Tracking rapidly changing dynamical systems using a non‐parametric statistical method based on wavelets

A non‐parametric statistical method of tracking rapidly changing dynamical systems is introduced. The method implements the non‐linear wavelet thresholding estimator to estimate the evolutionary transfer function of the system. The consistency and optimality of the resulting estimate of our method on rapidly changing systems are stated, while no linear estimator can achieve the same optimality. Two examples of real data are studied using this non‐parametric method, including two vertical seismic array case studies and a series of seismic slope experiments. The analysis results are consistent with previous research for the vertical seismic array data. Moreover, our estimate is superior to those from previous research in the sense that our estimate is neither over‐smoothed nor under‐smoothed. The ana‐ lysis results are also consistent with the experimental observations for the seismic slope experimental data. Copyright © 2003 John Wiley & Sons, Ltd.     

Trend analysis in Turkish precipitation data

This study aims to determine trends in the long‐term annual mean and monthly total precipitation series using non‐parametric methods (i.e. the Mann–Kendall and Sen's T tests). The change per unit time in a time series having a linear trend was estimated by applying a simple non‐parametric procedure, namely Sen's estimator of slope. Serial correlation structure in the data was accounted for determining the significance level of the results of the Mann–Kendall test. The data network used in this study, which is assumed to reflect regional hydroclimatic conditions, consists of 96 precipitation stations across Turkey. Monthly totals and annual means of the monthly totals are formed for each individual station, spanning from 1929 to 1993. In this case, a total of 13 precipitation variables at each station are subjected to trend detection analysis. In addition, regional average precipitation series are established for the same analysis purpose. The application of a trend detection framework resulted in the identification of some significant trends, especially in January, February, and September precipitations and in the annual means. A noticeable decrease in the annual mean precipitation was observed mostly in western and southern Turkey, as well as along the coasts of the Black Sea. Regional average series also displayed trends similar to those for individual stations. Copyright © 2005 John Wiley & Sons, Ltd.     

Hysteretic energy dissipation capacity and the cyclic to monotonic drift ratio for rectangular RC columns in flexure

Predictions of energy dissipation capacity and of the deterioration of deformation capacity due to cumulative damage have been made by means of a non‐parametric empirical approach, called the conditional average estimator method, using empirical data on rectangular reinforced concrete columns that failed in flexure. Five input parameters were used: axial load index, index related to confinement, shear span index, concrete compressive strength, and longitudinal reinforcement index. The energy capacity was expressed in three different normalized forms and the deterioration of deformation capacity was defined as the ratio of the cyclic to the monotonic ultimate drift. The longitudinal reinforcement index, the index related to confinement, and the axial load index are the most influential input parameters in the case of energy capacity, whereas the latter two indices exhibit the most significant influence in the case of the drift ratio. Energy capacity decreases with an increasing axial load index, whereas it increases with increasing longitudinal reinforcement and with better confinement. In the case of the shear span index, the trend is more complex. Normal concrete has a higher energy dissipation capacity than high‐strength concrete. Similar trends are observed for the drift ratio, with the exception of the influence of the axial load index, where the trend is opposite. The dispersion of the results is high. Copyright © 2008 John Wiley & Sons, Ltd.     

Flexural deformation capacity of rectangular RC columns determined by the CAE method

A non‐parametric empirical approach, called the conditional average estimator (CAE) method, has been implemented for the estimation of the flexural deformation capacity of reinforced concrete rectangular columns expressed in terms of the ultimate (‘near collapse’) drift. Two databases (PEER and Fardis), which represent subsets of the original databases, were used. Four input parameters were employed in the basic model: axial load index, index related to confinement, shear span index, and concrete compressive strength. The results of analyses suggest that, in general, ultimate drift decreases with increasing axial load index, and increases with better confinement. An increase in the shear span‐to‐depth ratio has a beneficial effect until a turning point is reached. After that the opposite trend can be observed, i.e. a decrease in the ultimate drift with further increasing of the shear span‐to‐depth ratio. No clear trend is observed in the case of concrete compressive strength. The predictions, obtained by using the Fardis database are in general somewhat larger than the predictions from the PEER database, due to the difference in the definition of ultimate drift. The scatter of results is large. The local coefficient of variation, which is a measure for dispersion, amounts to about 0.2–0.5. The ultimate drifts obtained by using the two databases, were compared with the values predicted by the Eurocode 8 empirical formula. Copyright © 2006 John Wiley & Sons, Ltd.     

Identification of a Point of Release by Use of Optimally Weighted Least Squares

This paper addresses the parametric inverse problem of locating the point of release of atmospheric pollution. A finite set of observed mixing ratios is compared, by use of least squares, with the analogous mixing ratios computed by an adjoint dispersion model for all possible locations of the release. Classically, the least squares are weighted using the covariance matrix of the measurement errors. However, in practice, this matrix cannot be determined for the prevailing part of these errors arising from the limited representativity of the dispersion model. An alternative weighting proposed here is related to a unified approach of the parametric and assimilative inverse problems corresponding, respectively, to identification of the point of emission or estimation of the distributed emissions. The proposed weighting is shown to optimize the resolution and numerical stability of the inversion. The importance of the most common monitoring networks, with point detectors at various locations, is stressed as a misleading singular case. During the procedure it is also shown that a monitoring network, under given meteorological conditions, itself contains natural statistics about the emissions, irrespective of prior assumptions.     

Magnetoelastic shear wave propagation in pre-stressed anisotropic media under gravity

The present study investigates the propagation of shear wave (horizontally polarized) in two initially stressed heterogeneous anisotropic (magnetoelastic transversely isotropic) layers in the crust overlying a transversely isotropic gravitating semi-infinite medium. Heterogeneities in both the anisotropic layers are caused due to exponential variation (case-I) and linear variation (case-II) in the elastic constants with respect to the space variable pointing positively downwards. The dispersion relations have been established in closed form using Whittaker’s asymptotic expansion and were found to be in the well-agreement to the classical Love wave equations. The substantial effects of magnetoelastic coupling parameters, heterogeneity parameters, horizontal compressive initial stresses, Biot’s gravity parameter, and wave number on the phase velocity of shear waves have been computed and depicted by means of a graph. As a special case, dispersion equations have been deduced when the two layers and half-space are isotropic and homogeneous. The comparative study for both cases of heterogeneity of the layers has been performed and also depicted by means of graphical illustrations.     

A distribution function based bandwidth selection method for kernel quantile estimation

The key problem in nonparametric frequency analysis of flood and droughts is the estimation of the bandwidth parameter which defines the degree of smoothing. Most of the proposed bandwidth estimators have been based on the density function rather than the cumulative distribution function or the quantile that are the primary interest in frequency analysis. We propose a new bandwidth estimator derived from properties of quantile estimators. The estimator builds on work by Altman and Léger (1995). The estimator is compared to the well-known method of least squares cross-validation (LSCV) using synthetic data generated from various parametric distributions used in hydrologic frequency analysis. Simulations suggest that our estimator performs at least as well as, and in many cases better than, the method of LSCV. In particular, the use of the proposed plug-in estimator reduces bias in the estimation as compared to LSCV. When applied to data sets containing observations with identical values, typically the result of rounding or truncation, the LSCV and most other techniques generally underestimates the bandwidth. The proposed technique performs very well in such situations.     

Prediction of the force–drift envelope for RC columns in flexure by the CAE method

A non‐parametric empirical approach, called the conditional average estimator (CAE) method, has been applied for the prediction of the normalized lateral force–drift envelope of reinforced concrete (RC) rectangular columns, as well as their characteristic drifts (effective yield drift, capping drift and ultimate drift), and drift‐related parameters (the ratio between the effective yield drift and elastic drift, and two ductility measures). A subset of the PEER RC column database was used. Five input parameters were employed: axial load index, index related to confinement, shear span index, concrete compressive strength, and longitudinal reinforcement index. The results suggest that the relations between the input and output parameters are complex, and that it is difficult to isolate the influence of a single parameter. Nevertheless, some trends were observed. The axial load index is the most influential input parameter. All the results decrease with an increasing axial load index, whereas they increase with an increasing longitudinal reinforcement index. An increase in the index related to confinement results in increases in the ultimate drift and in ductility. The influence of the shear span index is the most complex. The influence of the concrete strength is small with the exception of two output parameters related to elastic drift, which substantially decrease with increasing strength. The dispersion of the results is relatively large. The results of the predictions can be used for mathematical modelling of moment–rotation backbone curves for plastic hinges, and for the estimation of the deformation capacity of columns in seismic performance assessments. Copyright © 2007 John Wiley & Sons, Ltd.     

Bayesian improver of a distribution

 An estimate of a distribution obtained from a sample by any method of classical statistics may be erroneous when the sample is not representative of the population. A subjective distribution elicited from an expert may be miscalibrated when information is scanty and experience limited. The Bayesian Improver of a Distribution (BID) exploits a coherence principle and improves, in the ex ante sense, an initial estimate of a continuous distribution by using (i) the known distribution of a related variate and (ii) information about the dependence structure between the two variates. The theory of BID is developed into an applied (ABID) procedure. The ABID estimator is applicable to any continuous, monotone likelihood ratio dependent variates with arbitrary, strictly increasing marginal distributions, parametric or nonparametric; it is analytic in form and easy to implement via statistical or judgmental methods; it converges to the true distribution, provided the initial estimator does, as the sample size n→∞; it outperforms the initial estimator in the expected Kolmogorov–Smirnov distance for all n; and it offers the greatest gains when n is small – precisely when improved estimates are needed most.     

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.     

On the guided and surface shear horizontal waves in monoclinic transversely periodic layers and halfspaces with arbitrary variation of material properties across the unit cell

The paper outlines analytically and exemplifies numerically the basic aspects, characterizing dispersion spectra of the shear horizontal (SH) waves in transversely periodic layers and half-spaces with a monoclinic functionally graded unit cell. On introducing the background, the ’quasi-orthorhombic’ formulation is pointed out. Further analytical consideration bypasses explicit intricacy of the wave solutions in continuously varying media and relies only on a few basic traits of the governing equation of SH motion. An elementary reasoning pinpoints the key features of the Floquet eigenmodes and their link to the traction-free boundary conditions in question. This simple grounds suffices to generalize the remarkable property, previously restricted to the orthorhombic piecewise homogeneous periodic stacks, which implies that the SH dispersion spectrum for a unit cell, assumed free of traction at the faces, is embedded into the spectrum for the finite periodic structure of these unit cells and contains the locus of surface-wave solutions for the semi-infinite periodic structure. The conclusion is valid for an arbitrary continuous and/or discrete transverse periodic inhomogeneity. Numerical results, presented for the case of continuously inhomogeneous unit cell, are based on the Peano series of multiple integrals.     

A finite element method for the diffusion-convection equation with constant coefficients

Existing numerical methods for the solution of the diffusion-convection equation are unsatisfactory for convection dominated flow problems. A new finite element method incorporating the method of characteristics for the solution of the diffusion-convection equation with constant coefficients in one spatial dimensions is derived. This method is capable of solving diffusion-convection equation without any of the difficulties encountered in the existing numerical methods for the whole spectrum of dispersion from pure diffusion, through mixed dispersion, to pure convection. Several examples for the one-dimensional case are solved and results are compared with the exact solutions. The generalization of the method to variable coefficients and to the diffusion-convection equation in two space dimensions are discussed.     

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.     

A framework to estimate sediment loads using distributions with covariates: Beaurivage River watershed (Québec,Canada)

A framework to estimate sediment loads based on the statistical distribution of sediment concentrations and various functional forms relating distribution characteristics (e.g. mean and variance) to covariates is developed. The covariates are used as surrogates to represent the main processes involved in sediment generation and transport. Statistical models of increasing complexity are built and compared to assess their relative performance using available sediment concentration and covariate data. Application to the Beaurivage River watershed (Québec, Canada) is conducted using data for the 1989–2004 period. The covariates considered in this application are streamflow and calendar day. A comparison of different statistical models shows that, in this case, the log‐normal distribution with a mean value depending on streamflow (power law with an additive term) and calendar day (sinusoidal), a constant coefficient of variation for streamflow dependence and a constant standard deviation for calendar day dependence provide the best result. Model parameters are estimated using the maximum likelihood estimation technique. The selected model is then used to estimate the distribution of annual sediment loads for the Beaurivage River watershed for a selected period. A bootstrap parametric method is implemented to account for uncertainties in parameter values and to build the distributions of annual loads. Comparison of model results with estimates obtained using the empirical ratio estimator shows that the latter were rarely within the 0·1–0·9 quantile interval of the distributions obtained with the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.     

煤层厚度变化时地震槽波理论频散曲线计算方法及频散特征分析

槽波地震勘探利用槽波的频散特性反演煤层的结构特征,故理论频散曲线的计算是一个重要方面.使用水平层状模型假设下的面波频散曲线计算方法能够计算煤层厚度恒定模型地震槽波频散曲线;但当煤层厚度变化时该方法不再适用.基于前人水平层状均匀介质模型的面波理论频散曲线计算方法,对于含煤三层模型,本文发展了煤层厚度变化情况下的地震槽波理论频散曲线计算方法,并使用该方法计算分析了不同厚度函数模型的频散曲线形态特征.研究表明：与稳定厚度煤层相比,煤层厚度变化使得地震槽波群速度成为与频率及传播射线在水平面投影路径相关的二元函数;射线路径上煤层厚度的变化使得频散曲线在群速度方向上压缩,群速度变化范围变小,且使处于最小值位置的埃里相群速度增大;而煤层厚度的线性变化模型频散曲线只与射线首、尾处的煤层厚度有关,与煤层厚度恒定模型相比,曲线形态不发生改变;煤层厚度呈非线性变化时,频散曲线形态上可能发生改变.     

Evaluation of longitudinal and transverse dispersivities/distance ratios for tracer test in a radially convergent flow field with scale-dependent dispersion

This study presents a novel mathematical model for analysis of non-axisymmetrical solute transport in a radially convergent flow field with scale-dependent dispersion. A two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates is derived based on assuming that the longitudinal and transverse dispersivities increase linearly with the distance of the solute transported from its injected source. The Laplace transform finite difference technique is applied to solve the two-dimensional, scale-dependent advection–dispersion equation with variable-dependent coefficients. Concentration contours for different times, breakthrough curves of average concentration over concentric circles with a fixed radial distance, and breakthrough curves of concentration at a fixed observation point obtained using the scale-dependent dispersivity model are compared with those from the constant dispersivity model. The salient features of scale-dependent dispersion are illustrated during the non-axisymmetrical transport from the injection well into extraction well in a convergent flow field. Numerical tests show that the scale-dependent dispersivity model predicts smaller spreading than the constant-dispersivity model near the source. The results also show that the constant dispersivity model can produce breakthrough curves of averaged concentration over concentric circles with the same shape as those from the proposed scale-dependent dispersivity model at observation point near the extraction well. Far from the extracting well, the two models predict concentration contours with significantly different shapes. The breakthrough curves at observation point near the injection well from constant dispersivity model always produce lesser overall transverse dispersion than those from scale-dependent dispersivity model. Erroneous dimensionless transverse/longitudinal dispersivity ratio may result from parametric techniques which assume a constant dispersivity if the dispersion process is characterized by a distance-dependent dispersivity relationship. A curve-fitting method with an example is proposed to evaluate longitudinal and transverse scale-proportional factors of a field with scale-dependent dispersion.     

THE USE OF OBSERVATIONAL SPACE IN SEISMIC REFRACTION TECHNIQUES *

Attention is given in this paper to the display of intensive seismic refraction information in a convenient form similar to spatial display techniques being developed for resistivity methods. An observational space for first arrivals is defined and illustrated with respect to some simple structures and the possible use of such a representation in diagnosis is discussed. The spatial display is suitable both for steeply dipping and gently dipping structures. In the latter case, the low relief space may be subjected to further operations to produce apparent velocity distributions and values simply related to plus times. A practical example of the latter process and the subsequent interpretation is given.     

A note on near-field site amplification effects of ground motion from a radially inhomogeneous valley

To improve the understanding of the near-field soil and topographic amplification effects, an analytical solution by the authors for the scattering of plane SH waves by a radially inhomogeneous semi-cylindrical valley is extended to the case of a line source of cylindrical SH waves. Upon confirmation of its accuracy with past exact solutions for a homogeneous and an inhomogeneous semi-cylindrical valley under far-field plane SH waves, the extended solution is used to calculate the ground motion amplification factors for both the homogeneous and inhomogeneous valleys subjected to near-field waves. A comprehensive parametric study is conducted with respect to the location of the wave source, the dimensionless frequency of the incident waves, and the inhomogeneity degree of the covering soil layer. It is found that more amplifications and reductions of ground motions will occur within a certain range in and around the valley as the sources are located further. Consistent with the far-field case, it is confirmed that an increase of the degree of inhomogeneity of the covering soil layer generally amplifies the ground motions significantly.     

A central difference method with low numerical dispersion for solving the scalar wave equation

In this paper, we propose a nearly‐analytic central difference method, which is an improved version of the central difference method. The new method is fourth‐order accurate with respect to both space and time but uses only three grid points in spatial directions. The stability criteria and numerical dispersion for the new scheme are analysed in detail. We also apply the nearly‐analytic central difference method to 1D and 2D cases to compute synthetic seismograms. For comparison, the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method are used to model acoustic wavefields. Numerical results indicate that the nearly‐analytic central difference method can be used to solve large‐scale problems because it effectively suppresses numerical dispersion caused by discretizing the scalar wave equation when too coarse grids are used. Meanwhile, numerical results show that the minimum sampling rate of the nearly‐analytic central difference method is about 2.5 points per minimal wavelength for eliminating numerical dispersion, resulting that the nearly‐analytic central difference method can save greatly both computational costs and storage space as contrasted to other high‐order finite‐difference methods such as the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method.