Generalised scale invariance and multiplicative processes in the atmosphere

Many geophysical fields show highly intermittent fractal structures spanning wide ranges of scale. However, few are isotropic: texture, stratification, as well as variable (scale dependent) orientation of structures is far more common. To deal with such fractals, we must generalise the idea of scale invariance beyond the familiar self-similar (or even self-affine) notions. Taking the atmosphere as our primary example (however, we also model galaxies), we outline the necessary formalism (generalised scale invariance), and show how it can be used to deal with the strongly intermittent structures which result from multiplicative (cascade type) processes concentrating matter or energy into smaller and smaller scales.We illustrate these ideas with rain data from blotting paper and radar, showing first how to directly estimate the elliptical dimension characterising the stratification, and second, how to determine universal scale-independent (invariant) codimension functions that characterise the distribution of the intense rain regions.     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

An asperity model to simulate rupture along heterogeneous fault surfaces

A model has been developed to simulate the statistical and mechanical nature of rupture on a heterogeneous strike-slip fault. The model is based on the progressive failure of circular asperities of varying sizes and strengths along a fault plane subjected to a constant far-field shear displacement rate. The basis of the model is a deformation and stress intensity factory solution for a single circular asperity under a unidirectional shear stress. The individual asperities are unified through the fault stiffness and the far-field stress and displacement. During fault deformation asperities can fail and reheal, resulting in changes in the local stresses in the asperities, stress drops, and changes in the stiffness of the fault. Depending on how the stress is redistributed following asperity failure and on the strenghts of the neighboring asperities an earthquake event can be the failure of one or more asperities. Following an earthquake event seismic source parameters such as the stress drop, energy change, and moment magnitude are calculated. Results from the model show a very realistic pattern of earthquake rupture, with reasonable source parameters, the proper magnitude-frequency behavior, and the development of characteristic earthquakes. Also the progression ofb-values in the model gives some insight into the phenomenon of self-organized criticality.     

Walsh列率域中多维分形模型与GIS环境下地球物理信号处理

地球物理信号通常在多个尺度段表现尺度不变性,这些不变性起因于不同的地质、地球物理或成矿过程的自相似性. 利用这种在多个尺度段的尺度不变性可以设计多维分形滤波器,滤波所得信号表征了具尺度不变性的地质地球物理或成矿过程,可以用于成矿预测或环境评价. 本文研究了Walsh变换列率空间地球物理信号的列率功率谱密度与列率之间的分形与多维分形关系, 试验证实了大洋钻探、石油以及煤系地层地球物理测井资料在Walsh域的多维分形性质,提出了用于分解地球物理场,提取有用信号并用于矿产资源勘探或环境评价的多维分形W-A模型. 利用波列率域中的多维分形关系构造了W-A图解(W-A Plot). 借助W-A图解可以确定最小平方误差（LS）意义下Walsh功率谱变化的不同自相似性的频率分界点,从而用于设计W-A分形滤波器. 这种滤波器可将地球物理场分解成具有不同自相似性的局部场,并且保留原场的各向异性结构. 与通常使用的基于Fourier变换的滤波技术相比,W-A模型具有许多优点：W-A适用于检测地球物理信号中的突变、线性、环状、局部与纹理结构等弱信号. 同时,由于Walsh变换中只有简单的变号(加法与减法),其计算速度远快于建立在复数乘法之上的Fourier变换,所以W-A计算速度远快于Fourier域的滤波方法,可以用于地球物理信号的现场实时处理. 用加拿大Nova-cotia省西南地区的布格重力异常进行了W-A方法的试算,处理结果反映了地质、矿产分布规律,能够很好地进行矿产预测.     

On the time-scale invariance of the eruptive activity of Vesuvius

At Vesuvius, the intervals between final effusive eruptions since 1631 AD and those between successive plinian eruptions since 25 000 yr BP follow the same power law, when separately ranked in decreasing order. So, shifts from small to large scales are possible and this might be very helpful for the assessment of hazards of future strong volcanic activity.     

Limits to scale invariance in alluvial rivers

Assumptions about fluvial processes and process–form relations are made in general models and in many site-specific applications. Many standard assumptions about reach-scale flow resistance, bed-material entrainment thresholds and transport rates, and downstream hydraulic geometry involve one or other of two types of scale invariance: a parameter (e.g. critical Shields number) has the same value in all rivers, or doubling one variable causes a fixed proportional change in another variable in all circumstances (e.g. power-law hydraulic geometry). However, rivers vary greatly in size, gradient, and bed material, and many geomorphologists regard particular types of river as distinctive. This review examines the tension between universal scaling assumptions and perceived distinctions between different types of river. It identifies limits to scale invariance and departures from simple scaling, and illustrates them using large data sets spanning a wide range of conditions. Scaling considerations and data analysis support the commonly made distinction between coarse-bed and fine-bed reaches, whose different transport regimes can be traced to the different settling-velocity scalings for coarse and fine grains. They also help identify two end-member sub-types: steep shallow coarse-bed ‘torrents’ with distinctive flow-resistance scaling and increased entrainment threshold, and very large, low-gradient ‘mega rivers’ with predominantly suspended load, subdued secondary circulation, and extensive backwater conditions. © 2020 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd     

中国大陆中长期地震活动的群集特征

主要研究中国华北地区和南北地震带中长期地震活动的时间特征。选取华北地区和南北地震带二个地震活动地区1970年以来有仪器记录的完整地震目录。作相关地震删除后对于不同起始震级分别计算不同时-空尺度下观测地震数N_p 与泊松分布下理论地震数n_p 的比值(称为时-空“矩”),结果表明无论华北还是南北地震带,无论起始震级是ML 3. 0或ML 4. 0,所有的地震活动过程都显示群集特征N_p/n_p> 1。地震活动随时间呈指数分布,具有标度不变性,分维数在0. 2～0. 7之间变化。     

Describing near surface, transient flow processes in unconfined aquifers below irrigated lands: A deforming finite element model for heterogeneous aquifers

Hydrologic models of irrigated lands generally adopt either a basin-scale or a root-zone perspective. While basin-wide macro-scale models rely on the aggregation of important spatial and temporal data across large areas, micro-scale root-zone models depend on the definition of rigid boundaries around the zone of plant–soil–water interaction. In reality, irrigation management decisions are made on a field by field basis and can interact across field boundaries. This paper first describes a shallow water table model, based on deforming finite element (DFE) framework, to characterize the near-surface field-to-field hydrologic response to various irrigation and drainage management regimes along a gently sloping alluvial fan. The model is then enhanced through changing geometry of a fluctuating water table below a series of irrigated fields. Such an enhancement also offers computational flexibility relative to the saturated–unsaturated models commonly used in micro-scale studies. The model is designed with the alluvial fan aquifers of California’s western San Joaquin Valley as reference systems.     

Seismic regime of a single fault model

A special experimental technique enabled us to study in detail seismic events on a fault model in a uniaxial stress field. The recording system used made it possible to investigate the radiation pattern for all the events observed including the precise determination of the dislocation origin.The aim of the present paper is (1) to find the relation between shear and tensile displacements, (2) to check a possible influence of the process of tensile crack generation on the seismic energy release and (3) to compare the seismic regime of a single fault before and after the tensile crack generation, respectively.Results prove the dominating importance of shear mechanism for the seismic energy release. The tensile displacement can be seismoactive only under special contact conditions on the fault plane.The existence of tensile cracks at the fault tips changed the pattern of seismic energy radiation. This feature is probably caused by subsequent changes in contact conditions on the fault plane and in the stress field around the fault.A comparison of some results of the present model experiments with the already published results of geological and seismological measurements and investigations shows the analogous character in laboratory and in nature of the process of tectonic earthquake preparation, the displacement course on the fault during the earthquake and the manner of seismic energy release on faults.     

基于尺度变换随机共振的瞬变电磁弱信号检测

晚期瞬变电磁信号是大参数、多频率、且各频率分量未知的电磁信号,特别在晚期还是强噪声背景下的微弱信号.随机共振系统可以在极限信噪比的情况下提取微弱信号,但仅适用于小参数、单一频率或频率已知的高频信号的情况.为此,本文提出一种基于尺度变换的随机共振算法用于检测晚期瞬变电磁信号,该方法充分利用了随机共振检测弱信号的优势,通过引入尺度变换,消除了随机共振系统对待测信号频率的限制,在绝热近似理论下,实现了从强噪声中提取出微弱晚期瞬变电磁目标信号.理论分析和仿真结果表明:对埋在强噪声中的未知频率的晚期瞬变电磁信号,采用连续的压缩算法,以获得一个适当的输入信号到随机共振体系,根据输出信号共振谐振峰的变化,经反变换运算可得待测弱晚期瞬变电磁信号的各个未知频率,各个频率的叠加即可还原晚期瞬变电磁信号.与传统方法相比,本方法数据采集量和采集时间都减小到原来的几十分之一,能在极限信噪比(信噪比-50 dB)下,提取出目标信号,为获得深层目标信号提供了可能,从而提高了瞬变电磁探测仪器的探测深度和探测精度.     

地震研究中的断层流体动力学问题

流体在断裂带地震周期中具有重要作用。 在地震流体研究中, 该文建议在如下几个方面加强研究力度: ① 断层渗透结构和断裂带古水文地质旋回的研究; ② 断裂带流体循环的尺度效应; ③ 流体分布、 循环与构造展布关系; ④ 断裂带深浅部流体关系研究。 在断层流体动力学研究中, 建议就某一发震断裂带开展系统研究, 并优先解决以下问题: ① 断裂带流体的起源和成分; ② 产生和维持高孔隙压力的构造环境和水文地质条件; ③ 断裂带及邻近岩体流体运移及重新分布的机制; ④ 取得断裂带孔隙压力变化的数量知识; ⑤ 垂直方向和水平方向流体孔隙压力变化范围; ⑥ 地震周期中流体迁移与孔隙压变化规律。     

欧洲98版本地震烈度表简介

对欧洲98地震烈度表(EMS—98)作了筒要介绍，内容包括不同建筑结构类型的易损性等级、钢混建筑的破坏等级、砖石建筑的破坏等级、描述性术语的定量意义、欧洲98地震烈度表中考察的影响因素及烈度表全文。     

A scale‐invariant property of the water retention curve in weakly heterogeneous vadose zones

Abrupt changes of hydraulic properties in a vadose zone are modelled within a stochastic framework, which regards the saturated conductivity and parameters related to the distribution of soil pores as stationary, log‐normally distributed, random space functions. As a consequence, flow variables become random fields, and we aim at deriving an effective Richards equation. To obtain the latter, we adopt a perturbation expansion truncated at the first order (weakly heterogeneous media), which leads to the effective hydraulic conductivity and water retention curves. Overall, the effective properties are scale dependent. However, within the proposed framework, we demonstrate that the inflection point of the laboratory scale retention curve is not affected by the heterogeneity of the vadose zone. Finally, to illustrate the quantitative implications of our results, we consider a monitoring experiment at field scale, and we show how our approach leads to an effective water retention curve, which differs significantly from that which would be obtained without accounting for the above scale‐invariance property.     

On the free oscillations of a laterally heterogeneous earth model

Summary The equations of motion for the free oscillations of a heterogeneous spherical earth model are derived. It is found that the lateral variations of density and elastic moduli couple the odd(even) harmonics of the spheroidal oscillations with themselves as well as with the even (odd) harmonics of the torsional oscillations.List of symbols r, , Spherical coordinates;r is the radial distance from earth's center, is the co-latitude, and is the east longitude - r Space vector denoting a point with coordinatesr, , and - Gradient operator - ² Laplacian operator - _ij Kronecker's delta function - I Identity matrix - i      

A simplified stochastic model for infiltration into a heterogeneous soil forced by random precipitation

A simplified stochastic infiltration model is presented, aimed at representing the rainfall-runoff transformation in the presence of heterogeneity in the soil and with precipitation as a random variable with complex temporal evolution. Such a model is based on a simple water mass balance of a surface soil layer, considered as a non-linear reservoir. The explicit inclusion of spatial heterogeneity allows the model to be used in sub-grid parametrizations at a variety of scales, from the distributed modelling of the hydrological response of small watersheds to the representation of surface mass fluxes in General Circulation Models. An approximate solution procedure is developed, which allows the estimation of statistical moments of the soil effective saturation and runoff inside discrete time steps where the hydraulic saturated conductivity and the rainfall intensity are taken as random variables with known probability density functions. As a first test of the proposed model, two different simulations, relative to two soils with different hydraulic conductivity distributions, are presented and discussed. A year long record of hourly averaged rainfall intensities, as measured by a tipping bucket gauge in Central Italy, is taken as the main input. The main finding is that the non-linear nature of the soil filter is such that, for random precipitation intensity, the coefficient of variation of the runoff is always higher than that of precipitation. Such a non-linear variability enhancement, due mainly to the threshold character of the soil mass balance equation, tends to be slightly dampened by the variability of the hydraulic saturated conductivity.     

Graphical characterisation of probability distribution tails

 The purpose of this paper is to present a graphical method to characterise the nature of a distribution (exponential or algebraic). In the algebraic case, this statistical tool provides an estimation procedure of the parameter characterising the decrease of the survival function. The realizations of the random variable under study being available in the form of time series, this method is based on the relationship between the duration of exceeding an intensity threshold and the accumulation of the realizations of the random variable during this length of time. The behaviour of the duration-accumulation graphs (when the threshold of reference increases indefinitely) results in a function, the limit of which only depends on the parameter characterising the algebraic decrease of the probability distribution. The estimate of this parameter is biased but can be corrected effectively by numerical methods. We applied this method to two rainfall series differing by their geographical origin (Dédougou in Burkina Faso and a station on the Island of La Réunion) and their time step (respectively 1 day and 76 seconds). For both of them, the behaviour of tail distributions is shown to be algebraic and the values of the parameter characterizing the algebraic decrease of the probability distribution of the two series are very close. This would tend to justify the assumption of a multifractal nature for these series. This work was achieved as part of the National Programme of Research in Hydrology of the INSU (project 99 PNRH 27). The authors are grateful to A. Barcello for providing them the data of the Island of La Réunion Island.     

Scale factor for lengths of the geopotential model

Summary Using the data in[1], the scale factor for lengths is derived of the geopotential model R ₀ =GM/W ₀ (W ₀ is the potential on a generalized geoid). The resulting value, R ₀ ==6 363 672.9 m, which is2 m less than the original value[5], is practically the same as that in[6].     

弹性波传播中由微结构相互作用导致的尺度效应分析

相比于传统弹性波动方程,非对称弹性波动方程增加的独立自由项,包含有介质特征尺度参数.基于非对称弹性波动方程,可以分析弹性波传播中,由介质内微孔缝隙结构相互作用所导致的地震波传播尺度效应.本文从介质应变能密度函数出发,并结合几何方程和平衡方程,给出修正偶应力理论下的非对称弹性波动方程以及对应的非对称SH型横波波动方程的数...     

断层地震孕育和发生的不稳定性模型

在以前的工作中,考虑直立走滑型断层地震,假设断层面微元破裂强度遵循Weibull概率分布,由细观力学方法推导出断层面的宏观本构关系是一个非线性函数,表现为弹性-软化塑性特征,在此基础上用稳定性理论研究了地震稳定性问题.而实际断层大多是倾斜的,为此,本文首先建立了由围岩和倾斜断层构成的平面地震力学模型,采用宏观的断层载荷-变形的全过程曲线,详细讨论了倾斜断层地震的不稳定性问题.结果表明,远场一旦施加位移,断层也同时错动,这可能与实际情况不符合.为了更好的模拟断层的初始能量累计过程,进一步对断层本构模型进行改进.考虑断层面破裂强度,采用Coulomb破裂准则,则断层表现为刚塑性本构关系,只有当断层面剪应力达到一个临界值时,断层才开始错动.研究表明,对于倾斜断层地震,与直立走滑型断层地震一样,系统刚度比β（围岩切线刚度与断层刚度最大值之比）是决定地震失稳的重要参数,只有当ββ≥1时,仅仅是断层无震滑动,不会发生地震.在远场应施以位移形式边界条件,以致地震失稳发生在平衡路径的位移转向点并伴有应力突跳.