A review of numerical experiments on seismic wave scattering

This paper reviews applications of the finite-difference and finite-element methods to the study of seismic wave scattering in both simple and complex velocity models. These numerical simulations have improved our understanding of seismic scattering in portions of the earth where there is significant lateral heterogeneity, such as the crust. The methods propagate complete seismic wavefields through highly complex media and include multiply scattered waves and converted phases (e.g.,P toSV, SV toP, body wave to surface wave). The numerical methods have been especially useful in cases of moderate and strong scattering in complex media where multiple scattering becomes important. Progress has been made with numerical methods in understanding how near-surface, low-velocity basin structures scatter surface waves and vertically-incident body waves. The numerical methods have proven useful in evaluating scattering of surface waves and body waves from topography of both the free surface and interfaces buried at depth. Numerical studies have demonstrated the importance of conversions from body waves to surface waves (andvice versa) when lateral heterogeneities and topographic relief are present in the uppermost crust. Recently, several investigations have applied numerical methods to study seismic wave propagation in velocity models which vary randomly in space. This stochastic approach seeks to understand the effects of small-scale complexity in the earth which cannot be resolved deterministically. These experiments have quantified the relationships between the statistical properties of the random heterogeneity and the measurable properties of high-frequency (1 Hz) seismograms. These simulations have been applied to the study of many features observed in actual high-frequency seismic waves, including: the amplitude and time decay of seismic coda, the apparent attenuation from scattering, the dispersion of waveforms, and the travel time and waveform variations across arrays of receivers.     

Dimension and entropy in the soil-covered landscape

Both the Hausdorff dimension and the K-entropy supply a measure of the irregularity of the landspace surface. The relationship between the two measures is investigated over a variety of terrains in Britain and a method of calculating the entropy is checked against an independent estimate of the dimension with reasonable agreement. The calculation of the K-entropy requires that the landscape surface be represented by an homogenous ergodic random field. This condition is satisfied by the tendency of soil-covered terrains to progressively approximate to a form well represented by a Gaussian field. Gaussian random fields can either be very smooth, possessing derivatives of all orders at every point or they are highly irregular and non-differentiable everywhere. Within the regular conceptualization the Rice-Kac theory is used to predict the numbers of crossing points and the extent of excursion sets. These predictions are tested against an example terrain from the High Weald of East Sussex with very good agreement, apart from predictions of local maxima. A worked example of the calculation of the K-entropy is given as an appendix. The potential role of information theory in geomorphology extends beyond the use made of entropy in this investigation. In particular ergodic theory has important practical and theoretical implications.     

随机误差传递与合成的蒙特卡罗模拟

对分析测试过程中随机误差的传递与合成进行了蒙卡罗模拟，并偏制了ＢＡＳＩＣ程序。该方法解决了当被合成误差不相互独立时泰勒级数公式所遇到的困难，当被合成误差相互独立时，与泰勒公式计算结果一致。     

Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Entropy estimation of hydrological extremes

The method of Relative Entropy with Fractile constraints (REF method) is explained and applied to model extreme compound hydrological phenomena, such as extreme sea levels under storm conditions. Also presented is a simple method of Tail Entropy Approximation (TEA), which amounts to a correction of traditional statistical estimates for extreme observations.Distribution assumptions are necessary but downplayed in the REF method, relegating the prior distribution to the role of an extrapolation function. The estimates are objective in an information-theoretical sense. They also satisfy a strict requirement of self-consistency that is generally not satisfied by standard statistical methods: invariance under monotonic transformations of the random variable.Historical records of storm surge levels in the Netherlands and annual maximum tidal heights for Sheerness, UK, are used as examples. Comparison is made with distributions obtained using other methods.It is concluded that the tail entropy approximation provides simple, objective estimates of extremes in the tail beyond the range of observations.     

Estimation of a multidimensional covariance function in case of anisotropy

A model of a multivariate covariance function with an ellipsoidal directional correlation scale has been developed. The axes of the ellipsoidal scale are related to the eigenvalues and eigenvectors of a matrix B which characterizes the ellipsoid of the range of influence. The matrix B is found to be related to a matrix T which can be estimated directly from sparse sampling data and can be used to determine estimates of the matrix B. The method has been applied to both two-dimensional and three-dimensional cases. The numerical results show that the satisfactory accuracy is obtained with sparse sampling data from an anisotropic random function.     

Time effects on dynamic soil properties under random excitation conditions

Two types of clayey soils, a kaolinite and a bentonite, were tested using a resonant column apparatus under random excitation conditions. The concept of root mean square (rms) strain was utilized for the purpose of strain calculations during random loading. The conventional estimator of the transfer function was used for random vibration analysis. The effect of confinement duration (at a constant pressure) on dynamic soil properties, namely damping and shear modulus, was evaluated. The results indicate that for both cohesive soils, the effect of time was less pronounced during random vibration than sinusoidal loading at the same rms strain. This effect is however more pronounced when the peak shearing strain of sinusoidal loading is considered. Furthermore, time effects were more pronounced at low strain levels than at high strain levels.