Spatial and temporal variability of pore‐water pressures in residual soil slopes in a tropical climate

It is critical to understand and quantify the temporal and spatial variability in hillslope hydrological data in order to advance hillslope hydrological studies, evaluate distributed parameter hydrological models, analyse variability in hydrological response of slopes and design efficient field data sampling networks. The spatial and temporal variability of field‐measured pore‐water pressures in three residual soil slopes in Singapore was investigated using geostatistical methods. Parameters of the semivariograms, namely the range, sill and nugget effect, revealed interesting insights into the spatial structure of the temporal situation of pore‐water pressures in the slopes. While informative, mean estimates have been shown to be inadequate for modelling purposes, indicator semivariograms together with mean prediction by kriging provide a better form of model input. Results also indicate that significant temporal and spatial variability in pore‐water pressures exists in the slope profile and thereby induces variability in hydrological response of the slope. Spatial and temporal variability in pore‐water pressure decreases with increasing soil depth. The variability decreases during wet conditions as the slope approaches near saturation and the variability increases with high matric suction development following rainfall periods. Variability in pore‐water pressures is greatest at shallow depths and near the slope crest and is strongly influenced by the combined action of microclimate, vegetation and soil properties. Copyright © 2002 John Wiley & Sons, Ltd.     

Multifractal modeling and spatial statistics

In general, the multifractal model provides more information about measurements on spatial objects than a fractal model. It also results in mathematical equations for the covariance function and semivariogram in spatial statistics which are determined primarily by the second-order mass exponent. However, these equations can be approximated by power-law relations which are comparable directly to equations based on fractal modeling. The multifractal approach is used to describe the underlying spatial structure of De Wijs 's example of zinc values from a sphalerite-bearing quartz vein near Pulacayo, Bolivia. It is shown that these data are multifractal instead of fractal, and that the second-order mass exponent (=0.979±0.011 for the example) can be used in spatial statistical analysis.     

Fitting the Linear Model of Coregionalization by Generalized Least Squares

In geostatistical studies, the fitting of the linear model of coregionalization (LMC) to direct and cross experimental semivariograms is usually performed with a weighted least-squares (WLS) procedure based on the number of pairs of observations at each lag. So far, no study has investigated the efficiency of other least-squares procedures, such as ordinary least squares (OLS), generalized least squares (GLS), and WLS with other weighing functions, in the context of the LMC. In this article, we compare the statistical properties of the sill estimators obtained with eight least-squares procedures for fitting the LMC: OLS, four WLS, and three GLS. The WLS procedures are based on approximations of the variance of semivariogram estimates at each distance lag. The GLS procedures use a variance–covariance matrix of semivariogram estimates that is (i) estimated using the fourth-order moments with sill estimates (GLS₁), (ii) calculated using the fourth-order moments with the theoretical sills (GLS₂), and (iii) based on an approximation using the correlation between semivariogram estimates in the case of spatial independence of the observations (GLS₃). The current algorithm for fitting the LMC by WLS while ensuring the positive semidefiniteness of sill matrix estimates is modified to include any least-squares procedure. A Monte Carlo study is performed for 16 scenarios corresponding to different combinations of the number of variables, number of spatial structures, values of ranges, and scale dependence of the correlations among variables. Simulation results show that the mean square error is accounted for mostly by the variance of the sill estimators instead of their squared bias. Overall, the estimated GLS₁ and theoretical GLS₂ are the most efficient, followed by the WLS procedure that is based on the number of pairs of observations and the average distance at each lag. On that basis, GLS₁ can be recommended for future studies using the LMC.     

Assessment of regional air pollution distribution by point cumulative semivariogram method at Erzurum urban center, TURKEY

In the assessment of air quality, regional distribution and dispersion with distance are important, together with the variations of pollutants in time. On this occasion, the point cumulative semi-variogram (PCSV) method is used in order to find simply regional distribution of pollutants of Erzurum urban centre. This method is based simply on the summation of square differences in air pollutant concentrations between different sites. Monthly regional variation maps of Erzurum are constructed by finding radius of influence (for SO₂, from 1000 m to 3500 m and, for TSP, 1000–2000 m) and PCSV scattering diagram data at different levels by using monthly average sulphur dioxide (SO₂) and total suspended particulate (TSP) matter concentrations in 2001–2002 winter season. Consequently, the air pollution distribution of Erzurum is assessed.     

Improving kriging of groundwater level data using nonlinear normalizing transformations—a field application

Abstract

The present research study investigates the application of nonlinear normalizing data transformations in conjunction with ordinary kriging (OK) for the accurate prediction of groundwater level spatial variability in a sparsely-gauged basin. We investigate three established normalizing methods, Gaussian anamorphosis, trans-Gaussian kriging and the Box-Cox method to improve the estimation accuracy. The first two are applied for the first time to groundwater level data. All three methods improve the mean absolute prediction error compared to the application of OK to the non-transformed data. In addition, a modified Box-Cox transformation is proposed and applied to normalize the hydraulic heads. The modified Box-Cox transformation in conjunction with OK is found to be the optimal spatial model based on leave-one-out cross-validation. The recently established Spartan semivariogram family provides the optimal model fit to the transformed data. Finally, we present maps of the groundwater level and the kriging variance based on the optimal spatial model.

Editor D. Koutsoyiannis; Associate editor A. Montanari

Citation Varouchakis, E.A., Hristopoulos, D.T., and Karatzas, G.P., 2012. Improving kriging of groundwater level data using nonlinear normalizing transformations—a field application. Hydrological Sciences Journal, 57 (7), 1404–1419.     

The equivalence of predictions from universal kriging and intrinsic random-function kriging

A proof is provided that the predictions obtained from kriging based on intrinsic random functions of orderk are identical to those obtained from anappropriate universal kriging model. This is a theoretical result based on known variability measures. It does not imply that people performing traditional universal kriging will get the same predictions as those using intrinsic random functions, because traditionally these methods differ in how variability is modeled. For intrinsic random functions, the same proof shows that predictions do not depend on the specific choice of the generalized covariance function. It is argued that the choice between these methods is really one of modeling and estimating the variability in the data.     

Fractal properties of simulated bed profiles in coarse-grained channels

Bed roughness characteristics in coarse-grained channels are fairly complex. A hierarchy of roughness elements can be observed, ranging from variable particle sizes and shapes and small-scale sedimentary structures, to large-scale bedforms such as riffle-pool sequences. The effects of these scales of roughness on the flow geometry still remain to be thoroughly investigated. The semivariogram has been suggested in the past as a means of quantifying bed roughness effects on streamflow, as well as for distinguishing between scales of roughness. However, field measurements are rather time-consuming. The low number of bed profiles measured in the field precludes the identification of generally applicable relationships between the statistical properties derived from the semivariograms (such as the Hausdorff dimensions and the scale of autocorrelation corresponding to each fractal band) and the bed configuration itself (geometrical and sedimentological properties). Simulation results of gravel-bed profiles are, therefore, presented in order to complement the original investigation of Robert (1988a). The simulation experiments, based on grain characteristics of sizes and shapes and on morphological properties of small-scale bedforms, yield very significant information on boundary roughness at the microscale and give insight into the interpretation of empirical semivariograms (derived from field measurements). Bed-material sorting, variable grain shapes, and height and spacing of cluster bedforms control the fractal dimensions obtained from the semivariograms, as well as the location of the break of slope and the range of the process.     

Efficiency of kriging estimation for square, triangular, and hexagonal grids

Although several researchers have pointed out some advantages and disadvantages of various soil sampling designs in the presence of spatial autocorrelation, a more detailed study is presented herein which examines the geometrical relationship of three sampling designs, namely the square, the equilateral triangle, and the regular hexagon. Both advantages and disadvantages exist in the use of these designs with respect to estimation of the semivariogram and their effect on the mean square error or variance of error. This research could be used to design optimal sampling strategies; it is based on the theory of regionalized variables, in which the intrinsic hypothesis is satisfied. Among alternative designs, an equilateral triangle design gives the most reliable estimate of the semivariogram. It also gives the minimum maximum mean square error of point estimation of the concentration over the other two designs for the same number of measurements when the nugget effect is small relative to the variance. If the nugget effect is large (.90 ² or more), and the linear sampling density is >0.85r where r is the range, the hexagonal design is best. This study computes and compares the maximum mean square error for each of these designs.     

Characterization of the Structure of River-Bed Gravels Using Two-Dimensional Fractal Analysis

This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.     

Point Cumulative Semivariogram for Identification of Heterogeneities in Regional Seismicity of Turkey

Geological events are neither isotropic nor homogeneous in their occurrences. These two properties present difficulties for spatial modeling of regionalized variables. This paper presents a point cumulative semivariogram (PCSV) technique for quantifying the heterogeneity characteristics of the phenomenon concerned. The basis of the methodology is to obtain experimental PCSVs for each measurement point which led to estimation of radius of influence around each site. In addition, the experimental PCSVs provide basic information about the heterogeneity of the geological variable in the region, and furthermore many useful interpretations can be made concerning the regional variability of the variable. It provides the measure of cumulative similarity of a regional variable around any measurement site. Because PCSV is a means of measuring total similarity, maps at fixed similarity levels are provided in order to document the regional heterogeneity. Identification of heterogeneities depends on the comparison of fixed PCSV values at a multitude of irregularly scattered sites. The PCSV methodology has been applied to the regional seismic data of Turkey.