Groundwater Contaminant Transport Modeling Using Multiple Adaptive Data Assimilation Techniques

Advances in Water Resource and Protection, Volume 3, 2016 www.seipub.org/awrp doi: 10.14355/awrp.2016.03.001

Groundwater Contaminant Transport Modeling Using Multiple Adaptive Data Assimilation Techniques Elvis Boamah Addai1, PhDstudent, Shoou‐Yuh Chang2, PhD, and Godwin Appiah Assumaning3, PhD (corresponding author), Amirul Islam Rajib4, MSc. Student.

1. Research Assistant, 1601 East Market Street, Department of Civil and Environmental Engineering, North Carolina Agricultural &Technical State University, Greensboro, NC 27411, USA. ebaddai@aggies.ncat.edu; Phone: (336) 609‐4544, Fax: (336) 334‐7126.

2. DOE Samuel Massie Chair of Excellence Professor, 1601 East Market Street, Dept. of Civil and Environmental Engineering, North Carolina A&T State University, Greensboro, NC 27411, USA. chang@ncat.edu, Phone: (336) 285‐3674, Fax: (336) 334‐7126.

3. Postdoctoral Research Associate, 1601 East Market Street, Dept. of Civil and Environmental Engineering, North Carolina A&T State University, Greensboro, NC 27411, USA. godwinappiah30@yahoo.com; Phone: (336) 285‐2452, Fax: (336) 334‐7126(Corresponding author).

4. Research Assistant, 1601 East Market Street, Department of Civil and Environmental Engineering, North Carolina Agricultural &Technical State University, Greensboro, NC 27411, USA. rajibamirul@gmail.com; Phone: (336) 255‐1893, Fax: (336) 334‐7126.

Abstract In modeling the behavior of contaminants in a subsurface environment using data assimilation schemes, accurate assignment of model and observation errors are significant for the successful application of the techniques. In this study, a three‐dimensional transport model was used to simulate the advection and dispersiontransport of contaminant in the subsurface. Stochastic data assimilation schemes were coupled with the subsurface contaminant transport model to predict the state of the contaminant.

Three data assimilation techniques namely the conventional Ensemble KalmanFilter, the Adaptive Ensemble Kalman Filter and the Hybrid Adaptive Ensemble Kalman Filter were adopted to improve the prediction of the contaminant fate and transport in the groundwater. The Ensemble Kalman Filter applies a Monte Carlo approach to the filtering problem. The adaptive filtering technique employs the diagnostic approach to fine tune the model and observation covariance matrix. The hybrid technique uses combination of the forecast covariance matrix and the invariant background covariance matrix to explore the Ensemble Kalman filter.

The impact of the filters on the numerical model is examined by using the Normalized Root Mean Square Error (NRMSE), Average Absolute Bias (AAB) metric, and Maximum Absolute Deviation (MAD) techniques. The AAB evaluation of Adaptive Ensemble Kalman Filter and Hybrid Adaptive Ensemble Kalman Filter shows error reduction of 93% and 90%, respectively, while the MAD assessment recorded 94% and 89% improvement respectively, in the numerical model. The results of simulations show that the prediction accuracy of the filters is better than numerical model. The proposed Adaptive Ensemble Kalman Filter and Hybrid Adaptive Ensemble Kalman Filter, takes advantage of the adaptive factor and the weighting factor, respectively to improve the prediction efficiency of the Ensemble Kalman filter. Sensitivity analysis performed on Adaptive Ensemble Kalman Filter and Hybrid Adaptive Ensemble Kalman Filter using NRMSE indicates that the adaptive factor does not affect the prediction accuracy of the former whereas the weighting factor has influence on the later.

Keywords Ensemble Kalman Filter (EnKF); Adaptive EnKF; Hybrid Adaptive EnKF; Data Assimilation; Groundwater Contamination; Modeling; Prediction

Introduction Groundwater is an important natural resource which is useful for different purposes, such as drinking, irrigation, and industrial processes. As a convenient substitute to compensate for insufficient surface water resources, groundwater may be used to provide additional water supply (Nampak et al., 2014). The contamination of the

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groundwater poses a widespread threat due to its degradation of water quality, the effect it has on the ecosystem and its toxicity towards humans (Aral, 2010; Langwaldt and Puhakka, 2000). Therefore, groundwater should be protected from contamination by implementing effective measures. The risk assessment of groundwater pollution and modelling of affected areas is an effective tool to protect the groundwater.

Numerous mathematical models have been developed and successfully applied to the fate and transport of contaminants in the subsurface environment (Chang and Jin, 2005; McLaughlin, 2002; Zou and Par 1995). However, predictions by the conventional numerical models normally lead to unavoidable deviation from the truefield because of uncertainties in model mechanisms, the numerical schemes and theinaccurate model parameters used (Johnson et al., 2006).A lot of research on the fate and transport of contaminant in the subsurface using mathematical models have been done as a result of these issues (Lee 2004; Yan et al. 2006). Hence,data assimilation (DA) techniques have been combined with numerical models to provide a more accurate representation of the transport processto determine the best possible estimate of the state of a dynamical system (Ghil and Malanotte‐ Rizzoli, 1991).

Kalman filter (KF) algorithm, which is a typical example of DA, was introduced by Kalman (1960) as an optimal solution for state estimation of dynamic linear systems under Gaussian noise. When both the measurement model and the dynamic model are linear KF provides optimal solution. If any of the model turn out to be non‐linear in nature problem arises. The filter constitutes an optimal data assimilation scheme given an exact knowledge of all sources of error, linear dynamics, and a number of other conditions (Maybeck, 1979). The algorithm has been used extensively in the fields of signal processing and navigation. The filter is very effective in estimation of past, present, and future states even when the precise nature of the modeled system is unknown.

In case of high dimensional or non‐linear models the KF suffersfrom computational issues despite its efficiency in solving discrete linear filtering problem. To prevent the KFfilter divergence it is paramount to estimate the model error and adjust the corresponding term in the forecast‐error evolution equation (Dee, 1995). The KF provides a framework for assimilating observations and also for estimating the effect of observations on forecast error covariance. Even though the KF was effective in many real life cases, but it is not an optimal estimator as it falls short to account for the full non‐linear dynamics. The Ensemble Kalman filter (EnKF) technique was introduced by Evensen (1994) to solve nonlinear state estimation. For high‐dimensional weather forecasting systems where models are extremely nonlinear in nature, where large numbers of observations are available and initial states are very noisy, the EnKF is very popular in such cases. The EnKF developed by Evensen (1994) and later clarified by Burgers et al. (1998) gives a more accurate estimate of the state than the KF and the Extended Kalman filter (EKF) because of the efficient representation of the background error covariance. The EnKF is easy to implement, that is, no linearization of system equations are needed (Reichle et al., 2002). A Monte Carlo solution is provided by the EnKF to the problems arising from high dimensional non‐linear state‐space models. Gaussian approximation is assumed by the EnKF technique in the linearization of state and measurement equations to implement the error evolution in the analysis scheme to obtain accurate solution (Zhou et al., 2006). The EnKF functions in two main steps, a forecast step and an analysis step. The EnKF was used to assimilate hydraulic head data from ninety locations during two year period of groundwater flow modeling (Franssen et al., 2008). The EnKF was combined with polynomial chaos methodology to improve the forecasting ability of a reservoir model (Saad, 2007). The EnKF was used to match production rate with injection rates and to analyze the facies locations in the reservoir model by Liu and Oliver (2005). In another research, Chen et al. (2009) applied the EnKF concept with reparameterization in history matching problems of multiphase flow in a heterogeneous reservoir. Van Delft et al. (2009) used the EnKF as a data assimilation scheme to reduce the amount of uncertainties in a flood forecasting problem. A first‐order reaction rate was estimated using the EnKF in a synthetic reactive transport system (Bailey and Baú, 2011).

Different EnKF methodologies for estimating hydraulic conductivity and porosity in groundwater flow and transport systems have been implemented (Chen and Zhang, 2006; Gharamti and Hoteit, 2013; Li et al., 2012; Schoniger et al., 2012; Zhou et al., 2011). Although, the EnKF and its variants have successfully been implemented in many hydrology and reservoir applications for state and parameters estimation, it has some limitations when it comes to high degree nonlinearity problems (Slivinski et al., 2015). Also, the EnKF analysis scheme may have problems in cases where the number of measurements is larger than the number of members in the ensemble (Evensen, 2003).

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In a study conducted by Houtekamer and Mitchell (1998), as the ensemble size increases, the accuracy of the calculation increases. The use of a limited number of ensemble members gives rise to a rank problem in the estimate of the covariance matrix of the innovations. In three dimensional atmospheric data assimilation the use of innovations for covariance estimation is a well‐established practice, where one estimates a few parameters that determine the complete description of the forecast‐error covariances (Hollingsworth and Lonnberg, 1986). The use of KF, three‐dimensional variational assimilation (3DVAR) and direct insertion data assimilation schemes, respectively, within the context of simple models containing data voids of one sort or another, have all shown that targeted observations reduce forecast error more than fixed observations (Fischer et al., 1998; Lorenz and Emanuel, 1998; Morss et al., 2001). Hamill et al. (2000) conducted studies providing an interesting discussion of the statistical properties of the ensembles produced by various ensemble generation techniques.

An adaptive filter which is a computational tool models the relationship between two signals in real time in an iterative manner (Douglas, 1999). The same concept is adopted and applied in the subsurface environment. The adaptive ensemble Kalman filter gives relatively good estimates of the parameters and accounts rather well for the model error. It is found that, with temporal smoothing of the model‐error parameter estimates, its performance in a data assimilation cycle is almost as good as that of a cycle in which the correct model error parameters are used to increase the spread in the ensemble (Mitchell and Houtekamer, 2000). Akita et al. (2012) used an adaptive estimation method of space craft thermal mathematical model with EnKF to handle the nonlinearities contained in the thermal model. The concept of adaptive assimilation approaches have been developed that estimate the model and observation error parameters based on innovations (Dee et al., 1985; Gelb, 1974; Mehra, 1970; Moghaddamjoo and Kirlin, 1993). Dee and da Silva (1999) applied the adaptive methods in atmospheric data assimilation following a covariance matching approach. Mitchell and Houtekamer (2000) developed an adaptive technique for estimating model error that can be used in conjunction with the EnKF. The technique involves parameterizing the model error and using innovations to estimate the model‐error parameters. In another study, Anderson (2007) used the adaptive approach in ensemble data assimilation.

Hamill and Snyder (2000) introduced a hybrid adaptive EnKF‐3DVar as a way to mitigate the background limitations of the EnKF. The hybrid approach formulation was extended to accommodate parameter estimation through a dual filtering technique (Gharamti et al., 2013; Moradkhani et al., 2005; Samuel et al., 2014). The hybrid adaptive scheme uses a background Covariance in the EnKF estimated from an ensemble and a predefined static covariance from a 3DVar system (Hamill and Snyder, 2000). Gharamti et al. (2014) applied a hybrid EnKF‐ optimal interpolation assimilation scheme to mitigate background covariance limitations due to ensemble under‐sampling and neglected model errors.

In this paper, an Adaptive Ensemble Kalman Filter (AEnKF) and Hybrid AEnKF (HAEnKF) schemes are developed with an adaptive parameterto fine tune the system error covariance, measure menterror covariance and the optimal Kalman gain in order to reduce the errors in the system and measurement noise statistics during the filtering operation. The objective is to use the two data assimilation schemes to improve the prediction accuracy of groundwater contaminant fate and transport in a subsurface environment by exploring the strength of the EnKF and also to examine the performance of the techniques using different hydrologic parameter values.

In the next section, an overview of all the techniques is given. Further, in Section 3 will be the results and discussion of the performance of the techniques. Thereafter the focus will be on the summary of the research.

Methodology Model Description The techniques employed in this research involve numerical scheme, which is an explicit finite difference method, the EnKF scheme, the AEnKF scheme and finally the HAEnKF scheme. Comparisons are made between the results of the four schemes relative to the true solution (the numerical scheme with added noise) to find their effectiveness in state estimation in subsurface contaminant transport modeling.

Various mathematical models and the applicability of this approach that described solute transport and

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groundwater flow in fractured porous media have been developed in previous studies (Schwartz and Smith, 1988; McKay, 1991). Sudicky (1990) developed numerical model that employ the double‐porosity approach to describe transport in porous and fractured porous media. Rasmussen and Evans (1989) developed a three‐dimensional model based on the boundary element method to investigate variably‐saturated flow and solute transport in a fractured medium. Transport processes including advection, dispersion, molecular diffusion and sorption in the porous medium were accounted for since these processes have been shown to be non‐negligible for many groundwater flow and solute transport scenarios (Sudicky and McLaren, 1992). The advection‐dispersion equation for a three‐dimensional subsurface transport model was used to generate the numerical results (Assumaning and Chang, 2012). Due to the presence of a first‐order decay rate parameter, the model is used to simulate non‐ conservative pollutant. Domenico and Schwartz (1998) applied the model. The advection‐dispersion equation for subsurface contaminant transport with absorption and decay mechanisms inclusive is represented in a partial differential equation form as(Cheng, 2000). C Dx  2C Dy  2C Dz  2C V C kC , (1)      t R x 2 R y 2 R z 2 R x R

R  1  b

Kd 

Kd



, (2)

S , (3) C

where C is the concentration of contaminant (mg/L); Dx, Dy and Dz are the dispersion coefficients in the x, y and z directions, (m2/day), respectively; R is retardation factor (which has no unit);  b is the bulk density of the porous

medium (mg/L); represents porosity;

is the distribution coefficient; S is the sorbed contaminant concentration

(mg/L); V is the linear velocity (m/day); k is the first‐order decay rate parameter (1/day); x, y and z are the Cartesian coordinates; and t is time in days. A boundary condition was assumed and the initial conditionswere stated based on an instantaneous point source in the model implementation. The initial and boundary conditions of the subsurface transport model with instantaneous point source are given in Equations (2) and (3), respectively.

C  x, y, z, t t 0  C0  x0 , y0 , z0  , (4)

C  x, y, z, t t   0 , (5) where  represents the boundary of the domain in this study and (x0,y0,z0) is the initial pulse input coordinates. A specified initial concentration in the model acts as a source providing solute mass to the domain and any predicted concentration out of the simulated domainboundary should be equal to zero.

Numerical Model The advection‐dispersion model in (Equation 1) is solved by using Forward Time and Central‐Space (FTCS) method (Chang and Jin, 2005) to develop the state‐space form of the partial differential equation. The resulting numerical model after discretizing Equation 1 is shown as;

C  x, y, z, t  1  1C  x 1, y, z, t   2C  x, y  1, z, t   3C  x  1, y, z, t  4C  x, y  1, z, t   5C  x, y  1, z, t   6C  x, y  1, z, t   7C  x, y, z  1, t  , (6) where 1 , 2 , 3 , 4 , 5 , 6 and 7 are coefficients of hydrogeological functions defining contaminant transport in the porous medium, they are used to build the state transition matrix (STM) that transitions one state to another with time. A detailed account on how to discretize the governing equation has previously been done (Assumaning and Chang, 2012). The FTCS scheme is checked by satisfying the convergence and stability criterion for the following conditions V

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t  1.0 , (7) x

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Dx

t t t  Dy 2  Dz 2  0.5 , (8) x 2 y z

where t and x are temporal and spatial step sizes, respectively. Equation 6 can be written in a state‐space form as;

xt  Axt 1 , (9) where xt is the contaminant state vector at all nodes in the problem domain at time step t; xt‐1 is the contaminant state vector at all nodes in the problem domain at time step t‐1; and A is the state transition matrix (STM) containing the parameters which gives a finite difference scheme or driving operator to advance a time step state to the next time step. Data assimilation schemes can simulate transport processes with uncertain sources and inaccurate transport parameters by incorporating a random noise term in the deterministic dynamics (Saad, 2007).

Because of the heterogeneous and stochastic nature of the subsurface environment, random error is introduced into the numerical model to provide an accurate representation of the transport process. The process equation is given in the general state‐space form as;

xt  Axt 1  wt , t = 0,1,2,.…tn, (10) Where wt is the model system error which is the difference between the optimal estimate of the true state and the model prediction with covariance Qt and zero mean, the errors in this study are assumed to be white Gaussian; tn is the final time step. The observation equation for the entire domain is expressed as;

zt  HxtT   t , (11) Where H is the measurement sensitivity matrix which reflects the observations in the field and it is constructed as T

identity matrix; zt is the state vector for the observed values at all nodes at time step t; xt is the simulated true value of the state for all nodes at time step t; and  t is the observation error which is assumed to have zero mean and covariance matrix Rt. Data Assimilation with Ensemble Kalman Filter (EnKF) The EnKF was introduced to solve the problems associated with the KF and it uses Monte Carlo method through an ensemble of states to predict the error statistics (Evensen, 1994). The covariance matrix is replaced by the sample covariance computed from the ensemble. In the treatment of errors in model dynamics and parameters, the EnKF is noted to be flexible. The EnKF works recursively by performing in turn a model forecast and a filter update. The EnKF has become a popular choice for data assimilation (Durand and Margulis, 2008). The forecast step for ensemble member i can be expressed as;

xtf,i  Axta1,i  wt ,i , (12) f

a

where xt ,i represents the forecast state vector at time t; xt 1,i is the analysis state vector at time t‐1; wt ,i is the model a

system error with covariance Qtand zero mean. At time t, the filter update produces the analyzed state vector xt ,i and can be expressed as;

xta,i  xtf,i  Kt ( zt ,i  H t xtf,i ) , (13) whereHtis the measurement sensitivity matrix (observation operator); zt,i denotes the state vector for the observed values at all nodes at time step tand Kt is the Kalman gain matrix.The Kalman gain matrix can be determined by;



K t  Pt f H tT H t Pt f H tT  Rt f

x t ,i  Pt f 



1 N

N

x

f 1 N xtf,i  x t ,i  N  1 i 1

i 1

f t ,i

 x

f t ,i



1

, (14)

, (15) f

 x t ,i



T

 UU T , (16)

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f

where xt ,i is the forecast mean; U is state covariance anomaly formed to avoid computing the full covariance matrices; T denotes the matrix transpose; and Pt f is the forecast covariance matrix. a

The analysis error covariance matrix Pt is given as; Pt a   I  K t H t  Pt f , (17)

Data Assimilation with Adaptive Ensemble Kalman Filter (AEnKF) In the case of the AEnKF, the main idea behind the data assimilation system of the adaptive filtering technique is that internal diagnostics of the assimilation system should be consistent with the values that are expected from input parameters provided to the system (Desroziers and Ivanov, 2001). The difference between observation and forecast innovations encountered during the application of equation (13) is the most commonly used diagnostics for adaptive filtering. Alternative diagnostics are based on the analysis departures and the analysis increments (observation space). The EnKF starts with initial guesses for model and observation error covariances denoted with Qt and Rt, respectively. The AEnKF starts with the conventional EnKF forecast and update step from equation (12) to equation (17), which involves the propagate model, forecast error covariance, Kalman gain and the analysis update after appropriate initialization, along with the computation of the innovations, the analysis departures, and the analysis increments (Shi et al., 2010).

bt  E{zt ,i  H t xtf,i } , (18) st  E{zt ,i  H t xta,i } , (19) et  E{H t ( xta,i  xtf,i )} , (20) wherebtis the innovations; st is the analysis departures; etis the analysis increments and E{} representsthe ensemble mean. Desroziers et al. (2005) suggest a similar way of estimating the model and observation error covariances separately. The approach employed here has been reported to have good results. After the computation of the of the innovations, the analysis departures, and the analysis increments, the adaptive module follows by combining the alternative diagnostics with the innovation to yield two separate constrains (Desroziers et al., 2005). For linear systems operating under optimal conditions we have; E[et btT ]  H t Pt f H tT , (21) E[ st btT ]  Rt , (22)

whereE[.] is the expectation operator. The two separate constrains in Equations (21) and (22) are attractive because they suggest a simple way of estimating the model and observation error covariances separately. An operational procedure for tuning of Q and R to meet the constraints imposed by equations (21) and (22) was derived by Reichle et al. (2008). Selecting values of Q and R that conform to these two constraints should lead to optimized filter performance (Shi et al., 2010). Equations (21) and (22) are typically determined by averaging across the domain; MAVG[ebT ]t  (1   ) MAVG[ebT ]t 1   et btT

, (23)

MAVG[ sbT ]t  (1   ) MAVG[ sbT ]t 1   st btT , (24) MAVG[ HP f H T ]t  (1   ) MAVG[ HP f H T ]t 1   H t Pt f H tT , (25)

MAVG[ R ]t  (1   ) MAVG[ R ]t 1   Rt , (26)

Where MAVG[ ] denotes exponential moving average; and  is an ad hoc choice value which ranges from zero to one for this study. Reichle et al. (2008) defined a moving average operator of the form indicated in Equations (23) to (26) with a specified ad hoc choicevalue of 0.02. In this study we used  =0.02. After computing the time‐filtered estimates of Equation (23) to (26), the ratio of actual covariances and expected covariances are determined using these operators;

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gQ ,t 

 MAVG[ebT ]t MAVG[ HP f H T ]t

g R ,t 

, (27)

MAVG[ sbT ]t , (28) MAVG[ R]t

where  is the tuning parameter which addresses the generally nonlinear relationship between the model error covariance Q and the forecast error covariance Pf; g Q , t and g R , t are the ratios use to update the adaptive tuning factor ( ) for Q and R, respectively. Reichle et al. (2008) found that unless  is introduced, the ratio g Q , t will be underestimated for perfect input error parameters and the adaptive algorithm will tend to produce slightly worse results. In another study, three  values (0.98, 1.02 and 1.06.) were selected for the comparative analysis and based on the analysis,  was set to 1.02 in the computing process (Shi et al., 2010). The ratios g Q , t and g R , t were restricted to the interval [(1  n) 1 , (1  n)] , with n = 0.005, because of residual noise in the moving average estimates in equation (23) to (26). The restriction is necessary for stabilizing the algorithm. The adaptive tuning factors are updated as follows;

 Q ,t   Q ,t 1 max(min( g Q , g max ), g min ) , (29)  R ,t   R ,t 1 max(min( g R , g max ), g min ) , (30)  Q ,t  max(min( Q ,t ,  max ),  min ) , (31)  R ,t  max(min( R ,t ,  max ),  min ) , (32) where g min  (1  n) 1 ; g max  (1  n) ;  min  0.01 ;  max  100 . The current estimates of Q and R are adjusted with the latest estimates of  Q and  R , respectively.

Qt 1   Q ,t Qt , (33) Rt 1   R ,t Rt , (34)

this means Qt and Rt are decreased or increased by only a relatively small fraction in each update cycle. The latest Q and R are substituted back into analysis stage to update the Kalman gain in (Equation14), observation in (Equation11), and the analysis error covariance in (Equation17), to predict the next time step of the forecast state vector and the forecast error covariance recursively until the last time step.

Data Assimilation with Hybrid Adaptive Ensemble Kalman Filter (HAEnKF) The hybrid technique was introduced to mitigate the impact of the ensemble sampling errors in the EnKF by Hamill and Snyder (2000). The filtering algorithm developed in Hybrid AEnKF is based on the EnKF and optimal interpolation (OI), which is essentially a KF based on a prescribed invariant background covariance matrix schemes in which solute concentration measurements are assimilated through contaminant state variables(Etherton and Bishop, 2004; Hamill and Snyder, 2000; Wang et al., 2009). The hybrid scheme uses a background covariance in the EnKF estimated from an ensemble and a predefined static covariance from an OI system (Hamill and Snyder, 2000).

Like the AEnKF, HAEnKF starts with the traditional EnKF forecast and update step from equation (12) to (17). The background covariance for the hybrid method is determined as a linear combination of the forecast covariance matrix of the ensemble and stationary covariance matrix typically used in an OI assimilation system. The redefined error covariance is shown in equation (35).

PHf ,t  (1   ) Pt f   Pb , (35) wherePb is an invariant background covariance matrix; PHf ,t is the forecast covariance matrix of the hybrid ensemble;

and  is a weighting factor between the two covariance matrices. The logic behind the hybrid technique is to filter those background structures that have large errors relative to more accurately known background features and observations (Lorenc, 1986). Pb is mostly determined from a large inventory of historical forecast errors sampled

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over large windows and is assumed to be invariant in time (Wang et al., 2009; and Etherton and Bishop, 2004).

Empirical orthogonal function (EOF) technique can also be applied in finding Pb(Gharamti et al., 2012; Gharamti and Hoteit, 2013;Song et al., 2010). Parrish and Derber (1992)introduced a method also known as the NMC (National Meteorological Center)method to estimate the background covariance coefficients as averagedforecast difference statistics.This method is adopted in this research and is given as;

Pb  ( xtf  xtr )( xtf  xtr )T   T , (36) Pb   b bT , (37)

where xtr is the true state of the system; the over‐bar represents an average over time and/or space;  is the background error; and  b is a transformation matrix of spectral coefficients derived after LDL decomposition (where L is lower unit triangular matrix and D is a diagonal matrix). The selection of the weighting factors of each EnKF update in most assimilation studies that are based on the hybrid scheme resort to trial and errors techniques (Etherton and Bishop, 2004; Wang et al., 2007). This technique affects the accuracy and computational efficiency because it systematically enumerates all possible candidates for  and checking which candidate provides the better update.

In this research, we adopt the hybrid adaptive scheme to optimize the weighting factor  at every assimilation step to ensure a reasonable reduction of the forecast error variance. The following cost function is maximized at every analysis step of the state filter; f

a

f

f

T

J ( )  tr[ P t  P t ]  tr[ P t  ( I  K t H t ) P t ]  tr[ K t  ( H t U t  ) U t  ] , (38)

U t  [ 1  U t  b ] , (39) where Ut is the weighted state covariance anomaly formed to avoid computing the full covariance matrices;the over‐bar represents the weighted sum; and J ( ) is the objective function which represents the sum of the difference

between the forecast and the analysis variance. The Optimization of  is done using quasi‐Newton technique (Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) algorithm).

The HAEnKF scheme based on the forecast covariance matrix, invariant background covariance matrix and the optimized weighted factor can be performed by the following assigned equations

xtf,i  Axta1,i  wt ,i , (40) PHf ,t  (1   ) Pt f   Pb , (41)



K t  PHf ,t H tT H t PHf ,t H tT  Rt



1

, (42)

xta,i  xtf,i  K t ( zt ,i  H t xtf,i ) , (43)





PHa ,t  I  K t H t PHf ,t , (44)

where PHa ,t is the analysis covariance matrix of the hybrid ensemble; and Kt is the corrected Kalman gain matrix for the HAEnKF scheme.

Experimental Data Setup The model parameters used for the contaminant transport simulation in the subsurface was adopted from previous research done by various researcherson groundwater contaminant modeling (Cheng, 2000; Zou and Parr, 1995; Choi et al., 2005; Li, 2006; Jin and Chang, 2009). A three‐dimensional plane with 2700 grid points (30 x 30 x 3) and grid intervals, dx = 5m, dy = 5m and dz = 5m, in the x, y, and z directions, respectively was defined as the testing environment for the numerical scheme. The hydrodynamic dispersions Dx, Dy and Dz were assumed to be 4.7 m2/day, 3.7 m2/day and 2.6 m2/day, respectively. The first‐order decay rate parameter (k) is set at 0.04/day. The

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Retardation factor R was specified as 1.7 and the linear velocity V was given as 1.8 m/day. A contaminant producing an initial concentration C0 of 15,000 mg/l, is injected into the grid point with coordinate (15, 10, 1). A 35% Gaussian noise was introduced into the process model to create a random noise condition in the modeling process.

The simulated true value was considered to be unknown to the filtering process and it is designed by decreasing the linear velocity of the numerical model by 25%. MATLAB code is formulated and used as the computational tool to carry out the simulation. The total simulation period is 40 days with a time interval of 1 day.

Error Estimation The Normalized Root mean Square Error (NRMSE), Average Absolute Bias (AAB) metric, and Maximum Absolute Deviation (MAD) methods were used as an indicator to measure the performance of all the prediction techniques used in the simulation. Also, the variabilities of the techniques vis‐à‐vis the true solution are ascertained through the calculation of R‐Squared value (R2). R‐squared represents the amount of variation in the outcome that can be explained by the independent variables in the model and is related to correlation coefficient. The effectiveness of the data assimilation techniques and the numerical model is measured by comparing the model predicted results with the truevalue.The RMSE, AAB, MADand R2 are given by the equation as follows

NRMSE (t ) 

1  [CP ( x, y, z, t )  CE ( x, y, z, t )]2 N 1 , (45) C E max ( x, y , z , t )  C E min ( x, y , z , t )

AAB  t    MAD  

R

2

C P  x, y , z , t   C E  x, y , z , t  n

C P  x, y , z , t   C E  x, y , z , t 

 C (t )  1   C

, (46)

, (47)

C E max ( x, y , z , t ) p

( x, y , z , t )  C E ( x , y , z , t ) 

E

( x , y , z , t )  C E ( x, y , z , t )



2 2

, (48)

where CP(x,y,z) and CE(x,y,z) are the predicted contaminant concentrationsand reference true values, respectively in the three‐dimensional plane (x,y,z) for a particular time t; CEmax is the maximum reference true value at time t and CEmin is the minimum reference true value at time t. In this case, the total number of sampling nodes is n; CE is mean reference true value at time t.

Results and Discussions Based on the specified initialand boundary conditions, the numerical model was simulated, while the data assimilation techniques combine observation information to give accurate predictions of the contaminant transport.

Simulations were conducted for all the prediction techniques until the last time stepin quasi‐three dimensional form. In all three layers and at each grid point, simulated results of contaminant concentration were estimated.

Fig. 1 compares the numerical solution, the EnKF, AEnKF and HAEnKF models with the true solution inquasi‐ three dimensional form at time step 40. The space and time steps chosen for the numerical solution satisfy the stability and convergence criteria of peclet number.The contours predicted by the numerical solution lags behind the true contours (due to the velocity used in the estimation) and do not coincide with it. This reveals the fact that the numerical solution does not represent the true field successfully. The poor performance of the numerical model can be attributed to errors associated with it which includes distance‐related dispersion errors which originate from finite difference models, instability, truncation errors, round off errors and other types of numerical errors (Spitz and Moreno, 1996). The hydrogeologic parameters used affect the contour profile prediction. The movement and the spread of the contaminant are as a result of the advection and dispersionterms. Because of the advection, dispersion, decay and sorption in the subsurface porous medium, the contaminant concentration decreases with increase in time as the contaminant migrates from the top to the bottom layer in the assimilation process.

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FIG. 1 CONTOUR PLOT OF CONTAMINANT CONCENTRATION FOR LAYER 1 OF ALL TECHNIQUES AT TIME STEP 40

FIG. 2 MESH PLOT OF CONTAMINANT CONCENTRATION FOR LAYER 3 OF ALL TECHNIQUES AT TIME STEP 20

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In terms of the contour description, the EnKF, AEnKF and HAEnKF models (data assimilation techniques) are able to predict the irregular shapes of the natural stochastic system than the FTCS because the data assimilation techniques employ random Gaussian error and the error covariances in the technique to account for the variability in the model parameters and the heterogeneity in the porous medium, whereas the FTCS does not take these factors into account. In Fig. 2, the data assimilation techniques predicted peak concentrations closer to that of the simulated true value as compared to the numeral scheme in the third layer at day 20. The efficiency of the data assimilation technique relative to the numerical model is as a result of the fact that it uses the observation data to adjust the system model in predicting the concentration of the contaminant. This explains why the EnKF, AEnKF and HAEnKF techniques perform better than the FTCS technique. Error estimation analysis are performed on the schemes using NRMSE, AAB and MAD plot shown in Fig. 3, Fig. 4 and Fig. 5, respectively.

FIG. 3 NORMALISED ROOT MEAN SQUARE ERROR FOR ALL TECHNIQUES AT DIFFERENT TIME STEPS

The essence of using these error statistics is to compare the forecast generated by all the techniques to the simulated true value and assess their performance.The error reductions in the analyses are with respect to the performance of the numerical model. In Fig. 3, the NRMSE plot shows that the maximum error was predicted by the numerical method at all time step with exception of time step 4. At the end of the simulation the numerical recorded 17.9% error. Among the data assimilation techniques, the EnKF was more erractic due to the high level of noise in the system. The EnkF had 5% error at the end of the simulation. The HAEnKF and AEnKF predicted errors in the neighbourhood of 1%, even though the HAEnKF had a marginal advantage from time step 5 to time step 30. Both the HAEnKF and AEnKF performs better than the EnKF, because the former uses weighting factor whereas the later uses an adaptive factor to fine tune the system error covariance, measurement error covariance and the optimal Kalman gain to reduce the noise in the simulation. Validation of the techniques done by AAB metric and the MAD in Fig. 4 and Fig. 5, respectively gave a consistent results at the end of the simulation. From Fig. 4, the error for the the numerical, EnKF, HAEnKF and AEnKF at the end of the assimilation period was 1.48 mg/L, 0.4 mg/L, 0.15 mg/L and 0.1 mg/L, respectively. A further comparison in Fig. 4 indicates that the, EnKF, HAEnKF and AEnKF were successful by reducing the model error by 73%, 90% and 93%, respectively. Relative to the numerical solution, the MAD profile in Fig. 5 shows that the EnKF improved the prediction accuracy by 78%, the HAEnKF achieved error reduction of 89%, whereas the AEnKF improved the prediction by 94%.

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FIG. 4 AVERAGE ABSOLUTE BIAS FOR ALL TECHNIQUES AT DIFFERENT TIME STEPS

FIG. 5 MAXIMUM ABSOLUTE DEVIATION FOR ALL TECHNIQUES AT DIFFERENT TIME STEPS

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To measure how well the reference true values are replicated by the techniques, R square profile is plotted in Fig. 6 to assess the variability at every time step.

FIG. 6 R SQUARE PROFILE FOR ALL TECHNIQUES AT DIFFERENT TIME STEP

The values of R square for the numerical model, EnKF, HAEnKF and AEnKF are obtained for each time step as shown in Fig. 6. At every time step the HAEnKF and AEnKF performs better than the EnKF by approaching unity, whereas the EnKF alternates from 1 to 0.6.Hence, the ability of HAEnKF and AEnKF techniques to account for the uncertainty within the transport medium is high. Although all the data assimilation techniques are guided by the observation in estimating the likely states, the HAEnKF and AEnKF employ a tuning factor (weighting factor and adaptive factor, respectively) mechanism to reduce the error within the modeling. Unlike the data assimilation techniques, the numerical method which is deterministic propagates the error within the model with time because it is not guided by the observation in estimating the states. Sensitivity analyses were conducted to examine the impact of the adaptive and weighting factorson the model predictions by using the AAB metric. The performance of the adaptive factor in the AEnKF was investigated by setting it at 1, 0.7, 0.3 and 0.1, whereas the effect of the weighting factor on the performance of the HAEnKF was set at 0.1, 0.4, 0.8, and 1. The trend in Fig. 7 suggests that decreasing the adaptive factor does not degrade the performance and stability of the AEnKF. At time step 6, adaptive factor 1.0 had a prediction error of 0.7 mg/L, adaptive factor 0.7 had a prediction error of 0.63 mg/L, adaptive factor 0.3 had a prediction error of 0.55 mg/L, whereas that of 0.1 recorded 0.8 mg/L prediction error.

After time step 10, all the adaptive factors converge to give the same results. This suggests that irrespective of the adaptive factor value selected to initialize the simulation process, the AEnKF is likely to give a similar results at the last time step. Fig. 8 shows the effect of weighting factor values on the performance of the HAEnKF. Weighting factor 1.0 had the highest prediction error of 0.17 mg/L and very unstable until time step 25. Weighting factor 0.8 and 0.1 both had a prediction error of 0.15 mg/L at the end of the simulation even though there some tradeoff at some time steps. Using a weighting factor of 0.4 improved the performance of the HAEnKF with a prediction error of 0.12 mg/L. The investigation shows that the weighting factor selected to initialize the modelling process has effect on the prediction accuracy of the HAEnKF. The simulations were performed using MATLAB R2015a on a computer with 2.93 GHz processor and 8.0 GB RAM.

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FIG. 7 SENSITIVITY ANALYSIS ON THE ADAPTIVE FACTOR

FIG. 8 SENSITIVITY ANALYSIS ON THE WEIGHTING FACTOR

In Fig. 9 and Fig. 10, the influence of the ensemble size on the AEnKF and the HAEnKF respectively, were investigated.

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FIG.9 COMPARISON OF AEnKF PERFORMANCE WITH DIFFERENT ENSEMBLE SIZES.

FIG.10 COMPARISON OF HAEnKF PERFORMANCE WITH DIFFERENT ENSEMBLE SIZES.

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The trend in the result suggests that increasing the ensemble size tends to improve the AEnKF and the HAEnKF performance and stability. As shown in Fig. 9, an ensemble size of 20 had the highest prediction error whereas ensemble size of 50, 80 and 100 were characterized with a lower prediction error. But beyond ensemble size of 80, the ensemble size has no greater impact on the adaptive factor use in tunning the AEnKF. In Fig. 10, an ensemble size of 100 had the least prediction error of 0.004 mg/L whereas ensemble size of 20 was characterized with an unstable profile and a much higher prediction error of 0.018 mg/L. Also, ensemble size of 80 had better prediction accuracy than that of 50. This shows that using a large ensemble size may affect the optimality of the HAEnKF. Conclusion In this research, conventional EnKF, HAEnKF and AEnKF were used as data assimilation techniques for predicting contaminant concentration in a three‐dimensional subsurface porous environmentwith respect to time andspace.

The numerical solution was derived by discretizing the three‐dimensionalcontaminant transport modelin partial differential formusing FTCS. The numerical method which is deterministic scheme assumes homogeneity and constant parameters in generating results whereas the data assimilation techniques take into consideration predictionand the correction in generatingthe likely states. A comparison is made between contaminant transport predictions by using the EnKF, HAEnKF and AEnKF. The proposed HAEnKF and AEnKF, take advantage of the weighting and adaptive factor technique, respectively, to minimize the residuals between the estimated states and the simulated true value. From the error estimation analysis performed on the techniques, the simulation results show that the prediction deviation of the numerical model was significantly higher than that of the EnKF, HAEnKF and AEnKF methods.

The NRMSE recorded 17.9%, 5%, 1% and 1% error for the numerical, EnKF, HAEnKF and AEnKF, respectively at the end of the simulation in predicting the simulated true value. At the end of the simulation period, the AAB analysis shows that the EnKF, HAEnKF and AEnKF were successful in reducing the error prediction in the numerical by 73%, 90% and 93%, respectively whereas the MAD analysis indicates that the implementation of the EnKF, HAEnKF and AEnKF methods successfully improved the accuracy of the numerical model by 78%, 89% and 94%, respectively. The R square profile plotted to account for the uncertainty within the subsurface medium shows that the data assimilation techniques have the ability to give a more accurate prediction than the numerical model.

The sensitivity analysis performed on the adaptive factor suggest that, the adaptive factor specified to start simulation does not affect the prediction accuracy of the AEnKF whereas that of the weighting factor has influence on the HAEnKF. ACKNOWLEDGMENT

This work was sponsored by the Department of Energy Samuel Massie Chair of Excellence Program under grant number DE‐NA0000718. The views and conclusions contained herein are those of the writers and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the funding agency.

NOTATION

The following symbols are used in this paper A

=

state transition matrix (STM);

bt

=

innovations;

b

=

transformation matrix of spectral coefficient;

C

=

concentration of contaminant;

C0

=

initial concentration;

Dx, Dy, Dz

=

dispersion coefficient in the x, y, and z direction respectively;

dx, dy,dz

=

grid interval in x, y, and z coordinates respectively;

et

=

analysis increments;

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ɛt

=

observation error vector at time t;



=

background error;

H

=

measurement sensitivity matrix;

=

distribution coefficient;

K

=

ensemble Kalman gain matrix;

Kt

=

corrected Kalman gain matrix;

k

=

first order decay rate;

N

=

number of samples;

ɳ

=

porosity;

 

=

tuning parameter;

=

adaptive factor;

=

invariant background covariance matrix;

PHf ,t

=

forecast covariance matrix of the hybrid ensemble;

Pt f

=

forecast covariance matrix;

Pt a

=

analysis error covariance matrix;

Qt

=

process model covariance matrix at time step t;

R

=

retardation factor;

Rt

=

observation covariance matrix at time step t;

S

=

sorbed contaminant concentration;

st

=

analysis departures;

t

=

time in days;

U

=

state covariance anomaly;

Ut

=

weighted state covariance anomaly;

V

=

linear velocity;

x, y, z

=

Cartesian coordinates

ρb

=

bulk density;



=

weighting factor

wt

=

model system error at time t;

xt

=

concentration state vector at all nodes at time t;

xt‐1

=

concentration state vector at all nodes at time t‐1;

xta,i

=

analysis ensemble of concentration vector at time t;

xtf,i

=

forecast ensemble of the state vector at time t;

x t ,i

=

forecast mean;

zt

=

observed state vector at all nodes at time t;

Kd e t

Pb

f

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