Markov chain model to study the occurrence of pre-monsoon thunderstorms over Bhubaneswar,India

The present work deals with pre-monsoon thunderstorms over Bhubaneswar belonging to the state of Orissa, India. A Markovian approach has been adopted to discern the probabilistic behavior of the time series of the occurrence and non-occurrence of this hazardous weather event by introducing a dichotomy within the time series. After a painstaking analysis through chi-square tests, we have identified serial independence in a few years and first-order two-state Markovian dependence in a few years (2000, 2001, 2004 and 2006). Finally, for the years of first-order two-state Markovian dependence, it has been observed that the probability of occurrence or non-occurrence of thunderstorm gets higher if the state of the previous day is similar to that of the current day. Furthermore, the probability of getting non-thunderstorm day followed by non-thunderstorm day is higher than the probability of getting thunderstorm day followed by thunderstorm day. It has been also observed that the unconditional climatological probability of the occurrence of severe pre-monsoon thunderstorm implied by the Markov chain is closely in agreement with the observed relative frequencies. However, it could be revealed that Markov chain cannot, in general, be suggested as a predictive tool for pre-monsoon thunderstorms under study without investigating the serial dependence inherent in the time series.     

Evaluating slope stability uncertainty using coupled Markov chain

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. Uncertainty in factor of safety (FS) and probability of failure (Pf) of slope induced by the geological uncertainty is not well studied in the past. This paper presents one approach to evaluate the uncertainty in FS and Pf of slope in the presence of geological uncertainty using borehole data. The geological uncertainty is simulated by an efficient coupled Markov chain (CMC) model. Slope stability analysis is then conducted based on the simulated heterogeneous soils. Effect of borehole layout schemes on uncertainty evaluation of FS and Pf is investigated. The results show that borehole within influence zone of the slope is essential for a precise evaluation of FS statistics and Pf. The mean of FS will converge to the correct answer as the borehole number increases.     

Inverse trishear modeling of bedding dip data using Markov chain Monte Carlo methods

We present a method for fitting trishear models to surface profile data, by restoring bedding dip data and inverting for model parameters using a Markov chain Monte Carlo method. Trishear is a widely-used kinematic model for fault-propagation folds. It lacks an analytic solution, but a variety of data inversion techniques can be used to fit trishear models to data. Where the geometry of an entire folded bed is known, models can be tested by restoring the bed to its pre-folding orientation. When data include bedding attitudes, however, previous approaches have relied on computationally-intensive forward modeling. This paper presents an equation for the rate of change of dip in the trishear zone, which can be used to restore dips directly to their pre-folding values. The resulting error can be used to calculate a probability for each model, which allows solution by Markov chain Monte Carlo methods and inversion of datasets that combine dips and contact locations. These methods are tested using synthetic and real datasets. Results are used to approximate multimodal probability density functions and to estimate uncertainty in model parameters. The relative value of dips and contacts in constraining parameters and the effects of uncertainty in the data are investigated.     

湖北刘家场地区奥陶系地层沉积相及马尔科夫链分析

根据野外实测剖面分析刘家场地区奥陶系从西陵峡组到临湘组的地层特征、古生物类型和沉积相演化规律,表明该区奥陶系以碳酸盐岩台地相序为主,主要由局限台地、开阔台地、台地边缘生物礁和沉没台地组成。在此基础上,运用马尔科夫链随机过程建立研究区沉积序列概率模式。以定量的数学地质方法估计了岩相在垂直剖面上的转移概率,克服了传统的定性沉积相模式建立方法中存在的由人为因素导致的差异,研究结果表明这是一种行之有效的将复杂地质问题简化的处理方法。     

Influence of the 2008 Wenchuan earthquake (Mw 7.9) on the occurrence probability of future earthquakes on neighboring faults

The Longquan–Shan fault and the Huya fault are two major neighboring faults of the Longmen–Shan fault zone where the 12 May 2008 Wenchuan earthquake (M_w 7.9) occurred. To study the influence of the Wenchuan event on these two active faults, we calculate changes of Coulomb stress on the Longquan–Shan fault and the Huya fault caused by the Wenchuan mainshock. Our results indicate that the Coulomb stress in the northern section (Zone A) of the Longquan–Shan fault is increased by 0.07–0.10 bars, that in the middle section (Zone B) by 0.04–0.11 bars, and that in the southern section (Zone C) shows almost no change. For the Huya fault, the Coulomb stress is decreased by 0.01–0.03 bars in the northern section (Zone A), 0.10–0.35 bars in the middle section (Zone B), and nearly 0.5 bars in the southern section (Zone C). The epicenter distribution of small earthquakes (M_L ⩾ 1.5) on the Longquan–Shan fault and the Huya fault after the Wenchuan earthquake is consistent with the distribution of the Coulomb stress change. This implies that the Wenchuan earthquake may have triggered small events on the Longquan–Shan fault, but inhibited those on the Huya fault. We then use the rate/state friction law to calculate the occurrence probability of future earthquakes in the study region for the next decade. They include the distribution of b-values, magnitude of completeness (M_c), the background seismicity rate, a value of Aσ_n and the duration for the transient effect (t_a) in the study region. We also estimate the earthquake occurrence probabilities on the neighboring faults after the Wenchuan earthquake. Our results show that, the occurrence probability of future earthquakes in the Longquan–Shan has a slight increase, being 7% for M ⩾ 5.0 shocks during the next decade, but the earthquake probability in the Huya region is reduced obviously, being 5–20%, 7–26% and 3–9% for M ⩾ 5.0 shocks during the next decade in sections A, B and C of the Huya fault, respectively.     

含水层非均质结构的马尔可夫链地质统计方法及应用

针对含水层非均质性对地下水流动和溶质运移模拟的准确性有着重要的影响,但非均质性的刻画十分困难的问题,介绍了马尔可夫转移概率理论方法及其在含水层非均质结构研究中的应用,利用Walther定律建立了三维马尔可夫链,并将该方法用于研究位于山西省介休市龙凤河冲积扇岩相的空间分布规律.结果表明该地区细颗粒具有在垂直方向上向上沉积、水平方向上向外沉积的趋势,细颗粒岩相与粗颗粒岩相之间的关联性强而粗颗粒岩相之间的关联性差.该方法的缺点在于Walther定律的应用可能会增加不确定性.     

边坡失效概率估计的高斯过程动态响应面法

针对传统响应面法在求解具有高度非线性隐式功能函数边坡可靠性问题上的局限性,采用适用于处理高维度、小样本、非线性回归问题的高斯过程回归模型构建隐式功能函数的响应面,将高斯过程响应面与蒙特卡罗模拟法相结合,通过构造合理的迭代方式,在利用高斯过程回归模型的不确定性评价功能获取最优采样点的基础上,实现了高斯过程响应面动态更新,由此提出了边坡失效概率快速估计的高斯过程动态响应面法。利用数值算例验证了该方法的有效性,在此基础上对3个边坡算例进行了可靠性分析。结果表明,与传统响应面法相比较,该方法计算精度与计算效率明显较高,易于与既有的边坡分析软件相结合,且实现容易,适用于边坡可靠性的快速分析。     

砂土地震液化的模糊概率评判方法

利用模糊数学中的模糊概率理论,建立了砂土地震液化的模糊概率综合评判模型。在此模型中,提出了模糊权重的概念,可充分考虑权重的模糊性,从而避免权重取值带来的不确定性。结合砂土地震液化特点,选取地震烈度、标准贯入击数、平均粒径和上覆有效压力作为主要评价影响因子,同时将液化程度划分为不液化、轻微液化、中等液化和严重液化4个等级,进而使其评判结果更为精细化。通过算例分析,表明文中方法对砂土液化评判的合理性与有效性。     

基于权马尔可夫链模型的庐江县降水量预测

由于降水过程的随机性与不确定性,使得降水量预测存在一定的难度。对安徽省庐江县1952~2009年逐年降水资料进行了分析,采用样本-标准差分级法将这58年的逐年降水量序列分为枯水年,偏枯水年,平水年,偏丰水年,丰水年5个状态,采用权马尔可夫链模型预测了2010年的降水量,预测结果与实测结果相吻合。     

基于概率积分法的潜山型地裂缝空间预测方法研究

苏南地区锡北镇地裂缝灾害活跃,且为典型的基岩潜山型抽水沉降地裂缝。以无锡锡北镇杨墅里地裂缝为研究对象（31.703174°~31.705488°N,120.452707°~120.453410°E）,基于非连续介质理论,运用概率积分方法建立基岩潜山条件下的抽水沉降裂缝空间预测数学模型。运用该理论模型计算得出基岩潜山上覆岩土体的差异性沉降规律、地表倾斜程度及地表曲率变化规律,并指出地表倾斜程度与地表曲率为地裂缝易发位置的重要评价指标。将地裂缝空间预测数学模型计算出的沉降规律、地裂缝发生位置等结果与物理模型试验结果进行对比研究,结果显示：1）当W"（x）=0时,地表倾斜函数取得极值,此时x=±0.8 m,表明距模型左边界2.6 m与4.2 m处地表倾斜程度最大,差异性沉降最为明显。2）当W"'（x）=0时,地表曲率函数取得极值,x=0,表明在距模型左边界3.4 m处地表曲率最大,为拉应力集中位置。3）模型试验地裂缝集中发生2.4~2.6 m、3.3~3.5 m和3.9~4.2 m这3处;基于概率积分方法的潜山型地裂缝空间预测结果与物理模型试验结果基本一致。预测模型能够合理解释物理模型试验中潜山山腰及潜山山顶三处地裂缝的发生位置,验证了其正确性,为地裂缝空间预测方法提供了新的研究思路。

     

基于Markov chain理论的科技成果转化的市场预测

科技成果成功转化的主要标志是其产品的市场畅销度。产品的畅销度由于受各种内外部不确定因素影响，具有随机不确定性。基于这种随机不确定性，运用Markov chain对其产品进行市场预测，并通过实证分析说明其有效性，为科技成果的转化提供具体可行的定量依据。     

A Fixed-Path Markov Chain Algorithm for Conditional Simulation of Discrete Spatial Variables

The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics. In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables (i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation and spatial uncertainty analysis of discrete spatial variables.     

Markov chain Monte Carlo methods for conditioning a permeability field to pressure data

Generating one realization of a random permeability field that is consistent with observed pressure data and a known variogram model is not a difficult problem. If, however, one wants to investigate the uncertainty of reservior behavior, one must generate a large number of realizations and ensure that the distribution of realizations properly reflects the uncertainty in reservoir properties. The most widely used method for conditioning permeability fields to production data has been the method of simulated annealing, in which practitioners attempt to minimize the difference between the ’ ’true and simulated production data, and “true” and simulated variograms. Unfortunately, the meaning of the resulting realization is not clear and the method can be extremely slow. In this paper, we present an alternative approach to generating realizations that are conditional to pressure data, focusing on the distribution of realizations and on the efficiency of the method. Under certain conditions that can be verified easily, the Markov chain Monte Carlo method is known to produce states whose frequencies of appearance correspond to a given probability distribution, so we use this method to generate the realizations. To make the method more efficient, we perturb the states in such a way that the variogram is satisfied automatically and the pressure data are approximately matched at every step. These perturbations make use of sensitivity coefficients calculated from the reservoir simulator.     

A test of independence for stratigraphic sequences with respect to embedded Markov chains

Identification of cyclic sequences gives valuable insight into depositional associations of stratigraphic facies. An embedded Markov chain is a reasonable general model for facies transitions. But a model with independent random occurrences of facies is not an appropriate null hypothesis to be tested to show the presence of cycles because of definitional restriction in transition observations to only those between different facies. This is a common stratigraphic situation and the problem has been raised recently by several authors. We present here a test statistic for null hypothesis derived from the concept of partial independence and inherent to the model of embedded Markov processes.     

基于集对分析和模糊马尔可夫链的滑坡变形预测新方法研究

滑坡体的变形表现出复杂的非线性演化特征,为克服传统分析方法在处理系统不确定性方面的不足,引入集对分析（SPA）理论,并结合模糊马尔可夫（fuzzy-Markov）理论来对滑坡SPA模型中的不确定系数进行二次预测,提出了滑坡变形SPA-fuzzy-Markov预测新模型。针对fuzzy-Markov子系统,分别提出了基于相关系数和特征值方差的多维空间模糊马尔可夫链拟合效果检验的两个新方法,并据此定量考察fuzzy-Markov子系统对参数响应的敏感性,用以确定最佳的模型参数组合。运用上述理论模型对刘家沱滑坡变形监测成果进行分析,综合定量考察了模型对模糊区间重合等级和马尔可夫状态转移步数这两个参数的敏感程度。结果表明：该算例条件下状态转移步数为敏感因子;与单纯SPA模型相比,复合模型能够进一步提高整体预测精度,在岩土监测分析领域中具有良好的实用价值;同时fuzzy-Markov理论的引入也为深化SPA基础理论研究提供了一条途径,不仅在岩土监测领域,在其他相关领域也具有理论和实用价值。     

Modification of the pattern informatics method for forecasting large earthquake events using complex eigenfactors

Recent studies have shown that real-valued principal component analysis can be applied to earthquake fault systems for forecasting and prediction. In addition, theoretical analysis indicates that earthquake stresses may obey a wave-like equation, having solutions with inverse frequencies for a given fault similar to those that characterize the time intervals between the largest events on the fault. It is therefore desirable to apply complex principal component analysis to develop earthquake forecast algorithms. In this paper we modify the Pattern Informatics method of earthquake forecasting to take advantage of the wave-like properties of seismic stresses and utilize the Hilbert transform to create complex eigenvectors out of measured time series. We show that Pattern Informatics analyses using complex eigenvectors create short-term forecast hot-spot maps that differ from hot-spot maps created using only real-valued data and suggest methods of analyzing the differences and calculating the information gain.     

A way to synchronize models with seismic faults for earthquake forecasting: Insights from a simple stochastic model

Numerical models are starting to be used for determining the future behaviour of seismic faults and fault networks. Their final goal would be to forecast future large earthquakes. In order to use them for this task, it is necessary to synchronize each model with the current status of the actual fault or fault network it simulates (just as, for example, meteorologists synchronize their models with the atmosphere by incorporating current atmospheric data in them). However, lithospheric dynamics is largely unobservable: important parameters cannot (or can rarely) be measured in Nature. Earthquakes, though, provide indirect but measurable clues of the stress and strain status in the lithosphere, which should be helpful for the synchronization of the models.The rupture area is one of the measurable parameters of earthquakes. Here we explore how it can be used to at least synchronize fault models between themselves and forecast synthetic earthquakes. Our purpose here is to forecast synthetic earthquakes in a simple but stochastic (random) fault model. By imposing the rupture area of the synthetic earthquakes of this model on other models, the latter become partially synchronized with the first one. We use these partially synchronized models to successfully forecast most of the largest earthquakes generated by the first model. This forecasting strategy outperforms others that only take into account the earthquake series. Our results suggest that probably a good way to synchronize more detailed models with real faults is to force them to reproduce the sequence of previous earthquake ruptures on the faults. This hypothesis could be tested in the future with more detailed models and actual seismic data.     

A Probability-Based Renewal Rainfall Model for Flow Forecasting

In real-time flood warning systems, sufficient lead-time is important for people to take suitable actions. Rainfall forecasting is one of the ways commonly used to extend the lead-time for catchments with short response time. However, an accurate forecast of rainfall is still difficult for hydrologists using the present deterministic model. Therefore, a probability-based rainfall forecasting model, based on Markov chain, was proposed in this study. The rainfall can be forecast one to three hours in advance for a specified nonexceeding probability using the transition probability matrix of rainfall state. In this study, the nonexceeding probability, which was hourly updated on the basis of development or decay of rainfall processes, was taken as a dominant variable parameter. The accuracy of rainfall forecasting one to three hours in advance is concluded from the application of this model to four recording rain gauges. A lumped rainfall-runoff forecasting model derived from a transfer function was further applied in unison with this rainfall forecasting model to forecast flows one to four hours in advance. The results of combination of these two models show good performance with agreement between the observed and forecast hydrographs.     

Integration of RADARSAT and GIS modelling for estimating future Red River flood risk

A new geomatics-based approach for flood prediction was developed and used to model the magnitude and spatial extent of a future Red River flood in southern Manitoba. This approach combines the statistical modelling capabilities of Markov (non-spatial) analysis and logistic regression (spatial) within a geographic information system (GIS) environment, utilizing modelling inputs derived from remotely sensed RADARSAT imagery and other digital geographic data. The 1997 Red River flood was the second largest in recorded history, and the largest for which accurate data are available. The results indicate: (i) a flood “one time interval-in terms of 3 days time unit measurement- larger in area” than the 1997 flood is expected to affect 17.6% more land (an additional 47.6 km²) within the study area compared to 1997 levels based on Markovian probability derived from observations from the 1997 event; and (ii) the majority of this excess flooding will take place on agricultural land; no additional communities are expected to be at risk. Quantitative assessment verified the capability of this modelling approach for producing statistically significant results. The methodology used in this research would be easily transferable to other areas, and may provide the basis for a viable alternative to conventional hydrologic-based flood prediction approaches This revised version was published online in August 2006 with corrections to the Cover Date.     

A method for determining the reversibility of a Markov sequence

This paper describes, given a tally matrix with strictly positive entries, a method to determine whether the associated Markov process is reversible, and (for reversible Markov processes) methods to compute the reversibility matrix from the tally matrix. If the tally matrixN is symmetric, then it is shown that the Markov process must be reversible and the reversibility matrixC equalss (R ^–1NR^–1), whereR is the diagonal matrix whosei ^th diagonal entry is the sum of the entries of thei ^th row ofN (for everyi) ands denotes the sum of all the entries ofN. Because a symmetric tally matrix is of special importance in applications, a ² test is proposed for determining, in the presence of experimental errors, whether such a matrix is symmetric.