Can we improve the spatial predictions of seabed sediments? A case study of spatial interpolation of mud content across the southwest Australian margin

Spatially continuous data of environmental variables is often required for marine conservation and management. However, information for environmental variables is usually collected by point sampling, particularly for the marine region. Thus, methods generating such spatially continuous data by using point samples to estimate values for unknown locations become essential tools. Such methods are, however, often data- or even variable-specific and it is difficult to select an appropriate method for any given dataset. In this study, 14 methods (37 sub-methods) are compared using samples of mud content with five levels of sample density across the southwest Australian margin. Bathymetry, distance-to-coast, slope and geomorphic province were used as secondary variables. Ten-fold cross validation with relative mean absolute error (RMAE) and visual examination were used to assess the performance of these methods. A total of 1850 prediction datasets are produced and used to assess the performance of the methods and the effects of other factors considered. Considering both the accuracy and the visual examination, we found that a combined method (i.e., random forest and ordinary kriging: RKrf) is the most robust method. This method is novel, with a RMAE up to 17% less than that of the control. No threshold in sample density was detected in relation to prediction accuracy. No consistent patterns are observed between the performance of the methods and data variation. The RMAE of three most accurate methods is about 30% lower than that of the best methods in previous publications, highlighting the robustness of the methods selected in this study. The implications and limitations of this study are discussed and a number of suggestions are provided for further studies.     

Evaluation of a water equivalent of snow cover map in a small catchment area using a geostatistical approach

We describe an objective method for evaluating the spatial distribution of water equivalents of the snow cover within a small catchment. Regression analysis is used to quantify the relationship between elevation, presence or absence of forest, and potential direct solar radiation as independent variables and water equivalent as measured at a number of sites. First, this regression relationship is used to interpolate water equivalent data all over the basin area. Then we interpolate the residuals of the regression using a geostatistical approach. Superimposing the results obtained by interpolating the regression relationship and the interpolated residuals eventually yields the water equivalent distribution over the test area. The advantages of the interpolation method used lie in the optimal (effective, unbiased) estimation of the interpolated values as well as in the possibility to quantify the associated estimation variances.     

Geostatistical modelling of spatial and depth variability of SPT data for Bangalore

The purpose of this study is to develop a geostatistical model to evaluate the spatial and depth variability of Standard Penetration Test (SPT) data from Bangalore, India. The database consists of 766 boreholes spread over a 220 km² area, with several SPT values (N) in each of them. The geostatistical analysis is done for N corrected (N _corrected) values. The N _corrected value has been corrected for different parameters such as overburden stress, size of the bore hole, type of sampler, hammer energy and length of the connecting rod. The knowledge of the semivariogram of the SPT data is used with kriging theory to estimate the values at points in the subsurface of Bangalore where field measurements are not available. The model is used to compute the variance of estimated data. The model predicts reasonably well the SPT data. The geostatistical model provides valuable results that can be used for seismic hazard analysis, site response and liquefaction studies for the development of microzonation maps. The predicted N values from the developed model can also be used to estimate the subsurface information, allowable bearing pressure of soils and elastic modulus of soils.     

Quantifying the transferability of hydraulic parameters using geostatistics: The Boom Clay Case

The Boom Clay is currently investigated as potential host formation for the deep disposal of high-level and/or long-lived radioactive waste in Belgium. As such, the formation is expected to play a role of natural barrier, to slow the migration of radionuclides towards the biosphere for a sufficiently long time when the man-made barriers are no longer effective. In this context, the Boom Clay aquitard requires to be precisely characterized in terms of hydrogeological parameters, to confirm its role of geological barrier between its surrounding aquifers. Therefore, hydraulic conductivity and migration parameters have been intensively measured over the years in a few boreholes in Belgium; the latter are mainly located in the Mol-Dessel area, assuming historically a good lateral continuity of the geology. Combining these measurements with more densely acquired geophysical information allows quantifying their spatial variability and consolidating the continuity assumption. From a methodological point of view, the modeling of hydrogeological parameters requires to solve several issues. First, it is required to find a consistent geo-reference coordinate system allowing to laterally correlate thin observations derived from boreholes separated by several tens of kilometers. Then, in order to provide a reliable 3D model, it is compulsory to integrate the correlation between the scarcely sampled target parameters (core measurements) and numerous geophysical logs (gamma ray, resistivity). Geostatistics provides a suitable framework to analyze and solve these issues. Finally, a 3D model of the target parameters is proposed, together with an uncertainty envelope. This uncertainty quantification is of significant added value to assess the efficiency of the geological barrier. Besides the actual modeling of target parameters, the paper also presents sampling recommendations for forthcoming boreholes.     

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h  ) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity (Y=lnK)$(Y=\ln K)$, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.     

Geostatistical radar–raingauge merging: A novel method for the quantification of rain estimation accuracy

Compared to other estimation techniques, one advantage of geostatistical techniques is that they provide an index of the estimation accuracy of the variable of interest with the kriging estimation standard deviation (ESD). In the context of radar–raingauge quantitative precipitation estimation (QPE), we address in this article the question of how the kriging ESD can be transformed into a local spread of error by using the dependency of radar errors to the rain amount analyzed in previous work. The proposed approach is implemented for the most significant rain events observed in 2008 in the Cévennes-Vivarais region, France, by considering both the kriging with external drift (KED) and the ordinary kriging (OK) methods. A two-step procedure is implemented for estimating the rain estimation accuracy: (i) first kriging normalized ESDs are computed by using normalized variograms (sill equal to 1) to account for the observation system configuration and the spatial structure of the variable of interest (rainfall amount, residuals to the drift); (ii) based on the assumption of a linear relationship between the standard deviation and the mean of the variable of interest, a denormalization of the kriging ESDs is performed globally for a given rain event by using a cross-validation procedure. Despite the fact that the KED normalized ESDs are usually greater than the OK ones (due to an additional constraint in the kriging system and a weaker spatial structure of the residuals to the drift), the KED denormalized ESDs are generally smaller the OK ones, a result consistent with the better performance observed for the KED technique. The evolution of the mean and the standard deviation of the rainfall-scaled ESDs over a range of spatial (5–300 km²) and temporal (1–6 h) scales demonstrates that there is clear added value of the radar with respect to the raingauge network for the shortest scales, which are those of interest for flash-flood prediction in the considered region.     

Local and Spatial Joint Frequency Uncertainty and its Application to Rock Mass Characterisation

Stability is a key issue in any mining or tunnelling activity. Joint frequency constitutes an important input into stability analyses. Three techniques are used herein to quantify the local and spatial joint frequency uncertainty, or possible joint frequencies given joint frequency data, at unsampled locations. Rock quality designation is estimated from the predicted joint frequencies. The first method is based on kriging with subsequent Poisson sampling. The second method transforms the data to near-Gaussian variables and uses the turning band method to generate a range of possible joint frequencies. The third method assumes that the data are Poisson distributed and models the log-intensity of these data with a spatially smooth Gaussian prior distribution. Intensities are obtained and Poisson variables are generated to examine the expected joint frequency and associated variability. The joint frequency data is from an iron ore in the northern part of Norway. The methods are tested at unsampled locations and validated at sampled locations. All three methods perform quite well when predicting sampled points. The probability that the joint frequency exceeds 5 joints per metre is also estimated to illustrate a more realistic utilisation. The obtained probability map highlights zones in the ore where stability problems have occurred. It is therefore concluded that the methods work and that more emphasis should have been placed on these kinds of analyses when the mine was planned. By using simulation instead of estimation, it is possible to obtain a clear picture of possible joint frequency values or ranges, i.e. the uncertainty.     

Evaluating and classifying contaminated areas based on loss functions using annealing simulations

This paper presents a methodology based on geostatistical theory for quantifying the risks associated with heavy-metal contamination in the harbor area of Santana, Amapá State, Northern Brazil. In this area there were activities related to the commercialization of manganese ore from Serra do Navio. Manganese and arsenic concentrations at unsampled sites were estimated by postprocessing results from stochastic annealing simulations; the simulations were used to test different criteria for optimization, including average, median, and quantiles. For classifying areas as contaminated or uncontaminated, estimated quantiles based on functions of asymmetric loss showed better results than did estimates based on symmetric loss, such as the average or the median. The use of specific loss functions in the decision-making process can reduce the costs of remediation and health maintenance. The highest global health costs were observed for manganese.     

Spatial distribution of metals in ground/surface waters in the Chandrapur district (Central India) and their plausible sources

This study addresses a framework to evaluate and map environmental hazard with reference to spatial distribution of major and trace metal contamination and its relationship with lithology in Chandrapur district of Maharashtra, India using geospatial, statistical and GIS tools. In all, 208 ground water and 35 surface water samples were collected using global positioning system (GPS) synoptically with satellite imagery IRS P6 LISS III and were analyzed in ICP-AES. Analytical results reflect the presence of major and trace metals in ground water in terms of % as Fe (48%), Mn (12%), Zn (9%), Al (8%), Pb (7%), Cu (6%), Ni (4%), Cd (3%) and Cr (3%) of the total average concentration. The contamination is attributed to weathering of rocks and also to mining activities. Similarly, surface water contribution of major and trace metals was found as Al (47.8%), Fe (42.8%), Mn (5.5%), Zn (2.3%), Pb (0.56%), Ni (0.42%), Cu (0.16%), Cr (0.16%) and Cd (0.10%) of the total average concentration. Ordinary kriging interpolation method was adopted to assess the spatial distribution of different major and trace metals in groundwater samples with their best model fit variogram Classical statistical method like principal component analysis (PCA) was carried out in order to establish correlation between spatial pattern of metal contamination and geology of the area in GIS environment. Various surface and subsurface aspects like landuse/land cover, structural features, hydrogeology, topography etc were also considered to ascertain their impact to supplement the inference of the study.     

A three-dimensional high-resolution reservoir model of the Eocene Shahejie Formation in Bohai Bay Basin,integrating stratigraphic forward modeling and geostatistics

Reservoir characterization based on geostatistics method requires well constraints (e.g. seismic data with high quality) to predict inter-well reservoir quality that is conformed to geological laws. Nevertheless, the resolution of seismic data in multiple basins or reservoirs is not high enough to recognize the distribution of different types of sand bodies. In this paper, we propose a new method to improve the precision of reservoir characterization: reservoir modeling with the constrains of sedimentary process model and sedimentary microfacies. We employed stratigraphic forward modeling, a process-based method, to constrain the reservoir modeling in one oil-bearing interval of the third member of Eocene Shahejie Formation in J-Oilfield of Liaoxi Sag, Bohai Bay Basin.We divide reservoir modeling into two orders using different types of constrains. In the first order, we use the simulated shale model from stratigraphic forward model that is corrected by wells data as a 3D trend volume to constrain the reservoir sand-shale modeling. In the second order, different types of sedimentary microfacies in the sandy part of the model are further recognized and simulated within the constrains of sedimentary microfacies maps. Consequently, the porosity, permeability and oil saturation are modeled under the control of precise sedimentary microfacies model. The high-resolution reservoir model shows that the porosity, permeability and oil saturation of distributary channel is generally above 20%, 10md and 50%, respectively, which are much higher than that of other types of sedimentary microfacies. It can be concluded that comparing to other types of sedimentary facies, distributary channel has better physical properties and more oil accumulation in the fan-delta front and therefore is the most favorable zones for petroleum development in the research area.