An efficient multivariate random field generator using the fast Fourier transform

It is often convenient to use synthetically generated random fields to study the hydrologic effects of spatial heterogeneity. Although there are many ways to produce such fields, spectral techniques are particularly attractive because they are fast and conceptually straightforward. This paper describes a spectral algorithm for generating sets of random fields which are correlated with one another. The algorithm is based on a discrete version of the Fourier-Stieltjes representation for multidimensional random fields. The Fourier increment used in this representation depends on a random phase angle process and a complex-valued spectral factor matrix which can be readily derived from a specified set of cross-spectral densities (or cross-covariances). The inverse Fourier transform of the Fourier increment is a complex random field with real and imaginary parts which each have the desired coveriance structure. Our complex-valued spectral formulation provides an especially convenient way to generate a set of random fields which all depend on a single underlying (independent) field, provided that the fields in question can be related by space-invariant linear transformations. We illustrate this by generating multi-dimensional mass conservative groundwater velocity fields which can be used to simulate solute transport through heterogeneous anisotropic porous media.     

Conditional components for simulation of vector random fields

Conditional component random fields (CC) based on Cholesky decomposition of the multivariate spectra are introduced in this study to develop a new method for conditional simulation of vector attributes in environmental and geological phenomena. The CC are independent random fields with covariance models obtained from projections and conditioning in the frequency domain. The approach is to simulate one attribute in the physical space and use the results to estimate the other attributes in the frequency domain. Then, a CC for the next attribute is simulated and projected on the other attributes. In general, any attribute is built as the sum of inverse Fourier transform of the orthogonal projection of previous simulated CC plus a last CC simulated in the physical space. This simulation approach continues in this fashion for several attributes and the order of them may be changed for different realizations. This method allows for data conditioning and simulation. A simplified version for intrinsically correlated random fields allows for an approach that avoids the frequency domain.     

Conditional simulation of spatially variable seismic ground motions based on evolutionary spectra

This paper proposes a computational procedure for the conditional simulation of spatially variable seismic ground motions for long span bridges with multiple supports. The seismic ground motions, with part of their time histories measured at some supports, are regarded as zero‐mean nonstationary random processes characterized by predefined evolutionary power spectral density. To conditionally simulate unknown seismic ground motion time histories at other supports, the Kriging method is first described briefly for the conditional simulation of a random vector comprised of zero‐mean Gaussian variables. The multivariate oscillatory processes characterized by the evolutionary power spectral density matrix are then introduced, and the Fourier coefficients of the oscillatory processes and their covariance matrix are derived. By applying the Kriging method to the random vector of the Fourier coefficients and using the inverse Fourier transform, unknown nonstationary seismic ground motion time histories can be simulated. A numerical example is selected to demonstrate capabilities of the proposed simulation procedure, and the results show that the procedure can ensure unbiased time‐varying correlation functions, especially the cross correlation between known and unknown time histories. The procedure is finally applied to the Tsing Ma suspension bridge in Hong Kong to generate ground accelerations at its multiple supports using limited seismic records. Copyright © 2012 John Wiley & Sons, Ltd.     

Stochastic simulation of bivariate gamma distribution: a frequency-factor based approach

A frequency-factor based approach for stochastic simulation of bivariate gamma distribution is proposed. The approach involves generation of bivariate normal samples with a correlation coefficient consistent with the correlation coefficient of the corresponding bivariate gamma samples. Then the bivariate normal samples are transformed to bivariate gamma samples using the well-known general equation of hydrological frequency analysis. We demonstrate that the proposed bivariate gamma simulation approach is capable of generating random sample pairs which not only have the desired marginal densities of component random variables but also their correlation coefficient. Scatter plots of simulated bivariate sample pairs also exhibit appropriate linear patterns (dependence structure) that are commonly observed in environmental and hydrological applications. Caution should also be exercised when specifying combinations of coefficients of skewness and the correlation coefficient for bivariate gamma simulation.     

CRITICAL CROSS POWER SPECTRAL DENSITY FUNCTIONS AND THE HIGHEST RESPONSE OF MULTI-SUPPORTED STRUCTURES SUBJECTED TO MULTI-COMPONENT EARTHQUAKE EXCITATIONS

The highest response of multi-supported structures subjected to partially specified multi-component earthquake support motions is considered. The seismic inputs are modelled as incompletely specified vector Gaussian random processes with known autospectral density functions but unknown cross spectral densities and these unknown functions are determined such that the steady state response variance of a given linear system is maximized. The resulting cross power spectral density functions are shown to be dependent on the system properties, autospectra of excitation and the response variable chosen for maximization. It emerges that the highest system response is associated neither with fully correlated support motions, nor with independent motions, but, instead, specific forms of cross power spectral density functions are shown to exist which produce bounds on the response of a given structure. Application of the proposed results is demonstrated by examples on a ground based extended structure, namely, a 1578 m long, three span, suspension cable bridge and a secondary system, namely, an idealized piping structure of a nuclear power plant.     

Broadband Kinematic Stochastic Simulation of an Earthquake Source: a Refined Procedure for Application in Seismic Hazard Studies

To carry out a realistic simulation of earthquake strong ground motion for applied studies, one needs an earthquake fault/source simulator that can integrate most relevant features of observed earthquake ruptures. A procedure of this kind is proposed that creates a broadband kinematic source model. At lower frequencies, the source is described as propagating slip pulse with locally variable velocity. The final slip is assumed to be a two-dimensional (2D) random function. At higher frequencies, radiation from the same running strip is assumed to be random and incoherent in space. The model is discretized in space as a grid of point subsources with certain time histories. At lower frequencies, a realistic shape of source spectrum is generated implicitly by simulated kinematics of slip pulse propagation. At higher frequencies, the original approach is used to generate signals with spectra that plausibly approximate the prescribed smooth far-field source spectrum. This spectrum is set on the basis of the assumedly known regional empirical spectral scaling law, and subsource moment rate time histories are conditioned so as to fit this expected spectrum. For the random function that describes final slip over the fault area, lognormal probability distribution of amplitudes is assumed, on the basis of exploratory analysis of inverted slip distributions. Similarly, random functions that describe local slip rate time histories are assumed to have lognormal distribution of envelope amplitudes. In this way one can effectively emulate expressed ??asperities?? of final slip and occasional occurrence of large spikes on near-source accelerograms. A special procedure is proposed to simulate the spatial coherence of high-frequency fault motion. This approach permits the simulation of fault motion plausibly at high spatial resolution, fulfilling the prerequisite for simulation of strong motion in the vicinity of a fault. A particular realization (sample) of a source created in a simulation run depends on several random seeds, and also on a considerable number of parameters. Their values can be selected so as to take into account expected source features; they can also be perturbed to examine the source-related component of uncertainty of strong motion. The proposed approach to earthquake source specification is well adapted to the study of deterministic seismic hazard: it may be used for simulation of individual scenario events, or suites of such events, as well as for analysis of uncertainty for expected ground motion parameters from a particular class of events. Examples are given of application of the proposed approach to strong motion simulations and related uncertainty estimation.     

Semigroup solutions to stochastic unsteady groundwater flow subject to random parameters

Two methods for the solution of partial differential equations (PDE) for the general case of random in time physical parameters are presented and their application to the solution of unsteady regional groundwater flow equations are illustrated. The first method is the semigroup approach which directly offers a solution without resorting to closure approximations (hierarchy techniques), perturbation techniques, or Montecarlo simulation techniques. The semigroup approach can also handle the general stochastic problem when randomness also appears as initial conditions, boundary conditions or forcing terms. The second method is an approximation scheme to obtain the semigroup solution in complex cases and permits the solution of equations with more than one random coefficient.     

Spatial prediction of categorical variables: the Bayesian maximum entropy approach

 Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.     

On a continuous spectral algorithm for simulating non-stationary Gaussian random fields

This paper presents an algorithm for simulating Gaussian random fields with zero mean and non-stationary covariance functions. The simulated field is obtained as a weighted sum of cosine waves with random frequencies and random phases, with weights that depend on the location-specific spectral density associated with the target non-stationary covariance. The applicability and accuracy of the algorithm are illustrated through synthetic examples, in which scalar and vector random fields with non-stationary Gaussian, exponential, Matérn or compactly-supported covariance models are simulated.     

Goodness-of-fit tests for the spatial spectral density

Detection and modeling the spatial correlation is an important issue in spatial data analysis. We extend in this work two different goodness-of-fit testing techniques for the spatial spectral density. The first approach is based on a smoothed version of the ratio between the periodogram and a parametric estimator of the spectral density. The second one is a generalized likelihood ratio test statistic, based on the log-periodogram representation as the response variable in a regression model. As a particular case, we provide tests for independence. Asymptotic normal distribution of both statistics is obtained, under the null hypothesis. For the application in practice, a resampling procedure for calibrating these tests is also given. The performance of the method is checked by a simulation study. Application to real data is also provided.     

Application of a Monte‐Carlo simulation approach for the probabilistic assessment of seismic hazard for geographically distributed portfolio

The conventional integral approach is very well established in probabilistic seismic hazard assessment (PSHA). However, Monte‐Carlo (MC) simulations can become an efficient and flexible alternative against conventional PSHA when more complicated factors (e.g. spatial correlation of ground shaking) are involved. This study aims at showing the implementation of MC simulation techniques for computing the annual exceedance rates of dynamic ground‐motion intensity measures (GMIMs) (e.g. peak ground acceleration and spectral acceleration). We use multi‐scale random field technique to incorporate spatial correlation and near‐fault directivity while generating MC simulations to assess the probabilistic seismic hazard of dynamic GMIMs. Our approach is capable of producing conditional hazard curves as well. We show various examples to illustrate the potential use of the proposed procedures in the hazard and risk assessment of geographically distributed structural systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Non-stationary random seismic responses of non-uniform beams

In this paper, a direct Ritz method is presented for analysing the responses of non-uniform shear beams subjected to the action of an evolutionary random seismic excitation. Because of the use of a pseudo ground acceleration, time-dependent random responses can conveniently be calculated by solving a set of deterministic equations of motion. These responses include power spectral densities, variances and higher spectral moments of any required responses of the beam. A non-uniform beam example was solved and the time-dependent variances and second spectral moments of both the beam-top displacement and the beam-base shear force are given to show the simplicity and effectiveness of the method, which can also easily be applied to other continuous elastic structures.     

Variance component estimation for partial errors-in-variables models

An iterative algorithm for variance component estimation based on partial errors-invariables (PEIV) model is proposed. Correction of observation vector and random elements of the coefficient matrix is taken as one kind of posterior information. Variance components in the observation vector and the random elements of the coefficient matrix are estimated according to Helmert estimation method. During the estimating process, the correction factors are used to modify the initial weight matrix, so as to make it more accurate. At the same time, a method for determining correction factors is given. Through examples of linear fitting and numerical simulation experiment of coordinate transformation, the practical effect of this algorithm is verified.     

平稳地震动过程具分段均布随机频率的谱拟合模型

本文首先针对平稳随机地震动谱拟合问题建立了一个统一的谱拟合模型,在此基础上,提出了一新的具分段均布随机频率的拟合模型,并与几类常用的谱拟合模型进行了比较研究,指出拟合模型的频率特性对拟合功率谱与反应谱具有显著的影响。算例表明,当结构的自振频率与拟合模型的分量频率接近时,计算的响应值明显偏大,对于小阻尼情况,这一现象更为明显。     

Simulation of intrinsic random fields of order <Emphasis Type="Italic">k</Emphasis> with a continuous spectral algorithm

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance.     

Seismic intensity measures for above‐ground liquid storage tanks

A series of scalar and vector intensity measures is examined to determine their suitability within the seismic risk assessment of liquid storage tanks. Using a surrogate modelling approach on a squat tank that is examined under both anchored and unanchored support conditions, incremental dynamic analysis is adopted to generate the distributions of response parameters conditioned on each of the candidate intensity measures. Efficiency and sufficiency metrics are used in order to perform the intensity measure evaluation for individual failure modes, while a comparison in terms of mean annual frequency of exceedance is performed with respect to a damage state that is mutually governed by the impulsive and convective modes of the tank. The results reveal combinations of spectral acceleration ordinates as adequate predictors, among which the average spectral acceleration is singled out as the optimal solution. The sole exception is found for the sloshing‐controlled modes of failure, where mainly the convective period spectral acceleration is deemed adequate to represent the associated response due to their underlying linear relationship. A computationally efficient method in terms of site hazard analysis is finally proposed to serve in place of the vector‐valued intensity measures, providing a good match for the unanchored tank considered and a more conservative one for the corresponding anchored system.     

Propagation of partially coherent non-stationary random waves in a viscoelastic layered half-space

This paper presents a highly accurate method based on the precise integration method (PIM) and on the pseudo excitation method (PEM). The method computes the propagation behaviour of partially coherent non-stationary random waves in a viscoelastic, transversely isotropic solid, which consists of a multi-layered soil resting on a homogeneous semi-infinite space. The excitation source is a local rupture between two layers, which causes a partially coherent non-stationary random field. The analysis of non-stationary random wave propagation is transformed into that for deterministic waves by using PEM. The resulting governing equations in the frequency-wavenumber domain are linear ordinary differential equations, which are solved very precisely by using PIM. The evolutionary power spectral densities of the ground level responses are investigated and some typical earthquake phenomena are explained.     

大跨度桥梁风场模拟方法对比研究

本文将基于线性滤波器的ARMA模型应用于大跨度桥梁的风场模拟,推导出自回归(AR)阶数P和滑动回归(MA)阶数q不等情况下,ARMA模型用于模拟多变量稳态随机过程的公式,将ARMA风场模拟方法与目前广泛应用于大跨度桥梁风场模拟的谐波合成法应用于一座实际大跨度斜拉桥的风场模拟,通过对比研究得出一些有意义的结论,并证实了ARMA法能够在保证模拟精度的前提下,大大提高风场模拟的效率。     

Fully probabilistic seismic displacement analysis of spatially distributed slopes using spatially correlated vector intensity measures

Earthquake‐induced slope displacement is an important parameter for safety evaluation and earthquake design of slope systems. Traditional probabilistic seismic hazard analysis usually focuses on evaluating slope displacement at a particular location, and it is not suitable for spatially distributed slopes over a large region. This study proposes a computationally efficient framework for fully probabilistic seismic displacement analysis of spatially distributed slope systems using spatially correlated vector intensity measures (IMs). First, a spatial cross‐correlation model for three key ground motion IMs, that is, peak ground acceleration (PGA), Arias intensity, and peak ground velocity, is developed using 2686 ground motion recordings from 11 recent earthquakes. To reduce the computational cost, Monte Carlo simulation and data reduction techniques are utilized to generate spatially correlated random fields for the vector IMs. The slope displacement hazards over the region are further quantified using empirical predictive equations. Finally, an illustrative example is presented to highlight the importance of the spatial correlation and the advantage of using spatially correlated vector IMs in seismic hazard analysis of spatially distributed slopes. Copyright © 2013 John Wiley & Sons, Ltd.