A semi-analytical method for predicting the outflow hydrograph due to dam-break in natural valleys

The paper presents a semi-analytical method for predicting the flow rate hydrograph due to a hypothetical sudden and total dam failure in a natural valley. The method generalizes the approach proposed by Hunt for the dam-break problem in a rectangular frictionless sloping channel to a valley with a cross-section area expressed by a power-law function of water depth, in order to take into account the most common shapes of natural valleys. The parameters of the deriving model can be set by exploiting data usually available concerning the dam section geometry and the reservoir storage-depth curve. The application of the technique to three different reservoirs is discussed. The results show that the flow rate hydrographs obtained at the dam site agree with the ones calculated by means of a finite volume numerical code based on two-dimensional shallow water equations. The method requires moderate computational and data collecting effort, so it can be regarded as a useful alternative to other procedures commonly adopted in the practice.     

Representation and Analysis of the Earthquake Size Distribution: A Historical Review and Some New Approaches

—The size distribution of earthquakes has been investigated since the early 20th century. In 1932 Wadati assumed a power-law distribution n(E) = kE ^?w for earthquake energy E and estimated the w value to be 1.7 ～ 2.1. Since the introduction of the magnitude-frequency relation by Gutenberg and Richter in 1944 in the form of log n(M) = a?bM, the spatial or temporal variation (or stability) of b value has been a frequently discussed subject in seismicity studies. The log n(M) versus M plots for some data sets exhibit considerable deviation from a straight line. Many modifications of the G-R relation have been proposed to represent such character. The modified equations include the truncated G-R equation, two-range G-R equation, equations with various additional terms to the original G-R equation. The gamma distribution of seismic moments is equivalent to one of these equations.¶In this paper we examine which equation is the most suitable to magnitude data from Japan and the world using AIC. In some cases, the original G-R equation is the most suitable, however in some cases other equations fit far better. The AIC is also a powerful tool to test the significance of the difference in parameter values between two sets of magnitude data under the assumption that the magnitudes are distributed according to a specified equation. Even if there is no significant difference in b value between two data sets (the G-R relation is assumed), we may find a significant difference between the same data sets under the assumption of another relation. To represent a character of the size distribution, there are indexes other than parameters in the magnitude-frequency distribution. The η value is one of such numbers. Although it is certain that these indexes vary among different data sets and are usable to represent a certain feature of seismicity, the usefulness of these indexes in some practical problems such as foreshock discrimination has not yet been established.     

Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method

Seismic wave propagation in transversely isotropic (TI) media is commonly described by a set of coupled partial differential equations, derived from the acoustic approximation. These equations produce pure P‐wave responses in elliptically anisotropic media but generate undesired shear‐wave components for more general TI anisotropy. Furthermore, these equations suffer from instabilities when the anisotropy parameter ε is less than δ. One solution to both problems is to use pure acoustic anisotropic wave equations, which can produce pure P‐waves without any shear‐wave contaminations in both elliptical and anelliptical TI media. In this paper, we propose a new pure acoustic transversely isotropic wave equation, which can be conveniently solved using the pseudospectral method. Like most other pure acoustic anisotropic wave equations, our equation involves complicated pseudo‐differential operators in space which are difficult to handle using the finite difference method. The advantage of our equation is that all of its model parameters are separable from the spatial differential and pseudo‐differential operators; therefore, the pseudospectral method can be directly applied. We use phase velocity analysis to show that our equation, expressed in a summation form, can be properly truncated to achieve the desired accuracy according to anisotropy strength. This flexibility allows us to save computational time by choosing the right number of summation terms for a given model. We use numerical examples to demonstrate that this new pure acoustic wave equation can produce highly accurate results, completely free from shear‐wave artefacts. This equation can be straightforwardly generalized to tilted TI media.     

An empirical method for approximating canopy throughfall

Rainfall replenishes surface and subsurface water but is partially intercepted by a canopy. However, it is challenging to quantify the rainfall passing through the canopy (i.e. throughfall). This study derives simple‐to‐use empirical equations relating throughfall to canopy and rainfall characteristics. Monthly throughfall is calculated by applying a mass balance model on weather data from Singapore; Vancouver, Canada; and Stanford, USA. Regression analysis is then performed on the calculated throughfall with three dependent variables (i.e. maximum canopy storage, average rainfall depth and time interval between two consecutive rainfall in a month) to derive the empirical equations. One local equation is derived for each location using data from that particular location, and one global equation is derived using data from all three locations. The equations are further verified with calculated monthly throughfall from other weather data and actual throughfall field measurements, giving an accuracy of about 80–90%. The global equation is relatively less accurate but is applicable worldwide. Overall, this study provides a global equation through which one can quickly estimate throughfall with only information on the three variables. When additional weather data are available, one can follow the proposed methodology to derive their own equations for better estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

Fractional conservation of mass

The traditional conservation of mass equation is derived using a first-order Taylor series to represent flux change in a control volume, which is valid strictly for cases of linear changes in flux through the control volume. We show that using higher-order Taylor series approximations for the mass flux results in mass conservation equations that are intractable. We then show that a fractional Taylor series has the advantage of being able to exactly represent non-linear flux in a control volume with only two terms, analogous to using a first-order traditional Taylor series. We replace the integer-order Taylor series approximation for flux with the fractional-order Taylor series approximation, and remove the restriction that the flux has to be linear, or piece-wise linear, and remove the restriction that the control volume must be infinitesimal. As long as the flux can be approximated by a power-law function, the fractional-order conservation of mass equation will be exact when the fractional order of differentiation matches the flux power-law. There are two important distinctions between the traditional mass conservation, and its fractional equivalent. The first is that the divergence term in the fractional mass conservation equation is the fractional divergence, and the second is the appearance of a scaling term in the fractional conservation of mass equation that may eliminate scale effects in parameters (e.g., hydraulic conductivity) that should be scale-invariant.     

高阶方程偏移的分裂算法

对于用高阶偏微分方程进行地震偏移,本文提出一种分裂算法。有限差分法目前只能解波动方程的二阶和三阶的近似方程的偏移问题,更高阶的近似方程还没有一种有效的解法。利用高阶方程进行偏移会提高偏移的精度和效果。本文就是为了解决这个问题所进行的探索。 本文阐述了分裂算法的原理,并给出了应用该算法解决偏移问题的实例。     

Empirical shear wave velocity equations in terms of characteristic soil indexes

An investigation to systematize empirical equations for the shear wave velocity of soils was made in terms of four characteristic indexes. The adopted indexes are the N-value of the Standard Penetration Test, depth where the soil is situated, geological epoch and soil type. As some of these indexes are variates belonging to interval scales while others belong to nominal or ordinal scales, the technique known as a multivariate analysis cannot be employed. A new approach to the theory of quantification, after C. Hayashi, was introduced and developed for solving this difficulty. Fifteen sets of empirical equations to estimate low strain shear wave velocity theoretically may be obtained by combining the above four indexes. All of these sets were derived by use of about 300 data, and their accuracies were evaluated by means of correlation coefficients between the measured and estimated shear wave velocities. The best equation was found to be the one which included all the indexes, and its correlation coefficient was 0.86. The empirical equation relating the standard penetration N-value to the shear wave velocity provided a correlation of only 0.72, and is one of the lowest ranking among the 15 sets of equations.     

POLYNOMIAL AND POWER FUNCTIONS FOR GLACIAL VALLEY CROSS-SECTION MORPHOLOGY

An empirical evaluation of glacial trough cross-section shape is performed on seven vertical cross-sections in three Sierra Nevada valleys glaciated during the late Quaternary. Power and second-order polynomial functions are fitted by statistical regression. Power functions are very sensitive to subtle valley-bottom topographic features and require precise specification of the valley-bottom-centre location. This dependency is problematic given under-representation of valley bottoms by conventional contour-sampling methods, and the common alteration of valley-bottom morphology by non-glacial processes. Power function exponents vary greatly in response to these and other non-genetic factors and are not found to be reliable indicators of overall valley morphology. Second-order polynomials express overall valley shape in a single robust function. They are applied to both bedrock- and sediment-floored glacial valleys with negligible statistical bias except where side-slopes are stepped or convex-upward or where valley form is asymmetrical. They can describe alluviated or severely eroded valleys, and can objectively identify indi-vidual components of polymorphic valleys, because valley bottom and centre locations need not be specified. Mathematical expressions of parameters useful for geomorphic measurements and glaciological modelling are analytically derived from the polynomials as functions of the three polynomial coefficients. These parameter equations provide estimates of valley side-slopes, mean and maximum depth, midpoint location, width, area, boundary length, form ratio and symmetry.     

Clasmatic seismodynamics—Oxymoron or pleonasm?

Clasmatic means fractional in a physical and a mathematical sense, seismic means shaking. Truncated Lévy-processes and related fractional differential equations are introduced by means of statistics and balances. Isoclasmatic balance equations are proposed for sand samples, they imply energy-based hypoplastic and hypoelastic relations for the subcritical range and can be extended for rock. Balances of conserved and not conserved quantities with isoclasmatic distributions in spacetime are represented by coupled partial fractional differential equations, these can be transformed into classical balance equations for the subcritical range. Micro-seismic power-law spectra of sand are obtained with a fractional Schrödinger equation, its extension for polar effects and rock is indicated. Critical phenomena beyond the verge of energetic convexity are related with a degeneration of the fractional wave propagation. Due to them the lithosphere is polyclasmatic. Focussing on qualitative aspects, we show that clasmatic seismodynamics is no oxymoron, but rather a pleonasm and a promising new paradigm.     

Site-dependent shear-wave velocity equations versus depth in California and Japan

Shear-wave velocity of the near-surface ground is an important soil property in earthquake and civil engineering. Using the data from 643 boreholes from the KiK-net in Japan and 135 boreholes from California where the shear-wave velocity profile reaches at least 30 m, firstly, we classify sites by building code, then build site-dependent relationships between travel time and depth by regression utilizing the logarithmic model and power-law model, lastly, we get shear-wave velocity equations versus depth based on the mathematical relationship between travel time and velocity. The results show that: (1) the travel time is strongly correlated with depth, and the Pearson correlation coefficients range between 0.867 and 0.978. (2) there is a certain difference between linear velocity equations and power-law velocity equations, and the power-lower equations are generally more close to measured data than linear equations except for class E in Japan and class D in California. (3) the velocities are similar at the sites of each class for different regions but that the gradient of velocity with depth vary between different regions.     

A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media

In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.     

含气介质临界点、流体和骨架弹性参数的线性数值计算方法(英文)

临界孔隙度及其孔隙介质的研究均为测试分析的实验方法,如何运用数值计算方法求取孔隙流体介质临界点、流体和骨架的弹性参数是备受关注的课题。本文提出了求取临界点、流体和骨架弹性参数的数值计算公式及方法,并结合含气样品测试数据实现了这种数值计算。首先,基于孔隙度≯为白变量,而密度ρ,密度与横波速度平方的乘积ρVs^2和密度与纵波速度平方的乘积ρVp^2均为函数的三个线性方程,从每个线性方程中的两个系数的有机组合得到求取有关弹性参的线性数值计算公式。然后,详细阐述了数值计算的步骤和方法以及需要注意的问题,把室内含气砂岩数的线性数值计算公式。然后,详细阐述了数值计算的步骤和方法以及需要注意的问题,把室内含气砂岩样品测试的整体介质密度、纵波速度和横波速度作为数值计算的输入数据,求取了临界点和流体及骨架共计11个弹性参数的具体数值。通过数学方法的计算数据与实验方法的测试数据的比较分析,表明了本文数值计算公式的正确性和实现方法的有效性。本文提出的求取含气介质临界点、流体和骨架弹性参数的“数值计算公式及方法”含义清晰且形式简洁,为孔隙流体介质的数值计算分析和流体属性研究提供了可能的新方法。     

Evaluation and generalization of radiation‐based methods for calculating evaporation

Eight radiation‐based equations for determining evaporation were evaluated and expressed in five generalized forms. Five evaporation equations (Abtew, Hargreaves, Makkink, Priestley and Taylor and Turc), where each represents one generalized form, were then compared with pan evaporation measured at Changins station in Switzerland. The comparison was first made using the original constant values involved in each equation, and then using the recalibrated constant values. Evaluation of the Priestley and Taylor equation requires net radiation data as input, in this study, net radiation was estimated using Equation (16) owing to the lack of observation data. The results showed that when the original constant values were used, large errors resulted for most of the equations. When recalibrated constant values were substituted for the original constant values, four of the five equations improved greatly, and all five equations performed well for determining mean annual evaporation. For seasonal and monthly evaporation, the Hargreaves and Turc equations showed a significant bias, especially for cold months. With properly determined constant values, the Makkink and modified Priestley and Taylor equations resulted in monthly evaporation values that agreed most closely with pan evaporation in the study region. The simple Abtew equation can also be used when other meteorological data except radiation are not available. Copyright © 2000 John Wiley & Sons, Ltd.     

几种反射波时距方程的比较

在地震资料处理中，速度分析和成像技术极为重要.常规地震资料处理方法是利用双曲线方程来描述反射波时距曲线规律，此方程随着地层非均质性、各向异性和排列长度的增加，其误差变大.目前发展的反射波非双曲时距方程，主要有基于层状各向同性模型的非双曲时距方程、基于均匀弱各向异性模型的时距方程、基于速度随炮检距变化模型的时距方程、基于线性连续速度模型的时距方程.本文针对三个典型的模型：层状均匀各向同性模型、层状弱各向异性模型和层状非均匀模型，对这几种时距方程进行了精度比较与分析，得出了一些有益的结论.最后，从不同角度说明了应用这几种方程的合理性.     

Non-homogeneous elastic waves in soils: Notes on the vector decomposition technique

Geological media are invariably non-homogeneous, which complicates considerably the analysis of seismically induced wave propagation phenomena. Thus, closed-form solutions in the form of Green's functions are difficult to construct, but are quite valuable in their own right and often play the role of kernels in boundary integral equation formulations that are used for the solution of complex boundary-value problems of engineering importance. In this work, we examine in some detail the types of wave-like equations that result from vector decomposition of the equations of motion for the infinitely extending non-homogeneous continuum, which would be a first step for evaluating Green's functions. Specifically, an eigenvalue analysis is first performed, followed by computations using the finite difference method for a specific example involving a soil layer with quadratically varying material parameters. The aforementioned wave-like equations, defined in terms of dilatational and rotational strains, are originally coupled. Their uncoupling involves use of algebraic transformations, which are in turn valid for certain restricted categories of non-homogeneous materials. Numerical solution of these equations clearly shows attenuation patterns and phase changes that are manifested as the incoming wave disturbance is continuously scattered by non-constant material stiffness values encountered along the propagation path.     

Roughness coefficients and velocity estimation in well-inundated sheet and rilled overland flow without strongly eroding bed forms

Even with the flow of water over a soil surface in which roughness elements are well inundated, and in less erosive situations where erosional bed forms are not pronounced, the magnitude of resistance coefficients in equations such as those of Darcy–Weisbach, Chezy or Manning vary with flow velocity (at least). Using both original laboratory and field data, and data from the literature, the paper examines this question of the apparent variation of resistance coefficients in relation to flow velocity, even in the absence of interaction between hydraulics and resulting erosional bed forms. Resistance equations are first assessed as to their ability to describe overland flow velocity when tested against these data sources. The result is that Manning's equation received stronger support than the Darcy–Weisbach or Chezy equations, though all equations were useful. The second question addressed is how best to estimate velocity of overland flow from measurements of slope and unit discharge, recognizing that the apparent flow velocity variation in resistance coefficients is probably a result of shortcomings in all of the listed resistance equations. A new methodology is illustrated which gives good agreement between estimated and measured flow velocity for both well-inundated sheet and rill flow. Comments are given on the predictive use of this methodology. Copyright © 1999 John Wiley & Sons, Ltd.     

Correcting circulation biases in a lower-resolution global general circulation model with data assimilation

In this study, we aim at developing a new method of bias correction using data assimilation. This method is based on the stochastic forcing of a model to correct bias by directly adding an additional source term into the model equations. This method is presented and tested first with a twin experiment on a fully controlled Lorenz ’96 model. It is then applied to the lower-resolution global circulation NEMO-LIM2 model, with both a twin experiment and a real case experiment. Sea surface height observations are used to create a forcing to correct the poorly located and estimated currents. Validation is then performed throughout the use of other variables such as sea surface temperature and salinity. Results show that the method is able to consistently correct part of the model bias. The bias correction term is presented and is consistent with the limitations of the global circulation model causing bias on the oceanic currents.     

ACCURACY AND STABILITY OF FINITE DIFFERENCE EQUATIONS FOR SHALLOW WATER FLOWS IN ORTHOGOPNAL COORDINATES

IINTRODUCTIONNumericalmethodsasatooltosimulateflowsandpollutanttransportareincreasinglyimportantinhydraulicandenvironmentalengineering.AveryusefulapplicationofthenumericalmethodologyinengineeringproblemswouldbetosolvethesystemofZDdepth-integratedshallowwaterequations.ManysolutionsofthegoverningequationsarederivedusingtraditionalfinitedifferencemethodonCartesianregulargrids.ThedisadvantageofthismethodseemstobetheinflexibilityofCartesiangridstocomplywithirregularorcurvedperimeterswhichsur…     

Power-Law Testing for Fault Attributes Distributions

This paper is devoted to statistical analysis of faults’ attributes. The distributions of lengths, widths of damage zones, displacements and thicknesses of fault cores are studied. Truncated power-law (TPL) is considered in comparison with commonly used simple power-law (PL) (or Pareto) distribution. The maximal likelihood and the confidence interval of the exponent for both PL and TPL are estimated by appropriate statistical methods. The Kolmogorov–Smirnov (KS) test and the likelihood ratio test with alternative non-nested hypothesis for exponential distribution are used to verify the statistical approximation. Furthermore, the advantage of TPL is proved by Bayesian information criterion. Our results suggest that a TPL is more suitable for describing fault attributes, and that its condition is satisfied for a wide range of fault scales. We propose that using truncated power laws in general might eliminate or relax the bias related to sampling strategy and the resolution of measurements (such as censoring, truncation, and cut effect) and; therefore, the most reliable range of data can be considered for the statistical approximation of fault attributes.