Kalman filtering in groundwater flow modelling: problems and prospects

The popularity of applying filtering theory in the environmental and hydrological sciences passed its first climax in the 1970s. Like so many other new mathematical methods it was simply the fashion at the time. The study of groundwater systems was not immune to this fashion, but neither was it by any means a prominent area of application. The spatial-temporal characteristics of groundwater flow are customarily described by analytical or, more frequently, numerical, physics-based models. Consequently, the state-space representations associated with filtering must be of a high order, with an immediately apparent computational over-burden. And therein lies part of the reason for the but modest interest there has been in applying Kalman filtering to groundwater systems, as reviewed critically in this paper. Filtering theory may be used to address a variety of problems, such as: state estimation and reconstruction, parameter estimation (including the study of uncertainty and its propagation), combined state-parameter estimation, input estimation, estimation of the variance-covariance properties of stochastic disturbances, the design of observation networks, and the analysis of parameter identifiability. A large proportion of previous studies has dealt with the problem of parameter estimation in one form or another. This may well not remain the focus of attention in the future. Instead, filtering theory may find wider application in the context of data assimilation, that is, in reconstructing fields of flow and the migration of sub-surface contaminant plumes from relatively sparse observations. Received: October 27, 1997     

Data assimilation in groundwater modelling: ensemble Kalman filter versus ensemble smoothers

Groundwater modelling calls for an effective and robust data integrating method to fill the gap between the model and observation data. The ensemble Kalman filter (EnKF), a real‐time data assimilation method, has been increasingly applied in multiple disciplines such as petroleum engineering and hydrogeology. In this approach, a groundwater model is updated sequentially with measured data such as hydraulic head and concentration. As an alternative to the EnKF, the ensemble smoother (ES) has been proposed for updating groundwater models using all the data together, with much less computational cost. To further improve the performance of the ES, an iterative ES has been proposed for continuously updating the model by assimilating measurements together. In this work, we compare the performance of the EnKF, the ES, and the iterative ES using a synthetic example in groundwater modelling. Hydraulic head data modelled on the basis of the reference conductivity field are used to inversely estimate conductivities at unsampled locations. Results are evaluated in terms of the characterization of conductivity and groundwater flow predictions. It is concluded that (a) the iterative ES works better than the standard ES because of its continuous updating and (b) the iterative ES could achieve results comparable with those of the EnKF, with less computational cost. These findings show that the iterative ES should be paid much more attention for data assimilation in groundwater modelling.     

Analysis of recharge conceptualization in inverse groundwater modelling

ABSTRACT

This study presents a systematic illustration quantifying how misleading the calibration results of a groundwater simulation model can be when recharge rates are considered as the model parameters to be estimated by inverse modelling. Three approaches to recharge estimation are compared: autocalibration (Model 1), the empirical return coefficient method (Model 2), and distributed hydrological modelling using the Soil and Water Assessment Tool, SWAT (Model 3). The methodology was applied in the Dehloran Plain, western Iran, using the MODFLOW modular flow simulator and the PEST method for autocalibration. The results indicate that, although Model 1 performed the best in simulating water levels at observation wells in the calibration stage, it did not perform satisfactorily in real future scenarios. Model 3, with SWAT-based recharge rates, performed better than the other models in the validation stage. By not evaluating the model performance solely on calibration results, we demonstrate the relative significance of using more accurate recharge estimates when calibrating groundwater simulation models.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR M. Besbes     

Simultaneous identification of parameter,initial condition,and boundary condition in groundwater modelling

In this paper, we perform an inverse method to simultaneously estimate aquifer parameters, initial condition, and boundary conditions in groundwater modelling. The parameter estimation is extended to a complete inverse problem that makes the calibrated groundwater flow model more realistic. The adjoint state method, the gradient search method, and the least square error algorithm are combined to build the optimization procedure. Horizontal two‐dimensional groundwater flow in a confined aquifer is exemplified to demonstrate the correlation between unknowns, the contribution of observation, as well as the suitability of applying the inverse method. The correlation analysis shows the connection between storage coefficient and initial condition. Besides, transmissivity and boundary conditions are also highly correlated. More observations at different location and time are necessary to provide sufficient information. A time series of unsteady head is requested for estimation of storage coefficient and initial condition. Observation near boundary is very effective for boundary condition estimation. The observation at pumping well mostly contributes to the estimation of transmissivity. According to all observations, it is possible to identify parameters, initial condition, and boundary condition simultaneously. Furthermore, the results not only illustrate the traditional assumption of known boundary condition but also initial condition, which may cause an incorrect estimation. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling and prediction of environmental data in space and time using Kalman filtering

 The Kalman filter is used in this paper as a framework for space time data analysis. Using Kalman filtering it is possible to include physically based simulation models into the data analysis procedure. Attention is concentrated on the development of fast filter algorithms to make Kalman filtering feasible for high dimensional space time models. The ensemble Kalman filter and the reduced rank square root filter algorithm are briefly summarized. A new algorithm, the partially orthogonal ensemble Kalman filter is introduced too. We will illustrate the performance of the Kalman filter algorithms with a real life air pollution problem. Here ozone concentrations in a part of North West Europe are estimated and predicted.     

Sensitivity of soil parameters in unsaturated zone modelling and the relation between effective,laboratory and in situ estimates

Simulation of soil moisture content requires effective soil hydraulic parameters that are valid at the modelling scale. This study investigates how these parameters can be estimated by inverse modelling using soil moisture measurements at 25 locations at three different depths (at the surface, at 30 and 60 cm depth) on an 80 by 20 m hillslope. The study presents two global sensitivity analyses to investigate the sensitivity in simulated soil moisture content of the different hydraulic parameters used in a one‐dimensional unsaturated zone model based on Richards' equation. For estimation of the effective parameters the shuffled complex evolution algorithm is applied. These estimated parameters are compared to their measured laboratory and in situ equivalents. Soil hydraulic functions were estimated in the laboratory on 100 cm³ undisturbed soil cores collected at 115 locations situated in two horizons in three profile pits along the hillslope. Furthermore, in situ field saturated hydraulic conductivity was estimated at 120 locations using single‐ring pressure infiltrometer measurements. The sensitivity analysis of 13 soil physical parameters (saturated hydraulic conductivity (K_s), saturated moisture content (θ_s), residual moisture content (θ_r), inverse of the air‐entry value (α), van Genuchten shape parameter (n), Averjanov shape parameter (N) for both horizons, and depth (d) from surface to B horizon) in a two‐layer single column model showed that the parameter N is the least sensitive parameter. K_s of both horizons, θ_s of the A horizon and d were found to be the most sensitive parameters. Distributions over all locations of the effective parameters and the distributions of the estimated soil physical parameters from the undisturbed soil samples and the single‐ring pressure infiltrometer estimates were found significantly different at a 5% level for all parameters except for α of the A horizon and K_s and θ_s of the B horizon. Different reasons are discussed to explain these large differences. Copyright © 2004 John Wiley & Sons, Ltd.     

Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

Geochemical evolution of groundwater in hard-rock aquifers of South India using statistical and modelling techniques

ABSTRACT

Multivariate statistical analysis and inverse geochemical modelling techniques were employed to deduce the mechanism of groundwater evolution in the hard-rock terrain of Telangana, South India. Q-mode hierarchical cluster analysis (HCA) and principal component analysis (PCA) were used to extract the hydrogeochemical characteristics and classify the groundwater samples into three principal groups. Use of thermodynamic stability diagrams and inverse geochemical modelling in PHREEQC identified the chemical reactions controlling hydrogeochemistry of each of the groups obtained from statistical analysis. The model output showed that a few phases are governing the water chemistry in this area and the geochemical reactions responsible for evolution of groundwater chemistry along the flow path are (i) dissolution of evaporite minerals (dolomite, halite); (ii) dissolution of primary silicate minerals (albite, anorthite, K-feldspar, biotite); (iii) precipitation of secondary silicate minerals (kaolinite, quartz, gibbsite, Ca-montmorillonite) along with anhydrite and calcite; and (iv) reverse ion exchange processes.     

Uncertainty and multiple objective calibration in regional water balance modelling: case study in 320 Austrian catchments

We examine the value of additional information in multiple objective calibration in terms of model performance and parameter uncertainty. We calibrate and validate a semi‐distributed conceptual catchment model for two 11‐year periods in 320 Austrian catchments and test three approaches of parameter calibration: (a) traditional single objective calibration (SINGLE) on daily runoff; (b) multiple objective calibration (MULTI) using daily runoff and snow cover data; (c) multiple objective calibration (APRIORI) that incorporates an a priori expert guess about the parameter distribution as additional information to runoff and snow cover data. Results indicate that the MULTI approach performs slightly poorer than the SINGLE approach in terms of runoff simulations, but significantly better in terms of snow cover simulations. The APRIORI approach is essentially as good as the SINGLE approach in terms of runoff simulations but is slightly poorer than the MULTI approach in terms of snow cover simulations. An analysis of the parameter uncertainty indicates that the MULTI approach significantly decreases the uncertainty of the model parameters related to snow processes but does not decrease the uncertainty of other model parameters as compared to the SINGLE case. The APRIORI approach tends to decrease the uncertainty of all model parameters as compared to the SINGLE case. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of Proper Orthogonal Decomposition (POD) to inverse problems in saturated groundwater flow

We develop a new Proper Orthogonal Decomposition (POD) reduced order model for saturated groundwater flow, and apply that model to an inverse problem for the hydraulic conductivity field. We use sensitivities as the POD basis. We compare the output when the optimizer uses the reduced order model against results obtained with a full PDE based model. The solutions generated using the POD reduced model are comparable in residual norm to the solutions formed using only the full-scale model. The material parameters are similarly comparable. The time to solution when using the reduced model is reduced by at least an order of magnitude, as are the number of calls to the full model.     

Anisotropy and depth‐related heterogeneity of hydraulic conductivity in a bog peat. II: modelling the effects on groundwater flow

Anisotropy and heterogeneity of hydraulic conductivity (K) are seldom considered in models of mire hydrology. We investigated the effect of anisotropy and heterogeneity on groundwater flow in bog peat using a steady‐state groundwater model. In five model simulations, four sets of K data were used. The first set comprised measured K values from an anisotropic and heterogeneous bog peat. These data were aggregated to produce the following simplified data sets: an isotropic and heterogeneous distribution of K; an isotropic and homogeneous distribution; and an anisotropic and homogeneous distribution. We demonstrate that, where anisotropy and heterogeneity exist, groundwater flow in bog peat is complex. Fine‐scale variations in K have the potential to influence patterns and rates of groundwater flow. However, for our data at least, it is heterogeneity and not anisotropy that has the greater influence on producing complex patterns of groundwater flow. We also demonstrate that patterns and rates of groundwater flow are simplified and reduced when measured K values are aggregated to create a more uniform distribution of K. For example, when measured K values are aggregated to produce isotropy and homogeneity, the rate of modelled seepage is reduced by 28%. We also show that when measured K values are used, the presence of a drainage ditch can increase seepage through a modelled cross‐section. Our work has implications for the accurate interpretation of hydraulic head data obtained from peat soils, and also the understanding of the effect of drainage ditches on patterns and rates of groundwater flow. Copyright © 2002 John Wiley & Sons, Ltd.     

Inverse modelling for freshwater lens in small islands: Kish Island,Persian Gulf

A number of challenges including instability, nonconvergence, nonuniqueness, nonoptimality, and lack of a general guideline for inverse modelling have limited the application of automatic calibration by generic inversion codes in solving the saltwater intrusion problem in real‐world cases. A systematic parameter selection procedure for the selection of a small number of independent parameters is applied to a real case of saltwater intrusion in a small island aquifer system in the semiarid region of the Persian Gulf. The methodology aims at reducing parameter nonuniqueness and uncertainty and the time spent on inverse modelling computations. Subsequent to the automatic calibration of the numerical model, uncertainty is analysed by constrained nonlinear optimization of the inverse model. The results define the percentage of uncertainty in the parameter estimation that will maintain the model inside a user‐defined neighbourhood of the best possible calibrated model. Sensitivity maps of both pressure and concentration for the small island aquifer system are also developed. These sensitivity maps indicate higher sensitivity of pressure to model parameters compared with concentration. These sensitivity maps serve as a benchmark for correlation analysis and also assist in the selection of observations points of pressure and concentration in the calibration process. Copyright © 2012 John Wiley & Sons, Ltd.     

Conceptual and numerical models for groundwater flow in an arid inland river basin

The Heihe River Basin (HRB) is an inland watershed in northwest China with a total area of approximately 130,000 km², stretching from the Qilian Mountains in the south to the oases and agricultural fields in the middle and further to the Gobi desert in the north bordering Mongolia. As part of a major ecohydrological research initiative to provide a stronger scientific underpinning for sustainable water management in arid ecosystems, a regional‐scale integrated ecological and hydrological model is being developed, incorporating the knowledge based on the results of environmental isotope tracer analysis and the multiscale observation datasets. The first step in the model development effort is to construct and calibrate a groundwater flow model for the middle and lower HRB where the oases and vegetation along the Heihe river corridor are highly dependent on groundwater. In this study, the software tool ‘Arc Hydro Groundwater’ is used to build and visualize a hydrogeological data model for the HRB that links all relevant spatiotemporal hydrogeological data in a unified geodatabase within the ArcGIS environment. From the conceptual model, a regional‐scale groundwater flow model has been developed using MODFLOW‐2005. Critical considerations in developing the flow model include the representation of mountainous terrains and fluvial valleys by individual model layers, treatment of aquifer heterogeneities across multiple scales and selection of proper observation data and boundary conditions for model calibration. This paper discusses these issues in the context of the Heihe River Basin, but the results and insights from this study will have important implications for other large, regional groundwater modelling studies, especially in arid and semiarid inland river basins. Copyright © 2014 John Wiley & Sons, Ltd.     

Characterization of groundwater quality in Kinmen Island using multivariate analysis and geochemical modelling

Kinmen is located in the south‐west of Mainland China and mainly has two islands, Large Kinmen and Leiyu. Residents in Kinmen have drunk groundwater over several decades. This work characterized the quality of groundwater in Kinmen using factor analysis (FA), cluster analysis (CA) and geochemical simulation. The factor scores were plotted spatially to illustrate the groundwater quality and were used to discuss the grouped relationship using CA. Salinization, redox and organic matter factors are identified from 17 hydrochemical measurements in 18 wells. Acidic and oxidizing groundwater with nitrate‐N pollution is distributed mainly in the west of Large Kinmen. Saline groundwater is distributed to the north‐east of Large Kinmen and in the south of Leiyu. Groundwater with organic matter is present throughout Leiyu. Five groups of the groundwater quality divided by CA can be interpreted according to one to three factors. The grouped characteristics of the groundwater quality help the local government and industries to plan the use and protection of groundwater resources. Furthermore, a geochemical simulation was used to demonstrate the formation processes of the acidic and oxidizing groundwater properties in granitic sediments. A large amount of the precipitation of the kaolinite and magnetite releases concentrations of hydrogen ion and raises the redox potential in the aquifers. Copyright © 2007 John Wiley & Sons, Ltd.     

Identifying the flow pattern and natural recharge at a strategic groundwater resource in the Dornogobi Province,Mongolia

The strategic project of economic development in the Dornogobi Province in Mongolia is dependent on water supply. Thus a comprehensive hydrogeological characterization was focused on the Upper Cretaceous multi-aquifer system north of Sainshand city. A conceptual model was developed to discover the groundwater flow pattern essential to correct the setting of the numerical model of groundwater flow created using MODFLOW to assess the natural recharge of the aquifer. The conceptualization was based on geological and hydrogeological characterization. However, the evaluation of hydrochemistry proved to be the key factor revealing the principal feature of the groundwater flow pattern, which is the presence of preferential flow zones. These zones allow for intensive transfer of relatively fresh Na(Mg,Ca)?HCO₃-dominated groundwater into discharge areas, where it leaks into the Quaternary aquifer. The numerical model suggested an enormous natural recharge of 22 100 m³/d, originating in 64% of the preferential flow zones.     

A novel general methodology for studying and remedying finite precision error with application in Kalman filtering

 Least squares (LS) techniques, like Kalman filtering, are widely used in environmental science and engineering. In this paper, a new general approach is introduced for the study of the generation, propagation and accumulation of the quantization error in any algorithm. This methodology employs a number of fundamental propositions demonstrating the way the four operations addition, multiplication, division and subtraction, influence quantization error generation and transmission. Using these, one can obtain knowledge of the exact number of erroneous digits with which all quantities of any algorithm are computed at each step of it. This methodology offers understanding of the actual cause of the generation and propagation of finite precision error in any computational scheme. Application of this approach to all Kalman type LS algorithms shows that not all their formulas are equivalent concerning the quantization error effects. More specifically, few generate the greater amount of quantization error. Finally, a stabilization procedure, applicable to all Kalman type algorithms, is introduced that renders all these algorithms very robust.     

Overexploitation and continuous drought effects on groundwater yield and marine intrusion: considerations arising from the modelling of Mamora coastal aquifer,Morocco

This paper studies the effect of drought and pumping discharge on groundwater supplies and marine intrusion. The investigation concerns the Mamora coastal aquifer, northwest of Morocco. A large‐scale groundwater model was established to model (a) the amount of freshwater discharge towards the ocean and the sea water volumes flowing inland as a consequence of the inverse hydraulic gradient, (b) the impact of drought and pumping discharge on the water table level and, as a consequence, on marine water intrusion. In fact, the simulated submarine groundwater discharge (SGWD) would decrease from 864 m³/d/km in 1987 to 425 m³/d/km in 2000. The simulated volumes of sea water intruding the aquifer as a result of inverse hydraulic gradient would increase from 0·25 Mm³/y in 1987 to 0·3 Mm³/y in 2000. As a consequence of a negative rainfall gradient of −5 mm/y, the simulated SGWD would decline to 9 m³/d/km and the sea water intrusion (SWI) would increase to 0·35 Mm³/y since the year 2010. Due to insufficient data on the trend of pumping discharge, a hypothetical increase of this latter from 38·3 Mm³/y to 53·2 Mm³/y is simulated to induce an increase of marine water intrusion from 0·25 Mm³/y to 0·9 Mm³/y. Consequently, to optimally exploit this seemingly fragile coastal aquifer, a plan of future actions to implement is proposed. Copyright © 2005 John Wiley & Sons, Ltd.     

Hydrological Outlook UK: an operational streamflow and groundwater level forecasting system at monthly to seasonal time scales

This paper describes the development of the first operational seasonal hydrological forecasting service for the UK, the Hydrological Outlook UK (HOUK). Since June 2013, this service has delivered monthly forecasts of streamflow and groundwater levels, with an emphasis on forecasting hydrological conditions over the next three months, accompanied by outlooks over longer time horizons. This system is based on three complementary approaches combined to produce the outlooks: (i) national-scale modelling of streamflow and groundwater levels based on dynamic seasonal rainfall forecasts, (ii) catchment-scale modelling where streamflow and groundwater level models are driven by historical meteorological forcings (i.e. the Ensemble Streamflow Prediction, ESP, approach), and (iii) a catchment-scale statistical method based on persistence and historical analogues. This paper provides the background to the Hydrological Outlook, describes the various component methods in detail and then considers the impact and usefulness of the product. As an example of a multi-method, operational seasonal hydrological forecasting system, it is hoped that this overview provides useful information and context for other forecasting initiatives around the world.     

Sensitivity analysis of prior model probabilities and the value of prior knowledge in the assessment of conceptual model uncertainty in groundwater modelling

A key point in the application of multi‐model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi‐model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three‐dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi‐model approach, expressed by reductions of the multi‐model prediction variances by up to 60% compared with a non‐informative case. However, the ratio between‐model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Stream response to precipitation variability: a spectral view based on analysis and modelling of hydrological cycle components

Hydrological processes commonly exhibit long‐term persistence, also known as the ‘Hurst phenomenon’. Here, we examine long‐term precipitation and streamflow time series from the Elbe River Basin to quantify differences in the spectral properties and in the Hurst parameter estimates () of the individual hydrological cycle components. Precipitation‐runoff modelling is performed for the Elbe River sub‐catchment Striegis using the Soil and Water Assessment Tool (SWAT). For 38 daily 50 years long streamflow time series from the Elbe River Basin, baseflow separation and spectral analysis is performed. The results show a spectral shift towards low‐frequency scales (>2 years) from precipitation to baseflow, with a parallel increase of from 0.52 (precipitation) to 0.65 (baseflow). The SWAT model is able to reproduce both, the main low‐frequency mode (≈7 yr.) and the (0.62) of the observed Striegis River flow time series. The baseflow appears to be the main component which shapes the low‐frequency response and of streamflow in the Elbe River Basin to the input precipitation. This conclusion is further confirmed through PMWIN‐MODFLOW groundwater modelling of a hypothetic phreatic stream‐connected aquifer system that consists of various soils (sand, loamy sand and silt). A power shift towards lower frequencies and an increase of for the hydraulic heads is obtained, as the aquifer's lateral dimensions increase and its hydraulic conductivity decreases. The average of the groundwater heads is 0.80, 0.90 and 1.0 for sand, loamy sand and silt aquifers, respectively. Copyright © 2014 John Wiley & Sons, Ltd.