industrial pollution. Therefore, it is important to estimate the distribution of groundwater pollution. One of the estimation algorithm is Kalman filter. Kalman filter is.
Canadian Journal on Science and Engineering Mathematics Vol. 2 No. 2, June 2011
The Groundwater Pollution Estimation by The Ensemble Kalman Filter Erna Apriliani, Bandung Arry Sanjoyo, and Dieky Adzkiya Department Mathematic, Intitut Teknologi Sepuluh Nopember-Surabaya-Indonesia
In this paper, we estimate the groundwater pollution concentration by using the Ensemble Kalman filter. We compare the accuracy and the computational time between the Ensemble Kalman filter and Kalman filter.
Abstract — In the industrial area in the city such as Surabaya, the groundwater quality is influenced by the industrial pollution. Therefore, it is important to estimate the distribution of groundwater pollution. One of the estimation algorithm is Kalman filter. Kalman filter is method to estimate the state variable of stochastic dynamic system. Kalman filter combines the data measurement with the mathematical modeling. The ensemble Kalman filter (EnKF) is one of the Kalman filter modification. In the Kalman filter, the initial estimation is chosen as one value, but in the EnKF, the initial estimation are generated ensemble value from the normal distribution. Here we estimate the groundwater pollution distribution by the Kalman filter and the EnKF. At first step, we make mathematical model of groundwater pollution, we discretize the system respect to time and posistion. Kalman filter and the EnKF need less measurent data to estimate the distribution of pollution in hole are We compared the accuracy and the computational time between the Kalman filter with the En KF.
II. THE GROUNDWATER POLLUTION MODEL The groundwater pollution distribution for non reactive dissolved substance, can be written as advection-dispersion equation
Dx
∂ 2C ∂ 2C ∂C ∂C ∂C + D − vx − vy = y 2 2 ∂x ∂y ∂x ∂y ∂t
(1)
With boundary condition:
C (( x, y ), 0) = 0, x ≥ 0, y ≥ 0; C (( x, y ), t ) = C0 , t ≥ 0 C ((∞, ∞), t ) = 0, t ≥ 0
C (( x, y ), t ) is the concentration of pollution in position x, y and time t. Parameter Dx , Dy and vx , v y are Where
Key Words — Grounwater pollution, Ensemble , Kalman filter
coefficient of diffusion and groundwater flow velocities in the x, y direction respectively.
I. INTRODUCTION
Before we apply Kalman filter algorithm or the Ensemble Kalman filter to the groundwater pollution problem, we discretize the groundwater pollution equation. Here we use the forward difference method respect to time and we use the central difference method respect to space. The discrete form of equation (1) is
The groundwater quality is important for the human healthy. In the industrial area in the city such as Surabaya, the groundwater quality is influenced by the industrial pollution. Therefore, it is important to estimate the distribution of the groundwater pollution. The concentration of groundwater pollution has been estimated by Kalman filter [Apriliani, 2006]. Kalman filter is an algorithm to estimate the state variable of the stochastic dynamical linear system. This algorithm combines the mathematical model with the measurement data[Lewis, 1986]. Kalman filter has been applied in various problems such as the estimation of the water level in the river, wave of ocean and the tide [Heemink, 2001]. The estimation of pollution distribution in the air. The other side, there are many modification of Kalman filter algorithm. This modification has been done to avoid the convergence of algorithm, to reduce the computation time, to decrease the error of estimation and other. One of the modification of Kalman filter is the Ensemble Kalman filter.
Ci, j ,k +1 = ( pDx + 2r vx ) Ci−1, j ,k + ( qDy + 2s vy ) Ci, j −1,k +
(1− 2 pD − 2 pD ) C ( qD − v ) C x
y
s 2
y
y
i, j ,k
+ ( pDx − 2r vx ) Ci+1, j ,k +
i , j +1,k
(2)Where p = ∆t , q = ∆t , r = ∆t , and 2 2
∆x
∆y
∆x
s=
∆t . ∆y
We arrange and add the system noise such that we have the stochastic dynamical system
Ck +1 = ACk + Gwk 60
(3)
Canadian Journal on Science and Engineering Mathematics Vol. 2 No. 2, June 2011 Where
wk is Gaussian white noise and wk
The relation between state space
N (0, Q ) .
1 N
xˆk− =
xk with the measurement
N
∑ xˆ
− k, j
j =1
data, we define the measurement equation
zk = Hxk + vk Where
Covariance of error time update estimation is
(4)
Pk− =
N (0, R ) .
vk is measurement noise and vk
Kalman filter and the ensemble Kalman filter are applied in the stochastic dynamical system with the measurement system in equation (3)-(4).
Where
N (0, R ) are the ensemble of measurement
(
xˆk , j = xˆk−, j + K k zk , j − Hxˆk−, j
)
−1
)
The mean of measurement update estimation is
xˆk =
1 N
N
∑ xˆ
k, j
j =1
With covariance error
Pk = [ I − K k H ] Pk− IV. SIMULATION AND RESULT As a study case, we applied the EnKF and Kalman Filter to estimate the concentration of groundwater pollution. We assumed the area is 238 m x 252 m. We divided into 10 x 10 sub area, Dx = Dy = 0.001 , Vx = Vy = 0.001 . We
(6)
measured the concentration of pollution in 10 locations as the measurement data. Here we generate the “real system” based on equation (3), and the measurement data based on equation (4). In this simulation, the number of ensemble is taken 100, 200 and 300, respectively. The computational time and the accuracy of estimation are compared between Kalman filter and Ensemble Kalman Filter.
zk and the states variable xk . And wk , vk
are system white noise and measurement white noise with mean zeros and covariance Q, R respectively.
The algorithm of EnKF to estimate the state variable is i. Initial estimation Generate the N-ensemble of initial condition
x0, N
N ( x0 , P0 ) .
ii. Time update Generate the N-ensemble of time update estimation
xˆk−, j = f ( xˆk −1, j , u ) + wk , j wk , j
T
)
Measurement update estimation are
zk is measurement data, H is matrix, which connects between
Where
vk , j
(
xk is a state variable in time k, uk is input vector,
x0,2 . . . x0, N −1
− xˆk−
K k = Pk− H T HPk− H T + R
The measurement equation is
x0 = x0,1 Where x0,i
− k, j
noises Kalman gain is defined as
The EnKF is proposed by Evensen (1994). In the Kalman filter , we choose a scalar/ vector for the initial estimation but in the EnKF, we generate the ensemble of scalar/vector for initial estimation. Suppose we have a dynamical stochastic system xk +1 = f ( xk , uk ) + wk (5)
measurements data
)( xˆ
zk , j = zk + vk , j
Kalman filter is one of data assimilation method. In Kalman filter, we estimate the state variable based on the combination between the measurement data with mathematical modeling. The Kalman filter is an optimal estimation for linear dynamical stochastic system. For non linear dynamical stochastic system, we cannot apply Kalman filter but we can use Extended Kalman filter (EKF), Unscented Kalman filter (UKF) or Ensemble Kalman filter (EnKF). In this paper we use the EnKF to estimate the concentration of groundwater pollution.
Where,
(
iii. Measurement update Generate the ensemble of measurement data
III. THE ENSEMBLE KALMAN FILTER
zk = Hxk + vk
1 N − ∑ xˆk , j − xˆk− N − 1 j =1
N (0, Q ) are the ensemble of system
noises. Mean of time up date estimation is
61
Canadian Journal on Science and Engineering Mathematics Vol. 2 No. 2, June 2011 The 3-D of Concentration Estimation by KF
3.6
Concentration
3.4 3.2 3 2.8 2.6 10 8
10 8
6 6
4 2 Position Y
4 2 Position X
Figure 1. The concentration estimation by Kalman Filter
Figure 3. The error of estimation by Kalman Filter and Ensemble Kalman Filter, N=200 Figure 1 shows the estimation of pollution concentration by using Kalman filter and Figure 2 shows the estimation of pollution concentration by using Ensemble Kalman filter in three dimension with the number of ensemble is 200. . Figure 3 shows the error of estimation by Kalman filter and Ensemble Kalman filter in all positions with number of ensembles N=200. Figure 4 shows the error of estimation between Kalman filter and Ensemble Kalman filter with number of ensemble N=300. The error of estimation and computational time for number of ensemble 100, 200 and 300 are given in Table 1. We do ten simulations for each number of ensembles. From table 1, it is showed that the estimation by Ensemble Kalman filter is more accurate than Kalman filter, but the Ensemble Kalman filter need more computational time. More number of the ensemble gives more accurate estimation result, and need more computational time.
The Concentration Estimation by Ensemble Kalman Filter
4.5
Concentration
4 3.5 3 2.5 2 1.5 10 8
10 8
6 6
4 2 Position Y
4 2 Position X
Figure 2. The concentration estimation by Ensemble Kalman Filter
Figure 4. The error of estimation by Kalman filter and Ensemble Kalman Filter N=300 Table 1. The error of estimation and the computational time
62
Canadian Journal on Science and Engineering Mathematics Vol. 2 No. 2, June 2011 Iter ation
60
Number of Error of Error of Ensemble KF EnKF
100
Average
60
200
Average
60
300
Average
Time of KF
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BIOGRAPHIES
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.
V. CONCLUSION From this research we conclude that • Kalman filter and Ensemble Kalman filter are can be used to estimate the groundwater pollution in 100 positions based on the 15 data measurements. • The Ensemble Kalman filter is more accurate than Kalman filter • More the number of ensemble give more accurate estimation but need more computational time . 63
Erna Apriliani, (Surabaya, Indonesia, 14 – 4 - 1966) Education: Bachelor of Mathematics, ITS, 1989, Master of Mathematics, ITB Indonesia,1992, Doctor of Mathematics, ITB Indonesia, 2002. Institution: Mathematics Department ITS Indonesia. Professional society: Indonesia Mathematics Society. Research interest: Data assimilation, applied analysis
