Turning Bands Method Research Articles

Probabilistic Slope Stability Analysis with Stochastic Soil Hydraulic Conductivity

The effects of stochastic hydraulic conductivity on the slope stability of an embankment dam are investigated using a combination of random field simulation, seepage analysis, and slope stability analysis. The hydraulic conductivity distribution is treated as a spatially stationary random field following a lognormal distri- bution. The turning band method is used to generate the spatial variability of the saturated hydraulic conductivity Ks in the domain. Different standard deviations of log hydraulic conductivity are investigated. For each sln Ks value of various realizations of hydraulic conductivity were generated and combined with a numerical s , ln Ks model to simulate water flow in an earth dam with variable Ks. The first-order second-moment reliability index b was employed to characterize the influence of the variability of Ks, and hence, pore-water pressures, on the stability of the downstream slope. A linear relationship between and the standard deviation of the factor sln Ks of safety sF was obtained from the simulation results. A relationship between b and , in which every 0.1 sln Ks increment of results in a decrease of 1.0 in b, is deduced based on the simulation results. Results of a sln Ks Shapiro-Wilk test for goodness of fit indicate that the factor of safety can be assumed to be normally or lognormally distributed when the saturated hydraulic conductivity follows a lognormal distribution and is sln Ks small (#0.5). When is large (>0.5), neither normal nor lognormal distributions provide a reasonable s ln Ks approximation of the factor of safety. Simulation results show that neither standard deviation nor coefficient of variation of the factor of safety is constant when only the variability of hydraulic conductivity is considered. While the results presented are directly applicable only to the particular earth dam geometry and boundary conditions studied, the methodology is general and may be extended to embankments with different boundary conditions. INTRODUCTION AND BACKGROUND Natural soils are highly variable and heterogeneous (Lamb 1966; Peck 1967; Vanmarcke 1977a; DeGroot and Baecher 1993). In practice, methods used for the analysis of slope sta- bility typically are deterministic, with soil properties charac- terized as constants for a given soil layer and each specified layer assumed to be homogeneous. The variability of soil properties presumably is taken into account indirectly for de- sign purposes by selecting a factor of safety, which is chosen based on experience and/or some design criteria (Griffiths and Fenton 1993; Duncan 1996). Many authors have pointed out the limitations of deterministic methods because they do not account explicitly for variability and uncertainty related to soil parameters (Vanmarcke 1977b; Oka and Wu 1990; Soulie et al. 1990; Kulhawy et al. 1991). Methods are needed to quan- tify uncertainties associated with soil properties and to include them in slope stability analysis (Anderson et al. 1982; Whit- tlestone et al. 1995). By incorporating uncertainties quantita- tively, reliability analysis can enhance engineering judgment and facilitate improved decision making (Li and Lumb 1987). Many water-retaining structures in North America are em- bankment dams (Fenton and Griffiths 1996) and the prediction of dam slope stability is of great importance. Hydraulic con- ductivity of the embankment soils is used to calculate all of the hydraulic parameters associated with dam performance, in- cluding seepage rates and amounts, pore-water pressures, and therefore, the state of effective stress. It follows that uncer- tainty associated with the spatial variability of hydraulic con- ductivity leads to uncertainty in estimating pore-water pres-

The Correlation Bias for Two-Dimensional Simulations by Turning Bands

The turning bands method (TBM) generates realizations of isotropic Gaussian random fields by summing contributions from line processes. We consider two-dimensional simulations and study the correlation bias attributable to the use of only a finite number L of lines. Our analytical and numerical results confirm that the maximal bias is of order 1/L, and that L = 64 lines suffice for excellent covariance reproduction. The notorious banding observed in simulations with an insufficient number of lines is a related but different phenomenon and depends strongly on the choice of the line simulation technique. Clear-cut recommendations for the number of lines necessary to avoid the effect can only be based on practical experience with the specific code at hand.

Fractal generation of surface area of porous media

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.

A space‐time rainfall disaggregation model adapted to Sahelian Mesoscale Convective Complexes

A model adapted to the disaggregation of rainfields associated to Sahelian mesoscale convective complexes (MCCs) at small time (≈5 min) and space (1–10 km) scales is presented. Spatial and temporal disaggregation have been decoupled in order to simplify the problem. The average rain depth over the study area for each storm is first disaggregated in space by using the turning band method, a geostatistical technique able to generate spatially correlated fields, with a known covariance function. The method has been adapted in order to simulate non‐Gaussian rainfall fields including in particular a probability of zero rainfall. The movement of the rain system is described by mapping the starting time of the rain event as the convective front moves through the domain. Finally, at each location, the temporal disaggregation of the storm rain depth is obtained by using a standard hyetogram model reproducing the typical sequence of a convective front followed by a stratiform trail. In its present form the only input of the disaggregation model is the average rain depth over the domain which can be provided either by a general circulation model output or a satellite estimate. To perform the various steps of the spatial and temporal disaggregation, some knowledge on the climatology of the zone is needed. Over the Sahel, this knowledge is provided by the EPSAT‐Niger data set, which contains 170 mesoscale convective systems (MCSs) observed over a 4‐year period by a dense recording raingauge network and a weather radar in the Niamey area. Half of these MCSs were identified as MCCs. The analysis of the MCCs' data set allowed the derivation of a probability distribution and a covariance function of the point storm rain depth. It also allowed the calculation of statistics for the average speed and direction of storm displacement and the inference of parameters describing the standard hyetograms. A complete example of the use of the model is presented showing that it produces realistic rainfall fields. Perspectives for further development and validation of the model are given in the conclusion.

SPATIAL VARIABILITY OF CONDUCTWITY AS APPLIED TO DISTRIBUTED PARAMETER INFILTRATION MODELS1

ABSTRACT: Two infiltration models, called VVSIM (variable variance stochastic infiltration model) and EVVSIM (enhanced variable variance stochastic infiltration model), are developed in this study. A distributed parameter infiltration model can estimate the amount of infiltration over a field area by computing the infiltration over zones of the field area. Hydraulic conductivity is the most important parameter determining infiltration in simulations by infiltration models. The performance of an infiltration model depends on how well the model accommodates the complicated spatial distribution of hydraulic conductivity. The two proposed models include the effects of spatial correlation of the conductivity distribution. Virtual conductivity fields are generated using the turning bands method. Monte Carlo simulations show that the proposed models give infiltration estimates more accurate than those obtained by the other models employed.

Monte Carlo Treatment of Random Fields: A Broad Perspective

A review of a number of methods for random fields simulation in conjunction with Monte Carlo studies of probabilistic mechanics problems is presented from a broad perspective. This article complements some of the previous review articles in that it compares various simulation algorithms, assesses their relative computational efficiency and versatility, discusses the properties of generated field samples, and incorporates some of the recent developments. Collectively, a comprehensive discussion of the covariance decomposition method, the spectral method, the ARMA method, the noise shower method, the scale refinement methods, and the turning band method is attempted. For tutorial effectiveness univariate, uni-dimensional, Gaussian, and homogeneous fields are discussed, primarily in connection with various simulation methods. Nevertheless, appropriate references are included addressing the simulation of more general fields. This review article contains 110 references.

Closed Form Solutions of the Two-Dimensional Turning Bands Equation

The turning bands method generates realizations of isotropic Gaussian random fields by means of appropriately summed line processes. For two-dimensional simulations the relation between the isotropic correlation function of the random field and the correlation function to be simulated along the lines is given by an integral equation of Abel type. We present closed form solutions of this integral equation for almost all two-dimensional correlation models encountered in practice and discuss their numerical implementation. As an additional benefit, our tables and illustrations serve as a concise guide to correlation models useful in geostatistics.

Simulated Recovery of Light, Nonaqueous Phase Liquid from Unconfined Heterogeneous Aquifers

AbstractA multiphase flow model (ARMOS) was used to evaluate the effects of subsurface heterogeneities on the recovery of light, nonaqueous phase liquids (LNAPLs or simply “oil”). Stochastic inputs for the model included the saturated hydraulic conductivity (Ksw), van Genuchten α and n, the water saturation at field capacity (Sm), and the maximum residual oil saturations in the saturated (Sor) and the unsaturated zones (Sog). The turning bands method was used to generate stochastic soil parameters representing three hypothetical sandy loam soils. Oil recovery in the three heterogeneous cases was compared to an “equivalent” homogeneous soil with effective parameters computed as the geometric means of the stochastic parameters. Distributions of the free oil plumes were described over time using statistical and spatial moment analyses. Due to the smoothing effect of the flow process, the predicted well oil thickness (Ho) and free oil specific volume (Vof) were less variable than the input stochastic parameter In(Ksw). However, Ho and Vof became more variable as free oil volume diminished and the oil distribution was controlled more by soil variability than gradients in the fluid levels. At the onset of oil recovery, the free oil area was greater in all three heterogeneous soils than the homogeneous soil. Nevertheless, soil heterogeneities did not greatly affect oil recovery or trapping in the saturated and unsaturated zones. Heterogeneities may have had a greater influence on oil recovery if a smaller spill had been studied or if the vertical dimension had been simulated explicitly. The results suggest that the geometric mean soil properties provide a useful estimate of the potential for oil recovery from oil spills that span a large number of correlation scales.

Spatial and temporal variability of nitrate in irrigated salad crops

The objective of this study was to analyze the spatial and seasonal variations in NO3 –-N concentration in soil samples and solution samplers and the N leaching of an irrigated crop cultivated intensively in the Mediterranean zone. Although much information is available from controlled field experiments concerning N concentration and its spatial variability, quantitative estimates of nitrate fluxes under normal farming conditions and when the field is directly managed by farmers are rare. This is particularly true for gardening crops in the Mediterranean zone, where high evapotranspiration rates lead to intensive irrigation and may be responsible for N leaching. A field experiment was conducted in the Departement du Gard under agricultural conditions. Salads (Cichorium endivia, Lactuca sativa) were planted in three consecutive periods. The field was irrigated with sprinklers. Local measurements with a neutron probe were made at two sites (row, interrow), and an experimental plot (95 m×25 m) was surveyed at 36 points located on a 10 m×10 m equilateral grid to analyze the spatial variability of water and NO3 –-N balances. To analyze the basic statistical properties of our sampling scheme, random fields of soil concentration were simulated with the turning-bands method. Sampling strategy simulations indicated that when a spatial structure exists, sampling according to a regular grid was more efficient than a purely random sampling strategy. Global trends indicated high spatial variability for nitrate leaching with differences between periods of different irrigation intensity (97 kg ha–1 NO3 –-N leaching during the spring and summer, and 199 kg ha–1 NO3 –-N leaching during autumn and winter). Leaching caused temporal variations in the spatial distributions of NO3 –-N. The origin of the spatial variability of N leaching was explained by first, the variability in NO3 –-N concentration in the soil profile, and second, by spatial variability in irrigation. Furthermore, the spatial distribution of the NO3 –-N concentration was time dependent, and NO3 –-N spatial distributions became independent after approximately 2 or 3 months under our conditions. Our results show that better management of irrigation and fertilizer in spring and summer may reduce N leaching and, thus, improve ground water quality.

Estimation and simulation of lognormal random fields

This paper describes several techniques for estimating and simulating lognormal random fields. The lognormal maximum entropy estimator is presented and compared with simple lognormal kriging. Conditional simulation is performed using the turning bands method combined with simple lognormal kriging. For Gaussian random fields, kriging and the maximum entropy estimator are shown to be equivalent. For lognormal random fields, however, simple lognormal kriging and the lognormal maximum entropy estimator differ. Conditional simulation via simulated annealing is also demonstrated but with generally unsatisfactory results. A hybrid approach involving the turning bands method and simple lognormal kriging yields satisfactory results in the sense that it generates conditional simulations that consistently reproduce conditional marginal distributions.

Comment on “A Simple and Efficient Space Domain Implementation of the Turning Bands Method” by C. R. Dietrich

Comment on “A Simple and Efficient Space Domain Implementation of the Turning Bands Method” by C. R. Dietrich

Reply [to “Comment on ‘A Simple and Efficient Space Domain Implementation of the Turning Bands Method’ by C. R. Dietrich”

Reply [to “Comment on ‘A Simple and Efficient Space Domain Implementation of the Turning Bands Method’ by C. R. Dietrich”

Effect of subsurface heterogeneity on free-product recovery from unconfined aquifers

Free-product record system designs for light-hydrocarbon-contaminated sites were investigated to evaluate the effects of subsurface heterogeneity using a vertically integrated three-phase flow model. The input stochastic variable of the areal flow analysis was the log-intrinsic permeability and it was generated using the Turning Band method. The results of a series of hypothetical field-scale simulations showed that subsurface heterogeneity has a substantial effect on free-product recovery predictions. As the heterogeneity increased, the recoverable oil volume decreased and the residual trapped oil volume increased. As the subsurface anisotropy increased, these effects together with free- and total-oil contaminated areas were further enhanced. The use of multiple-stage water pumping was found to be insignificant compared to steady uniform pumping due to reduced recovery efficiency and increased residual oil volume. This observation was opposite to that produced under homogeneous scenarios. The effect of subsurface heterogeneity was enhanced at relatively low water pumping rates. The difference in results produced by homogeneous and heterogeneous simulations was substantial, indicating greater attention should be paid in modeling free-product recovery systems with appropriate subsurface heterogeneity.

Modelling the effects of spatial variability in rainfall on catchment response. 1. Formulation and calibration of a stochastic rainfall field model

The relationship between the spatial variability of rainfall and catchment response is investigated by conducting experiments with a stochastic rainfall field model and a physically based distributed modelling system, the Système Hydrologique Européen (SHE), both of which are calibrated for a small upland catchment. The development and calibration of the rainfall field model is described in Part 1 of this paper, and the experiments with simulated rainfall fields and the SHE catchment model are described in Part 2. The rainfall field model is based on the use of the Turning Bands Method (TBM) incorporating a fractionally differenced line process to generate Gaussian random fields with a specified space-time correlation structure which can be isotropic or anisotropic. A transformation is then applied to the Gaussian field to reproduce the non-stationary temporal structure and skewed marginal distribution of observed rainfall. The transformed field is then propagated in space with the required velocity. The model is calibrated using hourly data for ten storms observed at three sites in the upper Wye catchment (area 10.55 km 2) at Plynlimon, Wales. As rainfall over the catchment exhibits significant variation with altitude (and other factors), an altitude correction factor is applied to the simulated rainfall fields. Comparisons of the means, variances, skewnesses, cross- and auto-correlation functions of observed and simulated storms at the sampling points show good agreement, and realistic spatial patterns are observed in the simulated fields. A procedure is developed and applied for generating conditional simulations whereby the historical storm rainfall at the sampling points can be reproduced exactly in simulated fields, thus allowing several realizations of the unobserved spatial variability of rainfall at all other points in the catchment to be generated for any historical storm.

Monte Carlo Simulation of Radon Emanation from Dry Building Materials

Prediction of radon release from building materials is a general problem. Since it is impractical to follow each recoil radon path and measure the surface area of porous texture by experimental methods, a powerful computer simulation approach is conducted in this paper. The Monte Carlo simulation program TRIM, based on the momentum and energy conservation of the ion transport in matter, is modified to simulate the recoil path in a 3-D frame. A follow-up Monte Carlo program was established further to calculate the detailed recoil range distribution and the recoil probability (emanating power). The powerful Turning Bands Method (TBM) of random field is applied to simulate the 3-D porous texture based on the porosity and the correlation function of the porous texture. Based on the simulated porous texture, the fractal dimension of the surface is calculated and used to calculate the effective surface area for radon recoil. The relations between the air buffer thickness, embedding effect of recoil, and the measuring scale of the surface area for the porous texture are discussed and numerically calculated. After these calculations are performed, the relations between the emanating rate, surface area, material density, and porosity are established to calculate the radon emanation rate from porous materials. This paper provides a clear theoretical picture of the mechanism of radon release from the building materials. It has also potential application to the recoil release calculations of other radioactive elements from solid materials.

Infrared extinction of the powder of brass 70Cu/30Zn calculated by the FDTD and turning bands methods [military smoke screening applications

The finite-difference time-domain (FDTD) method is used to calculate the specific extinction cross section of the powder of brass 70Cu/30Zn with 10/sup 3/ to 2.16-10/sup 5/ cubical particles for cell sizes in the range of 0.025 to 0.5 /spl mu/m at infrared frequency. The digitized models with a random process using the turning bands method are simulated for the powder of brass 70Cu/30Zn. From theoretical calculations, the value of the specific extinction cross section of the powder of brass 70Cu/30Zn is between 0.1 to 4.6 m/sup 2//g. While from the experimental measurement, the value of the specific extinction cross section is between 0.58 to 3.78 m/sup 2//g. Most of the theoretical results make a good agreement with those obtained from the experimental measurements for the cell sizes of particles in the range of 0.925 to 0.5 /spl mu/m. From the numerical calculations, it is also found that there is a resonant extinction value occuring at the resonant particle size d/sub 0/ which is approximately 2.54-n/sub p//sup -0.293/ /spl mu/m determined by a least square curve fitting method, where n/sub p/ is the number of particles. The resonant value calculated by the numerical solution is larger than the maximum value obtained from the experimental measurement. The work has applications to military smoke screening.

A comparative study of simulation techniques for two dimensional data honouring specified exponential semivariograms

AbstractThe effectiveness of four techniques for producing wide sense stationary data with exponential semivariograms is examined. Comparison is made primarily on the basis of the observed semivariograms. The LU decomposition of the covariance matrix appears to most accurately model specified semivariograms, whilst the more computationally efficient Matrix Polynomial approximation and Turning Bands methods may be more useful in practice.

A Simple and Efficient Space Domain Implementation of the Turning Bands Method

Given a correlation function c(x) with x in Rn, the turning bands method (TBM( for n = 2 and 3 constructs realizations of an n‐dimensional and stationary process Y ∼ N(O, c(x)) from appropriately summed line processes. Therefore an implementation of TBM calls naturally for fast and accurate generations of line realizations. These have generally been generated by a spectral approach since the Fourier transform of the line covariances is linked in a very simple fashion to the n‐dimensional Fourier transform of c(x). However, we show that in many instances a space domain implementation of TBM, with line realizations obtained from a circulant embedding method, has advantages over a spectral implementation in that without increase in computational costs (1) the line realizations display exactly the required covariance structure, (2) knowledge of the spectral density of the process Y is not required, and (3) fine tuning of line process parameters is not needed.

Stochastic simulation of the regional pesticide transport including the unsaturated and the saturated zone

Abstract New drinking water standards in the European Community allow a maximum total pesticide concentration of 0.5 μg/l. Water works and environmental agencies at the local, state and federal level are forced to rigidly assess the contamination potential of pesticides under a great variety of natural conditions. A large number of pesticide studies were therefore initiated, mostly either at laboratory scale using column or lysimeter experiments or at catchment scale using groundwater quality data. However, there is also a need for coupled approaches, where laboratory data are integrated into regional catchment studies taking into account the spatial and temporal variability and/or uncertainty of the governing transport parameters. This paper presents a stochastic approach where a numerical, one-dimensional pesticide transport model has been coupled with a numerical, three-dimensional groundwater flow and transport model. The variability of the soil parameters is taken into account through multiple realizations of the one-dimensional parameter sets at a scale of individual agricultural fields. The regional heterogeneity of the aquifer (saturation zone) can be described from borehole and structural data either deterministically or as shown here using a random field generator (e.g. Turning Bands method). Based on a set of soil data from the literature, the importance of different soil parameter distributions was analysed with respect to concentration breakthrough curves at an assumed groundwater extraction well. This analysis is aimed towards an optimization of data collection strategies for future pesticide leachate risk assessments. Using the described coupled stochastic modelling approach, the sensitivity of pesticide concentrations in the production well was estimated as a result of varying soil and aquifer transport properties.

Generating correlated gaussian random fields by orthogonal polynomial approximations to the square root of the covariance matrix

Monte Carlo simulations in geostatistics often require the generation of realizations of multi-dimensional Gaussian processes with prescribed mean and covariance over a specified grid Ω of sampling points in R:d In principle, given a factorization C = AAT of the process’ covariance matrix C, realizations can always be constructed from the product Aϵ where ϵ is a white noise vector with unit variance. However, this approach generally has a high computational cost when compared with techniques such as the spectral or turning bands methods. Nevertheless, matrix factorization has advantages in that it does not require any special structure, such as isotropy or stationarity of the covariance model, o r regularity of the sampling grid Ω. Therefore, in an effort to speed up the approach at the cost of some loss in accuracy, Davis (1987b) proposed approximating the square root A by a low order matrix polynomial in C. This paper explores the construction of such polynomial approximations in more detail The paper f...