Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

A survey of numerical methods for stochastic differential equations

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.     

A survey of numerical methods for stochastic differential equations

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.     

Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Development of stochastic partial differential equations for subsurface hydrology

The use of stochastic models in subsurface hydrology is growing at a logistic pace. To tie together a number of different stochastic methodologies for deriving subsurface transport equations, we have put together a brief review of some of the more common techniques. Our attention is confined to a few select methodologies so that we might delve in detail into assumptions required by the various approaches and their strengths and weaknesses. The methods reviewed include: Martingale, stochastic-convective, stochastic-relativist, spectral-integral, perturbative, statistical-mechanical, and generalized hydrodynamics. Within this list, we also have included a few stochastic methodologies which have been used solely to develop expressions for the dispersion tensor.     

A stochastic differential equation approach to soil moisture

The dynamics of water within the unsaturated root zone of the soil are represented by a pair of stochastic differential equations (SDE's), one representing the so-called surplus state of the moisture and the other the deficit condition. The inputs to the model are the climatically controlled random infiltration events and evapotranspiration which are modeled as a compound Poisson process and a Wiener (Brownian motion) process, respectively.The solutions to these SDE's are not in close-form but sample functions are obtained by numerical integration. The moment properties of the soil moisture evolution process have also been derived analytically including the mean, variance, covariance and autocorrelation functions.To illustrate the model, climatic parameters representing the surplus and deficit cases and properties of clay loam soil have been used to numerically derived the corresponding sample functions. With proper selection of all the parameters, physically realistic sample trajectories can be obtained for the model.     

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h  ) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity (Y=lnK)$(Y=\ln K)$, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.     

CHAMP型卫星定轨顾及非线性改正的轨道扰动方程

本文针对CHAMP型卫星建立了顾及非线性改正的轨道扰动方程定轨理论与方法.首先从卫星运动的二阶微分方程出发,引入了正常引力位以及相应的参考轨道,然后分别推导了线性化轨道扰动方程与顾及非线性改正的轨道扰动方程,同时说明了建立的线性化轨道扰动方程与目前处理CHAMP卫星数据的动力学定轨方法是等价的.其次分别对线性化轨道扰动方程与顾及非线性改正的轨道扰动方程的精度进行了估计,在卫星定位精度为3 cm与非惯性力测量精度为3×10^-10m·s^-2的前提下证明了下列结论：当参考轨道与实际轨道之间的距离ρ≤4.7 m时线性化轨道扰动方程的精度能达到非惯性力的测量精度以及当ρ≤4.14×10³ m时顾及非线性改正的轨道扰动方程能达到非惯性力的测量精度.由此便可得出结论：相对于线性化轨道扰动方程,顾及非线性改正的轨道扰动方程具有更高的精度,且适合在更长的时间弧段上建立关于引力场位系数的法方程组,特别是针对CHAMP卫星计划进行的模拟计算也完全验证了该结论.最后利用叠加原理,给出了顾及非线性改正的轨道扰动方程的求解方法.此外,还针对GRACE卫星计划利用顾及非线性改正的轨道扰动方程进行了恢复引力场的模拟计算,结果表明：分段建立位系数的法方程组时子弧段分别取值2 h、1 d、6 d对恢复引力场的结果几乎不产生影响,这表明在处理GRACE数据时能够以6 d的弧长来建立法方程组.

     

A preliminary study on vegetation-erosion dynamics and its applications

The development of vegetation in mountainous and hilly areas depends on the local climate, precipitation, soil texture, parent material, topography, soil erosion, types of land use and human activities. Among them, soil erosion is an important natural factor and impacts the vegetation development directly and indirectly, changing the morphology and even changing the cli-mate. An erosion-induced efflux of carbon is about 1.14 billion tons per year totally from soil to the at-mosphere, which gen…     

Modeling the ionosphere wind dynamo: A review

This paper reviews the current status of research concerned with modeling the ionospheric wind dynamo. Simulation models have been reasonably successful in reproducing the types of magnetic perturbations that are produced by the dynamo. Ionospheric electric fields are less well simulated, particularly at night. The primary areas of research needed to improve our ability to simulate realistically the ionospheric wind dynamo are in the modeling of nighttime conditions, hemispherical asymmetries of thermospheric tides, and mutual dynamic coupling among winds, conductivities, electric fields, and electric currents.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

刚毛藻(Cladophora)生态学研究进展

刚毛藻(Cladophora)在全球海洋及淡水中普遍存在,它为沿岸带微生物和无脊椎动物提供了生产力和微生存空间.随着水体富营养化程度的加剧,刚毛藻在沿岸带出现暴发现象,造成严重的环境污染,同时刚毛藻的存在也影响了湖泊生态修复过程中沉水植被的恢复.本文从刚毛藻形态与种类鉴定、生长与分布影响因素、生长周期与种群动态、种间关系及生物去除等研究前沿方面进行综述,并对相关研究领域进行展望,为进一步研究刚毛藻生态生理特征、有效控制和去除提供科学依据.     

A Free Boundary Problem Related to the Location of Volcanic Gas Sources

A mathematical model of general gas emitting systems is derived, and a sample of relevant mathematical results is offered. The present paper indicates that shallow subsurface gas sources in typical volcanic areas can be located if appropriate physico-chemical measurements are made on the Earths surface and put to use.     

基于偏微分方程的定向插值在偏移成像中的应用

对于射线类偏移成像来说,求解射线追踪系统中所涉及的属性值不在网格节点上的插值计算问题是一个非常重要的环节,它影响到求解走时、路径和振幅信息的计算效率和精度,进而影响到整个偏移成像的质量和效率.本研究根据速度模型的空间梯度特点,考虑被插值点处速度的梯度在横向和纵向的分布特征,构建基于速度梯度空间变化的偏微分方程算法,将近几年发展起来的基于偏微分方程的定向插值算法引入到射线类偏移成像当中,实现射线追踪当中涉及的属性值不在网格节点上的插值计算.由于偏微分方程法本身固有的特性(局部特征不变性、解的唯一性和线性叠加性),因此,该算法可以实现不破坏原始速度模型空间梯度结构的非网格节点属性的插值计算.通过在常用的速度模型上的插值计算对比、不同速度模型上射线路径对比分析以及复杂介质模型上最后的偏移成像结果分析可以得出,应用基于速度梯度构建的偏微分方程插值算法在进行插值计算的过程当中可以实现不破坏原始速度模型空间速度梯度结构的属性计算,同时应用该算法可以最终提高射线类偏移成像的质量.

     

基于随机地震反演的Russell流体因子直接估算方法

本文研究了一种基于随机地震反演的Russell流体因子直接估算方法,该方法是一种基于蒙特卡罗的非线性反演,能够有效地融合测井资料中的高频信息,提高反演结果的分辨率.本文应用贝叶斯理论框架,首先通过测井数据计算井位置处的Russell流体因子,利用序贯高斯模拟方法(sequential Gaussian simulation, SGS)得到流体因子的先验信息;然后构建似然函数;最后利用Metropolis抽样算法对后验概率密度进行抽样,得到反演的Russell流体因子.其中对每道数据进行序贯高斯模拟时,采用一种新的逐点模拟方式,具有较高的计算速度.数值试验表明:反演结果与理论模型和实际测井数据吻合较好,具有较高的分辨率,对于判识储层含流体特征具有较好的指示作用.     

Estimating the size of an earthquake using short-period seismograms of the first three seconds: A simulated experiment using the 1999 Chi-Chi earthquake sequence

Most of the present earthquake early warning systems are based on broadband or strong motion recordings. How-ever, the short-period instruments are still deployed. It is well-known that short-period recordings have saturation problems for large earthquakes when estimating the size of an earthquake. Thus, it is necessary to make clear the magnitude at which saturation starts to occur for the commonly used τc and Pd measurements, respectively. To investigate the possibility of using short-period seismic recordings for earthquake early warning, we conducted a simulated experiment using the strong motion data of the 1999 Chi-Chi earthquake sequence including its main shock and 31 aftershocks, with magnitude span from 4 to 7.6. The strong motion acceleration recordings were convolved with the instrument response of short-period seismographs in northern China to simulate short-period seismograms. Parameters τc and Pd from the first-three-second seismograms were calculated for the simulated short-period recordings and compared with that obtained by the original strong ground motion recordings. The result showed that to some extent, short-period recordings can be used for threshold earthquake early warning, while the magnitude saturation of Pd estimation can be up to 6.5, better than τc estimation.     

The discrete wave number formulation of boundary integral equations and boundary element methods: A review with applications to the simulation of seismic wave propagation in complex geological structures

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.     

A stochastic model for daily subsurface CO2 concentration and related soil respiration

Near-surface soil CO₂ gas-phase concentration (C) and concomitant incident rainfall (P_i) and through-fall (P_t) depths were collected at different locations in a temperate pine forest every 30 min during the 2005 and 2006 growing seasons (and then averaged to the daily timescale). At the daily scale, C temporal variations were well described by a sequence of monotonically decreasing functions interrupted by large positive jumps induced by rainfall events. A stochastic model was developed to link rainfall statistics responsible for these jumps to near-surface C dynamics. The model accounted for the effect of daily rainfall variability, both in terms of timing and amount of water, and permitted an analytical derivation of the C probability density function (pdf) using the parameters of the rainfall pdf. Given the observed positive correlation between daily C and soil CO₂ fluxes to the atmosphere (F_s), the effects of various rainfall regimes on the statistics of F_s can be deduced from the behavior of C under different climatic conditions. The predictions from this analytical model are consistent with flux measurements reported in manipulative experiments that varied rainfall amount and frequency.     

A scattering model for seismic attenuation and its global applications

The attenuation characteristics of Indian lithosphere and its comparison with different tectonic settings in the world are determined from the observations of the Q for L_g(Q_Lg)-, and S(Q_S)-waves in the 1-30 Hz frequency range. The scattering is approximated with a Gaussian distribution of spherical scatterers. To approximate single scattering, we use Dainty's [Geophy. Res. Lett. 8 (11) (1981) 1126] model that attenuation is given by 1/Q(ω) = 1/Q_i + g(ω)v/ω, where Q_i is intrinsic Q due to anelastic attenuation, v is shear wave velocity, ω is angular frequency, g = ∫n(a)σ da is the total scattering coefficient for S-to-S scattering, n(a) da is the number of scattering spheres of radius a per unit volume, and σ is the scattering cross-section for the sphere. We find that if n(a) is described by a simple two parameter (a₀ and c) Gaussian of amplitude c and standard deviation and mean a₀, the attenuation data for different regions of the world are well approximated over the frequency band of seismic observations. Our major findings are: (1) the maximum effect of scattering on attenuation occurs at 0.84 Hz or a wavelength of 4.16 km; (2) the values of g are frequency dependent. Values of g are of the order of 10⁻³ km⁻¹ at 1-30 Hz, varying from 0.0031 to 0.01 and 0.001 to 0.0083 km⁻¹ for tectonically active and stable regions, respectively; (3) regions of active tectonics and seismicity generally have lower Q_i values (1000) than that in stable regions (2000); and (4) regions of high Q_i value exhibit low intensity of scattering.