Normal-score Ensemble Kalman Filter Research Articles

Contaminant Spill in a Sandbox with Non-Gaussian Conductivities: Simultaneous Identification by the Restart Normal-Score Ensemble Kalman Filter

The joint identification of the parameters defining a contaminant source and the heterogeneous distribution of the hydraulic conductivities of the aquifer where the contamination took place is a difficult task. Previous studies have demonstrated the applicability of the restart normal-score ensemble Kalman filter (rNS-EnKF) in synthetic cases making use of sufficient hydraulic head and concentration data. This study shows an application of the same technique to a non-synthetic case under laboratory conditions and discusses the difficulties found on its application and the avenues taken to solve them. The method is first tested using a synthetic case that mimics the sandbox experiment to establish the minimum number of ensemble members and the best technique to prevent the filter collapsing. The synthetic case shows that among different techniques based on update damping and covariance inflation, the Bauser’s covariance inflation method works best in preventing filter collapse. Its application to the sandbox data shows that the rNS-EnKF can benefit from Bauser’s inflation to reduce the number of ensemble realizations substantially in comparison with a filter without inflation, arriving at a good joint identification of both the contaminant source and the spatial heterogeneity of the conductivities.

Simultaneous identification of a contaminant source and hydraulic conductivity via the restart normal-score ensemble Kalman filter

Detecting where and when a contaminant entered an aquifer from observations downgradient of the source is a difficult task; this identification becomes more challenging when the uncertainty about the spatial distribution of hydraulic conductivity is accounted for. In this paper, we have implemented an application of the restart normal-score ensemble Kalman filter (NS-EnKF) for the simultaneous identification of a contaminant source and the spatially variable hydraulic conductivity in an aquifer. The method is capable of providing estimates of the spatial location, initial release time, the duration of the release and the mass load of a point-contamination event, plus the spatial distribution of hydraulic conductivity together with an assessment of the estimation uncertainty of all the parameters. The method has been applied in synthetic aquifers exhibiting both Gaussian and non-Gaussian patterns. The identification is made possible by assimilating in time both piezometric head and concentration observations from an array of observation wells. The method is demonstrated in three different synthetic scenarios that combine hydraulic conductivities with unimodal and bimodal histograms, and releases in high and low conductivity zones. The results prove that the specific implementation of the EnKF is capable of recovering the source parameters with some uncertainty and of recovering the main patterns of heterogeneity of the hydraulic conductivity fields by assimilating a sufficient number of state variable observations. The proposed approach is an important step towards contaminant source identification in real aquifers, which may have logconductivity spatial distributions with either Gaussian or non-Gaussian features, yet, it is still far from practical applications since the transport parameters, the external sinks and sources and the initial and boundary conditions are assumed known.

Identification of high-permeability subsurface structures with multiple point geostatistics and normal score ensemble Kalman filter

Alluvial aquifers are often characterized by the presence of braided high-permeable paleo-riverbeds, which constitute an interconnected preferential flow network whose localization is of fundamental importance to predict flow and transport dynamics. Classic geostatistical approaches based on two-point correlation (i.e., the variogram) cannot describe such particular shapes. In contrast, multiple point geostatistics can describe almost any kind of shape using the empirical probability distribution derived from a training image. However, even with a correct training image the exact positions of the channels are uncertain. State information like groundwater levels can constrain the channel positions using inverse modeling or data assimilation, but the method should be able to handle non-Gaussianity of the parameter distribution. Here the normal score ensemble Kalman filter (NS-EnKF) was chosen as the inverse conditioning algorithm to tackle this issue. Multiple point geostatistics and NS-EnKF have already been tested in synthetic examples, but in this study they are used for the first time in a real-world case study. The test site is an alluvial unconfined aquifer in northeastern Italy with an extension of approximately 3km2. A satellite training image showing the braid shapes of the nearby river and electrical resistivity tomography (ERT) images were used as conditioning data to provide information on channel shape, size, and position. Measured groundwater levels were assimilated with the NS-EnKF to update the spatially distributed groundwater parameters (hydraulic conductivity and storage coefficients). Results from the study show that the inversion based on multiple point geostatistics does not outperform the one with a multiGaussian model and that the information from the ERT images did not improve site characterization. These results were further evaluated with a synthetic study that mimics the experimental site. The synthetic results showed that only for a much larger number of conditioning piezometric heads, multiple point geostatistics and ERT could improve aquifer characterization. This shows that state of the art stochastic methods need to be supported by abundant and high-quality subsurface data.

Characterization of non‐Gaussian conductivities and porosities with hydraulic heads, solute concentrations, and water temperatures

AbstractReliable characterization of hydraulic parameters is important for the understanding of groundwater flow and solute transport. The normal‐score ensemble Kalman filter (NS‐EnKF) has proven to be an effective inverse method for the characterization of non‐Gaussian hydraulic conductivities by assimilating transient piezometric head data, or solute concentration data. Groundwater temperature, an easily captured state variable, has not drawn much attention as an additional state variable useful for the characterization of aquifer parameters. In this work, we jointly estimate non‐Gaussian aquifer parameters (hydraulic conductivities and porosities) by assimilating three kinds of state variables (piezometric head, solute concentration, and groundwater temperature) using the NS‐EnKF. A synthetic example including seven tests is designed, and used to evaluate the ability to characterize hydraulic conductivity and porosity in a non‐Gaussian setting by assimilating different numbers and types of state variables. The results show that characterization of aquifer parameters can be improved by assimilating groundwater temperature data and that the main patters of the non‐Gaussian reference fields can be retrieved with more accuracy and higher precision if multiple state variables are assimilated.

Characterisation of river–aquifer exchange fluxes: The role of spatial patterns of riverbed hydraulic conductivities

Interactions between surface water and groundwater play an essential role in hydrology, hydrogeology, ecology, and water resources management. A proper characterisation of riverbed structures might be important for estimating river–aquifer exchange fluxes. The ensemble Kalman filter (EnKF) is commonly used in subsurface flow and transport modelling for estimating states and parameters. However, EnKF only performs optimally for MultiGaussian distributed parameter fields, but the spatial distribution of streambed hydraulic conductivities often shows non-MultiGaussian patterns, which are related to flow velocity dependent sedimentation and erosion processes. In this synthetic study, we assumed a riverbed with non-MultiGaussian channel-distributed hydraulic parameters as a virtual reference. The synthetic study was carried out for a 3-D river–aquifer model with a river in hydraulic connection to a homogeneous aquifer. Next, in a series of data assimilation experiments three different groups of scenarios were studied. In the first and second group of scenarios, stochastic realisations of non-MultiGaussian distributed riverbeds were inversely conditioned to state information, using EnKF and the normal score ensemble Kalman filter (NS-EnKF). The riverbed hydraulic conductivity was oriented in the form of channels (first group of scenarios) or, with the same bimodal histogram, without channelling (second group of scenarios). In the third group of scenarios, the stochastic realisations of riverbeds have MultiGaussian distributed hydraulic parameters and are conditioned to state information with EnKF. It was found that the best results were achieved for channel-distributed non-MultiGaussian stochastic realisations and with parameter updating. However, differences between the simulations were small and non-MultiGaussian riverbed properties seem to be of less importance for subsurface flow than non-MultiGaussian aquifer properties. In addition, it was concluded that both EnKF and NS-EnKF improve the characterisation of non-MultiGaussian riverbed properties, hydraulic heads and exchange fluxes by piezometric head assimilation, and only NS-EnKF could preserve the initial distribution of riverbed hydraulic conductivities.

Inverse sequential simulation: A new approach for the characterization of hydraulic conductivities demonstrated on a non‐Gaussian field

AbstractInverse sequential simulation (iSS) is a new inverse modeling approach for the characterization of hydraulic conductivity fields based on sequential simulation. It is described and demonstrated in a synthetic aquifer with non‐Gaussian spatial features, and compared against the normal‐score ensemble Kalman filter (NS‐EnKF). The new approach uses the sequential simulation paradigm to generate realizations borrowing from the ensemble Kalman filter the idea of using the experimental nonstationary cross‐covariance between conductivities and piezometric heads computed on an ensemble of realizations. The resulting approach is fully capable of retrieving the non‐Gaussian patterns of the reference field after conditioning on the piezometric heads with results comparable of those obtained by the NS‐EnKF.

The power of transient piezometric head data in inverse modeling: An application of the localized normal-score EnKF with covariance inflation in a heterogenous bimodal hydraulic conductivity field

The localized normal-score ensemble Kalman filter (NS-EnKF) coupled with covariance inflation is used to characterize the spatial variability of a channelized bimodal hydraulic conductivity field, for which the only existing prior information about conductivity is its univariate marginal distribution. We demonstrate that we can retrieve the main patterns of the reference field by assimilating a sufficient number of piezometric observations using the NS-EnKF. The possibility of characterizing the conductivity spatial variability using only piezometric head data shows the importance of accounting for these data in inverse modeling.

Groundwater flow inverse modeling in non-MultiGaussian media: performance assessment of the normal-score Ensemble Kalman Filter

Abstract. The normal-score ensemble Kalman filter (NS-EnKF) is tested on a synthetic aquifer characterized by the presence of channels with a bimodal distribution of its hydraulic conductivities. This is a clear example of an aquifer that cannot be characterized by a multiGaussian distribution. Fourteen scenarios are analyzed which differ among them in one or various of the following aspects: the prior random function model, the boundary conditions of the flow problem, the number of piezometers used in the assimilation process, or the use of covariance localization in the implementation of the Kalman filter. The performance of the NS-EnKF is evaluated through the ensemble mean and variance maps, the connectivity patterns of the individual conductivity realizations and the degree of reproduction of the piezometric heads. The results show that (i) the localized NS-EnKF can characterize the non-multiGaussian underlying hydraulic distribution even when an erroneous prior random function model is used, (ii) localization plays an important role to prevent filter inbreeding and results in a better logconductivity characterization, and (iii) the NS-EnKF works equally well under very different flow configurations.

Pattern Recognition in a Bimodal Aquifer Using the Normal-Score Ensemble Kalman Filter

The ensemble Kalman filter (EnKF) is now widely used in diverse disciplines to estimate model parameters and update model states by integrating observed data. The EnKF is known to perform optimally only for multi-Gaussian distributed states and parameters. A new approach, the normal-score EnKF (NS-EnKF), has been recently proposed to handle complex aquifers with non-Gaussian distributed parameters. In this work, we aim at investigating the capacity of the NS-EnKF to identify patterns in the spatial distribution of the model parameters (hydraulic conductivities) by assimilating dynamic observations in the absence of direct measurements of the parameters themselves. In some situations, hydraulic conductivity measurements (hard data) may not be available, which requires the estimation of conductivities from indirect observations, such as piezometric heads. We show how the NS-EnKF is capable of retrieving the bimodal nature of a synthetic aquifer solely from piezometric head data. By comparison with a more standard implementation of the EnKF, the NS-EnKF gives better results with regard to histogram preservation, uncertainty assessment, and transport predictions.

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).