Stochastic dynamic stiffness of a surface footing for offshore wind turbines: Implementing a subset simulation method to estimate rare events

The purpose of this study, which concerns the stochastic dynamic stiffness of foundations for large offshore wind turbines, is to quantify uncertainties related to the first natural frequency of a turbine supported by a surface footing and to estimate the low event probabilities. Herein, a simple model of a wind turbine structure with equivalent coupled springs at the base is calibrated with the mean soil property values. A semianalytical solution, based on the Green׳s function for a layered half-space is utilized for estimation of foundation responses. Soil elastic modulus and layer depth are considered as random variables with lognormal distributions. The uncertainties are quantified, and the estimation of rare events of the first natural frequency is discussed through an advanced reliability approach based on subset simulation. This analysis represents a first step in the estimation of the safety with respect to the failure of a turbine in the fatigue limit state.     

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.     

海陆风及沿海风速廓线在风电场风速预报中的应用

为了建立沿海风功率预报系统,本文探讨了中国沿海风电场风速预报问题,并利用数值模式RAMS对海陆风进行了模拟研究.发现海陆风发生时,海风和陆风阶段风速廓线存在较大差异,海风阶段风速的垂直切变明显小于陆风阶段.海陆风发生时,风速会呈现有规律的变化,即海风和陆风分别有两个时段：风速增加时段和风速减少时段.在为沿海风电场提供风速预报时,当模式预报到海陆风发生时,可以利用海陆风的这种特点,使用统计方法对预报出的风速进行有效的订正.并发现即使没有海陆风发生,当风向为海洋吹向陆地时,风速随高度的垂直切变同样小于陆地吹向海洋的时段.利用统计方法根据不同风向时风速廓线的特性,把数值模式计算高度上的预报结果,精确地插值到风机涡轮高度,会很大程度上减少风速预报的误差及风功率预报环节的误差.     

General solution to random advective-dispersive equation in porous media

This series of articles present general applications of functional-analytic theory to the solution of the partial differential equation describing solid transport in aquifers, when either the evolution of the system, the sources, the parameters and/or the boundary conditions are prescribed as stochastic processes in time or in space. This procedure does not require the restricting assumptions placed upon the current particular solutions on which today's stochastic transport theory is based, such as small randomness assumptions (perturbation techniques), Montecarlo simulations, restriction to small spatial stochasticity in the hydraulic conductivity, use of spectral analysis techniques, restriction to asymptotic steady state conditions, and restriction to variance of the concentration as the only model output among others. Functional analysis provides a rigorous tool in which the concentration stochastic properties can be predicted in a natural way based upon the known stochastic properties of the sources, the parameters and/or the boundary conditions. Thus the theory satisfies a more general modeling need by providing, if desired, a systematic global information on the sample functions, the mean, the variance, correlation functions or higher-order moments based on similar information of any size, anywhere, of the input functions. Part I of this series of articles presents the main relevant results of functional-analytic theory and individual cases of applications to the solution of distributed sources problems, with time as well as spatial stochasticity, and the solution subject to stochastic boundary conditions. It was found that the stochastically-forced equation may be a promising model for a variety of random source problems. When the differential equation is perturbed by a time and space stochastic process, the output is also a time and space stochastic process, in contrast with most of the existing solutions which ignore the temporal component. Stochastic boundary conditions seems to quickly dissipate as time increases.     

Stochastic delineation of well capture zones

In this work, we describe a stochastic method for delineating well capture zones in randomly heterogeneous porous media. We use a moment equation (ME) approach to derive the time-dependent mean capture zones and their associated uncertainties. The mean capture zones are determined by reversely tracking the non-reactive particles released at a small circle around each pumping well. The uncertainty associated with the mean capture zones is calculated based on the particle displacement covariances for nonstationary flow fields. The flow statistics are obtained either by directly solving the flow moment equations derived with a first-order ME approach or from Monte Carlo simulations (MCS) of flow. The former constitutes a full ME approach, and the latter is a hybrid ME-MCS approach. This hybrid approach is invoked to examine the validity of the transport component of the stochastic method by ensuring that the ME and MC transport approaches have the same underlying flow statistics. We compared both the full ME and the hybrid ME-MCS results with those obtained with a full MCS approach. It has been found that the three approaches are in excellent agreement when the variability of hydrologic conductivity is small (_Y²=0.16). At a moderate variability (_Y²=0.5), the hybrid ME-MCS and the full MCS results are in excellent agreement whereas the results from the full ME approach deviate slightly from the full MCS results. This indicates that the (first-order) ME transport approach renders a good approximation at this level of variability and that the first-order ME flow approximation may not be sufficiently accurate at this variability in the case of divergent/convergent flow. The first-order ME flow approach may need to be corrected with higher-order terms even for moderate _Y² although the literature results reveal that the first-order ME flow approach is robust for uniform mean flow (i.e., giving accurate results even with _Y² as large as four).     

Stochastic analysis of estuarine hydraulics

A new methodology is presented for the solution of the stochastic hydraulic equations characterizing steady, one-dimensional estuarine flow. The methodology is predicated on quasi-linearization, perturbation methods, and the finite difference approximation of the stochastic differential operators. Assuming Manning's roughness coefficient is the principal source of uncertainty in the model, stochastic equations are presented for the water depths and flow rates in the estuarine system. Moment equations are developed for the mean and variance of the water depths. The moment equations are compared with the results of Monte Carlo simulation experiments. The results confirm that for any spatial location in the estuary that (1) as the uncertainty in the channel roughness increases, the uncertainty in mean depth increases, and (2) the predicted mean depth will decrease with increasing uncertainty in Manning'sn. The quasi-analytical approach requires significantly less computer time than Monte Carlo simulations and provides explicit     

Stochastic generation of hourly rainstorm events

Occurrence of rainstorm events can be characterized by the number of events, storm duration, rainfall depth, inter-event time and temporal variation of rainfall within a rainstorm event. This paper presents a Monte-Carlo based stochastic hourly rainfall generation model considering correlated non-normal random rainstorm characteristics, as well as dependence of various rainstorm patterns on rainfall depth, duration, and season. The proposed model was verified by comparing the derived rainfall depth–duration–frequency relations from the simulated rainfall sequences with those from observed annual maximum rainfalls based on the hourly rainfall data at the Hong Kong Observatory over the period of 1884–1990. Through numerical experiments, the proposed model was found to be capable of capturing the essential statistical features of rainstorm characteristics and those of annual extreme rainstorm events according to the available data.     

随机有限断层法及其工程应用中的问题分析

随机有限断层法作为半经验半理论的地震动合成方法,综合考虑了震源、传播路径以及场地条件对地震动的影响,可在工程关注的频率范围内模拟地震动时程,为实现较大区域地震动模拟提供了一种有效的方法。本文系统阐述了随机有限断层法的基本原理、静力学拐角频率模型及动拐角频率模型的发展;分析了主要模型参数（断层尺度、破裂速度、剪切波速及应力降）、Kappa因子和介质品质因子的取值原则,并结合中国大陆的实际情况给出了相应的取值范围;最后讨论了现有模拟方法在考虑参数取值、子断层划分和子断层之间相互作用等方面的不足以及能量处理方式上的缺陷,明确了随机有限断层法发展的方向。     

热带气旋风场模型构造及特征参数估算

探讨了利用气旋风场分布的经验模型估算热带气旋尺度(8级大风圈半径)的方法.用美国联合台风警报中心整编的2001年西北太平洋热带气旋的“最佳尺度”资料,确定了各模型的经验常数,并计算了各模型的估算精度.结果表明,“VBogus”模型能获得热带气旋(Tropical Cyclone,简称TC)尺度的较好估算.基于“VBogus”模型,通过拟合热带气旋尺度的非对称分布,构造了能描述热带气旋非对称风场的"修正VBogus"模型,并估算了该模型中各参数在不同季节和不同地理区域的取值,为热带气旋尺度变化和非对称结构机制等问题的研究和应用提供新依据.     

Recursive cross-entropy downscaling model for spatially explicit future land uses: A case study of the Heihe River Basin

Downscaling methods assist decision makers in coping with the uncertainty regarding sustainable local area developments. In particular, they allow investigating local heterogeneities regarding water, food, energy, and environment consistently with global, national, and sub-national drivers and trends. In this paper, we develop a conceptual framework that integrates a partial equilibrium Global Biosphere Management Model (GLOBIOM) with a dynamic cross-entropy downscaling model to derive spatially explicit projections of land uses at 1-km spatial resolution from 2010 to 2050 relying on aggregate land demand projections. The fusion of the two models is applied in a case study in Heihe River Basin to analyze the extent of potential cropland, grassland, and unused land transformations, which may exacerbate already extensive water consumption caused by rapid expansion of irrigated agriculture in the case study region. The outcomes are illustrated for two Shared Socioeconomic Pathway scenarios. The kappa coefficients show that the downscaling results are in agreement with the land use and land cover map of the Heihe River Basin, which indicates that the proposed approach produces realistic local land use projections. The downscaling results show that under both SSP scenarios the cropland area is expected to increase from 2010 to 2050, while the grassland area is projected to increase sharply from 2010 to 2030 and then gradually come to a standstill after 2030. The results can be used as an input for planning sustainable land and water management in the study area, and the conceptual framework provides a general approach to creating high-resolution land-use datasets.     

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.     

Stochastic modeling of Iranian earthquakes and estimation of ground motion for future earthquakes in Greater Tehran

Strong-motion data from eight significant well-documented earthquakes in Iran have been simulated using a stochastic modeling technique for finite faults proposed by Beresnev and Atkinson [Bull Seismol Soc Am 87 (1997) 67–84; Seism Res Lett 69 (1998) 27–32]. The database consists of 61 three-component records from eight earthquakes of magnitude ranging from M 6.3 to M 7.4, recorded at hypocentral distances up to 200 km. The model predictions are in good agreement with available Iranian strong-motion data as evidenced by near-zero average of differences between logarithms of the observed and predicted values for all frequencies. The strength factor, sfact, a quantity that controls the high-frequency radiation from the source is determined, on an event-by-event basis, by fitting simulated to observed response spectra.     

Modelling the influence of thermal discharge under wind on algae

Wind-driven processes exert an important impact on aquatic ecosystems, especially on shallow reservoirs. Flow and heat transport under wind in the Douhe reservoir in China were simulated by a two-dimensional mathematical model. Areas corresponding to different temperature rises were calculated for different wind speed conditions with high frequency. It is shown that high temperature rise areas increase for maximum wind speed conditions while low temperature rise areas keep constant for various wind speed conditions. The concentration of Chl.a decreases with the increase of wind speed, indicating that low wind speed is suitable for algae blooming in the Douhe reservoir. The effects of wind on Bacillariophyta biomass growth become more obvious with the increase of temperature rise areas. The influenced areas of lower temperature rise (0.2–1.49 °C) and higher temperature rise (1.5–2 °C) zone are 8.57 × 10⁶ m² and 5.18 × 10⁶ m², respectively, and corresponding total variation amounts of Bacillariophyta biomass are 2.24 × 10⁵/m² and 0.42 × 10⁵/m², respectively. Results show that wind has a significant impact on ecological effects due to thermal discharge from thermal power plant into shallow reservoirs.     

随机方法模拟地震动的应用研究

对随机方法模拟地震动的实际应用进行了研究。通过实际模拟验证,此方法简单有效,可为结构抗震设计提供验算依据,同时为设计反应谱的拟合提供参照数据,有一定的实用性,但仍存在一定的不足。     

Identification of wind stress on shallow water surfaces by optimal smoothing

Using the state space approach, an on-line filter procedure for combined wind stress identification and tidal flow forecasting is developed. The stochastic dynamic approach is based on the linear twodimensional shallow water equations. Using a finite difference scheme, a system representation of the model is obtained. To account for uncertainties, the system is embedded into a stochastic environment. By employing a Kalman filter, the on-line measurements of the water-level available can be used to identify and predict the shallow water flow. Because it takes a certain time before a fluctuation in the wind stress can be noticed in the water-level measurements, an optimal fixed-lag smoother is used to identify the stress.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT

Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.     

Semigroup solutions to stochastic unsteady groundwater flow subject to random parameters

Two methods for the solution of partial differential equations (PDE) for the general case of random in time physical parameters are presented and their application to the solution of unsteady regional groundwater flow equations are illustrated. The first method is the semigroup approach which directly offers a solution without resorting to closure approximations (hierarchy techniques), perturbation techniques, or Montecarlo simulation techniques. The semigroup approach can also handle the general stochastic problem when randomness also appears as initial conditions, boundary conditions or forcing terms. The second method is an approximation scheme to obtain the semigroup solution in complex cases and permits the solution of equations with more than one random coefficient.