Turbulent flow structure in a gravel-bed river: Markov chain analysis of the fluctuating velocity profile

Three electromagnetic current meter probes were deployed in a Canadian gravel-bed river to obtain simultaneous records at 10 Hz of streamwise (u) and vertical (v) velocity components at three heights above the bed. By looking at the positive and negative signs of the instantaneous fluctuations from the time-average values of each velocity component at each height, the fluctuating velocity profile of u or v can be treated as a Markov chain with eight states and its statistical properties can be tested against null hypotheses based on the absence of spatial structure. We report results of this novel approach. The most common states of the u profile were those with either higher-than-average or lower-than-average velocities at all heights; these ‘high speed’ and ‘low speed’ states persisted for up to 3 s. The most common v profiles were all-upwards or all-downwards, but these persisted for shorter times than the high speed and low speed u profiles. Analysis of transition probabilities shows statistically significant tendencies for acceleration from the low speed u profile, and change from all-upwards to all-downwards v profile, to take place progressively from the uppermost probe downwards, in a sweep-like way. Deceleration from the high speed to low speed u profile and change from all-downwards to all-upwards v profile (burst-like behaviour) do not show such clear patterns. The results are interpreted in terms of the advection of inverted wedges of relatively high-momentum fluid, followed by more chaotic structures. A separate set of flow visualization experiments over a mixed gravel bed in a flume supports the presence of advected wedge structures, the decelerating part of the sequence corresponding to irregular ejections of near-bed fluid.     

基于Markov链模型的相控随机建模

Markov链模型在储层随机建模中的作用越来越受到关注,但其多用于类型属性(岩相、沉积相、沉积亚相等)的模拟,对于连续型属性(孔隙度、渗透率、含油气饱和度等)的模拟还比较困难.本文提出用Markov链模型相控建模方法模拟连续型属性的思路,即首先用Markov链模型模拟出类型属性,其次在类型属性约束下模拟出连续型属性,从而解决连续型属性不能产生突变边界的问题.最后应用此方法进行了模拟实验,模拟结果显示不同岩相中孔隙度差异较大,而同种岩相中孔隙度变化较小,证明了此方法的可靠性和适用性.     

A study for improving precipitation occurrences modelling with a Markov chain

Daily precipitation occurrences and their monthly wet-days' sums of precipitation-measuring stations in Greece are modelled with a Markov chain. The order of the chain is taken to be seasonally varying in accordance with the precipitation station's meteorological conditions and geographical location. The modelling efficiency of the Markov chain is significantly improved when it is conjunctively used with a second-order autoregressive stochastic model fitted on the monthly wet-days' sums.     

Analysis of agricultural drought in Iiuni,Eastern Kenya: application of a Markov model

In semi‐arid Kenya, episodes of agricultural droughts of varying severity and duration occur. The occurrence of these agricultural droughts is associated with seasonal rainfall variability and can be reflected by seasonal soil moisture deficits that significantly affect crop performance and yield. The objective of this study was to stochastically simulate the behaviour of dry and wet spells and rainfall amounts in Iiuni watershed, Kenya. The stochastic behaviour of the longest dry and wet spells (runs) and largest rainfall amounts were simulated using a Markov (order 1) model. There were eight raingauge stations within the watershed. The entire analysis was carried out using probability parameters, i.e. mean, variance, simple and conditional probabilities of dry and rain days. An analysis of variance test (ANOVA ) was used to establish significant differences in rainfall characteristics between the eight stations. An analysis of the number of rain days and rainfall amount per rain day was done on a monthly basis to establish the distribution and reliability of seasonal rainfall. The graphic comparison of simulated cumulative distribution functions (Cdfs) of the longest spells and largest rainfall amounts showed Markovian dependence or persistence. The longest dry spells could extend to 24 days in the long rainy season and 12 in the short rainy season. At 50% (median) probability level, the largest rainfall amounts were 91 mm for the long rainy season and 136 mm for the short rainy season. The short rains were more reliable for crop production than the long rains. The Markov model performed well and gave adequate simulations of the spells and rainfall amounts under semi‐arid conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Transition probability/Markov chain analyses of DNAPL source zones and plumes

At sites where a dense nonaqueous phase liquid (DNAPL) was spilled or released into the subsurface, estimates of the mass of DNAPL contained in the subsurface from core or monitoring well data, either in the nonaqueous or aqueous phase, can be highly uncertain because of the erratic distribution of the DNAPL due to geologic heterogeneity. In this paper, a multiphase compositional model is applied to simulate, in detail, the DNAPL saturations and aqueous-phase plume migration in a highly characterized, heterogeneous glaciofluvial aquifer, the permeability and porosity data of which were collected by researchers at the University of Tübingen, Germany. The DNAPL saturation distribution and the aqueous-phase contaminant mole fractions are then reconstructed by sampling the data from the forward simulation results using two alternate approaches, each with different degrees of sampling conditioning. To reconstruct the DNAPL source zone architecture, the aqueous-phase plume configuration, and the contaminant mass in each phase, one method employs the novel transition probability/Markov chain approach (TP/MC), while the other involves a traditional variogram analysis of the sampled data followed by ordinary kriging. The TP/MC method is typically used for facies and/or hydraulic conductivity reconstruction, but here we explore the applicability of the TP/MC method for the reconstruction of DNAPL source zones and aqueous-phase plumes. The reconstructed geometry of the DNAPL source zone, the dissolved contaminant plume, and the estimated mass in each phase are compared using the two different geostatistical modeling approaches and for various degrees of data sampling from the results of the forward simulation. It is demonstrated that the TP/MC modeling technique is robust and accurate and is a preferable alternative compared to ordinary kriging for the reconstruction of DNAPL saturation patterns and dissolved-phase contaminant plumes.     

Estimating threshold-exceeding probability maps of environmental variables with Markov chain random fields

Estimating and mapping spatial uncertainty of environmental variables is crucial for environmental evaluation and decision making. For a continuous spatial variable, estimation of spatial uncertainty may be conducted in the form of estimating the probability of (not) exceeding a threshold value. In this paper, we introduced a Markov chain geostatistical approach for estimating threshold-exceeding probabilities. The differences of this approach compared to the conventional indicator approach lie with its nonlinear estimators—Markov chain random field models and its incorporation of interclass dependencies through transiograms. We estimated threshold-exceeding probability maps of clay layer thickness through simulation (i.e., using a number of realizations simulated by Markov chain sequential simulation) and interpolation (i.e., direct conditional probability estimation using only the indicator values of sample data), respectively. To evaluate the approach, we also estimated those probability maps using sequential indicator simulation and indicator kriging interpolation. Our results show that (i) the Markov chain approach provides an effective alternative for spatial uncertainty assessment of environmental spatial variables and the probability maps from this approach are more reasonable than those from conventional indicator geostatistics, and (ii) the probability maps estimated through sequential simulation are more realistic than those through interpolation because the latter display some uneven transitions caused by spatial structures of the sample data.     

Prediction of hydrological drought durations based on Markov chains: case of the Canadian prairies

Abstract

Hydrological drought durations (lengths) in the Canadian prairies were modelled using the standardized hydrological index (SHI) sequences derived from the streamflow series at annual, monthly and weekly time scales. The rivers chosen for the study present high levels of persistence (as indicated by values exceeding 0.95 for lag-1 autocorrelation in weekly SHI sequences), because they encompass large catchment areas (2210–119 000 km²) and traverse, or originate in, lakes. For such rivers, Markov chain models were found to be simple and efficient tools for predicting the drought duration (year, month, or week) based on annual, monthly and weekly SHI sequences. The prediction of drought durations was accomplished at threshold levels corresponding to median flow (Q50) (drought probability, q?=?0.5) to Q95 (drought probability, q?=?0.05) exceedence levels in the SHI sequences. The first-order Markov chain or the random model was found to be acceptable for the prediction of annual drought lengths, based on the Hazen plotting position formula for exceedence probability, because of the small sample size of annual streamflows. On monthly and weekly time scales, the second-order Markov chain model was found to be satisfactory using the Weibull plotting position formula for exceedence probability. The crucial element in modelling drought lengths is the reliable estimation of parameters (conditional probabilities) of the first- and second-order persistence, which were estimated using the notions implicit in the discrete autoregressive moving average class of models. The variance of drought durations is of particular significance, because it plays a crucial role in the accurate estimation of persistence parameters. Although, the counting method of the estimation of persistence parameters was found to be unsatisfactory, it proved useful in setting the initial values and also in subsequent adjustment of the variance-based estimates of persistence parameters. At low threshold levels corresponding to q < 0.20, even the first-order Markov chain can be construed as a satisfactory model for predicting drought durations based on monthly and weekly SHI sequences.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2012. Prediction of hydrological drought durations based on Markov chains in the Canadian prairies. Hydrological Sciences Journal, 57 (4), 705–722.     

Probabilistic forecasting of drought: a hidden Markov model aggregated with the RCP 8.5 precipitation projection

The creeping characteristics of drought make it possible to mitigate drought’s effects with accurate forecasting models. Drought forecasts are inevitably plagued by uncertainties, making it necessary to derive forecasts in a probabilistic framework. In this study, we proposed a new probabilistic scheme to forecast droughts that used a discrete-time finite state-space hidden Markov model (HMM) aggregated with the Representative Concentration Pathway 8.5 (RCP) precipitation projection (HMM-RCP). The standardized precipitation index (SPI) with a 3-month time scale was employed to represent the drought status over the selected stations in South Korea. The new scheme used a reversible jump Markov chain Monte Carlo algorithm for inference on the model parameters and performed an RCP precipitation projection transformed SPI (RCP-SPI) weight-corrected post-processing for the HMM-based drought forecasting to perform a probabilistic forecast of SPI at the 3-month time scale that considered uncertainties. The point forecasts which were derived as the HMM-RCP forecast mean values, as measured by forecasting skill scores, were much more accurate than those from conventional models and a climatology reference model at various lead times. We also used probabilistic forecast verification and found that the HMM-RCP provided a probabilistic forecast with satisfactory evaluation for different drought categories, even at long lead times. In a drought event analysis, the HMM-RCP accurately predicted about 71.19 % of drought events during the validation period and forecasted the mean duration with an error of less than 1.8 months and a mean severity error of <0.57. The results showed that the HMM-RCP had good potential in probabilistic drought forecasting.     

Multi-year drought frequency analysis at multiple sites by operational hydrology - A comparison of methods

This paper compares two generators of yearly water availabilities from sources located at multiple sites with regard to their ability to reproduce the characteristics of historical critical periods and to provide reliable results in terms of the return period of critical sequences of different length. The two models are a novel multi-site Markov mixture model explicitly accounting for drought occurrences and a multivariate ARMA. In the case of the multisite Markov mixture model parameter estimation is limited to a search in the parameter space guided by the value of parameter λ to show the sensitivity of the model to this parameter. Application to two of the longest time series of streamflows available in Sicily (Italy) shows that the models can provide quite different results in terms of estimated return periods of historic droughts, although they seem to perform more uniformly when it comes to simulate drought-related statistics such as drought length, severity and intensity. The role of parameter selection for the multisite Markov mixture model and of the marginal probability of generated flows in providing results consistent with the characteristics of the observed series is discussed. Both models are applied to the system of sources supplying the city of Palermo (Sicily) and its environs showing the applicability of the newly developed multisite Markov mixture model to medium-to-large scale water resources systems.     

Spatio-temporal drought analysis in the Trakya region,Turkey

Abstract

Since droughts are natural phenomena, their occurrence cannot be predicted with certainty and thus it must be treated as a random variable. Once drought duration and magnitude have been found objectively, it is possible to plan for the transport of water in known quantities to drought-stricken areas either from alternative water resources or from water stored during wet periods. The summation of deficits over a particular period is referred to as the drought magnitude. Drought intensity is the ratio of drought magnitude to its duration. These drought properties at different truncation levels provide significant hydrological and hydrometeorological design quantities. In this study, the run analysis and z-score are used for determining drought properties of given hydrological series. In addition, kriging is used as a spatial drought analysis for mapping. This study is applied to precipitation records for Istanbul, Edirne, Tekirdag and Kirklareli in the Trakya region, Turkey and then the drought period, magnitude and standardized precipitation index (SPI) values are presented to depict the relationships between drought duration and magnitude.     

Markov chain Monte Carlo ground‐motion selection algorithms for conditional intensity measure targets

Two new algorithms are presented for efficiently selecting suites of ground motions that match a target multivariate distribution or conditional intensity measure target. The first algorithm is a Markov chain Monte Carlo (MCMC) approach in which records are sequentially added to a selected set such that the joint probability density function (PDF) of the target distribution is progressively approximated by the discrete distribution of the selected records. The second algorithm derives from the concept of the acceptance ratio within MCMC but does not involve any sampling. The first method takes advantage of MCMC's ability to efficiently explore a sampling distribution through the implementation of a traditional MCMC algorithm. This method is shown to enable very good matches to multivariate targets to be obtained when the numbers of records to be selected is relatively large. A weaker performance for fewer records can be circumvented by the second method that uses greedy optimisation to impose additional constraints upon properties of the target distribution. A preselection approach based upon values of the multivariate PDF is proposed that enables near‐optimal record sets to be identified with a very close match to the target. Both methods are applied for a number response analyses associated with different sizes of record sets and rupture scenarios. Comparisons are made throughout with the Generalised Conditional Intensity Measure (GCIM) approach. The first method provides similar results to GCIM but with slightly worse performance for small record sets, while the second method outperforms method 1 and GCIM for all considered cases.     

布谷鸟马尔科夫链蒙特卡洛混合高斯地质统计学随机反演

地质统计学随机反演可以获得比常规反演更高分辨率的结果,目前已成为储层高分辨率预测的主流方法.地下不同岩相储层参数存在明显差异,本文在地质统计学反演框架下构建了岩相和储层参数同步反演目标函数,实现不同岩相条件下储层参数分布精细描述.在求解该高维数据多参数同步反演问题时,本文将可以动态调节搜索步长的布谷鸟算法与马尔科夫链蒙...     

An advanced data collection procedure in bivariate drought frequency analysis

In recent years, the bivariate frequency analysis of drought duration and severity using independent drought events and copula functions has been under extensive application. Meanwhile, emphasis on the procedure of independent drought data collection leads to the omission of the actual potential of short-term extreme droughts within a long-term independent drought. However, a long-term individual continuous drought as an Unconnected Drought Runs can be considered as a combination of short-term Connected Drought Runs. Thus, an advanced and new procedure of data collection in the bivariate drought characteristics analysis has been developed in this study. The results indicated a high relative advantage of the new proposed procedure in analysing bivariate drought characteristics (i.e., drought duration and severity frequency analysis). This advantage has been reflected in the more appropriate determination of the best copula and significant reduction in the uncertainty of bivariate drought frequency analysis.     

Spatial mapping of drought in Zambia using regional frequency analysis

Regional frequency analysis based on L-moments was applied to assess the spatial extent of meteorological droughts in tandem with their return periods in Zambia. Weather station monthly rainfall data were screened to form homogeneous sub-regions-, validated by a homogeneity criterion and fitted by a generalized extreme value distribution using goodness-of-fit test statistics. Predictor equations at regional scale for L-moment ratios and mean annual precipitation were developed to generate spatial maps of meteorological drought recurrences. The 80% of normal rainfall level and two thresholds of 60% and 70% were synonymous with moderate and severe droughts, respectively. Droughts were more severe in the south than in the north of Zambia. The return periods for severe and moderate droughts showed an overlapping pattern in their occurrence at many locations, indicating that in certain years droughts can affect the entire country. The extreme south of Zambia is the most prone to drought.     

Hydrologic data assimilation using particle Markov chain Monte Carlo simulation: Theory,concepts and applications

During the past decades much progress has been made in the development of computer based methods for parameter and predictive uncertainty estimation of hydrologic models. The goal of this paper is twofold. As part of this special anniversary issue we first shortly review the most important historical developments in hydrologic model calibration and uncertainty analysis that has led to current perspectives. Then, we introduce theory, concepts and simulation results of a novel data assimilation scheme for joint inference of model parameters and state variables. This Particle-DREAM method combines the strengths of sequential Monte Carlo sampling and Markov chain Monte Carlo simulation and is especially designed for treatment of forcing, parameter, model structural and calibration data error. Two different variants of Particle-DREAM are presented to satisfy assumptions regarding the temporal behavior of the model parameters. Simulation results using a 40-dimensional atmospheric “toy” model, the Lorenz attractor and a rainfall–runoff model show that Particle-DREAM, P-DREAM_(VP) and P-DREAM_(IP) require far fewer particles than current state-of-the-art filters to closely track the evolving target distribution of interest, and provide important insights into the information content of discharge data and non-stationarity of model parameters. Our development follows formal Bayes, yet Particle-DREAM and its variants readily accommodate hydrologic signatures, informal likelihood functions or other (in)sufficient statistics if those better represent the salient features of the calibration data and simulation model used.     

用于钢框架结构二阶分析的一种实用塑性区数值模型

提出了一种基于有限元的考虑初始缺陷的塑性区模型,详细介绍了这种模型的单元类型、网格密度、材料属性、边界条件、求解方法以及残余应力和初始几何缺陷的施加,并通过和经典的Vogel框架计算结果的对比,验证了模型的可靠性。并利用这种模型分析了初始缺陷对于钢框架二阶弹塑性受力性能的影响。     

Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data

This study aims to model the joint probability distribution of periodic hydrologic data using meta-elliptical copulas. Monthly precipitation data from a gauging station (410120) in Texas, US, was used to illustrate parameter estimation and goodness-of-fit for univariate drought distributions using chi-square test, Kolmogorov–Smirnov test, Cramer-von Mises statistic, Anderson-Darling statistic, modified weighted Watson statistic, and Liao and Shimokawa statistic. Pearson’s classical correlation coefficient r _n , Spearman’s ρ _n, Kendall’s τ, Chi-Plots, and K-Plots were employed to assess the dependence of drought variables. Several meta-elliptical copulas and Gumbel-Hougaard, Ali-Mikhail-Haq, Frank and Clayton copulas were tested to determine the best-fit copula. Based on the root mean square error and the Akaike information criterion, meta-Gaussian and t copulas gave a better fit. A bootstrap version based on Rosenblatt’s transformation was employed to test the goodness-of-fit for meta-Gaussian and t copulas. It was found that none of meta-Gaussian and t copulas considered could be rejected at the given significance level. The meta-Gaussian copula was employed to model the dependence, and these results were found satisfactory.     

Critical analysis of the alternating direction implicit method of aquifer analysis

The alternating direction implicit method, when used for aquifer analysis, is liable to lead to erroneous results. By a careful examination of the effect of various factors, such as external boundaries, variable mesh intervals, internal boundaries with fixed heads, re-entrant boundaries and leakage, the occasions when these errors are serious have been identified. Techniques are described which can avoid these errors and lead to reliable numerical solutions.     

A comparative study of nonlinear Markov chain models for conditional simulation of multinomial classes from regular samples

Simulating fields of categorical geospatial variables from samples is crucial for many purposes, such as spatial uncertainty assessment of natural resources distributions. However, effectively simulating complex categorical variables (i.e., multinomial classes) is difficult because of their nonlinearity and complex interclass relationships. The existing pure Markov chain approach for simulating multinomial classes has an apparent deficiency—underestimation of small classes, which largely impacts the usefulness of the approach. The Markov chain random field (MCRF) theory recently proposed supports theoretically sound multi-dimensional Markov chain models. This paper conducts a comparative study between a MCRF model and the previous Markov chain model for simulating multinomial classes to demonstrate that the MCRF model effectively solves the small-class underestimation problem. Simulated results show that the MCRF model fairly produces all classes, generates simulated patterns imitative of the original, and effectively reproduces input transiograms in realizations. Occurrence probability maps are estimated to visualize the spatial uncertainty associated with each class and the optimal prediction map. It is concluded that the MCRF model provides a practically efficient estimator for simulating multinomial classes from grid samples.