Transp Porous Med (2012) 91:133–152 DOI 10.1007/s11242-011-9837-3
Using an Ensemble Kalman Filter Method to Calibrate Parameters and Update Soluble Chemical Transfer From Soil to Surface Runoff Ju-Xiu Tong · Bill X. Hu · Jin-Zhong Yang
Received: 19 June 2010 / Accepted: 25 August 2011 / Published online: 15 September 2011 © Springer Science+Business Media B.V. 2011
Abstract A data assimilation method, an ensemble Kalman filter (EnKF), is applied to simultaneously calibrate parameters and update prediction for soluble chemical transfer from soil into surface runoff. The soluble chemical transfer is calculated using a two-layer analytical model with constant parameters, h mix (water depth of the soil-mixing layer), α and γ (surface runoff and infiltration-related incomplete mixing parameters). The model is presented as the forward model. Based on laboratory experimental results, the measured chemical concentrations in the surface runoff are assimilated into the calculation through the developed EnKF method to calibrate the parameters and update chemical concentration in the runoff. In comparison with the calculation without data assimilation method, the updated solute concentration results are much closer to the experimental observed data and the calibrated parameters, h mix , α and γ , are no longer constants, but time dependent, which are physically reasonable. The study results indicate that the EnKF method significantly improves the prediction for solute chemical transfer from soil into surface runoff, whereas the extended Kalman filter will not, and the ensemble size of 100 will be suitable for the chemical concentration calculation based on our trial. Keywords Chemical transfer from soil to surface runoff · Incomplete parameter · Data assimilation · EnKF · Analytical solution
J.-X. Tong · B. X. Hu (B) China University of Geosciences, 29 Xueyuan Road, Beijing 100083, People’s Republic of China e-mail:
[email protected] J.-X. Tong · B. X. Hu Department of Earth, Ocean and Atmospheric Sciences/Geological Sciences, Florida State University, 108 Carraway Building, Tallahassee, FL 32306, USA J.-X. Tong · J.-Z. Yang State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, People’s Republic of China
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1 Introduction Chemical loss from soil to surface runoff during rainfall and irrigation periods decreases the efficiency of the applied chemical (e.g., a pesticide or a fertilizer for agricultural soil) and also deteriorates the surface water quality. It is essential to use physically based accurate modeling of soil chemical transfer into runoff for management of non-point source pollution in farming land (Gao et al. 2004, 2005; Hesterberg et al. 2006; Mulqueen et al. 2004; Russo 1991; Wallach et al. 2001; Yoshinaga et al. 2007). The mixing layer concept is probably the most commonly used model for chemical transport from soil to overland flow (Ahuja et al. 1981; Ahuja and Lehman 1983; Emmerich et al. 1989; Steenhuis and Walter 1980; Tong and Yang 2008; Zhang et al. 1997, 1999). This theory assumes that there exists a region below soil surface, where surface water, soil solution and infiltrating water are assumed to mix completely and instantaneously. There is no chemical transfer into that region, and the mixing layer depth is constant. Tong et al. (2010) have applied the incomplete mixing theory to develop a two-layer analytical model for soluble chemical transfer from soil to surface runoff in an initially unsaturated soil, considering infiltration and a certain depth of ponding water on soil surface before the surface runoff. Their study results indicate that the analytical model is accurate and reliable, but they do not take into account observation error. All uncertainty sources, including parameters and output observations, need to be considered for improvement of forecast capability (Beven and Freer 2001; Wang et al. 2009). In recent years, newly developed sequential data assimilation techniques provide an integrated framework for analyzing all sources of uncertainties (Chen and Zhang 2006; Gabriëlle et al. 2007; Huang et al. 2009; Weerts and El Serafy 2006). The standard Kalman filter (KF) and its extensions are popular tools. The traditional KF is an efficient sequential data assimilation method for linear dynamics and measurement processes with Gaussian error statistics (Kalman 1960; Drecourt 2003; Drecourt et al. 2006; Tipireddy et al. 2008; Yangxiao et al. 1991; Zhang et al. 2007). To assimilate data for non-linear dynamics and measurement processes, the extended Kalman filter (EKF) was developed (Jazwinski 1970; Miller et al. 1994). However, the EKF is very unstable if the nonlinearities are strong. Furthermore, this method is not computationally feasible for large-scale environmental systems (Neef et al. 2006). To overcome these limitations, the ensemble Kalman filter (EnKF), in its native formulation as originally introduced by Evensen (1994) and Burgers et al. (1998), uses pure Monte Carlo sampling when generating the initial ensemble, the model noise and the measurement perturbations, where the ensemble of model state evolves in state space with the mean as the best estimate and the spreading of the ensemble as the error. EnKF has gained popularity because of its simple conceptual formulation and relative ease of implementation, e.g., it requires no derivation of a tangent linear operator or adjoint equations, and no integrations backwards in time (Evensen 2006; Franssen and Kinzelbach 2009). In hydrology and water resources, the EnKF has become more popular than KF and EKF. The method has also been successfully applied to soil moisture retrieval and estimation (Reichle et al. 2002), snow cover prediction (Andreadis and Lettenmaier 2006; Su et al. 2008), streamflow simulation (Clark et al. 2006), parameter estimation (Moradkhani et al. 2005), and flood forecasting (Weerts and El Serafy 2006). Recent developments in EnKF methods are no longer limited to updating system state variables only as in conventional data assimilation, it can simultaneously update state variables and calibrate model parameters to yield more accurate model predictions. Models usually have relevant physical laws or settings, but in sequential data assimilation techniques such as the KF and the EnKF, the state variables are updated by minimizing the mean square error rather than following physical principles, which may result in the violation
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of physical constraints. To solve the problem, many methods have been proposed (Wang et al. 2009). In this study, based on the analytical two-layer model of Tong et al. (2010) for predicting soluble chemical transfer from soil to surface runoff, we develop an EnKF method to update the system state variable, the chemical concentration of surface runoff and calibrate model parameters simultaneously through assimilating the observed data. This is the first time, from our review of the literature, that EnKF method is applied as a data assimilation tool to a model for chemical transfer from soil to runoff. In Sect. 2, we briefly introduce the analytical two-layer model of Tong et al. (2010) and corresponding analytical solutions for the sake of completeness. In Sect. 3, a data assimilation method based on EnKF is developed for the solute transfer from soil to surface runoff, where the solute concentration in the runoff is the state variable and the h mix , α and γ (mixing soil depth, surface runoff and infiltration-related parameters) are the parameters. In Sect. 4, the developed method is applied to a synthetic case to show how significant improvement in the model state variable and parameter calibration the EnKF method can do. Some constraints to the EnKF based on the physical meanings of the parameters are also added. In Sect. 5, the EnKF method is applied to analyze our former experimental results through calibrating the model parameters and improving the prediction for the soluble chemical transfer from soil to surface runoff by assimilating the measured data, and the results by the EKF method are also compared with that of the EnKF method based on some criterion. Finally, summary and conclusions are provided in Sect. 6.
2 Analytical Model and Solution The two-layer model is described in detail in Tong et al. (2010), we only briefly introduce it here for the sake of completeness. As shown in Fig. 1, the surface runoff system is composed of two layers, the whole-mixing layer and soil layer. The whole-mixing layer includes the soil-mixing zone and the surface ponding-runoff zone. The ‘net’ chemical flux from the soil-mixing zone into the underlying soil layer is expressed as, iγ Cw (γ ≤ 1), where i is the water infiltration flux (cm/min), Cw is the chemical concentration in the soil-mixing zone (μg/cm3 ). Here, we should point out that the Cw is a function of time and γ is the percentage of the ‘net’ chemical flux after abstracting the upwards mass diffusion. An incomplete mixing parameter α(0 < α) was adopted to describe the incomplete solute mixing in the pondingrunoff zone, so the solute concentration in this zone is αCw . To simplify the complicated chemical transport processes near the soil surface, the chemical concentration in each zone is assumed to be uniform, but different from one zone to another.
p the pondingrunoff zone
q αCw
the soil mixing zone
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Fig. 1 Sketch of the simple two-layer model
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In the whole-mixing layer, according to mass conservation, we obtain the following equation: Mw = Cw [α (h w − h mix · θs ) + h mix · θs ] ,
(1)
where Mw is the soluble chemical mass per unit area in the water phase (μg/cm2 ), h w is the water depth in the whole-mixing layer (cm), h mix is the depth of the soil-mixing zone (cm) and θs is the saturated volumetric water content in the soil-mixing zone (cm3 /cm3 ). If we assume that the solute chemical concentration in the rainfall water is zero, we can also obtain the following equation according to mass conservation: d [Mw (t)] (2) = −γ · i · Cw (t) − α · q · Cw (t) , dt where q is the specific discharge rate of the overland flow (cm/min) and t is the time (min). Equations (1) and (2) provide the mathematical modeling of mass conservations in the ‘static’ and kinetic conditions in the whole-mixing layer. The developed mathematical relationships are experimentally and numerically studied in the following. At the first time period, there is no ponding water on the soil surface, so the infiltration rate of soil was equal to the rainfall intensity p (cm/min). The initial saturated concentration was C0 , thus the Eq. (2) can be solved as γ · p · (t − tsa ) t0 < t < tp , Cw (t) = C0 · exp − (3) h mix · θs where t0 is the start time of rainfall, which is considered to be 0 (min); tsa is the time that the mixing zone will be saturated and tsa = h mix (θs − θ0 ) / p (min); tp is the ponding start time (min). After the water begins to pond on the soil surface, the analytical solutions are presented for different periods, which are described in detail in Tong et al. (2010), we only rewrite the main equations here. During the time from the ponding start (tp ) to the runoff (tr ), the average infiltration rate of soil was assumed to be i 1 , so the increasing depth of ponding water was h p (t) = ( p − i 1 ).(t − tp ). The analytical solution for the soluble chemical concentration of the surface runoff was: −γ ·i1 α · ( p − i 1 ) · t − tp + h mix · θs α·( p−i1 ) tp ≤ t < tr , Cw (t) = Cw tp · (4) h mix · θs where Cw (tp ) is the initial ponding concentration by substitute tp to Eq. 3. During the transition period between the start of runoff (tr ) and steady runoff (ts ), the average infiltration rate of soil was assumed to be i 2 , and the depth of ponding water was constant as h p (tr ). Initial concentration of mixing soil Cw (tr ) could be obtained by substituting tr in Eq. 4; therefore, the soluble chemical concentration in runoff was presented as: −γ · i 2 − α · ( p − i 2 ) tr ≤ t < ts (5) α · Cw (t) = α · Cw (tr ) · exp · (t − tr ) α · h p (tr ) + h mix · θs After the runoff became steady, infiltration rate was constant, i s in this period. Similarly, initial concentration of runoff (αCw (ts )) could be obtained according to Eq. 5, and the soluble chemical concentration in runoff was: −γ · i s − α · ( p − i s ) t > ts · (t − ts ) α · Cw (t) = α · Cw (ts ) · exp (6) α · h p (tr ) + h mix · θs
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The times tr , tr + t1 , tr + t1 + t2 , . . ., tr + t1 + t2 + · · · + tm are the different measurement times, and the corresponding incomplete parameters are γ j and α j , ir j are infiltration rates at the time stage, t j ( j = 1, 2, 3, . . . , m). We can get the solution for the time step tm as follows: C (tr + t1 + t2 + · · · + tm−1 ) C (t) = · αm αm−1 −γm · ir m − αm · ( p − ir m ) · exp · (t − tr − t1 − t2 − · · · − tm−1 ) αm · h p + (h mix )m · θs (7) tr + t1 + t2 + · · · + tm−1 < t ≤ tr + t1 + t2 + · · · + tm−1 + tm . We can only get the observed concentration data after runoff, so we assume that the parameters of h mix , α and γ are constant as h mix0 , α0 and γ0 before the surface runoff. In this study, we use the solute concentration data to calibrate the parameters and update the prediction through a data assimilation method.
3 Data Assimilation Method The data assimilation algorithm used in this study is the EnKF method, which uses observations of soluble chemical concentration in surface runoff to update the model state variables and parameters used in the analytical model. A data assimilation system is composed of a model operator, observation operator and a data assimilation algorithm. In this study, the model operator is the analytical model. The observation operator is used to build the relationship between model state (chemical concentration) and observations. The observations are soluble chemical concentration data of surface runoff. The observation is the chemical concentration in the surface runoff, which is the same as the model state. There are two major computational components involved in the EnKF method, forward forecast and data assimilation. The forecast procedure predicts the values of dependent variables for any particular sets of model parameters. The assimilation procedure combines and reconciles the information coming from the forecast step, and the observations such that the least squares estimation of the state vector is achieved (Evensen 2003). Before the start of the EnKF assimilation, a number of initial realizations must be generated for all uncertain model parameters. In the forward forecast step, forecasted model variables of each ensemble member are updated according to f
a X i,t+1 = M(X i,t ) + ui
u i ∼ N (0, Q),
(8)
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where X i,t+1 is the forecasted model variable (including chemical concentration and parama is the analyzed model variable of eters) of the ith ensemble member at time t + 1; X i,t the ith ensemble member at time t; M(.) is model operator, which is the analytical model; u i is the model error vector, which is assumed to satisfy Gaussian distribution with zero mean and covariance matrix, Q. Here we do not consider the model error. Let P f and P a denote the prediction-error and analysis-error covariance matrix, respectively, then
T T (9) P f = xf − xt xf − xt ∼ = Pef = x f − x f x f − x f , T T Pa = xa − xt xa − xt ∼ = Pea = x a − x a x a − x a ,
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Prediction Model
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Fig. 2 Flowchart of the EnKF
where the overbar denotes the average over the ensemble; the superscripts f, a and t represent forecast, analyzed and true states, respectively, and ensemble mean values are considered as true values. In the analysis step, the observation data are perturbed by adding random observation errors, and they can be expressed as follows f
Y = Yt+1 + εi = X i,t+1 + εi
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(11)
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where Yt+1 is the observation data at time t + 1, X i,t+1 is the mean of forecast state vector of ensemble members at time t+1 and εi is random error vector of observation with zero mean and covariance matrix R. The EnKF updates each ensemble member according to (Huang et al. 2009):
−1 f a f f f X i,t+1 = X i,t+1 + Pt+1 H T H Pt+1 HT + R , (12) (Yt+1 + εi ) − H X i,t+1 where H (.) is the linear observation operator used to convert the model state variables to f observations and Pt+1 is the forecast background covariance matrix at time t + 1. The analysis state estimate at time t + 1 is given by the mean of the ensemble members. In comparison with the commonly used inverse method (e.g., general least square and maximum likelihood methods), the EnKF can sequentially adjust system estimate, without reprocessing existing data when new observations become available (Huang et al. 2009). Every time we get the updated parameters through EnKF, we should make sure that the parameters conform to the physical law, such as α > 0, γ ≤ 1 or h mix > 0. If the updated data violate the physical law, such as α ≤ 0, γ > 1 or h mix ≤ 0, then we give some constraints such as α = 0.0001, γ = 1 and h mix = cons (cons is a small positive value) to make the parameters physically meaningful. The calculation procedure of the constraint EnKF and the process is shown in Fig. 2. Based on the EnKF method, the EKF method can be easily executed (Ribeiro 2004).
4 Verification of Algorithm A synthetic one-dimensional case is used to test the data assimilation method developed above. The forward model, the analytical solutions introduced in Sect. 2, is applied with known parameter values to generate a reference (or real) solute transfer process, from which we obtain the real solute concentrations in the runoff. The soil in this case is initially unsaturated and all the used parameter values are presented in Table 1. The real concentration
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Table 1 Model parameters used in the synthetic unsaturated soil
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values are perturbed by white noises and served as the observed concentration values. In this study, the noises have a mean equal to 0, which indicate unbiased observations, and the standard deviation is 1% of the concentration measurements. Because the synthetic case is one-dimensional, the average of the root mean square error (RMSE), the quantity Di f (mg/l) during the whole simulation time, is used to quantitatively analyze the error between the true concentrations and the simulations, M
M 2 1 1 f Di f = ci − cit = RMSE2 (13) M M i=1
i=1
where Di f is the criteria for the accuracy of the simulation, M is the total number of the f observed soluble chemical concentration data in the surface runoff and ci and cit stand for the simulated and true or observed chemical concentration of the surface runoff, respectively, at the ith observation time. For one-dimensional study case, another measure of the goodness is the Ensemble Spread, which represents the estimated uncertainty based on the ensemble: Nens 2 1 Ensemble spread = VAREn = (14) cj − c , Nens j=1
where VAREn denotes the ensemble variance at one time, Nens is the number of ensemble size, c j is the analysis of the jth ensemble at one time and c is the corresponding ensemble mean of surface concentration. As shown in Table 1, the true values of the parameters h mix , α and γ for the initially unsaturated soil are 1.5, 0.6 and 0.4, respectively. The ensemble size number is chosen to be 100. We calculate the chemical concentrations in surface runoff by the initial ensemble values without and with the constraint EnKF under three initial guess parameter values and compare the calculation results with the observations. In the first case, the initially guessed mean values of h mix , α and γ after surface runoff are chosen to be the same as their true values and the error variances are 1% of their corresponding ensemble mean values. Hereafter, we will name the observation error ratio as the value of the observation error variance to the corresponding ensemble mean value, so the observation error ratio is 1% in this case. The calculated results without and with the constraint EnKF and the real data are shown in Fig. 3a. From Fig. 3a, the calculated results through the two approaches seem both very close to the observation. However, when we do the quantitative analysis, the Dif for the simulation without data assimilation method is 4.59 mg/l, whereas the value is 1.54 mg/l for the simulation with EnKF method, which suggests that the data assimilation method significantly improves the prediction even though the improvement seems trivial in the figure. In reality, we do not know the true values of the parameters. In the second case, we arbitrarily choose the initially guessed mean values of h mix , α and γ to be 1.0, 0.8 and 0.6 cm respectively. Figure 3b presents the calculated results before and after data assimilation as well as the real observations. From Fig. 3b, we can see that the prediction of soluble chemical concentration in the surface runoff without data assimilation method is much higher than the true data at the early time of the surface runoff, whereas the prediction is much lower than the true data at the late time of the surface runoff. The simulated soluble chemical concentration of the surface runoff by assimilating the observed data via EnKF is much better than the prediction without data assimilation method in Fig. 3b. The prediction with data assimilation
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Fig. 3 Comparison of prediction without and with data assimilation method to the true data and prediction with initial ensemble mean as a h mix = 1.5 cm, α = 0.6, γ = 0.4, b h mix = 1.0 cm, α = 0.8, γ = 0.6 and c parameters vary with time for the theoretical initially unsaturated soil using the EnKF method of b. DA represents data assimilation
method slightly deviates from the true data at the early time, but the calculation results overlap with the true data after several time steps of assimilating the observed data. The Dif for the prediction without and with data assimilation method is 19.65 and 4.79 mg/l, respectively. Figure 3a and b indicates that the EnKF can efficiently improve the prediction. Figure 3c presents the variation of the updated parameters h mix , α and γ with time under the initial guesses of their ensemble means as 1.0, 0.8 and 0.6 cm respectively. All three parameters decrease at the first time, and then increase with time. In the above study, the ensemble size is chosen to be 100, and the error variance is 1% of the mean value. Here, we conduct a sensitivity study on the two factors to investigate the effects of their variations on the solute prediction accuracy. Instead of just using 100 realizations for the data assimilation method, several ensemble sizes are selected in the data assimilation method to calculate solute concentration and Dif values. It is shown in Fig. 4a that the Dif initially decreases with the increasing ensemble size, which means the prediction becomes better with the increase of the ensemble size. However, when the ensemble size increases to 100, further increase of the ensemble size hardly affects the Dif and the Dif almost becomes constant. Therefore, in this study, the ensemble size is chosen to be 100 as described in the following. To investigate the effect of observation error on the solute transfer prediction, several values of observation error variance (relative to its real value), ranging from 0.001 to 0.5, are chosen for the solute transfer calculation, and the calculated Dif are shown in Fig. 4b. From Fig. 4b, we can see that the Dif gradually increases with the increase of the observation error
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Fig. 4 Influences of different factors on the prediction a ensemble size and b observation error ratio to ensemble mean
variance before the variance reaches 0.4; however, after that number, the Dif dramatically increases with the increase of the variance, which means the calculation error is out of control. Because the solute concentration chemical analysis is generally accurate, the relative observation error variance is chosen to be 0.01 or 1% of the ensemble mean value in the following study.
5 Application of the EnKF to the Real Experiments In this section, we apply the data assimilation method developed above to analyze our previous experimental results (Tong et al. 2010). In the experiment, sand and loam are the two types of soils studied. We apply the EnKF to improve the prediction of the chemical KCl concentration in the surface runoff from the two soils with initially unsaturated and saturated conditions, respectively, and calibrate the corresponding parameters h mix , α and γ . We assimilate the observation data whenever they are available. The parameters for these soils are presented in Table 2. The cases 1 and 2 are for the experimental sand soils, whereas the cases 3 and 4 are for the experimental loam soils. 5.1 Experimental Sand Soil 5.1.1 Initially Unsaturated Experimental Soil The experimental sand is initially unsaturated in case 1. We have found that the optimal parameters of h mix , α and γ were constant and their values were 1.5, 1.0 and 0.7 cm respectively, through the whole simulation in Tong et al. (2010). Here, we still assume these parameters as 1.5, 1.0 and 0.7 cm before the surface runoff. After the surface runoff, the parameters vary and their values are unknown, so we arbitrarily select the initial guessed ensemble mean of the parameters h mix , α and γ as 1.0, 0.8 and 0.5 cm respectively. The EnKF is applied to this experimental soil with the two constraints: (1) When the updated parameter h mix ≤ 0.01, we let h mix = 0.01 and (2) the other two parameters have to satisfy the conditions, α > 0 and γ ≤ 1. The KCl concentration of the surface runoff is calculated through the analytical solution introduced in Sect. 2, and the updated predictions by assimilating the observation data are shown in Fig. 5a, and the calibrated values of h mix , α and γ are presented in Fig. 5b, and the corresponding RMSE and ensemble spread are presented in Fig. 5c and d.
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Table 2 Parameters used in the experimental soil and model
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Fig. 5 Comparisons and parameters for the initially unsaturated experimental sand soil of case 1 a comparison of the analytical solution and the prediction by the EnKF with the observations, b variation of the parameters α, γ and h mix produced by the EnKF, c RMSE and spread by the EnKF and d ensemble spread by the EnKF
From Fig. 5a, we can see that the updated prediction by the EnKF is much closer to the observed KCl concentration of the surface runoff than that by the analytical solution with constant parameters, especially for the high KCl concentration during the early time of the surface runoff. Moreover, the high KCl concentration in the early surface runoff plays an important role during the KCl solution loss from the soil into the surface runoff (Tong et al. 2010), which suggests that the result produced by the EnKF is much better than that by the analytical solution without calibration. This is because the prediction by the EnKF assimilates the observations to the model, calibrates the parameters and updates variables through every assimilation step. The calibrated parameters are no longer constant and vary with the assimilation steps; whereas the prediction by the analytical solution without data assimilation uses the constant values. Furthermore, the Dif of the prediction by the analyt-
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ical solution and the EnKF are 6.31 and 1.88 mg/l, respectively, which further indicates the significant improvement with the data assimilation method. We also calculate the inversed surface runoff concentration produced by the EKF method (Ribeiro 2004), and the corresponding Dif is 149.62 mg/l, which is far greater than that of the analytical solution and the EnKF method. The corresponding concentrations are also far away from the observations, so we do not present the inversed concentrations and parameters obtained through the EKF method here. This result suggests that the EKF method is not suitable to the study case. In Fig. 5b, the parameter h mix is constrained to be 0.01 cm by EnKF at all the simulated time steps. If we do not pose any constraint on h mix in the assimilation process, h mix could become negative in some time steps making the prediction mathematically optimal, but it would violate the physical reality. This phenomenon can be explained as that the KCl solution concentration in the soil-mixing layer decreases with the time and generates concentration gradient along the soil-mixing depth. To reduce the gradient, more chemicals need to move into the mixing zone across the bottom of the soil-mixing layer. The thinner the depth of the soil-mixing layer, the shorter path and sooner the chemicals can move into the mixing zone from bottom of the mixing layer. So the value of h mix becomes as small and becomes limited to the value of the constraint. The infiltration-related incomplete parameter γ is negative all the time, which means that the upward diffusion of the KCl solution into the soil-mixing layer is greater than the amount of the KCl solution leached out the bottom of the soil-mixing layer by the downward infiltration after the surface runoff starts. Furthermore, the parameter γ decreases with time, and the absolute value becomes larger. This is because that the KCl in the soil-mixing layer continuously decreases with the positive downward loss of the KCl solution by the infiltration water before the surface runoff occurs. This will increase the difference of the KCl concentrations between the soil-mixing layer and the soil below, so the upward diffusion also increases with time. The runoff-related incomplete mixing parameter, α, increases with the time. This can be explained as that the KCl solution in the soil-mixing layer is assumed to be the unique source of the KCl concentration in surface runoff, so the concentration of the KCl solution in the soil-mixing layer becomes very small after a long time of the surface runoff. However, there is upward diffusion of KCl solution into the soil-mixing layer from the bottom, and the KCl concentration in the soil-mixing layer is larger than the original assumed one, which leads to the greater difference of the KCl concentration between the soil-mixing layer and the surface runoff and more diffusion of the KCl solution from the soil-mixing layer to the surface runoff. So α increases and KCl concentration in surface runoff is much larger than the initially guessed value. Because we still use the assumed value of the KCl solution concentration in the soil-mixing layer, the diffusion of the KCl concentration into the surface runoff is even much more than the assumed concentration, this is why a value of α is even greater than 1 after the surface runoff is shown in Fig 5b. From Fig. 5c, we can see that although the RMSE increases at some assimilation steps, it has the trend to decrease during the whole assimilation time. Comparing RMSE with the spread, it is found that the ensemble spread systematically underestimate the discrepancy between mean and the observations as well as the underlying variation of the realizations around the ensemble mean. The spread decreases with the assimilation time, as shown in Fig. 5d. The mean of the ensemble spread during the whole simulation time is 0.024 mg/l, which is much smaller than the Dif (1.88 mg/l). This is maybe caused by the small ensemble size. However, if both the accuracy and efficiency are considered, the ensemble of size 100
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seems reasonable for estimating the mean field in this study case, which is also described in Sect. 4. 5.1.2 Initially Saturated Experimental Soil The experimental sand is initially saturated in case 2. In Tong et al. (2010), the optimal parameters of h mix , α and γ were 0.1, 0.134 and 1.0 cm, respectively, and kept unchanged in the whole process. The parameter values are assumed to be same as those before the surface runoff starts. We select the initial ensemble mean guessed values of h mix , α and γ to be 0.1, 0.134 and 1.0 cm, respectively. The value of h mix should not be less than 0.001 cm during the data assimilation process. The predicted results of the KCl concentration (in log form) in the surface runoff by the analytical solution with constant parameter values and by the EnKF method are presented in Fig. 6a. The updated values of h mix , α and γ through the EnKF are shown in Fig. 6b. The RMSE and spread by the EnKF method are shown in Fig. 6c. In Fig. 6a, both the analytical solution and the prediction by the EnKF agree quite well with the observations. Because the observed data of KCL solution concentration in the surface runoff are very large at the early time, but become very small at the later time of the surface runoff, which vary in many orders of magnitude from the early time to the late time of the surface runoff. Thus, we plot KCl solution concentrations with log form in the surface runoff (Fig. 6a). From the KCl concentrations in the later time shown in Fig. 6a, we can see that the
(b) 112
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analytical solution results significantly deviates from the observed data, and the prediction by the EnKF is almost identical to the observed data. Moreover, the values of Dif for the prediction by the analytical solution and the EnKF are 23.24 and 20.31 mg/l, respectively, which indicates that the EnKF slightly improves the prediction of the KCl solution transfer from the soil into the surface runoff. The Dif of the concentration produced by the EKF method is 1006.52 mg/l, which means the prediction is very far away from the observation. Same as we did in Sect. 5.1.1, we also do not present the results by the EKF method. The calculation results indicate that the EKF method is not suitable to our study case. In Fig. 6b, γ is a constant of 1 using the EnKF, which is the maximum value of the parameter. This is because of the large infiltration rate (0.032 cm/min in Table 2) during surface runoff, in comparison with the value of the infiltration rate for case 1. The upward diffusion of the KCl solution into the soil-mixing layer is ignored with this amount of KCl solution leached by the downward infiltration. The soil-mixing depth h mix increases with the time. This is because the downward infiltration of the KCl solution loss to the soil-mixing layer below and the incomplete upward diffusion of the KCl solution transfer to the surface runoff results in the KCl solution in the original soil-mixing layer decreasing continuously. There is no other source of the KCl solution from the bottom of soil-mixing layer, so the model extends the depth of the soil-mixing layer to include more mass of KCl solution to supply the KCl solute concentration in the surface runoff. The runoff-related incomplete parameter, α, increases with the time. The reason is same as that given for Fig. 5b, the KCl solution in the soil-mixing layer decreases with time if h mix is constant. However, h mix increases with time, and the concentration of the KCl solution in the new soil-mixing layer is more than initially present. So the KCl concentration difference between the new soil-mixing layer and the surface runoff becomes greater than the originally assumed one, which leads to the increase of α with the time as shown in Fig. 6b and the greater KCl concentration of the surface runoff than the initial value occurs at a later time for the surface runoff as shown in Fig. 6a. The RMSE and the spread of concentration through the EnKF method are presented in Fig. 6c, and they are almost the same as those in Fig. 5c. Both of them decrease with time, although the RMSE increases at some assimilation steps. However, the mean spread for the whole simulation time is 0.51, which is smaller than the RMSE. This result is consistent with that obtained by Chen and Zhang (2006) for groundwater flow in a saturated medium. From the above study results, we conclude that the EnKF method was successfully applied to calibrate parameters and update the KCl solution transfer from the soil into the surface runoff with both the initially unsaturated and saturated experimental sand soils. The calibrated parameters are no longer constant, but process-dependent, which is meaningful physically. 5.2 Experimental Loam Soil 5.2.1 Initially Unsaturated Experimental Soil The experimental loam is initially unsaturated in case 3. Parameters h mix , α and γ were considered to be constant as 0.44, 0.88 and 0.71 cm, respectively, before the surface runoff, and they are also chosen to be the initial guessed ensemble mean values for the parameters. Under the constraints of h mix ≥ 0.01 cm, α ≥ 0.0001 and γ ≤ 1, the prediction by the EnKF and the observed data of the KCl concentration in the surface runoff from the experimental
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soil is presented in Fig. 7a and the corresponding parameters for the EnKF are shown in Fig. 7b. If the values of the constant parameters h mix , α and γ are chosen arbitrarily, the concentration from the analytical solution decreases with time and the modeled concentrations obviously deviate from the observations. However, we still plot the predicted results of the KCl concentration in the surface runoff by the analytical solution with constant parameter values in Fig. 7a, and the constant values are the same as the initial guess ensemble mean values shown in Table 2. As shown in Fig. 7a, the calculated KCl concentration in the surface runoff by the EnKF method is close to the observations, whereas the prediction of the analytical solution with the constant parameters is larger than the observed data at the early time of runoff and less than the observed data at the late time. The Dif for the prediction by the EnKF and the analytical solution without data assimilation method is 0.88 and 3.10 mg/l, respectively. While the Dif predicted through the EKF is 1.04 mg/l, which is greater than that by the EnKF method. This further suggests that the EnKF can be used to accurately update the chemical concentration transfer from the soil in the surface runoff while the EKF cannot. Similar to the parameters variation in Fig. 6b, the infiltration-related incomplete parameter, γ , reaches the maximum value of 1, whereas h mix and α increase with time in Fig. 7b. The reasons for the increases are the same as explained above, the decrease of the KCl concentration in the soil-mixing layer leads to an increase in the depth of the mixing layer and corresponding greater concentration difference between the soil-mixing layer and the
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surface runoff, and results in the increase of the parameters h mix and α. The γ is constrained to be 1 by the EnKF for the same reason explained for the sand soil. Figure 7c indicates that the spread calculated by the EnKF method decreases with time, and the RMSE increases at first and then decreases with time. This is maybe because the simulation time for this study case is not very long. At long time the RMSE tends to become smaller than the spread. 5.2.2 Initially Saturated Experimental Soil
(a) 1240 observation with DA no DA
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31
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The experimental loam is initially saturated in case 4. The same as we did for the initially unsaturated sand soil, the guessed mean values of h mix , α and γ before the surface runoff are fixed to be 0.1, 0.084 and 1.0 cm respectively. The analytical solution with these fixed constant parameters does not match the observations well, and significantly deviate from the observations (Fig. 8a). But similar to Sect. 5.2.1, the predicted results of the KCl concentration in the surface runoff by the analytical solution with constant parameter values (the initial guess mean values) are still presented in Fig. 8a. The EnKF is applied to this experimental soil with the constraints that h mix ≥ 0.001cm, α ≥ 0.0001, γ ≤ 1. The results are shown in Fig. 8a and b. From Fig. 8a, we can see that the simulations by the EnKF agree well with the observed data, whereas the prediction of the analytical solution with constant parameters and without data assimilation method is less than the observations through the whole runoff time, which suggests that the EnKF can also be applied to the loam successfully. Moreover, the Dif for the results without data assimilation method, with the EnKF and EKF are 300.92,
-0.1 0
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time/min Fig. 8 Comparisons and parameters for the initially saturated experimental loam soil of case 4 a comparison of the analytical solution and the prediction by the EnKF with the observations and b variation of the parameters α, γ and h mix produced by the EnKF
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47.90 and 361 mg/l, respectively. AS in the former sections, we do not present the inversed results of the EKF method here. The calibrated results of h mix , α and γ by the EnKF are shown in Fig. 8b. All three parameters change only slightly throughout the surface runoff process, so they look like constants in Fig. 8b. The average values of these parameters produced by the EnKF are 0.35, 0.294 and -1.50 cm respectively. Therefore, we tried to apply these averaged values as the constant values to the analytical solution. However, the results significantly deviate from the observations and are identical to the analytical results without data assimilation method shown Fig. 8a, so the analytical results with these averaged values of parameters are not presented here. This further indicates that the EnKF method is efficient for the surface concentration prediction because the EnKF method assimilates the observed data and updates the prediction timely even if the updated parameters are constant. The negative value of the γ here means that the KCl solution diffuses from the bottom of the soil-mixing layer into the mixing layer. Furthermore, γ slightly decreases with time (meaning stronger upward diffusion), which is because of the large KCl concentration difference between the soil-mixing layer and the soil below. For the same reason as discussed above, h mix and α increase slightly during the surface runoff. From Fig. 8c, we can see that the RMSE and spread are almost the same as that in Sects. 5.1.1 and 5.1.2. So we do not make analysis to them in this section. From the above study results, we conclude that the EnKF can also be applied to the initially unsaturated and saturated experimental loam soils to improve the prediction and calibrate the corresponding parameters.
6 Conclusions In this study, based on the analytical two-layer model of Tong et al. (2010) for predicting soluble chemical transfer from soil to surface runoff, we develop an EnKF method to update the system state variables, the chemical concentration of surface runoff and calibrate model parameters simultaneously through assimilating the observed data. To avoid the violation of the physical laws produced by the EnKF, some constraints have been posed to the updated parameters h mix , α and γ after EnKF application. This EnKF method is verified by a synthetic case that it can successfully improve the prediction of the soluble chemical transfer from the soil to the surface runoff with the corresponding updated variable parameters h mix , α and γ . Compared with the analytical solution with constant parameters h mix , α and γ , the simulation by the EnKF is much closer to the observed data, which suggests that prediction by the EnKF is much more accurate than the analytical solution with constant parameters. Analysis for the ensemble size suggests that 100 realizations will be suitable enough for the prediction of the soluble chemical transfer from the soil into the surface runoff. The developed EnKF is also applied to experimental results for sand and loam soils under initially unsaturated and saturated conditions, respectively. Four cases are used to represent the different combinations of the soil and initial condition. In comparison with the calculation without data assimilation, the EnKF method significantly improves the calculation for the solute transfer from soil to runoff and much closer to the observations, whereas the EKF method does not. At the same time, calibrated parameters h mix , α and γ are more reasonable to the physical process. However, it still needs our further work to improve the model, where the parameters should be constant.
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Acknowledgements This work is partly supported by the CMG program of the National Science Foundation under grant numbers DMS-0620035 and partly supported by the Key National Nature Science Foundation of China (Grant No. 50639040).
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