evaluated through the traditional method may be either conservative or rather unconservative. The upper and lower bounds of the correction factor are studied with respect to kurtosis and skewness of the generated random process and the slope of S–N curve.

Nonstationary Response of Bridge Due to Eccentrically Moving Vehicles at Variable Velocity

The non-stationary response of a simply supported girder bridge induced by the vehicle travelling at variable speed has been studied. The deck unevenness has been considered as the realization of homogeneous random process described by power spectral density (PSD) function. The vehicle has been idealized as rigid beam connected by front and rear suspension elements moving at an eccentricity with the centre line. The bridge has been idealized as a continuous system with distributed mass, stiffness and damping properties having flexural and torsional degrees of freedom. Monte Carlo simulation technique has been used to simulate deck profile for generation of input samples in numerical integration of the system equations. A detailed parametric study has been conducted to examine the effect of variable speed, suspension characteristics, vehicle mass, eccentricity of vehicle path, torsional rigidity of the bridge and deck unevenness on the dynamic amplification factors of the response.

Characterisation of alumina-supported vanadium oxide catalysts by kinetic analysis of H 2-TPR data

Alumina-supported vanadium oxide catalysts with vanadium contents of 2, 5 and 11 wt.% were prepared by incipient wetness impregnation and characterised by XRD, Raman spectroscopy and H 2-TPR. The catalysts with the low vanadium contents contained vanadium mainly in the well-dispersed phase, but the catalyst with the highest vanadium content contained also some crystalline AlVO 4 according to XRD and Raman spectroscopic results. The reduction kinetics of the vanadium catalysts was modelled based on the hydrogen consumption during the TPR. The reduction kinetics could be described with a single-reducible-site random nucleation model for the catalyst containing the lowest amount of vanadium. The reduction kinetic models for the other catalysts required a combination of multiple processes to describe the experiments properly. In the catalyst with 5 wt.% V, a part of vanadium species possibly reduces as a homogeneous random nucleation process, but topochemical reduction by nuclei growth also takes place. In the catalyst with 11 wt.% V, reduction by nuclei growth seems to be the predominant reduction mechanism. The characterisation of the reduced catalysts by XRD and during reduction by Raman spectroscopy enabled the identification of the features of the TPR profiles.

Stochastic analysis of groundwater flow in aquifers with leakage

Stochastic analysis of one- and two-dimensional flow through a shallow semi-confined aquifer with spatially variable hydraulic conductivity K represented by a stationary (statistically homogeneous) random process is carried out by using the spectral technique. The hydraulic head covariance functions for flows in a semi-confined aquifer bounded by a leaky layer above and an impervious stratum below are derived by assuming that the randomness forcing the head variation to originate from the hydraulic conductivity field of the aquifer. The head covariance functions are studied using two convenient forms of the logarithmic hydraulic conductivity process. The results demonstrate the significant reduction in the head variances and covariances due to the presence of a leaky layer. The hydraulic head correlation distance is also reduced greatly due to the presence of the leaky layer.

Stochastic Analysis of Macrodispersion in a Semi‐Confined Aquifer

In this paper, the macroscopic dispersion resulting from one‐ and two‐dimensional flows through a semi‐confined aquifer with spatially variable hydraulic conductivity K which is represented by a stationary (statistically homogeneous) random process is analyzed using the spectral representation technique. Stochastic fluctuation equations of the steady flow and solute transport are solved to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of the leakage factor and input covariances describing the hydraulic conductivity in a semi‐confined aquifer bounded by a leaky layer above and an impervious stratum below. The macrodispersivity tensor is studied using some convenient forms of the log hydraulic conductivity process. The sensitivity of the resulting macrodispersivity to the input covariances is discussed along with the influence of the leakage factor for both one‐ and two‐dimensional flows. It is found that the longitudinal macrodispersivities are increased due to the presence of leakage, while the transverse macrodispersivities are reduced due to leakage.

Over‐the‐horizon radar target localization using a hidden Markov model estimated from ionosonde data

Uncertainty about the downrange ionospheric conditions is a well‐known source of localization errors in over‐the‐horizon radar. Statistical modeling of ionospheric parameters has recently been proposed in order to derive a maximum likelihood (ML) localization method which is more robust to ionospheric variability. Maximum likelihood coordinate registration consists of determining the most likely target ground coordinates over an ensemble of ionospheric conditions consistent with the data. For greater computational efficiency the likelihood function is approximated by a hidden Markov model (HMM) for the probability of a sequence of observed slant coordinates given a hypothesized target location. In previous work, estimation of the HMM parameters was achieved assuming that the statistics of the underlying ionosphere were known precisely. This paper addresses the problem of estimating the parameters of the HMM from contemporaneous ionospheric sounder measurements. The approach taken here is to treat the plasma frequency profile as a homogeneous random process over the region around the midpoint between the radar and the dwell illumination region. In particular, spatial sampling of a three‐dimensional (3‐D) ionospheric model, fitted to ionosonde measurements, is used to generate quasi 2‐D plasma frequency profile realizations. Estimates of the hidden Markov model parameters are then obtained by using smoothed bootstrap Monte Carlo resampling. A comparison of ML localization and conventional methods, using full 3‐D ionospheric modeling and 2‐D ray tracing, are given using real data from a known target at a ground range of 2192 km. Results for over 250 radar dwells indicate that the ML localization technique achieves better than a factor of 2 improvement over conventional methods.