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1.
Xavier Emery 《Stochastic Environmental Research and Risk Assessment (SERRA)》2010,24(2):211-219
The multi-Gaussian model is used in geostatistical applications to predict functions of a regionalized variable and to assess
uncertainty by determining local (conditional to neighboring data) distributions. The model relies on the assumption that
the regionalized variable can be represented by a transform of a Gaussian random field with a known mean value, which is often
a strong requirement. This article presents two variations of the model to account for an uncertain mean value. In the first
one, the mean of the Gaussian random field is regarded as an unknown non-random parameter. In the second model, the mean of
the Gaussian field is regarded as a random variable with a very large prior variance. The properties of the proposed models
are compared in the context of non-linear spatial prediction and uncertainty assessment problems. Algorithms for the conditional
simulation of Gaussian random fields with an uncertain mean are also examined, and problems associated with the selection
of data in a moving neighborhood are discussed. 相似文献
2.
《水文科学杂志》2013,58(6)
Abstract Abstract Characterization of heterogeneity at the field scale generally requires detailed aquifer properties such as transmissivity and hydraulic head. An accurate delineation of these properties is expensive and time consuming, and for many if not most groundwater systems, is not practical. As an alternative approach, stochastic representation of random fields is used and presented in this paper. Specifically, an iterative stochastic conditional simulation approach was applied to a hypothetical and highly heterogeneous pre-designed aquifer system. The approach is similar to the classical co-kriging technique; it uses a linear estimator that depends on the covariance functions of transmissivity (T), and hydraulic head (h), as well as their cross-covariances. A linearized flow equation along with a conditional random field generator constitutes the iterative process of the conditional simulation. One hundred equally likely realizations of transmissivity fields with pre-specified geostatistical parameters were generated, and conditioned to both limited transmissivity and head data. The successful implementation of the approach resulted in conditioned flow paths and travel-time distribution under different degrees of aquifer heterogeneity. This approach worked well for fields exhibiting small variances. However, for random fields exhibiting large variances (greater than 1.0), an iterative procedure was used. The results show that, as the variance of the ln[T] increases, the flow paths tend to diverge, resulting in a wide spectrum of flow conditions, with no direct discernable relationship between the degree of heterogeneity and travel time. The applied approach indicates that high errors may result when estimation of particle travel times in a heterogeneous medium is approximated by an equivalent homogeneous medium. 相似文献
3.
Xavier Emery 《Stochastic Environmental Research and Risk Assessment (SERRA)》2006,20(1-2):53-65
In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values
of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory
threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under
study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a
nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian
field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although
both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications
since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around
this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive
kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations.
The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian
kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated
estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when
the mean is unknown. 相似文献
4.
Multigaussian kriging technique has many applications in mining, soil science, environmental science and other fields. Particularly, in the local reserve estimation of a mineral deposit, multigaussian kriging is employed to derive panel-wise tonnages by predicting conditional probability of block grades. Additionally, integration of a suitable change of support model is also required to estimate the functions of the variables with larger support than that of the samples. However, under the assumption of strict stationarity, the grade distributions and important recovery functions are estimated by multigaussian kriging using samples within a supposedly spatial homogeneous domain. Conventionally, the underlying random function model is required to be stationary in order to carry out the inference on ore grade distribution and relevant statistics. In reality, conventional stationary model often fails to represent complicated geological structure. Traditionally, the simple stationary model neither considers the obvious changes in local means and variances, nor is it able to replicate spatial continuity of the deposit and hence produces unreliable outcomes. This study deals with the theoretical design of a non-stationary multigaussian kriging model allowing change of support and its application in the mineral reserve estimation scenario. Local multivariate distributions are assumed here to be strictly stationary in the neighborhood of the panels. The local cumulative distribution function and related statistics with respect to the panels are estimated using a distance kernel approach. A rigorous investigation through simulation experiments is performed to analyze the relevance of the developed model followed by a case study on a copper deposit. 相似文献
5.
Abstract This paper compares the performance of three geostatistical algorithms, which integrate elevation as an auxiliary variable: kriging with external drift (KED); kriging combined with regression, called regression kriging (RK) or kriging after detrending; and co-kriging (CK). These three methods differ by the way by in which the secondary information is introduced into the prediction procedure. They are applied to improve the prediction of the monthly average rainfall observations measured at 106 climatic stations in Tunisia over an area of 164 150 km2 using the elevation as the auxiliary variable. The experimental sample semivariograms, residual semivariograms and cross-variograms are constructed and fitted to estimate the rainfall levels and the estimation variance at the nodes of a square grid of 20 km?×?20 km resolution and to develop corresponding contour maps. Contour diagrams for KED and RK were similar and exhibited a pattern corresponding more closely to local topographic features when (a) the network is sparse and (b) the rainfall–elevation correlation is poor, while CK showed a smooth zonal pattern. Smaller prediction variances are obtained for the RK algorithm. The cross-validation showed that the RMSE obtained for CK gave better results than for KED or RK. Editor D. Koutsoyiannis; Associate editor C. Onof Citation Feki, H., Slimani, M., and Cudennec, C., 2012. Incorporating elevation in rainfall interpolation in Tunisia using geostatistical methods. Hydrological Sciences Journal, 57 (7), 1294–1314. 相似文献
6.
Performance of detrending models of crop yield risk assessment: evaluation on real and hypothetical yield data 总被引：2，自引：1，他引：1
Tao Ye Jianliang Nie Jun Wang Peijun Shi Zhu Wang 《Stochastic Environmental Research and Risk Assessment (SERRA)》2015,29(1):109-117
Detrending is a widely used technique for obtaining stationary time series data in residual analysis and risk assessment. The technique is frequently applied in crop yield risk assessment and insurance ratings. Although several trend models have been proposed in the literature, whether these models achieve consistent detrending results and successfully extract the true yield trends is rarely discussed. In the present article, crop insurance pricing is evaluated by different trend models using real and historical yield data, and hypothetical yield data generated by Monte Carlo simulations. Applied to real historical data, the linear, loglinear, autoregressive integrated moving average trend models produce different risk assessment results. The differences among the model outputs are statistically significant. The largest deviation in the county crop assessment reaches 6–8 %, substantially larger than the present countrywide gross premium rate of 5–7 %. In performance tests on simulated yield trends, popular detrending methods based on smoothing techniques proved overall superior to linear, loglinear, and integrated autoregression models. The best performances were yielded by the moving average and robust locally weighted regression models. 相似文献
7.
Non-stationarity in statistical properties of the subsurface is often ignored. In a classical linear Bayesian inversion setting of seismic data, the prior distribution of physical parameters is often assumed to be stationary. Here we propose a new method of handling non-stationarity in the variance of physical parameters in seismic data. We propose to infer the model variance prior to inversion using maximum likelihood estimators in a sliding window approach. A traditional, and a localized shrinkage estimator is defined for inferring the prior model variance. The estimators are assessed in a synthetic base case with heterogeneous variance of the acoustic impedance in a zero-offset seismic cross section. Subsequently, this data is inverted for acoustic impedance using a non-stationary model set up with the inferred variances. Results indicate that prediction as well as posterior resolution is greatly improved using the non-stationary model compared with a common prior model with stationary variance. The localized shrinkage predictor is shown to be slightly more robust than the traditional estimator in terms of amplitude differences in the variance of acoustic impedance and size of local neighbourhood. Finally, we apply the methodology to a real data set from the North Sea basin. Inversion results show a more realistic posterior model than using a conventional approach with stationary variance. 相似文献
8.
Jichun Wu Bill X. Hu 《Stochastic Environmental Research and Risk Assessment (SERRA)》2007,21(6):665-682
The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that
the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al.
in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head
to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework
for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve
analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional
conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment
for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity
and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements
of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary
flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the
hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both
the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for
reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant. 相似文献
9.
Ryan M. Pollyea Jerry P. Fairley Robert K. Podgorney Travis L. McLing 《Stochastic Environmental Research and Risk Assessment (SERRA)》2013,27(7):1735-1740
The stochastic continuum (SC) representation is one common approach for simulating the effects of fracture heterogeneity in groundwater flow and transport models. These SC reservoir models are generally developed using geostatistical methods (e.g., kriging or sequential simulation) that rely on the model semivariogram to describe the spatial variability of each continuum. Although a number of strategies for sampling spatial distributions have been published in the literature, little attention has been paid to the optimization of sampling in resource- or access-limited environments. Here we present a strategy for estimating the minimum sample spacing needed to define the spatial distribution of fractures on a vertical outcrop of basalt, located in the Box Canyon, east Snake River Plain, Idaho. We used fracture maps of similar basalts from the published literature to test experimentally the effects of different sample spacings on the resulting semivariogram model. Our final field sampling strategy was based on the lowest sample density that reproduced the semivariogram of the exhaustively sampled fracture map. Application of the derived sampling strategy to an outcrop in our field area gave excellent results, and illustrates the utility of this type of sample optimization. The method will work for developing a sampling plan for any intensive property, provided prior information for a similar domain is available; for example, fracture maps or ortho-rectified photographs from analogous rock types could be used to plan for sampling of a fractured rock outcrop. 相似文献
10.
Andrea Zanini Peter K. Kitanidis 《Stochastic Environmental Research and Risk Assessment (SERRA)》2009,23(5):565-577
The estimation of field parameters, such as transmissivity, is an important part of groundwater modeling. This work deals
with the quasilinear geostatistical inverse approach to the estimation of the transmissivity fields from hydraulic head measurements.
The standard quasilinear approach is an iterative method consisting of successive linearizations. We examine a synthetic case
to evaluate the basic methodology and some modifications and extensions. The first objective is to evaluate the performance
of the quasilinear approach when applied to strongly heterogeneous (or “high-contrast”) transmissivity fields and, when needed,
to propose improvements that allow the solution of such problems. For large-contrast cases, the standard quasilinear method
often fails to converge. However, by introducing a derivative-free line search as a polishing step after each Gauss–Newton
iteration, we have found that convergence can be practically assured. Another issue is that the quasilinear procedure, which
uses linearization about the best estimate to evaluate estimation variances, may lead to inaccurate estimation of the variance
of the estimated variable. Our numerical results suggest that this may not be a particularly serious problem, though it is
hard to say whether this conclusion will apply to other cases. Nevertheless, since the quasilinear approach is an approximation,
we propose a potentially more accurate but computer-intensive Markov Chain Monte Carlo (MCMC) procedure based on conditional
realizations generated through the quasilinear approach and accepted or rejected according to the Metropolis–Hastings algorithm.
Six transmissivity fields with increasing contrast were generated and one thousand conditional realizations were computed
for each studied case. The MCMC procedure proposed in this work gives an overall more accurate picture than the quasilinear
approach but at a considerably higher computational cost. 相似文献
11.
Performance assessments of repositories for the underground disposal of nuclear fuel and waste include models of ground water flow and transport in the host rocks. Estimates of hydraulic conductivity, K, based on field measurements may require adjustment (upscaling) for use in numerical models, but the choice of upscaling approach can be complicated by the use of nested modeling, large-scale fracture zones, and a high degree of heterogeneity. Four approaches to upscaling K are examined using a reference case based on exhaustive site data and an application of nested modeling to evaluate performance assessment of a waste repository. The upscaling approaches are evaluated for their effects on the flow balance between nested modeling domains and on simple measures of repository performance. Of the upscaling approaches examined in this study, the greatest consistency of boundary flows was achieved using the observed scale dependence for the rock domains, measured values from the large-scale interference test for the conductor domain, and a semivariogram regularization based on the Moye model for packer test interpretation. Making the assumption that large fracture zones are two-dimensional media results in the greatest changes to the median of travel time and improves the flow balance between the nested models. The uncertainty of upscaling methods apparently has a small impact on median performance measures, but a significant impact on the variances and earliest arrival times. 相似文献
12.
We analyze the impact of a linear trend in the mean log-conductivity on the transport of a conservative tracer in a bounded domain. The effects of such a linear trend on solute transport were analyzed in depth for unbounded domains (Rubin and Seong, Water Resour Res 30(11):2901–2911, 1994; Indelman and Rubin, Water Resour Res 31(5):1257–1265, 1995; Water Resour Res 32(5):1257–1265, 1996), whereas studies concerning this special case of medium nonstationarity in finite domains usually focus on head or flow statistics (Guadagnini et al., Stoch Environ Res Risk Assess, 17:394–407, 2003). In this study both ensemble and effective plume moments are provided for an instantaneous release of a solute through a linear source normal to the mean flow direction, by taking into account different sizes of the source. The analysis involving a steady velocity field spatially nonstationary is developed by using the stochastic finite element method. Results show that ensemble moments are affected by increasing trends both parallel and normal to the mean flow direction, but the impact on effective plume moments is very different. A parallel trend does not seem to influence the effective second moments; while a normal trend, although modifies the transverse effective moment only weakly, strongly increases the longitudinal one, especially for large initial sizes of the source. Furthermore, the increase of the particle displacement variance produced by a parallel trend in the finite domain disagrees with the results obtained in an unbounded domain, due to the boundary conditions here considered making both head and velocity moments nonstationary and nonsymmetric. 相似文献
13.
14.
The random function is a mathematical model commonly used in the assessment of uncertainty associated with a spatially correlated
attribute that has been partially sampled. There are multiple algorithms for modeling such random functions, all sharing the
requirement of specifying various parameters that have critical influence on the results. The importance of finding ways to
compare the methods and setting parameters to obtain results that better model uncertainty has increased as these algorithms
have grown in number and complexity. Crossvalidation has been used in spatial statistics, mostly in kriging, for the analysis
of mean square errors. An appeal of this approach is its ability to work with the same empirical sample available for running
the algorithms. This paper goes beyond checking estimates by formulating a function sensitive to conditional bias. Under ideal
conditions, such function turns into a straight line, which can be used as a reference for preparing measures of performance.
Applied to kriging, deviations from the ideal line provide sensitivity to the semivariogram lacking in crossvalidation of
kriging errors and are more sensitive to conditional bias than analyses of errors. In terms of stochastic simulation, in addition
to finding better parameters, the deviations allow comparison of the realizations resulting from the applications of different
methods. Examples show improvements of about 30% in the deviations and approximately 10% in the square root of mean square
errors between reasonable starting modelling and the solutions according to the new criteria. 相似文献
15.
A new approach is described to allow conditioning to both hard data (HD) and soft data for a patch- and distance-based multiple-point geostatistical simulation. The multinomial logistic regression is used to quantify the link between HD and soft data. The soft data is converted by the logistic regression classifier into as many probability fields as there are categories. The local category proportions are used and compared to the average category probabilities within the patch. The conditioning to HD is obtained using alternative training images and by imposing large relative weights to HD. The conditioning to soft data is obtained by measuring the probability–proportion patch distance. Both 2D and 3D cases are considered. Synthetic cases show that a stationary TI can generate non-stationary realizations reproducing the HD, keeping the texture indicated by the TI and following the trends identified in probability maps obtained from soft data. A real case study, the Mallik methane-hydrate field, shows perfect reproduction of HD while keeping a good reproduction of the TI texture and probability trends. 相似文献
16.
A fundamental decision to make during the analysis of geostatistical data is the modeling of the spatial dependence structure as stationary or non-stationary. Although second-order stationary modeling approaches have been successfully applied in geostatistical applications for decades, there is a growing interest in second-order non-stationary modeling approaches. This paper provides a review of modeling approaches allowing to take into account the second-order non-stationarity in univariate geostatistical data. One broad distinction between these modeling approaches relies on the way that the second-order non-stationarity is captured. It seems unlikely to prove that there would be the best second-order non-stationary modeling approach for all geostatistical applications. However, some of them are distinguished by their simplicity, interpretability, and flexibility. 相似文献
17.
18.
Using Lake Superior mean monthly elevations as recorded at five gauges around the lake, time series of elevations and differences in elevations between pairs of gauges were analysed for trends, periodicities and autoregressive components. It was found that the variance of the time series of elevations consisted of 4–12% linear trends, 35–44% periodicities in the mean, 0.23–0.66% periodicities in the variance, a 43–54% autoregressive component and a 5% random component. On the other hand, the time series of differences in lake elevations were found to consist of 30–52% linear trends, 5–35% periodicities in the mean and variance, up to a 30% autoregressive component and a random component of 21–31%. Rates of crustal movement were computed from the trends in gauge differences. 相似文献
19.
None of the processes of estimation currently available is fully acceptable to the geophysicist. Firstly, they all assume that the variable, be it seismic reflection time, rms velocities, Bouguer anomaly, etc.… is random, amenable to pure statistical considerations, and the processes all disregard the relationships which link the values of the variable in the different points of the domain under investigation. Secondly, they do not provide the geophysicist with any guideline for smoothing his data, as smoothing and estimation are considered two separate operations. Thirdly, they fail to offer a valid criterion of estimation and a measure of the estimation error. The krigeage process overcomes the above mentioned difficulties. It synthesizes the structural or “geostatistical’ characteristics of the variable by using a function called the variogram (variances of the increases of the variable with respect to distance and direction). It smoothes the variable, when necessary, as a function of the “nugget effect’ (value at the origin of the experimental variogram). It yields an optimum estimation of the variable by minimizing the estimation error, and it computes a measure of the reliability of the estimation, the variance of krigeage. The process is demonstrated herein with three examples of variograms on seismic and gravity data and an example of contouring of velocities, reflection times and depths of a productive layer in an oil field, with detection and correction of irregular data, smoothing of velocities, migration of depth points, and display of estimation error. 相似文献
20.
Geostatistical estimation (kriging) and geostatistical simulation are routinely used in ground water hydrology for optimal spatial interpolation and Monte Carlo risk assessment, respectively. Both techniques are based on a model of spatial variability (semivariogram or covariance) that generally is not known but must be inferred from the experimental data. Where the number of experimental data is small (say, several tens), as is not unusual in ground water hydrology, the model fitted to the empirical semivariogram entails considerable uncertainty. If all the practical results are based on this unique fitted model, the final results will be biased. We propose that, instead of using a unique semivariogram model, the full range of models that are inside a given confidence region should be used, and the weight that each semivariogram model has on the final result should depend on its plausibility. The first task, then, is to evaluate the uncertainty of the model, which can be efficiently done by using maximum likelihood inference. The second task is to use the range of plausible models in applications and to show the effect observed on the final results. This procedure is put forth here with kriging and simulation applications, where the uncertainty in semivariogram parameters is propagated into the final results (e.g., the prediction of ground water head). A case study using log-transmissivity data from the Vega de Granada aquifer, in southern Spain, is given to illustrate the methodology. 相似文献