Geostatistical interpolation of chemical concentration

Measurements of contaminant concentration at a hazardous waste site typically vary over many orders of magnitude and have highly skewed distributions. This work presents a practical methodology for the estimation of solute concentration contour maps and volume averages (needed for mass calculations) from data obtained from the analysis of water and soil samples. The methodology, which is an extension of linear geostatistics, produces a point estimate, i.e., a representative value, as well as a confidence interval, which contains the true value with a given probability. The approach uses a parsimonious model that accounts for the skewness by adding only one parameter to those used in linear geostatistics (variograms or generalized covariances). The resulting nonlinear kriging method is not substantially more difficult to use than linear geostatistics. The methodology is most appropriate when concentration measurements are available on a reasonably dense grid and no additional information (based on modeling flow and transport) can be used. We present and illustrate through an application, a practical approach to estimate all the parameters needed and to select and test the model.     

瞬变电磁采样函数优化法降噪技术

常用的双极性同步采样方法对瞬变电磁系统的观测噪声具有一定的抑制作用,但在实际工作中其噪声抑制能力仍显不足.本文在双极性同步采样方法基础上提出一种优化方法,以提升瞬变电磁系统的噪声抑制能力,即：通过对取样道加高斯窗函数的方式,赋予取样道内各采样点合适的累加权重,提升系统对取样道内高频噪声的抑制能力;使用高斯窗函数与梳状滤波器构造复合窗函数,通过对包含所有叠加周期的完整观测序列加复合窗函数的形式优化各周期的叠加权重,提升系统对实际工频噪声的抑制能力.对优化前后不同采样函数的谱特性及野外实测数据的处理效果进行比较,结果表明本文提出的方法较传统双极性同步采样方法具有更强的噪声抑制能力.

     

Geostatistical integration of near-surface geophysical data

Accurate statics calculation and near‐surface related noise removal require a detailed knowledge of the near‐surface velocity field. Conventional seismic surveys currently are not designed to provide this information, and 3D high‐resolution reflection/refraction acquisition is not feasible for large survey areas. Satellite images and vibrator plate attributes are dense low‐cost data, which can be used in spatially extrapolating velocities from sparse uphole data by geostatistics. We tested this approach in two different areas of Saudi Arabia and found that the optimal recipe depends on the local geology.     

Geostatistical inversing for large-contrast transmissivity fields

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.     

Mapping optimization based on sampling size in earth related and environmental phenomena

A critical sampling grid can be defined for an earth related natural variable distributed in space, according to established theoretical results and under certain mathematical conditions. Sampling above this critical limit does not substantially improve mapping results, while based on this limit the ideal process of reproducing the original phenomenon is theoretically defined. The aim of the present paper is, by using an innovative approach; to investigate the validity of commonly used interpolation algorithms, both stochastic and deterministic, below and above this critical sampling limit. When sampling is dense, application to a simulated spatial random field shows that the results are equally accurate with those derived with more sophisticated stochastic methods. On the other hand, when the sampling grid is sparse, deterministic methods produce less accurate results, therefore stochastic algorithms with minimum estimation error are a much better option. To further demonstrate these points, the interpolation algorithms were applied in three different sampling grid densities in a contaminated waste disposal site in Russia.     

Geostatistical traveltime tomography in elliptically anisotropic media

In geological materials, anisotropy may arise due to different mechanisms and can be found at different scales. Neglecting anisotropy in traveltime tomographic reconstruction leads to artefacts that can obscure important subsurface features. In this paper, a geostatistical tomography algorithm to invert cross‐hole traveltime data in elliptically anisotropic media is presented. The advantages of geostatistical tomography are that the solution is regularized by the covariance of the model parameters, that known model parameters can be used as constraints and fitted exactly or within a prescribed variance and that stochastic simulations can be performed to appraise the variability of the solution space. The benefits of the algorithm to image anisotropic media are illustrated by two examples using synthetic georadar data and real seismic data. The first example confirms suspected electromagnetic anisotropy in the vadose zone caused by relatively rapid water content variations with respect to wavelength at georadar frequencies. The second presents how sonic log data can be used to constrain the inversion of cross‐well seismic data and how geostatistical simulations can be used to infer parameter uncertainty. Results of both examples show that considering anisotropy yields a better fit to the data at high ray angles and reduces reconstruction artefacts.     

Effects of Spatiotemporal Constraints on Geostatistical Analysis of Groundwater Depth

Geostatistical evaluation of the groundwater depth (GWD) in California's South Coast hydrologic region, and its sensitivity to different spatiotemporal assumptions, is presented in this paper. We obtain a pseudo-stationary representation of the groundwater depth, using the publicly available, online database from the GAMA GeoTracker project, while tracking the associated uncertainty throughout the process. We create nine different sub-datasets, using different temporal constraints, such as seasonal partitioning and different long-term variability filtering criteria. The geostatistical analysis and comparison between the different maps highlight the trade-off between spatial and temporal accuracy. For example, when moving to stricter filtering criteria, despite removing a large number of sites from the interpolation, the root mean squared error (RMSE) calculated in the analysis either decreased or only slightly increased. This suggests that the long-term variability filter is a good representation of the GWD accuracy and that the cross-validation RMSE captures both the stability effect as well as spatial density of the measurement points. We further find that the point-specific standard error is strongly correlated with the associated GWD prediction and that the mean relative error is approximately 60% of the prediction. Hence, it is highly recommended to account for such error in a forward-engineering application, by introducing a GWD distribution rather than a single value into the analysis. Finally, we analyze seasonal fluctuations in the study region and find that they are on average 2.5 m with a standard deviation of 8 m.     

Geostatistical mapping of geomorphic variables in the presence of trend

Mapping geomorphic variables geostatistically, specifically by kriging, runs into difficulties when there is trend. The reason is that the variogram required for the kriging must be of residuals from any trend, which in turn cannot be estimated optimally by the usual method of trend surface analysis because the residuals are correlated. The difficulties can be overcome by the use of residual maximum likelihood (REML) to estimate both the trend and the variogram of the residuals simultaneously. We summarize the theory of REML as it applies to kriging in the presence of trend. We present the equations to show how estimates of the trend are combined with kriging of residuals to give empirical best linear unbiased predictions (E‐BLUPs). We then apply the method to estimate the height of the sub‐Upper‐Chalk surface beneath the Chiltern Hills of southeast England from 238 borehole data. The variogram of the REML residuals is substantially different from that computed by ordinary least squares (OLS) analysis. The map of the predicted surface is similar to that made from kriging with the OLS variogram. The variances, however, are substantially larger because (a) they derive from a variogram with a much larger sill and (b) they include the uncertainty of the estimate of the trend. Copyright © 2006 John Wiley & Sons, Ltd.     

Geostatistical facies simulation with geometric patterns of a petroleum reservoir

During exploration and pre-feasibility studies of a typical petroleum project many analyses are required to support decision making. Among them is reservoir lithofacies modeling, preferably using uncertainty assessment, which can be carried out with geostatistical simulation. The resulting multiple equally probable facies models can be used, for instance, in flow simulations. This allows assessing uncertainties in reservoir flow behavior during its production lifetime, which is useful for injector and producer well planning. Flow, among other factors, is controlled by elements that act as flow corridors and barriers. Clean sand channels and shale layers are examples of such reservoir elements that have specific geometries. Besides simulating the necessary facies, it is also important to simulate their shapes. Object-based and process-based simulations excel in geometry reproduction, while variogram-based simulations perform very well at data conditioning. Multiple-point geostatistics (MPS) combines both characteristics, consequently it was employed in this study to produce models of a real-world reservoir that are both data adherent and geologically realistic. This work aims at illustrating how subsurface information typically available in petroleum projects can be used with MPS to generate realistic reservoir models. A workflow using the SNESIM algorithm is demonstrated incorporating various sources of information. Results show that complex structures (e.g. channel networks) emerged from a simple model (e.g. single branch) and the reservoir facies models produced with MPS were judged suitable for geometry-sensitive applications such as flow simulations.     

Geostatistical modelling of the Wasia aquifer in Central Saudi Arabia

Aquifers in sedimentary basins provide a regional domain for the spatial variabilities in geologic, hydrologic, geomorphologic and hydrochemical phenomena. Their study should account for this spatial variability within the study area prior to any formal modelling. A cumulative semivariogram scheme is adopted in this paper for the spatial variability, which is then incorporated with the kriging technique to provide maps of regional variation concerning variables such as storativity, transmissivity, piezometric levels, total dissolved solids and groundwater flow velocity. It is shown that the classical semivariogram models are not capable of accounting for the spatial variability of the Wasia aquifer. Comparison between the cumulative and classical semivariograms are given on the basis of hydrogeologic variables observed in the field. It is concluded, in general, that the cumulative semivariogram modelling of the spatial variability is more effective and yields realistic regional variables.     

Impact of sampling depth and plant species on local environmental conditions, microbiological parameters and bacterial composition in a mercury contaminated salt marsh

We compare the environmental characteristics and bacterial communities associated with two rushes, Juncus maritimus and Bolboschoenus maritimus, and adjacent unvegetated habitat in a salt marsh subjected to historical mercury pollution. Mercury content was higher in vegetated than unvegetated habitat and increased with sampling depth. There was also a significant relationship between mercury concentration and bacterial composition. Habitat (Juncus, Bolboschoenus or unvegetated), sample depth, and the interaction between both, however, explained most of the variation in composition (~70%). Variation in composition with depth was most prominent for the unvegetated habitat, followed by Juncus, but more constrained for Bolboschoenus habitat. This constraint may be indicative of a strong plant-microbe ecophysiological adaptation. Vegetated habitat contained distinct bacterial communities associated with higher potential activity of aminopeptidase, β-glucosidase and arylsulphatase and incorporation rates of (14)C-glucose and (14)C-acetate. Communities in unvegetated habitat were, in contrast, associated with both higher pH and proportion of sulphate reducing bacteria.     

Assessing TCE Source Bioremediation by Geostatistical Analysis of a Flux Fence

Mass discharge across transect planes is increasingly used as a metric for performance assessment of in situ groundwater remediation systems. Mass discharge estimates using concentrations measured in multilevel transects are often made by assuming a uniform flow field, and uncertainty contributions from spatial concentration and flow field variability are often overlooked. We extend our recently developed geostatistical approach to estimate mass discharge using transect data of concentration and hydraulic conductivity, so accounting for the spatial variability of both datasets. The magnitude and uncertainty of mass discharge were quantified by conditional simulation. An important benefit of the approach is that uncertainty is quantified as an integral part of the mass discharge estimate. We use this approach for performance assessment of a bioremediation experiment of a trichloroethene (TCE) source zone. Analyses of dissolved parent and daughter compounds demonstrated that the engineered bioremediation has elevated the degradation rate of TCE, resulting in a two‐thirds reduction in the TCE mass discharge from the source zone. The biologically enhanced dissolution of TCE was not significant (～5%), and was less than expected. However, the discharges of the daughter products cis‐1,2, dichloroethene (cDCE) and vinyl chloride (VC) increased, probably because of the rapid transformation of TCE from the source zone to the measurement transect. This suggests that enhancing the biodegradation of cDCE and VC will be crucial to successful engineered bioremediation of TCE source zones.     

Geostatistical seismic Amplitude‐versus‐angle inversion

Seismic inversion plays an important role in reservoir modelling and characterisation due to its potential for assessing the spatial distribution of the sub‐surface petro‐elastic properties. Seismic amplitude‐versus‐angle inversion methodologies allow to retrieve P‐wave and S‐wave velocities and density individually allowing a better characterisation of existing litho‐fluid facies. We present an iterative geostatistical seismic amplitude‐versus‐angle inversion algorithm that inverts pre‐stack seismic data, sorted by angle gather, directly for: density; P‐wave; and S‐wave velocity models. The proposed iterative geostatistical inverse procedure is based on the use of stochastic sequential simulation and co‐simulation algorithms as the perturbation technique of the model parametre space; and the use of a genetic algorithm as a global optimiser to make the simulated elastic models converge from iteration to iteration. All the elastic models simulated during the iterative procedure honour the marginal prior distributions of P‐wave velocity, S‐wave velocity and density estimated from the available well‐log data, and the corresponding joint distributions between density versus P‐wave velocity and P‐wave versus S‐wave velocity. We successfully tested and implemented the proposed inversion procedure on a pre‐stack synthetic dataset, built from a real reservoir, and on a real pre‐stack seismic dataset acquired over a deep‐water gas reservoir. In both cases the results show a good convergence between real and synthetic seismic and reliable high‐resolution elastic sub‐surface Earth models.