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     

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     

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.     

An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering

The ensemble Kalman filter (EnKF) is a commonly used real-time data assimilation algorithm in various disciplines. Here, the EnKF is applied, in a hydrogeological context, to condition log-conductivity realizations on log-conductivity and transient piezometric head data. In this case, the state vector is made up of log-conductivities and piezometric heads over a discretized aquifer domain, the forecast model is a groundwater flow numerical model, and the transient piezometric head data are sequentially assimilated to update the state vector. It is well known that all Kalman filters perform optimally for linear forecast models and a multiGaussian-distributed state vector. Of the different Kalman filters, the EnKF provides a robust solution to address non-linearities; however, it does not handle well non-Gaussian state-vector distributions. In the standard EnKF, as time passes and more state observations are assimilated, the distributions become closer to Gaussian, even if the initial ones are clearly non-Gaussian. A new method is proposed that transforms the original state vector into a new vector that is univariate Gaussian at all times. Back transforming the vector after the filtering ensures that the initial non-Gaussian univariate distributions of the state-vector components are preserved throughout. The proposed method is based in normal-score transforming each variable for all locations and all time steps. This new method, termed the normal-score ensemble Kalman filter (NS-EnKF), is demonstrated in a synthetic bimodal aquifer resembling a fluvial deposit, and it is compared to the standard EnKF. The proposed method performs better than the standard EnKF in all aspects analyzed (log-conductivity characterization and flow and transport predictions).     

Impact of environmental variables on spatial and seasonal internal phosphorus loading in a mesoeutrophic lake

Strzeszynskie Lake was formerly a slightly eutrophic (meso-eutrophic) water body. The aim of the current research was to define variables on both spatial and seasonal internal phosphorus loading from bottom sediments at five stations located in zones varying in depth, oxygenation, macrophyte presence, and uses of the neighboring catchment area. Ex situ experiments done with the use of intact bottom sediment cores have shown that the highest phosphorus release occurred in the deepest part of the lake and reached 3.6?mg?P/m²d under anoxic conditions during summer thermal stratification. In turn, the internal loading from littoral sediments, which were well aerated all year round, was clearly lower. Furthermore, phosphorus accumulation in the bottom sediment was observed to reach a maximum of 1.45?mg?P/m² d in autumn. A comparison of the internal loading intensity in lake zones with different land uses of the neighboring catchment area has shown slightly higher values at stations adjacent to the forest catchment area than those used for recreation. Changes in the land use of the catchment area of Strzeszynskie Lake, especially the increase in impermeable surfaces, have led to an increased inflow of external loads after heavy rains, resulting in deterioration in water quality and a delayed increase in internal loading.     

Future high- and low-flow estimations for Central Vietnam: a hydro-meteorological modelling chain approach

We apply a complex hydro-meteorological modelling chain for investigating the impact of climate change on future hydrological extremes in Central Vietnam, a region characterized by limited data availability. The modelling chain consists of six General Circulation Models (GCMs), six Regional Climate Models (RCMs), six bias correction (BC) approaches, the fully distributed Water Flow and Balance Simulation Model (WaSiM), and extreme values analysis. Bias corrected and raw climate data are used as input for WaSiM. To derive hydrological extremes, the generalized extreme value distribution is fitted to the annual maxima/minima discharge. We identify limitations according to the fitting procedure and the BC methods, and suggest the usage of the delta change approach for hydrological decision support. Tendencies towards increased high- and decreased low flows are concluded. Our study stresses the challenges in using current GCMs/RCMs in combination with state-of-the-art BC methods and extreme value statistics for local impact studies.     

Hydrologic and sediment modelling studies in the environmental impact assessment of a major tropical dam project

A computer-based study of the impact of the proposed Wabo hydroelectric scheme on the Purari River, Papua New Guinea was carried out. The HEC-6 model, Scour and Deposition in Rivers and Reservoirs developed by the Hydrologic Engineering Centre was used to simulate the effect of the dam on sediment transport and erosion in the lower Purari. Two runs with the model were carried out. The first one was used to establish baseline conditions and the second modelled dam impact. Before the study was carried out, data had to be collected on channel geometry, sediment input, river bed material size composition and hydraulic conditions in the river. Supplementary models also had to be developed to fill in gaps in runoff records and to describe flow in the river during power generation. Results of the investigation indicate that limited erosion will occur because of bed-armouring and the river will adjust towards a new equilibrium condition quite rapidly. The sediment output of the river into the Purari delta will change, load in the clay, silt and sand/gravel fractions decreasing by 22, 53 and 78 per cent respectively.     

Spatio-temporal modelling of hydro-meteorological derived risk using a Bayesian approach: a case study in Venezuela

Extreme environmental events have considerable impacts on society. Preparation to mitigate or forecast accurately these events is a growing concern for governments. In this regard, policy and decision makers require accurate tools for risk estimation in order to take informed decisions. This work proposes a Bayesian framework for a unified treatment and statistical modeling of the main components of risk: hazard, vulnerability and exposure. Risk is defined as the expected economic loss or population affected as a consequence of a hazard event. The vulnerability is interpreted as the loss experienced by an exposed population due to hazard events. The framework combines data of different spatial and temporal supports. It produces a sequence of temporal risk maps for the domain of interest including a measure of uncertainty for the hazard and vulnerability. In particular, the considered hazard (rainfall) is interpolated from point-based measured rainfall data using a hierarchical spatio-temporal Kriging model, whose parameters are estimated using the Bayesian paradigm. Vulnerability is modeled using zero-inflated distributions with parameters dependent on climatic variables at local and large scales. Exposure is defined as the total population settled in the spatial domain and is interpolated using census data. The proposed methodology was applied to the Vargas state of Venezuela to map the spatio-temporal risk for the period 1970–2006. The framework highlights both high and low risk areas given extreme rainfall events.     

Stochastic modelling of seasonal and yearly rainfalls with low-frequency variability

Stochastic rainfall models are important for many hydrological applications due to their appealing ability to simulate synthetic series that resemble the statistical characteristics of the observed series for a location of interest. However, an important limitation of stochastic rainfall models is their inability to preserve the low-frequency variability of rainfall. Accordingly, this study presents a simple yet efficient stochastic rainfall model for a tropical area that attempts to incorporate seasonal and inter-annual variabilities in simulations. The performance of the proposed stochastic rainfall model, the tropical climate rainfall generator (TCRG), was compared with a stochastic multivariable weather generator (MV-WG) in various aspects. Both models were applied on 17 rainfall stations at the Kelantan River Basin, Malaysia, with tropical climate. The validations were carried out on seasonal (monsoon and inter-monsoon) and annual basis. The third-order Markov chain of the TCRG was found to perform better in simulating the rainfall occurrence and preserving the low-frequency variability of the wet spells. The log-normal distribution of the TCRG was consistently better in modelling the rainfall amounts. Both models tend to underestimate the skewness and kurtosis coefficient of the rainfall. The spectral correction approach adopted in the TCRG successfully preserved the seasonal and inter-annual variabilities of rainfall amounts, whereas the MV-WG tends to underestimate the variability bias of rainfall amounts. Overall, the TCRG performed reasonably well in the Kelantan River Basin, as it can represent the key statistics of rainfall occurrence and amounts successfully, as well as the low-frequency variability.     

Total and opportunistic algal cover in relation to environmental variables

Based on a large data set from the national Danish monitoring program, spatial and temporal variability in total algal cover and in the fraction of opportunistic macroalgae was analysed in relation to environmental variables. Variations in water clarity and salinity combined with information on geographical location of sampling areas were found to explain almost 80% of the large-scale variation in algal cover between areas. As water clarity was largely regulated by concentrations of total-nitrogen (TN), and TN-concentrations by TN-input from land, total algal cover at given water depths was partly related to TN-input from land. The fraction of opportunistic algae responded predominantly to differences in salinity, the highest fractions being found in the most brackish areas. Temporal variability in algal cover and fraction of opportunists over the 14-year investigation period was much smaller than the variability between areas and could not be predicted from variations in environmental variables. In order for macroalgal cover to become a more sensitive indicator of water quality it would be necessary to either increase the sensitivity of the method or identify and include supplementary regulating factors in the model.     

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.     

Mapping water table depth using geophysical and environmental variables

Despite its importance, accurate representation of the spatial distribution of water table depth remains one of the greatest deficiencies in many hydrological investigations. Historically, both inverse distance weighting (IDW) and ordinary kriging (OK) have been used to interpolate depths. These methods, however, have major limitations: namely they require large numbers of measurements to represent the spatial variability of water table depth and they do not represent the variation between measurement points. We address this issue by assessing the benefits of using stepwise multiple linear regression (MLR) with three different ancillary data sets to predict the water table depth at 100-m intervals. The ancillary data sets used are Electromagnetic (EM34 and EM38), gamma radiometric: potassium (K), uranium (eU), thorium (eTh), total count (TC), and morphometric data. Results show that MLR offers significant precision and accuracy benefits over OK and IDW. Inclusion of the morphometric data set yielded the greatest (16%) improvement in prediction accuracy compared with IDW, followed by the electromagnetic data set (5%). Use of the gamma radiometric data set showed no improvement. The greatest improvement, however, resulted when all data sets were combined (37% increase in prediction accuracy over IDW). Significantly, however, the use of MLR also allows for prediction in variations in water table depth between measurement points, which is crucial for land management.     

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.     

Multi-site bias correction of climate model outputs for hydro-meteorological impact studies: An application over a watershed in China

Bias correction methods remove systematic differences in the distributional properties of climate model outputs with respect to observations, often as a means of pre-processing model outputs for use in hydrological impact studies. Traditionally, bias correction is applied at each weather station individually, neglecting the dependence that exists between different sites, which could negatively affect simulations from a distributed hydrological model. In this study, three multi-variate bias correction (MBC) methods—initially proposed to correct the inter-variable correlation or multi-variate dependence of climate model outputs—are used to correct biases in distributional properties and spatial dependence at multiple weather stations. To reveal the benefits of correcting spatial dependence, two distribution-based single-site bias correction methods are used for comparison. The effects of multi-site correction on hydro-meteorological extremes are assessed by driving a distributed hydrological model and then evaluating the model performance in terms of several meteorological and hydrological extreme indices. The results show that the multi-site bias correction methods perform well in reducing biases in spatial correlation measures of raw global climate model outputs. In addition, the multi-site methods consistently reproduce watershed-averaged meteorological variables better than single-site methods, especially for extreme values. In terms of representing hydrological extremes, the multi-site methods generally perform better than the single-site methods, although the benefits vary according to the hydrological index. However, when applying the multi-site methods, the original temporal sequence of precipitation occurrence may be altered to some extent. Overall, all multi-site bias correction methods are able to reproduce the spatial correlation of observed meteorological variables over multiple stations, which leads to better hydrological simulations, especially for extremes. This study emphasizes the necessity of considering spatial dependence when applying bias correction to ccc outputs and hydrological impact studies.     

The relationship between soft-bottom macrobenthic communities and environmental variables in Singaporean waters

Univariate and multivariate methods were used to study soft-bottom macrobenthos collected in December 2002 from the coastal waters of Singapore. The univariate parameters and community structure of benthic communities were related to environmental variables. Three samples were taken with a 0.1 m² Van Veen grab (33 × 30 × 15 cm) at each station from 12 sampling stations of two different geographical areas. The water depth ranged from 6.5 m to 34.0 m. The mean values of species number, abundance and species diversity (H′) were 24 species/grab, 77 animals/grab and 3.35/grab, respectively. A total of 172 species was recorded. Total petroleum hydrocarbons were strongly negatively related to species number, abundance and species diversity, suggesting that petroleum hydrocarbons have harmful effects on macrobenthic communities. The BIO-ENV analyses for all stations identified median particle size, silt–clay content, salinity and Zn as the major environmental variables influencing the infaunal patterns. However, separate analyses for two areas produced stronger correlations and different best-correlated environmental variable combinations. Total petroleum hydrocarbons were the only common factor in both areas, showing the importance of petroleum contamination in determining the community structure of benthic infauna in Singaporean waters.     

Accounting for high-order correlations in probabilistic characterization of environmental variables,and evaluation

Probabilistic characterization of environmental variables or data typically involves distributional fitting. Correlations, when present in variables or data, can considerably complicate the fitting process. In this work, effects of high-order correlations on distributional fitting were examined, and how they are technically accounted for was described using two multi-dimensional formulation methods: maximum entropy (ME) and Koehler–Symanowski (KS). The ME method formulates a least-biased distribution by maximizing its entropy, and the KS method uses a formulation that conserves specified marginal distributions. Two bivariate environmental data sets, ambient particulate matter and water quality, were chosen for illustration and discussion. Three metrics (log-likelihood function, root-mean-square error, and bivariate Kolmogorov–Smirnov statistic) were used to evaluate distributional fit. Bootstrap confidence intervals were also employed to help inspect the degree of agreement between distributional and sample moments. It is shown that both methods are capable of fitting the data well and have the potential for practical use. The KS distributions were found to be of good quality, and using the maximum likelihood method for the parameter estimation of a KS distribution is computationally efficient.     

季节性缺氧水库甲基汞的产生及其对下游水体的影响

本文采用蒸馏-乙基化结合GC-CVAFS法对贵州红枫湖水库及其各入库和出库河流中的甲基汞时空分布和控制因素进行了研究.在春、秋、冬季节总甲基汞浓度和分布无明显时空变化,在0．053-0．333 ng／L之间.春季河流是水库甲基汞一个重要的输入源．夏季水库下层甲基汞显著升高,缺氧层最高值达0．923 ng／L.同时发现．缺氧层升高的甲基汞主要来自于水体自己产生或上层水体甲基汞的沉降,而不是来自于沉积物的释放．各季节湖水和河流样品的总甲基汞和溶解氧存在显著的负相关关系,Personal相关系数r为-0．81(n=78).在春、秋、冬季节溶解态甲基汞比例略低于颗粒态甲基汞,但在夏季,特别是缺氧层,甲基汞主要以溶解态存在．夏季河流入水经水库蓄水后,到再流出时已经富含甲基汞,出库河流中总甲基汞浓度已达到各入湖河流总甲基汞平均值的5.5倍,很明显在复季红枫湖已成为下游水体甲基汞的输入源,必将会对下游生态系统产生一定影响．