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1.
Graphical application of the Type 1 (Gumbel) extreme value distribution is very simple since the distribution inverse gives a linear x-y plot. In contrast, the Type 2 and Type 3 extreme value distributions have nonlinear functions with respect to the same axes. A simple three-point graphical estimation procedure is described for these two distributions. This approach allows the nonlinear flood magnitude prediction functions to be located in any desirable position relative to the plotted annual maxima, subject to the constraint of having an extreme value form. The computation is very simple and requires only the location of a unique zero of a one-parameter function within a defined interval.  相似文献   

2.
We consider a Bayesian model for inversion of observed amplitude variation with offset data into lithology/fluid classes, and study in particular how the choice of prior distribution for the lithology/fluid classes influences the inversion results. Two distinct prior distributions are considered, a simple manually specified Markov random field prior with a first-order neighbourhood and a Markov mesh model with a much larger neighbourhood estimated from a training image. They are chosen to model both horizontal connectivity and vertical thickness distribution of the lithology/fluid classes, and are compared on an offshore clastic oil reservoir in the North Sea. We combine both priors with the same linearized Gaussian likelihood function based on a convolved linearized Zoeppritz relation and estimate properties of the resulting two posterior distributions by simulating from these distributions with the Metropolis–Hastings algorithm. The influence of the prior on the marginal posterior probabilities for the lithology/fluid classes is clearly observable, but modest. The importance of the prior on the connectivity properties in the posterior realizations, however, is much stronger. The larger neighbourhood of the Markov mesh prior enables it to identify and model connectivity and curvature much better than what can be done by the first-order neighbourhood Markov random field prior. As a result, we conclude that the posterior realizations based on the Markov mesh prior appear with much higher lateral connectivity, which is geologically plausible.  相似文献   

3.
In this paper we derive score tests for spatial independence in mortality or incidence risk in the framework of hierarchical spatial models where different Gaussian Markov random field (MRF) priors are given for modelling the area random effects (specifically, two non-intrinsic Gaussian priors and a convolution Gaussian prior). The techniques used to test the practically relevant and important simplifying hypotheses of an absence of spatial variation in risk will provide a guidance for practitioners to select an adequate model (i.e., a model with an exchangeable-independent-prior, an intrinsic prior, a convolution prior or a non-intrinsic prior, for the area-specific random effects distribution). The proposed methodology is illustrated by analyzing the well-known data set of lip cancer in Scotland and female mortality due to cerebrovascular disease in Navarra, Spain.  相似文献   

4.
Despite their apparent high dimensionality, spatially distributed hydraulic properties of geologic formations can often be compactly (sparsely) described in a properly designed basis. Hence, the estimation of high-dimensional subsurface flow properties from dynamic performance and monitoring data can be formulated and solved as a sparse reconstruction inverse problem. Recent advances in statistical signal processing, formalized under the compressed sensing paradigm, provide important guidelines on formulating and solving sparse inverse problems, primarily for linear models and using a deterministic framework. Given the uncertainty in describing subsurface physical properties, even after integration of the dynamic data, it is important to develop a practical sparse Bayesian inversion approach to enable uncertainty quantification. In this paper, we use sparse geologic dictionaries to compactly represent uncertain subsurface flow properties and develop a practical sparse Bayesian method for effective data integration and uncertainty quantification. The multi-Gaussian assumption that is widely used in classical probabilistic inverse theory is not appropriate for representing sparse prior models. Following the results presented by the compressed sensing paradigm, the Laplace (or double exponential) probability distribution is found to be more suitable for representing sparse parameters. However, combining Laplace priors with the frequently used Gaussian likelihood functions leads to neither a Laplace nor a Gaussian posterior distribution, which complicates the analytical characterization of the posterior. Here, we first express the form of the Maximum A-Posteriori (MAP) estimate for Laplace priors and then use the Monte-Carlo-based Randomize Maximum Likelihood (RML) method to generate approximate samples from the posterior distribution. The proposed Sparse RML (SpRML) approximate sampling approach can be used to assess the uncertainty in the calibrated model with a relatively modest computational complexity. We demonstrate the suitability and effectiveness of the SpRML formulation using a series of numerical experiments of two-phase flow systems in both Gaussian and non-Gaussian property distributions in petroleum reservoirs and successfully apply the method to an adapted version of the PUNQ-S3 benchmark reservoir model.  相似文献   

5.
Knowledge about saturation and pressure distributions in a reservoir can help in determining an optimal drainage pattern, and in deciding on optimal well designs to reduce risks of blow‐outs and damage to production equipment. By analyzing time‐lapse PP AVO or time‐lapse multicomponent seismic data, it is possible to separate the effects of production related saturation and pressure changes on seismic data. To be able to utilize information about saturation and pressure distributions in reservoir model building and simulation, information about uncertainty in the estimates is useful. In this paper we present a method to estimate changes in saturation and pressure from time‐lapse multicomponent seismic data using a Bayesian estimation technique. Results of the estimations will be probability density functions (pdfs), giving immediate information about both parameter values and uncertainties. Linearized rock physical models are linked to the changes in saturation and pressure in the prior probability distribution. The relationship between the elastic parameters and the measured seismic data is described in the likelihood model. By assuming Gaussian distributed prior uncertainties the posterior distribution of the saturation and pressure changes can be calculated analytically. Results from tests on synthetic seismic data show that this method produces more precise estimates of changes in effective pressure than a similar methodology based on only PP AVO time‐lapse seismic data. This indicates that additional information about S‐waves obtained from converted‐wave seismic data is useful for obtaining reliable information about the pressure change distribution.  相似文献   

6.
The solar cycle induces strong periodicity in processes underlying monthly rainfall totals. Seasonally varying parameters of rainfall distributions can be estimated with reasonable reliability from relatively few years of monthly data. The distribution of annual totals or maxima in terms of these varying parameters can thus be used to predict long term annual characteristics from quite short records. Specification of seasonal variation of parameters as a harmonic process simplifies the derivations. Ignoring seasonal variation in the rainfall process leads to incorrect estimates of long-term extreme rainfalls when using traditional methodology.  相似文献   

7.
Stationarity is often assumed for frequency analysis of low flows in water resources management and planning. However, many studies have shown that flow characteristics, particularly the frequency spectrum of extreme hydrologic events, were modified by climate change and human activities. Thus, the conventional frequency analysis that fails to consider the nonstationary characteristics may lead to costly design. The analysis presented in this paper was based on the more than 100 years of daily flow data from the Yichang gauging station 44 km downstream of the Three Gorges Dam. The Mann–Kendall trend test under the scaling hypothesis showed that the annual low flows had a significant monotonic trend, whereas an abrupt change point was identified in 1936 by the Pettitt test. The climate‐informed low‐flow frequency analysis and the divided and combined method were employed to account for the impacts from related climate variables and nonstationarities in annual low flows. Without prior knowledge of the probability density function for the gauging station, six distribution functions including the generalized extreme values (GEV), Pearson Type III, Gumbel, Gamma, Lognormal and Weibull distributions have been tested to find the best fit, in which the local likelihood method is used to estimate the parameters. Analyses show that GEV had the best fit for the observed low flows. This study has also shown that the climate‐informed low‐flow frequency analysis is able to exploit the link between climate indices and low flows, which would account for the dynamic feature for reservoir management and provide more accurate and reliable designs for infrastructure and water supply. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

8.
Univariate and bivariate Gamma distributions are among the most widely used distributions in hydrological statistical modeling and applications. This article presents the construction of a new bivariate Gamma distribution which is generated from the functional scale parameter. The utilization of the proposed bivariate Gamma distribution for drought modeling is described by deriving the exact distribution of the inter-arrival time and the proportion of drought along with their moments, assuming that both the lengths of drought duration (X) and non-drought duration (Y) follow this bivariate Gamma distribution. The model parameters of this distribution are estimated by maximum likelihood method and an objective Bayesian analysis using Jeffreys prior and Markov Chain Monte Carlo method. These methods are applied to a real drought dataset from the State of Colorado, USA.  相似文献   

9.
It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.  相似文献   

10.
The dynamical responses of a shoreline over long-term (years or decades) is a non-linear and time-dependent random process. It is affected by both longshore and cross-shore sediment transports. The former tends to cause cumulative changes in the mean shoreline position while the latter usually only leads to beach profile fluctuations relative to the moving mean beach profile. Due to the time-dependency of the process the life-cycle approach is ideally suited to formulate the probability distribution of extreme shoreline erosion. A model based on such approach and using standard Monte Carlo simulation techniques has been reported by Dong and Chen (1999). In this paper a simplified procedure is developed by introducing the assumption that the longshore and cross-shore processes are statistically independent. This then allows the probability distribution of the extreme erosion to be calculated in terms of the marginal probability distributions of the maximum recessions due to purely longshore and purely cross-shore transport. This method was applied to two idealised shoreline configurations and its usefulness for engineering applications is evaluated by comparison with the full Monte Carlo method.  相似文献   

11.
Extreme value models are widely used in different areas. The Birnbaum–Saunders distribution is receiving considerable attention due to its physical arguments and its good properties. We propose a methodology based on extreme value Birnbaum–Saunders regression models, which includes model formulation, estimation, inference and checking. We further conduct a simulation study for evaluating its performance. A statistical analysis with real-world extreme value environmental data using the methodology is provided as illustration.  相似文献   

12.
Satellite‐based soil moisture data accuracies are of important concerns by hydrologists because they could significantly influence hydrological modelling uncertainty. Without proper quantification of their uncertainties, it is difficult to optimize the hydrological modelling system and make robust decisions. Currently, the satellite soil moisture data uncertainty has been limited to summary statistics with the validations mainly from the in situ measurements. This study attempts to build the first error distribution model with additional higher‐order uncertainty modelling for satellite soil moisture observations. The methodology is demonstrated by a case study using the Soil Moisture and Ocean Salinity satellite soil moisture observations. The validation is based on soil moisture estimates from hydrological modelling, which is more relevant to the intended data use than the in situ measurements. Four probability distributions have been explored to find suitable error distribution curves using the statistical tests and bootstrapping resampling technique. General extreme value is identified as the most suitable one among all the curves. The error distribution model is still in its infant stage, which ignores spatial and temporal correlations, and nonstationarity. Further improvements should be carried out by the hydrological community by expanding the methodology to a wide range of satellite soil moisture data using different hydrological models. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

13.
In flood frequency analysis, a suitable probability distribution function is required in order to establish the flood magnitude-return period relationship. Goodness of fit (GOF) techniques are often employed to select a suitable distribution function in this context. But they have been often criticized for their inability to discriminate between statistical distributions for the same application. This paper investigates the potential utility of subsampling, a resampling technique with the aid of a GOF test to select the best distribution for frequency analysis. The performance of the methodology is assessed by applying the methodology to observed and simulated annual maximum (AM) discharge data series. Several AM series of different record lengths are used as case studies to determine the performance. Numerical analyses are carried out to assess the performance in terms of sample size, subsample size and statistical properties inherent in the AM data series. The proposed methodology is also compared with the standard Anderson–Darling (AD) test. It is found that the methodology is suitable for a longer data series. A better performance is obtained when the subsample size is taken around half of the underlying data sample. The methodology has also outperformed the standard AD test in terms of effectively discriminating between distributions. Overall, all results point that the subsampling technique can be a promising tool in discriminating between distributions.  相似文献   

14.
Why do we need and how should we implement Bayesian kriging methods   总被引：1，自引：0，他引：1  
The spatial prediction methodology that has become known under the heading of kriging is largely based on the assumptions that the underlying random field is Gaussian and the covariance function is exactly known. In practical applications, however, these assumptions will not hold. Beyond Gaussianity of the random field, lognormal kriging, disjunctive kriging, (generalized linear) model-based kriging and trans-Gaussian kriging have been proposed in the literature. The latter approach makes use of the Box–Cox-transform of the data. Still, all the alternatives mentioned do not take into account the uncertainty with respect to the distribution (or transformation) and the estimated covariance function of the data. The Bayesian trans-Gaussian kriging methodology proposed in the present paper is in the spirit of the “Bayesian bootstrap” idea advocated by Rubin (Ann Stat 9:130–134, 1981) and avoids the unusual specification of noninformative priors often made in the literature and is entirely based on the sample distribution of the estimators of the covariance function and of the Box–Cox parameter. After some notes on Bayesian spatial prediction, noninformative priors and developing our new methodology finally we will present an example illustrating our pragmatic approach to Bayesian prediction by means of a simulated data set.  相似文献   

15.
Large observed datasets are not stationary and/or depend on covariates, especially, in the case of extreme hydrometeorological variables. This causes the difficulty in estimation, using classical hydrological frequency analysis. A number of non-stationary models have been developed using linear or quadratic polynomial functions or B-splines functions to estimate the relationship between parameters and covariates. In this article, we propose regularised generalized extreme value model with B-splines (GEV-B-splines models) in a Bayesian framework to estimate quantiles. Regularisation is based on penalty and aims to favour parsimonious model especially in the case of large dimension space. Penalties are introduced in a Bayesian framework and the corresponding priors are detailed. Five penalties are considered and the corresponding priors are developed for comparison purpose as: Least absolute shrinkage and selection (Lasso and Ridge) and smoothing clipped absolute deviations (SCAD) methods (SCAD1, SCAD2 and SCAD3). Markov chain Monte Carlo (MCMC) algorithms have been developed for each model to estimate quantiles and their posterior distributions. Those approaches are tested and illustrated using simulated data with different sample sizes. A first simulation was made on polynomial B-splines functions in order to choose the most efficient model in terms of relative mean biais (RMB) and the relative mean-error (RME) criteria. A second simulation was performed with the SCAD1 penalty for sinusoidal dependence to illustrate the flexibility of the proposed approach. Results show clearly that the regularized approaches leads to a significant reduction of the bias and the mean square error, especially for small sample sizes (n < 100). A case study has been considered to model annual peak flows at Fort-Kent catchment with the total annual precipitations as covariates. The conditional quantile curves were given for the regularized and the maximum likelihood methods.  相似文献   

16.
Hydrological frequency analysis is the most widely used method to estimate risk for extreme values. The most used statistical distributions to fit extreme value data in hydrology can be regrouped in three classes: class C of regularly varying distributions, class D of sub exponential and class E, Exponential depending on their tail behavior. The Halphen distributions (Halphen type A (HA), Halphen type B (HB)) are separated by the Gamma distribution; these three distributions belong to class D and can be displayed in the (δ1, δ2) moment-ratio diagram. In this study, a statistical test for discriminating between HA, HB and the Gamma distribution is developed. The methodology is based on: (1) the generation of N samples of different sizes n around the Gamma curve; (2) the determination of the confidence zones around the Gamma curve for each fixed couple (δ1, δ2) moment-ratios and finally; (3) the study of the power of the test developed and the calculation of the type 2 error β and the power of the test which is 1-β for a fixed significance level α. Results showed that the test is powerful especially for high coefficients of skewness. This test will be included in Decision Support System of the HYFRAN-PLUS software.  相似文献   

17.
Bayesian data fusion in a spatial prediction context: a general formulation   总被引：1，自引：1，他引：1  
In spite of the exponential growth in the amount of data that one may expect to provide greater modeling and predictions opportunities, the number and diversity of sources over which this information is fragmented is growing at an even faster rate. As a consequence, there is real need for methods that aim at reconciling them inside an epistemically sound theoretical framework. In a statistical spatial prediction framework, classical methods are based on a multivariate approach of the problem, at the price of strong modeling hypotheses. Though new avenues have been recently opened by focusing on the integration of uncertain data sources, to the best of our knowledges there have been no systematic attemps to explicitly account for information redundancy through a data fusion procedure. Starting from the simple concept of measurement errors, this paper proposes an approach for integrating multiple information processing as a part of the prediction process itself through a Bayesian approach. A general formulation is first proposed for deriving the prediction distribution of a continuous variable of interest at unsampled locations using on more or less uncertain (soft) information at neighboring locations. The case of multiple information is then considered, with a Bayesian solution to the problem of fusing multiple information that are provided as separate conditional probability distributions. Well-known methods and results are derived as limit cases. The convenient hypothesis of conditional independence is discussed by the light of information theory and maximum entropy principle, and a methodology is suggested for the optimal selection of the most informative subset of information, if needed. Based on a synthetic case study, an application of the methodology is presented and discussed.  相似文献   

18.
The temporal‐spatial resolution of input data‐induced uncertainty in a watershed‐based water quality model, Hydrologic Simulation Program‐FORTRAN (HSPF), is investigated in this study. The temporal resolution‐induced uncertainty is described using the coefficient of variation (CV). The CV is found to decrease with decreasing temporal resolution and follow a log‐normal relation with time interval for temperature data while it exhibits a power‐law relation for rainfall data. The temporal‐scale uncertainties in the temperature and rainfall data follow a general extreme value distribution and a Weibull distribution, respectively. The Nash‐Sutcliffe coefficient (NSC) is employed to represent the spatial resolution induced uncertainty. The spatial resolution uncertainty in the dissolved oxygen and nitrate‐nitrogen concentrations simulated using HSPF is observed to follow a general extreme value distribution and a log‐normal distribution, respectively. The probability density functions (PDF) provide new insights into the effect of temporal‐scale and spatial resolution of input data on uncertainties involved in watershed modelling and total maximum daily load calculations. This study exhibits non‐symmetric distributions of uncertainty in water quality modelling, which simplify weather and water quality monitoring and reducing the cost involved in flow and water quality monitoring. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

19.
Determination of uninformative prior distributions is essential in many branches of knowledge integration and system processing. The conceptual difficulties of this determination are due to lack of uniqueness and consequential lack of objectivity associated with the state of complete ignorance. The present work overcomes the above difficulty by considering a class of priors that are consistent with a physical invariance principle, namely, invariance with respect to a change in the system of dimensional units. These priors do not represent total ignorance and they do not suffer from the aforementioned conceptual difficulties. This Dimensional Invariance Requirement (DIR) leads to a class of prior densities, which constitute a generalization of Jeffrey’s proposal concerning priors of inherently positive variables. This generalization possesses certain important features, from a formal as well as an interpretive viewpoint, which involve the notion of a knowledge-based natural reference point of physical random variables (RV). Conceptual difficulties associated with uninformative priors are resolved, whereas well-established results are derived as special cases of the DIR. Application of this requirement to a system of RV yields the familiar result that at the prior knowledge stage these variables should be considered as independent. Prior distributions for non-dimensional physical quantities are obtained by defining these variables in terms of dimensional quantities. A logarithmic transformation carries the physical prior into a uniform (flat) density that is convenient in certain applications. In a companion paper we examine the improvements gained in the maximum entropy context by means of the proposed class of physical prior densities.  相似文献   

20.
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