Characterizing the spatial variability of groundwater quality using the entropy theory: I. Synthetic data

This paper, the first in a series of two, applies the entropy (or information) theory to describe the spatial variability of synthetic data that can represent spatially correlated groundwater quality data. The application involves calculating information measures such as transinformation, the information transfer index and the correlation coefficient. These measures are calculated using discrete and analytical approaches. The discrete approach uses the contingency table and the analytical approach uses the normal probability density function. The discrete and analytical approaches are found to be in reasonable agreement. The analysis shows that transinformation is useful and comparable with correlation to characterize the spatial variability of the synthetic data set, which is correlated with distance. Copyright © 2004 John Wiley & Sons, Ltd.     

Capturing the essential spatial variability in distributed hydrological modelling: Infiltration parameters

Selecting the correct resolution in distributed hydrological modelling at the watershed scale is essential in reducing scale-related errors. The work presented herein uses information content (entropy) to identify the resolution which captures the essential variability, at the watershed scale, of the infiltration parameters in the Green and Ampt infiltration equation. A soil map of the Little Washita watershed in south-west Oklahoma, USA was used to investigate the effects of grid cell resolution on the distributed modelling of infiltration. Soil-derived parameters and infiltration exhibit decreased entropy as resolutions become coarser. This is reflected in a decrease in the maximum entropy value for the reclassified/derived parameters vis a vis the original data. Moreover, the entropy curve, when plotted against resolution, shows two distinct segments: a constant section where no entropy was lost with decreasing resolution and another part which is characterized by a sharp decrease in entropy after a critical resolution of 1209 m is reached. This methodology offers a technique for assessing the largest cell size that captures the spatial variability of infiltration parameters for a particular basin. A geographical information system (GIS) based rainfall-runoff model is used to simulate storm hydrographs using infiltration parameter maps at different resolutions as inputs. Model results up to the critical resolution are reproducible and errors are small. However, at resolutions beyond the critical resolution the results are erratic with large errors. A major finding of this study is that a large resolution (1209 m for this basin) yields reproducible model results. When modelling a river basin using a distributed model, the resolution (grid cell size) can drastically affect the model results and calibration. The error structure attributable to grid cell resolution using entropy as a spatial variability measure is shown.     

Groundwater Quality: Analysis of Its Temporal and Spatial Variability in a Karst Aquifer

This study develops an approach based on hierarchical cluster analysis for investigating the spatial and temporal variation of water quality governing processes. The water quality data used in this study were collected in the karst aquifer of Yucatan, Mexico, the only source of drinking water for a population of nearly two million people. Hierarchical cluster analysis was applied to the quality data of all the sampling periods lumped together. This was motivated by the observation that, if water quality does not vary significantly in time, two samples from the same sampling site will belong to the same cluster. The resulting distribution maps of clusters and box‐plots of the major chemical components reveal the spatial and temporal variability of groundwater quality. Principal component analysis was used to verify the results of cluster analysis and to derive the variables that explained most of the variation of the groundwater quality data. Results of this work increase the knowledge about how precipitation and human contamination impact groundwater quality in Yucatan. Spatial variability of groundwater quality in the study area is caused by: a) seawater intrusion and groundwater rich in sulfates at the west and in the coast, b) water rock interactions and the average annual precipitation at the middle and east zones respectively, and c) human contamination present in two localized zones. Changes in the amount and distribution of precipitation cause temporal variation by diluting groundwater in the aquifer. This approach allows to analyze the variation of groundwater quality controlling processes efficiently and simultaneously.     

Groundwater Depth Forecasting Using Configurational Entropy Spectral Analyses with the Optimal Input

Accurate groundwater depth forecasting is particularly important for human life and sustainable groundwater management in arid and semi-arid areas. To improve the groundwater forecasting accuracy, in this paper, a hybrid groundwater depth forecasting model using configurational entropy spectral analyses (CESA) with the optimal input is constructed. An original groundwater depth series is decomposed into subseries of different frequencies using the variational mode decomposition (VMD) method. Cross-correlation analysis and Shannon entropy methods are applied to select the optimal input series for the model. The ultimate forecasted values of the groundwater depth can be obtained from the various forecasted values of the selected series with the CESA model. The applicability of the hybrid model is verified using the groundwater depth data from four monitoring wells in the Xi'an of Northwest China. The forecasting accuracy of the models was evaluated based on the average relative error (RE), root mean square error (RMSE), correlation coefficient (R) and Nash-Sutcliffe coefficient (NSE). The results indicated that comparing with the CESA and autoregressive model, the hybrid model has higher prediction performance.     

Computational aspects in use of entropy theory in predicting water quality levels at discontinued stations

The computational aspects of using a new, entropy-based, theory to predict water quality values at discontinued water quality monitoring stations are discussed. The main computational issues addressed are the level of discretization used in converting the continuous probability distribution of water quality values to the discrete levels required for the entropy function, and the choice of the interval of time for which to assign the value of the water quality (period of time averaging) through the entropy function. Unlike most cases of entropy applications involving discretization of continuous functions the results of using entropy theory to predict water quality values at discontinued monitoring stations in this application appear to be insensitive to the choice of the level of discretization even down to the very coarse level discretization associated with only eight intervals. However, depending on the length of record available the choice of the time interval for which the water quality values are assigned (period for time averaging) appear to have a significant impact on the accuracy of the results.     

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.     

Spatial relationship of groundwater–phosphorus interaction in the Kissimmee river basin,South Florida

Fluctuations of groundwater levels were used to predict soluble phosphorus concentrations. In‐situ observations showed a decrease in soluble phosphorus during groundwater recession and an increase with groundwater rise. A spatial analysis of the simulated soluble phosphorus and groundwater levels indicated similarity of patterns (spatial correlation) 99% of the time. A geographically weighted multivariate analysis of soluble phosphorus using groundwater levels, phosphorus levels of the Kissimmee River, and distance from the Kissimmee River as predictors showed a goodness of fit (R²) ranging from 0.2 to 0.7. Results indicated no significant difference between the simulated and observed soluble phosphorus levels at a p value of 0.01. Among the parameters, the groundwater level explained 70% of the soluble phosphorus variability. The distance to surface waterbodies and their phosphorus levels had significant weights only within a 5‐km range from the waterbody. A model generalization is further required to simulate the spatiotemporal groundwater–phosphorus dynamics over meaningful temporal ranges – at least for 3 to 5 years – for conclusiveness of the data. Copyright © 2014 John Wiley & Sons, Ltd.     

Weather-induced temporal variations in nitrate concentrations in shallow groundwater

For the evaluation of policy action programs to improve groundwater quality, research institutes and governments intensively monitor nitrate concentrations in shallow or near surface groundwater. However, trend detection is often hampered by the large seasonal and multi-annual temporal variability in nitrate concentrations, especially in shallow groundwater within 0–5 m below the surface in relatively humid regions. This variability is mainly caused by variations in precipitation excess (precipitation minus evapotranspiration) that results in strong variability in groundwater recharge. The objective of this study was to understand and quantify this weather-induced variability in shallow groundwater nitrate concentrations.We present an example of measured weather related variations in shallow groundwater nitrate concentrations from De Marke, an intensively monitored experimental farm in The Netherlands. For the quantification of the weather-induced variability, concentration-indices were calculated using a 1D model for water and solute transport. The results indicate that nitrate concentrations in the upper meter of groundwater at De Marke vary between 55% and 153% of the average concentration due to meteorological variability. The concentration-index quantification method was successfully used to distinguish weather related variability from human-induced trends in the nitrate concentration monitoring data from De Marke. Our model simulations also shows that sampling from fixed monitoring wells produces less short term variability than measuring from open boreholes. In addition, using larger screen depths and longer screens filters out short term temporal variability at the cost of a more delayed detection of trends in groundwater quality.     

Spatial Quantification of Groundwater Abstraction in the Irrigated Indus Basin

Groundwater abstraction and depletion were assessed at a 1‐km resolution in the irrigated areas of the Indus Basin using remotely sensed evapotranspiration (ET) and precipitation; a process‐based hydrological model and spatial information on canal water supplies. A calibrated Soil and Water Assessment Tool (SWAT) model was used to derive total annual irrigation applied in the irrigated areas of the basin during the year 2007. The SWAT model was parameterized by station corrected precipitation data (R) from the Tropical Rainfall Monitoring Mission, land use, soil type, and outlet locations. The model was calibrated using a new approach based on spatially distributed ET fields derived from different satellite sensors. The calibration results were satisfactory and strong improvements were obtained in the Nash‐Sutcliffe criterion (0.52 to 0.93), bias (?17.3% to ?0.4%), and the Pearson correlation coefficient (0.78 to 0.93). Satellite information on R and ET was then combined with model results of surface runoff, drainage, and percolation to derive groundwater abstraction and depletion at a nominal resolution of 1 km. It was estimated that in 2007, 68 km³ (262 mm) of groundwater was abstracted in the Indus Basin while 31 km³ (121 mm) was depleted. The mean error was 41 mm/year and 62 mm/year at 50% and 70% probability of exceedance, respectively. Pakistani and Indian Punjab and Haryana were the most vulnerable areas to groundwater depletion and strong measures are required to maintain aquifer sustainability.     

Capturing the essential spatial variability in distributed hydrological modelling: Hydraulic roughness

The spatial variability of each parameter affecting storm runoff must be accounted for in distributed modelling. The objective of the work reported here is to assess the effects of using distributed versus lumped hydraulic roughness coefficients in the modelling of direct surface runoff. A spatially variable data set composed of Manning roughness coefficients is used to model direct surface runoff. To assess the information content (as measured by entropy) of spatially variable data and its significance in distributed modelling, various degrees of smoothing are applied. The error resulting from smoothing the hydraulic roughness coefficients is determined by modelling overland flow using a finite element solution. The Manning roughness coefficients were taken from field measurements of the Manning roughness coefficient at 0.6 m on a 14 m hillslope. These values were then used in a numerical simulation of outflow hydrographs to investigate the dependence of error on spatial variability. Our study focuses on the characteristics of spatial data used in distributed hydrological modelling. The field sites have fractal dimensions of ≈? 1.4, which is close to a Brownian variation. The sampling interval that captures the essential spatial variability of the Manning roughness coefficient does not seem to matter due to its Brownian variation in the field sites. Hence due to the nearly uniform random distribution, measurements at 0.6 m intervals are not necessary and larger intervals would yield results that are just as acceptable provided the mean value together with a uniformly random distribution is maintained for any size of finite element or sampling resolution. Because detailed measurements of hydraulic roughness are not practically available for deterministic catchment modelling, it is important to know that larger sampling resolutions may be used than 0.6 m.     

Mapping groundwater quality in the Netherlands

Maps of 25 groundwater quality variables were obtained by estimating 4 km × 4 km block median concentrations. Estimates were presented as approximate 95% confidence intervals related to four concentration levels mostly obtained from critical levels for human consumption. These maps were based on measurements from 425 monitoring sites of national and provincial groundwater quality monitoring networks. The estimation procedure was based on a stratification by soil type and land use. Within each soil-land use category, measurements were interpolated. Spatial dependence between measurements and regional differences in mean level were taken into account. Stratification turned out to be essential: no or partial stratification (using either soil type or land use) results in essentially different maps. The effect of monitoring network density was studied by leaving out the 173 monitoring sites of the provincial monitoring networks. Important changes in resulting maps were assigned to loss of information on short-distance variation, as well as loss of location-specific information. For 12 variables, maps of changes in groundwater quality were made by spatial interpolation of short-term predictions calculated for each well screen from time series of yearly measurements over 5–7 years, using a simple regression model for variation over time and taking location-specific time-prediction uncertainties into account.

From a policy point of view, the resulting maps can be used either for quantifying diffuse groundwater contamination and location-specific background concentrations (in order to assist local contamination assessment) or for input and validation of policy supporting regional or national groundwater quality models. The maps can be considered as a translation of point information obtained from the monitoring networks into information on spatial units, the size of which is used in regional groundwater models. The maps enable location-specific network optimization. In general, the maps give little reason for reducing the monitoring network density (wide confidence intervals).     

Internal catchment process simulation in a snow‐dominated basin: performance evaluation with spatiotemporally variable runoff generation and groundwater dynamics

Hydrologic models have increasingly been used in forest hydrology to overcome the limitations of paired watershed experiments, where vegetative recovery and natural variability obscure the inferences and conclusions that can be drawn from such studies. Models are also plagued by uncertainty, however, and parameter equifinality is a common concern. Physically‐based, spatially‐distributed hydrologic models must therefore be tested with high‐quality experimental data describing a multitude of concurrent internal catchment processes under a range of hydrologic regimes. This study takes a novel approach by not only examining the ability of a pre‐calibrated model to realistically simulate watershed outlet flows over a four year period, but a multitude of spatially‐extensive, internal catchment process observations not previously evaluated, including: continuous groundwater dynamics, instantaneous stream and road network flows, and accumulation and melt period spatial snow distributions. Many hydrologic model evaluations are only on the comparison of predicted and observed discharge at a catchment outlet and remain in the ‘infant stage’ in terms of model testing. This study, on the other hand, tests the internal spatial predictions of a distributed model with a range of field observations over a wide range of hydroclimatic conditions. Nash‐Sutcliffe model efficiency was improved over prior evaluations due to continuing efforts in improving the quality of meteorological data collection. Road and stream network flows were generally well simulated for a range of hydrologic conditions, and snowpack spatial distributions were well simulated for one of two years examined. The spatial variability of groundwater dynamics was effectively simulated, except at locations where strong stream–groundwater interactions exist. Model simulations overall were quite successful in realistically simulating the spatiotemporal variability of internal catchment processes in the watershed, but the premature onset of simulated snowmelt for one of the simulation years has prompted further work in model development. Copyright © 2011 John Wiley & Sons, Ltd.     

Groundwater Level Analysis Using Regional Kendall Test for Trend with Spatial Autocorrelation

Assessment of historical evolution of groundwater levels is essential for understanding the anthropogenic impact on groundwater exploitation and developing response policies. In this study, regional groundwater level trend was addressed based on the regional Kendall test with correlated spatial data. With a limited number of data at one location, an exponential relation was proposed to be used to approximate covariances of a variable as a function of distances between locations. The effectiveness of the method was demonstrated using synthetic data experiments. The regional Kendall method was applied to assess evolution of groundwater levels and their annual decline rates in Beijing, Tianjin, and Hebei in China based on county-level data in 1959, 1984, 2005, and 2013. Results indicated that a continuing declining regional trend was shown in groundwater levels, revealing generally higher groundwater recharge rates than withdrawal rates in the study region. The annual groundwater decline rates presented a firstly increasing then decreasing regional trend, which is consistent with the environmental Kuznets curve. The earlier accelerating groundwater decline rate was attributed to supply-driven water resources management, whereas the reversed trend in accelerating groundwater decline rate in the latter period was due to many measures implemented to relieve local water stresses.     

A gravity-spatial entropy model for the measurement of urban sprawl

Since the mid-twentieth century, most cities worldwide have undergone a rapid expansion in urban land use. Along with the expansion, several problems, such as excessive loss of prime agricultural land and increasing traffic congestion have arisen. Thus, understanding and measurements of the expansion scale and its speed are crucial to planners and officials during urban planning and management processes. To measure such geographic phenomena, Shannon first devised entropy theory, and then Batty developed it into spatial entropy. The recently developed spatial entropy model, which was used to measure urban sprawl, introduced area to represent spatial asymmetry. However, most models did not consider spatial discretization, particularly the impact of distance. This study attempted to construct an integrated gravity-spatial entropy model to delineate distance and spatial diffusion impacts on population distribution. Then, we tested the model using Shanghai’s temporal land use and community statistical data. Application results for the new gravity-spatial model show that it is a useful tool for identifying spatial and temporal variations of urban sprawl.     

Understanding the Geometry of Connected Fracture Flow with Multiperiod Oscillatory Hydraulic Tests

An understanding of the spatial and hydraulic properties of fast preferential flow pathways in the subsurface is necessary in applications ranging from contaminant fate and transport modeling to design of energy extraction systems. One method for the characterization of fracture properties over interwellbore scales is Multiperiod Oscillatory Hydraulic (MOH) testing, in which the aquifer response to oscillatory pressure stimulations is observed. MOH tests were conducted on isolated intervals of wells in siliciclastic and carbonate aquifers in southern Wisconsin. The goal was to characterize the spatial properties of discrete fractures over interwellbore scales. MOH tests were conducted on two discrete fractured intervals intersecting two boreholes at one field site, and a nest of three piezometers at another field site. Fracture diffusivity estimates were obtained using analytical solutions that relate diffusivity to observed phase lag and amplitude decay. In addition, MOH tests were used to investigate the spatial extent of flow using different conceptual models of fracture geometry. Results indicated that fracture geometry at both field sites can be approximated by permeable two‐dimensional fracture planes, oriented near‐horizontally at one site, and near‐vertically at the other. The technique used on MOH field data to characterize fracture geometry shows promise in revealing fracture network characteristics important to groundwater flow and transport.     

To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology

The multivariate Gaussian random function model is commonly used in stochastic hydrogeology to model spatial variability of log-conductivity. The multi-Gaussian model is attractive because it is fully characterized by an expected value and a covariance function or matrix, hence its mathematical simplicity and easy inference. Field data may support a Gaussian univariate distribution for log hydraulic conductivity, but, in general, there are not enough field data to support a multi-Gaussian distribution. A univariate Gaussian distribution does not imply a multi-Gaussian model. In fact, many multivariate models can share the same Gaussian histogram and covariance function, yet differ by their patterns of spatial continuity at different threshold values. Hence the decision to use a multi-Gaussian model to represent the uncertainty associated with the spatial heterogeneity of log-conductivity is not databased. Of greatest concern is the fact that a multi-Gaussian model implies the minimal spatial correlation of extreme values, a feature critical for mass transport and a feature that may be in contradiction with some geological settings, e.g. channeling. The possibility for high conductivity values to be spatially correlated should not be discarded by adopting a congenial model just because data shortage prevents refuting it. In this study, three alternatives to a multi-Gaussian model, all sharing the same Gaussian histogram and the same covariance function, but with different continuity patterns for extreme values, were considered to model the spatial variability of log-conductivity. The three alternative models, plus the traditional multi-Gaussian model, are used to perform Monte Carlo analyses of groundwater travel times from a hypothetical nuclear repository to the ground surface through a synthetic formation similar to the Finnsjön site in Sweden. The results show that the groundwater travel times predicted by the multi-Gaussian model could be ten times slower than those predicted by the other models. The probabilities of very short travel times could be severely underestimated using the multi-Gaussian model. Consequently, if field measured data are not sufficient to determine the higher-order moments necessary to validate the multi-Gaussian model — which is the usual situation in practice — other alternative models to the multi-Gaussian one ought to be considered.