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     

Calibration framework for a Kalman filter applied to a groundwater model

The paper presents a novel approach to the setup of a Kalman filter by using an automatic calibration framework for estimation of the covariance matrices. The calibration consists of two sequential steps: (1) Automatic calibration of a set of covariance parameters to optimize the performance of the system and (2) adjustment of the model and observation variance to provide an uncertainty analysis relying on the data instead of ad-hoc covariance values. The method is applied to a twin-test experiment with a groundwater model and a colored noise Kalman filter. The filter is implemented in an ensemble framework. It is demonstrated that lattice sampling is preferable to the usual Monte Carlo simulation because its ability to preserve the theoretical mean reduces the size of the ensemble needed. The resulting Kalman filter proves to be efficient in correcting dynamic error and bias over the whole domain studied. The uncertainty analysis provides a reliable estimate of the error in the neighborhood of assimilation points but the simplicity of the covariance models leads to underestimation of the errors far from assimilation points.     

A method for simulating sharp fluid interfaces in groundwater flow

We present a numerical model for two-phase porous media flow, where the phases are separated by a sharp interface. The model is based on a unified pressure equation, and an advection equation for tracking a pseudo-concentration function. The zero-level set of this function defines the interface between the fluids. The finite element method is used for spatial discretization, with local grid refinements in the vicinity of the interface. Examples on applications involving moving interface and steady-state seepage problems are investigated.     

Dual states estimation of a subsurface flow-transport coupled model using ensemble Kalman filtering

Modeling the spread of subsurface contaminants requires coupling a groundwater flow model with a contaminant transport model. Such coupling may provide accurate estimates of future subsurface hydrologic states if essential flow and contaminant data are assimilated in the model. Assuming perfect flow, an ensemble Kalman filter (EnKF) can be used for direct data assimilation into the transport model. This is, however, a crude assumption as flow models can be subject to many sources of uncertainty. If the flow is not accurately simulated, contaminant predictions will likely be inaccurate even after successive Kalman updates of the contaminant model with the data. The problem is better handled when both flow and contaminant states are concurrently estimated using the traditional joint state augmentation approach. In this paper, we introduce a dual estimation strategy for data assimilation into a one-way coupled system by treating the flow and the contaminant models separately while intertwining a pair of distinct EnKFs, one for each model. The presented strategy only deals with the estimation of state variables but it can also be used for state and parameter estimation problems. This EnKF-based dual state-state estimation procedure presents a number of novel features: (i) it allows for simultaneous estimation of both flow and contaminant states in parallel; (ii) it provides a time consistent sequential updating scheme between the two models (first flow, then transport); (iii) it simplifies the implementation of the filtering system; and (iv) it yields more stable and accurate solutions than does the standard joint approach. We conducted synthetic numerical experiments based on various time stepping and observation strategies to evaluate the dual EnKF approach and compare its performance with the joint state augmentation approach. Experimental results show that on average, the dual strategy could reduce the estimation error of the coupled states by 15% compared with the joint approach. Furthermore, the dual estimation is proven to be very effective computationally, recovering accurate estimates at a reasonable cost.     

Integrated surface–groundwater flow modeling: A free-surface overland flow boundary condition in a parallel groundwater flow model

Interactions between surface and groundwater are a key component of the hydrologic budget on the watershed scale. Models that honor these interactions are commonly based on the conductance concept that presumes a distinct interface at the land surface, separating the surface from the subsurface domain. These types of models link the subsurface and surface domains via an exchange flux that depends upon the magnitude and direction of the hydraulic gradient across the interface and a proportionality constant (a measure of the hydraulic connectivity). Because experimental evidence of such a distinct interface is often lacking in field systems, there is a need for a more general coupled modeling approach.     

A coupled model for simulating surface water and groundwater interactions in coastal wetlands

Coastal wetlands are characterized by strong, dynamic interactions between surface water and groundwater. This paper presents a coupled model that simulates interacting surface water and groundwater flow and solute transport processes in these wetlands. The coupled model is based on two existing (sub) models for surface water and groundwater, respectively: ELCIRC (a three‐dimensional (3‐D) finite‐volume/finite‐difference model for simulating shallow water flow and solute transport in rivers, estuaries and coastal seas) and SUTRA (a 3‐D finite‐element/finite‐difference model for simulating variably saturated, variable‐density fluid flow and solute transport in porous media). Both submodels, using compatible unstructured meshes, are coupled spatially at the common interface between the surface water and groundwater bodies. The surface water level and solute concentrations computed by the ELCIRC model are used to determine the boundary conditions of the SUTRA‐based groundwater model at the interface. In turn, the groundwater model provides water and solute fluxes as inputs for the continuity equations of surface water flow and solute transport to account for the mass exchange across the interface. Additionally, flux from the seepage face was routed instantaneously to the nearest surface water cell according to the local sediment surface slope. With an external coupling approach, these two submodels run in parallel using time steps of different sizes. The time step (Δt_g) for the groundwater model is set to be larger than that (Δt_s) used by the surface water model for computational efficiency: Δt_g = M × Δt_s where M is an integer greater than 1. Data exchange takes place between the two submodels through a common database at synchronized times (e.g. end of each Δt_g). The coupled model was validated against two previously reported experiments on surface water and groundwater interactions in coastal lagoons. The results suggest that the model represents well the interacting surface water and groundwater flow and solute transport processes in the lagoons. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical model for simulating Hortonian overland flow on tropical hillslopes with vegetation elements

This paper describes a two‐dimensional hydrodynamic model that characterizes surface runoff process resulting from a varying rainfall intensity event, on an infiltrating soil surface. The soil surface has spatially varied soil physical, hydraulic and microtopographic characteristics. Infiltration process is modelled with the Philip two‐term equation and the time before ponding approximated with the time compression algorithm. Vegetation is modelled as a dynamic component with the modified Gash model. The equation is solved with a modified second order Leapfrog explicit finite difference scheme with centred time and space derivatives. The model was validated with standard analytical solutions. Evaluation with results from field campaigns in the Volta Basin of West Africa during the 2002 rainfall season indicates good agreement, with r² values ranging from 0·89 to 0·96. The developed method will be useful in studying the dynamics of surface runoff generation under complex microtopographic conditions, spatially varying soil hydraulic characteristics and temporally dynamic rainfall intensity, as found in many tropical catchments. Copyright © 2007 John Wiley & Sons, Ltd.     

Assimilating transient groundwater flow data via a localized ensemble Kalman filter to calibrate a heterogeneous conductivity field

A localized ensemble Kalman filter (EnKF) method is developed to assimilate transient flow data to calibrate a heterogeneous conductivity field. To update conductivity value at a point in a study domain, instead of assimilating all the measurements in the study domain, only limited measurement data in an area around the point are used for the conductivity updating in the localized EnKF method. The localized EnKF is proposed to solve the problems of the filter divergence usually existing in a data assimilation method without localization. The developed method is applied, in a synthetical two dimensional case, to calibrate a heterogeneous conductivity field by assimilating transient hydraulic head data. The simulations by the data assimilation with and without localized EnKF are compared. The study results indicate that the hydraulic conductivity field can be updated efficiently by the localized EnKF, while it cannot be by the EnKF. The covariance inflation and localization are found to solve the problem of the filter divergence efficiently. In comparison with the EnKF method without localization, the localized EnKF method needs smaller ensemble size to achieve stabilized results. The simulation results by the localized EnKF method are much more sensitive to conductivity correlation length than to the localization radius. The developed localized EnKF method provides an approach to improve EnKF method in conductivity calibration.     

A hybrid framework for improving recharge and discharge estimation for a three-dimensional groundwater flow model

We employed the ArcGIS plug-in package PRO-GRADE (Lin et al. 2009), developed for zonation of recharge/discharge (R/D) for modeling two-dimensional aquifer systems, to develop alternative R/D zonations for an existing three-dimensional groundwater flow model of a complex hydrogeologic setting. Our process began by intersecting PRO-GRADE output with the existing model's 4-zone R/D representation to develop a model having 12 R/D zones (R12) and then calibrating the resulting model using PEST. We then revised the R12 zonation using supplementary GIS data to develop a 51-zone R/D zonation (R51). From R51, we developed a series of daughter models having 40, 30, 28, and 18 R/D zones by removing zones from R51 if calibration resulted in little change in the zone's starting R/D rate and/or if the model was insensitive to the zone's R/D rate. For these models (R40N, R30N, R28N, and R18N), we used the ArcGIS Nibble tool to rapidly and consistently reassign model cells within eliminated zones of R51 to the zone of the nearest model cell in a retained zone having the same starting value. R12, R51, R40N, R30N, R28N, and R18N are all more accurate than the original model (R4), although improvements relative to stream discharge targets exceeded improvements relative to head targets. The models also executed with better numerical stability and less mass balance discrepancy than R4. These improvements demonstrate that R/D estimation in a complex shallow three-dimensional steady-state model can be improved with PRO-GRADE estimates of R/D when guided by calibration statistics and supplemental geographic data.     

Comparison between gradient based UCODE_2005 and the ensemble Kalman Filter for transient groundwater flow inverse modeling

Gradient based UCODE_2005 and data assimilation based on the Ensemble Kalman Filter(EnKF) are two different inverse methods. A synthetic two-dimensional flow case with four no-flow boundaries is used to compare the UCODE_2005 with the Ensemble Kalman Filter(EnKF) for their efficiency to inversely calculate and calibrate a hydraulic conductivity field based on hydraulic head data. A zonal, random heterogeneous conductivity field is calibrated by assimilating the time series of heads observed in monitoring wells. The study results indicate that the two inverse methods, UCODE_2005 and EnKF, could be used to calibrate the hydraulic conductivity field to a certain degree. More available observations and information about the conductivity field, more accurate inverse results will be obtained for the UCODE_2005. On the other hand, for a realistic zonal heterogeneous hydraulic conductivity field, EnKF can only efficiently determine the hydraulic conductivity field at the first several assimilated time steps. The results obtained by the UCODE_2005 look better than those by the EnKF. This is possibly due to the fact that the UCODE_2005 uses observed head data at every time step, while EnKF can only use observed heads at first several steps due to the filter divergence problem.     

Efficient conceptual model for simulating the effect of aquifer heterogeneity on natural groundwater discharge to rivers

When linearity can be assumed (linear response of heads to stresses), stream–aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream–aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream–aquifer interaction.     

A variational principle for steady state groundwater flow with a free surface

Summary The velocity potential and the shape of the free surface of a steady state gravity flow of incompressible, homogeneous groundwater inside an inhomogeneous, anisotropic aquifer are to be determined as the solution of a non-linear boundary-value problem. In this paper an equivalent variational principle has been formulated which is useful as a base of variational methods solving this boundary-value problem approximately.

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Zusammenfassung Das Geschwindigkeitspotential und die Gestalt des freien Grundwasserspiegels einer stationären Strömung inkompressiblen, homogenen Grundwassers innerhalb eines anisotropen, inhomogenen Grundwasserträgers sind aus einem nichtlinearen Randwertproblem zu bestimmen. Es wird ein äquivalentes Variationsprinzip angegeben, welches als Basis für die Anwendung direkter Methoden der Variationsrechnung zur approximativen Lösung dieses Randwertproblemes dienen kann.

     

Mathematical models of groundwater flow with a horizontal drain

An analytical review is given to drainage theory, the foundation of which was laid down by researchers in our country in the first half of the 20th century. Later the author of this paper developed and thoroughly studied a complex of mathematical models describing two-dimensional groundwater flow in drained soils based on the boundary problems of the theory of analytical functions. These models were used to provide a hydrodynamic substantiation for the regularities in subsoil water dynamics that were established by ameliorators in relation to the leaching of salts from soils; to identify the specific features of groundwater flow with drainage in the presence in the flow of one or two unknown free boundaries; and to develop approaches to the analysis of such flows in a direct formulation.     

An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media

A lattice Boltzmann (LB) model is proposed for simulating fluid flow in porous media by allowing the aggregates of finer-scale pores and solids to be treated as ‘equivalent media’. This model employs a partially bouncing-back scheme to mimic the resistance of each aggregate, represented as a gray node in the model, to the fluid flow. Like several other lattice Boltzmann models that take the same approach, which are collectively referred to as gray lattice Boltzmann (GLB) models in this paper, it introduces an extra model parameter, n_s, which represents a volume fraction of fluid particles to be bounced back by the solid phase rather than the volume fraction of the solid phase at each gray node. The proposed model is shown to conserve the mass even for heterogeneous media, while this model and that model of Walsh et al. (2009) [1], referred to the WBS model thereafter, are shown analytically to recover Darcy–Brinkman’s equations for homogenous and isotropic porous media where the effective viscosity and the permeability are related to n_s and the relaxation parameter of LB model. The key differences between these two models along with others are analyzed while their implications are highlighted. An attempt is made to rectify the misconception about the model parameter n_s being the volume fraction of the solid phase. Both models are then numerically verified against the analytical solutions for a set of homogenous porous models and compared each other for another two sets of heterogeneous porous models of practical importance. It is shown that the proposed model allows true no-slip boundary conditions to be incorporated with a significant effect on reducing errors that would otherwise heavily skew flow fields near solid walls. The proposed model is shown to be numerically more stable than the WBS model at solid walls and interfaces between two porous media. The causes to the instability in the latter case are examined. The link between these two GLB models and a generalized Navier–Stokes model [2] for heterogeneous but isotropic porous media are explored qualitatively. A procedure for estimating model parameter n_s is proposed.     

Interbasin groundwater flow and groundwater interaction with surface water in a lowland rainforest, Costa Rica: A review

This paper reviews work related to interbasin groundwater flow (naturally occurring groundwater flow beneath watershed topographic divides) into lowland rainforest watersheds at La Selva Biological Station in Costa Rica. Chemical mixing calculations (based on dissolved chloride) have shown that up to half the water in some streams and up to 84% of the water in some riparian seeps and wells is due to high-solute interbasin groundwater flow (IGF). The contribution is even greater for major ions; IGF accounts for well over 90% of the major ions at these sites. Proportions are highly variable both among watersheds and with elevation within the same watershed (there is greater influence of IGF at lower elevations). The large proportion of IGF found in water in some riparian wetlands suggests that IGF is largely responsible for maintaining these wetlands. δ¹⁸O data support the conclusions from the major ion data. Annual water and major ion budgets for two adjacent watersheds, one affected by IGF and the other not, showed that IGF accounted for two-thirds of the water input and 92–99% of the major ion input (depending on the major ion in question) to the former watershed. The large (in some cases, dominating) influence of IGF on watershed surface water quantity and quality has important implications for stream ecology and watershed management in this lowland rainforest. Because of its high phosphorus content, IGF increases a variety of ecological variables (algal growth rates, leaf decay rate, fungal biomass, invertebrate biomass, microbial respiration rates on leaves) in streams at La Selva. The significant rates of IGF at La Selva also suggest the importance of regional (as opposed to small-scale local) water resource planning that links lowland watersheds with regional groundwater. IGF is a relatively unexplored and potentially critical factor in the conservation of lowland rainforest.     

自适应卡尔曼滤波在航空重力异常解算的应用研究

依据航空重力测量基本原理,构建了航空重力异常解算的卡尔曼滤波模型,将新息自适应卡尔曼滤波器(IAE,Innovation based Adaptive Estimation)应用于量测噪声未知的航空重力异常解算.针对IAE滤波器滑动窗口宽度难以准确确定的问题,通过对多个不同滑动窗口新息协方差估计的加权平均,获得改进的IAE滤波器,该IAE滤波器不仅具有量测噪声自适应估计能力,还能实现滑动采样窗口的优化选取.试验结果表明,IAE滤波器可以降低因量测噪声统计信息不明引起的解算误差,改进IAE解算的重力异常误差约为1 mGal.     

Toward a new conceptual model for groundwater flow in merokarst systems: Insights from multiple geophysical approaches

Merokarst aquifers — relatively thin (<1–2 m) karstified carbonate units interbedded between mudstone, shale, or sandstone — constitute a significant proportion of carbonate terrain and underlie a large portion of the west- and south-central USA, yet few advances have been made in our understanding of porosity development and flow-path generation in these complex systems in decades. Toward this end, we used a multi-geophysical approach at the well-studied Konza Prairie Biological Station (KPBS), a part of the larger Flint Hills (25,734 km²), underlain by thin limestone units (1–2 m thick) interbedded with mudstone/shale units (2–4 m thick), to elucidate hydrologic connectivity and potential controls on known groundwater flow directions. We combined electrical resistivity tomography (ERT), surface and borehole nuclear magnetic resonance (NMR), and ground penetrating radar (GPR) measurements across a low order catchment where over 25 boreholes and groundwater wells sampling perched aquifers could be used to constrain interpretation of lithology, potential flow paths, and permeability. Data revealed that groundwater export may be an unappreciated component of lateral-flow-dominated models used to represent merokarst in that: (a) potentiometric surfaces from two limestone units showed groundwater flows toward a hydrologic depression, opposite the direction of stream flow, in the upstream portion of the catchment, (b) long term measures of groundwater levels revealed a greater variance and overall water storage in this same upstream area compared to wells near the outlet, and (c) ERT and NMR results indicate pronounced lateral heterogeneity within a given unit, suggestive of a greater degree of vertical hydrological connectivity than usually considered for horizontally-layered merokarst. Our data suggest vertical connectivity can shunt water to depth in these “sandwiched” merokarst aquifers, yielding atypical groundwater flow directions and unrealized deep export of weathering solutes and carbon.