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.     

Ratio of generalized random variables with application

The generalized Pareto distribution is one of the popular models in the environmental sciences. Scientists in these areas are often interested in comparing the values of an environmental variable under two different conditions, locations, etc. This would require the study of the ratio X/(X+Y) where X and Y are independent generalized Pareto random variables. In this note, the exact distribution of X/(X+Y) is derived, which turns out to involve the Gauss hypergeometric function. An application of this result is provided to assess the relative extremity of rainfall for 14 locations in Florida. Some computer programs for use in the applications are also provided.     

Estimating the probability of river channel adjustment

River channels respond not only to natural external controls, and natural controls internal to individual drainage basins, but also to the influence of human activity. Although many site-specific instances of change have been documented, the complexity of the process interactions means that very little is known about the general nature of different styles of adjustment, or their relative sensitivity to drainage basin controls. Data obtained from the Thames Basin, southeast England, are used in a probabilistic approach to differentiate between four styles of river channel adjustment and a variety of drainage basin characteristics. Adopting a probabilistic approach quantifies the degree of confidence attributable to any prediction of river channel adjustment while acknowledging that certainties are difficult to obtain in studies of the natural environment. This approach could thus allow environmental planning decisions to be made with a quantified degree of uncertainty. Four multivariate logistic regression models are described which use a combination of continuous and categorical variables to associate drainage basin characteristics with four styles of river channel adjustment derived from a reconnaissance evaluation survey. In comparison, it is shown that laterally migrating river channels are the most common ‘natural’ channel type in the Thames Basin, and their probability of occurrence rises to 71 per cent in sand/gravel environments. In channels regulated by low weirs, deposition is the most likely channel activity where gradients are lower than 0·0040, whilst above this threshold the majority of channels are morphologically inactive. In urban channels, many of which are also lined by concrete, the likelihood of obtaining a stable channel is mostly in excess of 80 per cent. In channels straightened during this century, deposition is most likely in gradients below 0·0050, whereas erosional enlargement is most probable above this value. In channels which were initially channelized prior to this century, deposition gives way to stability at a threshold gradient of 0·0080.     

Linear combination of Gumbel random variables

The recent paper by Loaiciga and Leipnik (Stoch Environ Res Risk Environ 13:251–259, 1999) derived the probability distribution of the sum of two independent Gumbel random variables. The results given are of little practical use because they are given in terms of characteristic functions. In this note, we consider the more general problem of deriving the linear combination of two independent Gumbel random variables. Explicit expressions are given for the probability density function and the cumulative distribution function of the linear combination. Various particular cases are also considered.     

Approximation of Sahelian rainfall fields with meta-Gaussian random functions

For the purpose of numerically studying sahelian storm rainfields, a family of random functions is described with a characterization of its finite dimensional law. Some problems appearing when fitting its functional parameters are put forward and two solutions to bypass those problems are provided, according to the regularity properties of the marginal cumulative distribution function. An illustration of this method is implemented on a set of sahelian rainfields of event accumulation displaying a strong spatial intermittency.     

Drought modeling and products of random variables with exponential kernel

The distribution of products of random variables arises explicitly in many environmental problems. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this paper, we study the exact distribution of P=XY when X is an exponential random variable and Y comes from one of exponential, gamma, Weibull, Pareto or log-normal families. An application of the results is provided to drought data from Nebraska.     

An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities

This paper investigates the potential of Spartan spatial random fields (SSRFs) in real-time mapping applications. The data set that we study focuses on the distribution of daily gamma dose rates over part of Germany. Our goal is to determine a Spartan spatial model from the data, and then use it to generate “predictive” maps of the radioactivity. In the SSRF framework, the spatial dependence is determined from sample functions that focus on short-range correlations. A recently formulated SSRF predictor is used to derive isolevel contour maps of the dose rates. The SSRF predictor is explicit. Moreover, the adjustments that it requires by the user are reduced compared to classical geostatistical methods. These features present clear advantages for an automatic mapping system. The performance of the SSRF predictor is evaluated by means of various cross-validation measures. The values of the performance measures are similar to those obtained by classical geostatistical methods. Application of the SSRF method to data that simulate a radioactivity release scenario is also discussed. Hot spots are detected and removed using a heuristic method. The extreme values that appear in the path of the simulated plume are not captured by the currently used Spartan spatial model. Modeling of the processes leading to extreme values can enhance the predictive capabilities of the spatial model, by incorporating physical information.     

Joint facies and rock properties Bayesian amplitude‐versus‐offset inversion using Markov random fields

Seismic reflection pre‐stack angle gathers can be simultaneously inverted within a joint facies and elastic inversion framework using a hierarchical Bayesian model of elastic properties and categorical classes of rock and fluid properties. The Bayesian prior implicitly supplies low frequency information via a set of multivariate compaction trends for each rock and fluid type, combined with a Markov random field model of lithotypes, which carries abundance and continuity preferences. For the likelihood, we use a simultaneous, multi‐angle, convolutional model, which quantifies the data misfit probability using wavelets and noise levels inferred from well ties. Under Gaussian likelihood and facies‐conditional prior models, the posterior has simple analytic form, and the maximum a‐posteriori inversion problem boils down to a joint categorical/continuous non‐convex optimisation problem. To solve this, a set of alternative, increasingly comprehensive optimisation strategies is described: (i) an expectation–maximisation algorithm using belief propagation, (ii) a globalisation of method (i) using homotopy, and (iii) a discrete space approach using simulated annealing. We find that good‐quality inversion results depend on both sensible, elastically separable facies definitions, modest resolution ambitions, reasonably firm abundance and continuity parameters in the Markov random field, and suitable choice of algorithm. We suggest usually two to three, perhaps four, unknown facies per sample, and usage of the more expensive methods (homotopy or annealing) when the rock types are not strongly distinguished in acoustic impedance. Demonstrations of the technique on pre‐stack depth‐migrated field data from the Exmouth basin show promising agreements with lithological well data, including prediction accuracy improvements of 24% in and twofold in density, in comparison to a standard simultaneous inversion. Much clearer and extensive recovery of the thin Pyxis gas field was evident using stronger coupling in the Markov random field model and use of the homotopy or annealing algorithms.     

Wavelet-vaguelette decomposition of spatiotemporal random fields

A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample paths of Gaussian spatiotemporal random fields in the family studied.     

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.     

Comparing connected structures in ensemble of random fields

Very different connectivity patterns may arise from using different simulation methods or sets of parameters, and therefore different flow properties. This paper proposes a systematic method to compare ensemble of categorical simulations from a static connectivity point of view. The differences of static connectivity cannot always be distinguished using two point statistics. In addition, multiple-point histograms only provide a statistical comparison of patterns regardless of the connectivity. Thus, we propose to characterize the static connectivity from a set of 12 indicators based on the connected components of the realizations. Some indicators describe the spatial repartition of the connected components, others their global shape or their topology through the component skeletons. We also gather all the indicators into dissimilarity values to easily compare hundreds of realizations. Heat maps and multidimensional scaling then facilitate the dissimilarity analysis. The application to a synthetic case highlights the impact of the grid size on the connectivity and the indicators. Such impact disappears when comparing samples of the realizations with the same sizes. The method is then able to rank realizations from a referring model based on their static connectivity. This application also gives rise to more practical advices. The multidimensional scaling appears as a powerful visualization tool, but it also induces dissimilarity misrepresentations: it should always be interpreted cautiously with a look at the point position confidence. The heat map displays the real dissimilarities and is more appropriate for a detailed analysis. The comparison with a multiple-point histogram method shows the benefit of the connected components: the large-scale connectivity seems better characterized by our indicators, especially the skeleton indicators.     

Conditional components for simulation of vector random fields

Conditional component random fields (CC) based on Cholesky decomposition of the multivariate spectra are introduced in this study to develop a new method for conditional simulation of vector attributes in environmental and geological phenomena. The CC are independent random fields with covariance models obtained from projections and conditioning in the frequency domain. The approach is to simulate one attribute in the physical space and use the results to estimate the other attributes in the frequency domain. Then, a CC for the next attribute is simulated and projected on the other attributes. In general, any attribute is built as the sum of inverse Fourier transform of the orthogonal projection of previous simulated CC plus a last CC simulated in the physical space. This simulation approach continues in this fashion for several attributes and the order of them may be changed for different realizations. This method allows for data conditioning and simulation. A simplified version for intrinsically correlated random fields allows for an approach that avoids the frequency domain.     

Spatial mode estimation for functional random fields with application to bioturbation problem

This work provides a useful tool to study the effects of bioturbation on the distribution of oxygen within sediments. We propose here heterogeneity measurements based on functional spatial mode. To obtain the mode, one usually needs to estimate the spatial probability density. The approach considered here consists in looking each observation as a curve that represents the history of the oxygen concentration at a fixed pixel.     

Geostatistical simulation of random fields with bivariate isofactorial distributions by adding mosaic models

This work deals with the geostatistical simulation of a family of stationary random field models with bivariate isofactorial distributions. Such models are defined as the sum of independent random fields with mosaic-type bivariate distributions and infinitely divisible univariate distributions. For practical applications, dead leaf tessellations are used since they provide a wide range of models and allow conditioning the realizations to a set of data via an iterative procedure (simulated annealing). The model parameters can be determined by comparing the data variogram and madogram, and enable to control the spatial connectivity of the extreme values in the realizations. An illustration to a forest dataset is presented, for which a negative binomial model is used to characterize the distribution of coniferous trees over a wooded area.     

Cross-correlation of random fields: mathematical approach and applications

Random field cross‐correlation is a new promising technique for seismic exploration, as it bypasses shortcomings of usual active methods. Seismic noise can be considered as a reproducible, stationary in time, natural source. In the present paper we show why and how cross‐correlation of noise records can be used for geophysical imaging. We discuss the theoretical conditions required to observe the emergence of the Green's functions between two receivers from the cross‐correlation of noise records. We present examples of seismic imaging using reconstructed surface waves from regional to local scales. We also show an application using body waves extracted from records of a small‐scale network. We then introduce a new way to achieve surface wave seismic experiments using cross‐correlation of unsynchronized sources. At a laboratory scale, we demonstrate that body wave extraction may also be used to image buried scatterers. These works show the feasibility of passive imaging from noise cross‐correlation at different scales.     

Exact computation of Feynman-type integrals involving Gaussian random fields

We calculate explicitly some expectations (integrals over Hilbert spaces) of functionals of Gaussian random fields. Our work is inspired by the formulas for the harmonic oscillator. We also show how to do similar calculations by considering continuum limits of sequences of coupled harmonic oscillators. The results obtained in this work have applications in the study of random media, renormalization and scaling, and in engineering perturbation analysis.     

A stochastic model reduction method for nonlinear unconfined flow with multiple random input fields

In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR.     

Benthic macrofauna in Tunca River (Turkey) and their relationships with environmental variables

The numerical and proportional distributions of benthic macroinvertebrates in Tunca (Tundja, Tundzha) River (Edirne/Turkey) were determined from July 2002 to June 2003 at monthly intervals at four different stations. It was found that the benthic macrofauna consisted of 63% Oligochaeta, 24% Chironomidae larvae, and 13% Varia by numbers. According to the Shannon‐Wiener index, Tunca River had a diversity of 1.36; station 2 and September were found to have the highest diversity while station 4 and December to have the poorest. According to Bray‐Curtis similarity index, stations 2 and 3 and April and May were found to be the most similar to each other while stations 1 and 4 and August and January were found to be the most different from each other for the dynamics of the benthic macrofauna. Also some physicochemical parameters of the water (water temperature, electrical conductivity, pH, dissolved oxygen, chloride, total hardness, NO₃^–‐N, NO₂^–‐N, sulfate, phosphate, biochemical and chemical oxygen demands) were analyzed. Pearson correlation index supported the relationships between the dynamics of organisms and physicochemical variables. The relation between the number of macroinvertebrates and pH (r = +0.57, P < 0.05) was direct proportional while the relation between the number of macroinvertebrates and NO₃^–‐N (r = –0.99, P < 0.05) was inverse proportional. Furthermore, the Chironomidae larvae of Bryophaenocladius muscicola and Mesosmittia flexuella were new records for Turkish Thrace region. High pH and supersaturated oxygen levels, hard water quality, second quality levels of NO₃^–‐N, BOD, COD and fourth quality levels of NO₂^–‐N as well as the density of 490 individuals m^–2 for 124 taxa and the diversity of 1.36 showed that similar studies should be repeated periodically in Tunca to determine the future of the river.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.