Stationary Gaussian Random Process Research Articles

Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems

We consider physical situations where both the real and imaginary parts of the optical field obey Gaussian statistics. Within the limitations of the generalized ray-matrix method, an expression for the variance of intensity is obtained for arbitrary cylindrically symmetric optical systems in the presence of partially developed speckle, where the rms surface roughness is comparable to the optical wavelength. We assume beam illumination of reflective targets with arbitrary values of surface roughness whose surface-height fluctuations are taken as a zero-mean stationary Gaussian random process. In contrast to previous work, the present analysis is valid for an arbitrary complex optical system that can be characterized by an ABCD matrix (e.g., simple and complex imaging systems, free-space propagation in both the near- and far-field, and Fourier transform systems), including those that exhibit correlation between the real and imaginary parts of the optical field. As a direct application, we consider an optical system for probing angular deflections that is sensitive to the speckle dynamics, and derive the relationship between the surface roughness and the corresponding lateral scale length that yields the maximum AC signal strength.

Scour below pipelines and around vertical piles in random waves

An approach by which the scour depth and scour width below a fixed pipeline and scour depth around a circular vertical pile in random waves can be derived is presented. Here, the scour depth formulas by Sumer and Fredsøe [ASCE J. Waterw., Port, Coastal Ocean Eng. 116 (1990) 307] for pipelines and Sumer et al. [ASCE J. Waterw., Port, Coastal Ocean Eng. 114 (1992) 599] for vertical piles as well as the scour width formula by Sumer and Fredsøe [The Mechanics of Scour in the Marine Environment, World Scientific, Singapore, 2002] for pipelines combined with describing the waves as a stationary Gaussian narrow-band random process are used to derive the cumulative distribution functions of the scour depths and width. Comparisons are made between the present approach and random wave scour data. Tentative approaches to related random wave scour cases are also suggested.

An experimental analysis of fatigue crack growth under random loading

An overall 235 tests involving load histories representing seven stationary Gaussian random processes and featuring different power spectral density functions [psd, S( ω)] of different shapes for the loads and/or different root mean square were conducted with a view to determining the influence of various loading parameters (viz. history length, stress level and bandwidth) on variability in crack growth life. Tests were carried out on 2024-T351 aluminium compact tension (CT) specimens. The results obtained suggest a strong influence of history length (viz. the number of cycles in each history) on variability in crack growth life. The number of cycles a crack takes to grow from an initial value to a final one at a given load level is strongly dependent on the bandwidth of the process that causes the crack to grow. Moreover, several statistical distributions fit the test data reasonably well.

Efficient dynamic analysis of cable-stayed bridges under vehicular movement using space and time adaptivity

The effects of random road surface roughness on the impact effects on cable-stayed bridge due to moving vehicles are investigated. The random road surface roughness is described by a zero-mean stationary Gaussian random process. The bridge is modeled by finite element method as a planar structure. Each moving vehicle is idealized as a single degree-of-freedom lumped mass system, in which a mass is supported by a spring and a dashpot. The impact effects of random road surface roughness on a cable-stayed bridge vary a lot, depending on the location and the structural components. The entire computation has been carried out in an adaptive finite element environment. It has been clearly demonstrated that the dynamic amplification factor, a significant parameter for bridge design can become highly erroneous if the errors due to both space and time discretization are not kept within control.

ANALYSES OF DYNAMIC RESPONSE OF VEHICLE AND TRACK COUPLING SYSTEM WITH RANDOM IRREGULARITY OF TRACK VERTICAL PROFILE

A dynamic computational model for the vehicle and track coupling system is developed by means of finite element method in this paper. In numerical implementation, the vehicle and track coupling system is divided into two parts; lower structure and upper structure. The vehicle as the upper structure in the coupling system is a whole locomotive or rolling stock with two layers of spring and damping system in which vertical and rolling motion for vehicle and bogie are involved. The lower structure in the coupling system is a railway track where rails are considered as beams with finite length rested on a double layer continuous elastic foundation. The two parts are solved independently with an iterative scheme. Coupling the vehicle system and railway track is realized through interaction forces between the wheels and the rail, where the irregularity of the track vertical profile considered as stationary ergodic Gaussian random processes and simulated by trigonometry series is included. The amplitudes of vibrations, their velocities and the accelerations generated in the vehicle and rail and the interaction forces between the vehicle and the rail due to the random irregularity of the track vertical profile and different line grades and train speeds have been analyzed numerically by this model. Analyses of system responses are performed in time and frequency domains.

Calculation of free field response spectrum of a non-homogeneous soil deposit from bed rock response spectrum

In this paper, a method to determine free field response spectrum of a soil deposit from a specified bed rock response spectrum is presented. This method treats the earthquake motion as if it was a stationary Gaussian random process but to account for the non-stationary character, an approximate method is used. The soil deposit is assumed to have mechanical properties (strength, shear modulus, etc.) increasing with some power exponent of depth. This layer overlies either a compliant elastic half space or a layer with shear modulus increasing linearly with depth. To demonstrate the validity and usefulness of this approach, two examples are presented. The first one consists in the transfer, from bed rock to free surface of a given soil profile, of the response spectrum derived from an accelerogram used to generate the Eurocode 8 response spectrum. The obtained free field response spectrum is compared to the one obtained by using a wave propagation program. The second example consists in the validation of this method with experimental records.

Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

AbstractA random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen‐decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non‐stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second‐order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen‐solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non‐stationary processes with little additional effort. Copyright © 2001 John Wiley & Sons, Ltd.

Bottom friction in random waves plus current flow

Bottom friction in random waves plus current flow is presented. The model is an extension of the Myrhaug [Coastal Eng. 24 (1995) 259] approach for random waves alone. The effect of random waves on the bottom friction is studied by assuming the wave motion to be a stationary Gaussian narrow-band random process, and by using friction coefficient formulas for sinusoidal waves. The data used for comparison are obtained from statistical analysis of direct measurements of bottom shear stresses made in the UK Coastal Research Facility under combined random waves and orthogonal as well as near-orthogonal currents.

Bivariate simulation of non stationary and non Gaussian observed processes: Application to sea state parameters

A method for artificially generating operational sea state histories has been developed. This is a distribution free method to simulate bivariate non stationary and non Gaussian random processes. This method is applied to the simulation of the bivariate process ( H s, T p) of sea state parameters. The time series respects the physical constraints existing between the significant wave height and the peak period. Furthermore, we show that the persistence properties of the simulations match to those of the observations.

Human- and model-observer performance in ramp-spectrum noise: effects of regularization and object variability.

We consider detection of a nodule signal profile in noisy images meant to roughly simulate the statistical properties of tomographic image reconstructions in nuclear medicine. The images have two sources of variability arising from quantum noise from the imaging process and anatomical variability in the ensemble of objects being imaged. Both of these sources of variability are simulated by a stationary Gaussian random process. Sample images from this process are generated by filtering white-noise images. Human-observer performance in several signal-known-exactly detection tasks is evaluated through psychophysical studies by using the two-alternative forced-choice method. The tasks considered investigate parameters of the images that influence both the signal profile and pixel-to-pixel correlations in the images. The effect of low-pass filtering is investigated as an approximation to regularization implemented by image-reconstruction algorithms. The relative magnitudes of the quantum and the anatomical variability are investigated as an approximation to the effects of exposure time. Finally, we study the effect of the anatomical correlations in the form of an anatomical slope as an approximation to the effects of different tissue types. Human-observer performance is compared with the performance of a number of model observers computed directly from the ensemble statistics of the images used in the experiments for the purpose of finding predictive models. The model observers investigated include a number of nonprewhitening observers, the Hotelling observer (which is equivalent to the ideal observer for these studies), and six implementations of channelized-Hotelling observers. The human observers demonstrate large effects across the experimental parameters investigated. In the regularization study, performance exhibits a mild peak at intermediate levels of regularization before degrading at higher levels. The exposure-time study shows that human observers are able to detect ever more subtle lesions at increased exposure times. The anatomical slope study shows that human-observer performance degrades as anatomical variability extends into higher spatial frequencies. Of the observers tested, the channelized-Hotelling observers best capture the features of the human data.

Effects of random road surface roughness and long-term deflection of prestressed concrete girder and cable-stayed bridges on impact due to moving vehicles

The effects of random road surface roughness and long-term deflection of prestressed concrete bridges on the impact effects due to moving vehicles are investigated. The concrete bridges studied include multi-span girder bridges and cable-stayed bridges. The random road surface roughness is described by a zero-mean stationary Gaussian random process, while the long-term deflection of the concrete deck is represented as a kind of global roadway surface roughness in the study. The bridge is modelled by finite element method. Each moving vehicle is idealised as a one-foot dynamic system, in which a mass is supported by a spring and a dashpot. Numerical results show that the effect of random road surface roughness on the impact induced by moving vehicles is significant in the girder bridges, while that of the long-term deflection of concrete deck is small to moderate. The impact effects of the random road surface roughness and the long-term deflection of concrete deck on a cable-stayed bridge vary a lot depending on the location. In general, such effects on the bridge deck are more significant at sections closed to the bridge towers. Such effects on the bending moment at the tower base are also significant. The effects on the stay cables vary much, with significant effects on the short cables and negligible effects on the longest cables.

An improved method of calculating the peak stress distribution for a broad-band random process

An improved method of fast fatigue life prediction under broad-band random loading is proposed, which is based on the power spectral density of stress in the critical points of structures and the peak stress distribution of a stationary Gaussian random process. The improved method has higher precision than other existing approximate methods that are based on the peak stress distribution.

Transformation of the radiation fluctuations of a low-coherence source in a two-arm interferometer

We consider fluctuations of the radiation of a low-coherence source transmitted through a two-arm interferometer and transformed by a square-law detector. It is assumed that the output radiation of the source is a stationary Gaussian narrow-band random process. The fluctuation process at the detector output is presented as the sum of the intensity noise and the interference-component noise. We calculate the relationships between the spectral densities of these components and the degree of their correlation for different delays between the interferometer arms. We show theoretically and confirm experimentally that the power of interference-component fluctuations is nonzero even if the delay between the interferometer arms exceeds the correlation time of the process at the source output.

Is the squeezing of relic gravitational waves produced by inflation detectable?

Grishchuk has shown that the stochastic background of gravitational waves produced by an inflationary phase in the early Universe has an unusual property: it is not a stationary Gaussian random process. Due to squeezing, the phases of the different waves are correlated in a deterministic way, arising from the process of parametric amplification that created them. The resulting random process is Gaussian but non-stationary. This provides a unique signature that could in principle distinguish a background created by inflation from stationary stochastic backgrounds created by other types of processes. We address the question: could this signature be observed with a gravitational wave detector? Sadly, the answer appears to be "no": an experiment which could distinguish the non-stationary behavior would have to last approximately the age of the Universe at the time of measurement. This rules out direct detection by ground and space based gravitational wave detectors, but not indirect detections via the electromagnetic Cosmic Microwave Background Radiation (CMBR).

Stochastic analysis of the linear equivalent response of bridge piers with aseismic devices

The dynamic response of bridge piers with aseismic devices to earthquake excitation is evaluated by the stochastic equivalent linearization technique. The seismic acceleration is schematized through a Gaussian stationary random process. The pier is considered linear elastic, the span is idealized as a rigid mass, the restoring force of the device is represented through a non-linear differential model. The study of the complex modes of the linearized system gives an interpretation of the mechanical behaviour, leads to a formally elementary solution and highlights some phenomena which are typical of the hysteretic systems, particularly of those marked by weak hardening. Even though the solution is limited to the stationary field, it brings out several noteworthy considerations about the effective non stationary behaviour of the structure.

Validating a hypothesis of the form of a correlation function of a stationary random process

The problem of validating a statistical hypothesis stating that the correlation function of a stationary Gaussian random process possesses a specified form is considered for the case in which measurements of the process are performed at arbitrary moments of time.

Efficient Algorithm for Fatigue Life Calculations under Broad Band Loading Based on Peak Approximation

In this paper, a new method of fast fatigue life prediction under broad band (BB) random loading is proposed, which is based on statistical theory of the peak distribution of a stationary Gaussian random process and the concept of equivalent stress range. The equivalent stress range in the model can be calculated directly by frequency domain parameters, obviating the need for cycle-by-cycle counting and damage summation. The new method offers a great advantage when compared with any other existing approximate methods that are based on the peak stress distribution.

Continuous-time Wiener system identification

A continuous-time Wiener system is identified. The system consists of a linear dynamic subsystem and a memoryless nonlinear one connected in a cascade. The input signal is a stationary white Gaussian random process. The system is disturbed by stationary white random Gaussian noise. Both subsystems are identified from input-output observations taken at the input and output of the whole system. The a priori information is very small and, therefore, resulting identification problems are nonparametric. The impulse impulse of the linear part is recovered by a correlation method, while the nonlinear characteristic is estimated with the help of the nonparametric kernel regression method. The authors prove convergence of the proposed identification algorithms and examine their convergence rates.

Spline approximation of power spectra with measurements at random times

We propose and investigate algorithms for approximating the power spectra of stationary Gaussian random processes with first-order splines where the measurement points form a Poisson or recurrent event flow with known characteristics.

An Analytical Solution of Equivalent Stress for Structure Fatigue Life Prediction under Broad Band Random Loading*

ABSTRACT An analytical solution of an equivalent stress-range calculation, based on the power-spectral density of stress in critical points of structures, and a statistical theory for the peak distribution of a stationary Gaussian random process are presented in this paper for fatigue life assessment under broad-band random loading. This model has more advantages than similar existing models.