CHAPTER 2 METHODS
2.4 C ONTAMINATION CONCENTRATION
where the source release begins at (time) t0 and ends at t2 and the peak concentration occurs at t1. The dimensionless unit step release history function can be written as
( )
⎪⎩2.4 Contamination concentration
Based on Eq. (3), the concentration distribution of a contaminant plume can be estimated if the aquifer configuration and the source location, geometry, and release
τ
determined, the contamination concentration can be predicted by Eq. (3).
Conversely, if the contaminant concentrations are obtained from field measurements, one might treat the Cin(t) as an unknown and solve Eq. (3) as an inverse problem.
2.5 Future-Sequential Regularization Method
In fact, Eq. (3), which involves definite integral with a constant lower limits and a variable upper limit dependent on the time T, is the Volterra integral equation of the first kind (Press et al., 1992). If the upper limit of integration is also a constant; then Eq. (3) can be characterized as the Fredholm equation. The solution of Eq. (3) is extremely sensitive to arbitrarily small perturbations of the system.
The development of stable and reliable numerical methods particularly suited for the solution of Eq. (3) has therefore always been a challenge.
A reasonable way to compute a meaningful ‘smooth’ solution to Eq. (3), i.e., a solution which has some useful properties in common with the exact solution to the underlying and unknown-unperturbed problem, is to somehow filter out the high-frequency components associated with the small singular values. The classical way to filter out the high-frequency components associated with the small singular values is to apply regularization to the problem. It is standard terminology today to classify any method that seeks to compute a ‘smooth’ solution as a regularization method and regularization is commonly applied directly to solve the Volterra integral
equation of the first kind (Hansen, 1992).
The follows introduce how the FSRM solves Eq. (3) inversely. Lamm (1995) extended the theoretical context of the FSRM developed by Beck (1985) to solve the inverse heat conduction problem. For the application of FSRM in solving the groundwater plume source identification problem, Eq. (3) can be expressed as a first-kind Volterra equation with convolution kernel k and given data f. That is
) where u(s) is an unknown contaminant release history function. If an extra unknown function occurs on the left-hand side of Eq. (9), it is known as the Volterra
equation of the second kind. That is
∫
− =+ tk t s u s ds f t t
u( ) 0 ( ) ( ) ( ), t∈[0, t] (10) The right-hand side f(t) and the kernel function k are assumed to be known (Linz, 1985).
Lamm (1995) used a very effective stabilization method to analyze the inversion of linear Volterra operators of convolution type. The FSRM is a special case in a class of regularization methods in which the solution of an ill-posed, first-kind Volterra equation is found to be the limit of a sequence of solutions of well-posed, second-kind Volterra equation. A physical problem is considered as well-posed if there exists a unique solution that depends continuously on the
non-uniform data.
With the Volterra integral operator A, the solution of Eq. (9) starts with the following collocation equation
) some real ci, which are the unknown contaminant release history and χi is the characteristic function defined by χi(t) = 1 for ti-1 < t < ti, and χi(t) = 0 otherwise. matrix form. In fact, the ill-posed original problem leads to poor conditioning of the lower-triangular matrix AN, especially as ∆1 gets to zero. Therefore, there are errors introduced in calculating c1, c2, and so on. The nature of a Volterra equation is such that the output of c at time t is only influenced by the input data f at times prior to t. It is common for stabilizing the inversion process to impose additional constraints that bias the solution, a process referred to as regularization. Therefore, it makes sense to use future data f(ti+1), f(ti+2),…in computing ci. To illustrate, suppose that r has been fixed, and select c1 minimizing the least squares fit to data J1
as
2 period from t2 to tr+1 overlaps the function c1, thus the process amends the solutions and regularizes in the presence of data error to get the optimal solutions. After estimating the solution of c1, and hold c1 fixed, then the optimal solution of c2 is chosen by the same way based on minimizing the least square and so on. For this approach, when Ji(ci)=0, each ci is determined as the optimal value.
After a series of mathematical manipulation, Eq. (9) could be written as the regularized equation
( )
regularization parameter. Equation (14) is a well-posed, second-kind integral equation. The unknown u in Eq. (14) has to be solved sequentially with the lower-triangular matrix and appropriate regularization parameter r. Note that required total numbers of sampling data when applying the FSRM is N+r-1 and the recovered concentration is still N.2.6 Choice of regularization parameter in FSRM
The choice of the value of r is important. If r is too small, the solution will have numerical oscillation. In contrast, larger r gives a dispersed solution.
Different kinds of source release history may use different value of r, which actually may depend on the location of monitoring well, dispersion coefficient, average linear velocity, and the sampling time period. However, an appropriate value of r is usually found by trial-and-error. For recovering the source release history, Eq.
(14) is solved in matrix form using the observed concentrations. The solution of FSRM depends strongly on the value of the regularization parameter r, where r may equal 1, 2, …. This method uses no regularization if r = 1 and some regularization as r increases. One may start with r = 2 for recovering the release history. If the recovered release history exhibits obvious oscillation, then r should be increased until the oscillation is significantly diminished. When r gets larger, the curve of recovered history becomes dispersed or stabilized. FSRM utilizes future observed data if r ≥ 2 in recovering the release history. The value of r-1 represents the numbers of future measured data used in the analysis. In other words, a larger value of r requires more future sampled concentration data. Nevertheless, the number of r required to perform well in recovering the release history depends on the shape of the release pattern and source geometry.
2.7 Cubic spline
Field observed concentration data were usually not sampled uniformly. The implementation of FSRM requires that the time interval for sampling the temporal plume concentration data should be fixed. Therefore, a piecewise polynomial approximation such as the cubic spline can be used to interpolate the observation data from a non-uniform time interval into a uniform one.
Consider a set of third-degree polynomials, yi, between each pair of contiguous data points from xi to xi+1. The cubic spline constructs an interpolating polynomial that is smooth in the first derivative, and continuous in the second derivative, both within an interval and at its boundaries. A general expression for cubic spline is
( )
⎟⎟ interpolated data successfully for the observed drawdown data of non-uniform time intervals in order to facilitate the application of the Extend Kalman filter.2.8 Measurement errors
Field sampled concentration data inevitably contains measurement errors. A multiplicative error model is used to generate random measurement error on the
) , ( )
, ( )
,
(x T C x T C x T
Cmeas n = ext n +εδn ext n (16) where Cmeas(xn,T) denotes the measured concentration at location xn at time T, Cext(xn,T) represents the exact concentration (or simulation concentration) at location
xn at time T, xn is the location of the nth sample, ε is the error level, δn is the nth random deviate from a Gaussian standard population (standard normal), and the product εδnCext is equal to the measurement error at xn.
CHAPTER 3 CONCENTRATION DATA
3.1 Measured concentrations
Recovering the source release history of a groundwater contamination needs to be inferred from the plume concentration measurements. Therefore, to asses the performance of FSRM in recovering the source release history, the measured concentrations are generated by Eq. (3) using hypothetical release functions.
Consider an area contaminant source which has the dimensions of 5m × 5m. The concentration distribution is simulated based on Eq. (5) with v = 1 m/day, Dx = 0.5 m2/day, and Dy = 0.05 m2/day for a two dimensional contaminant transport. The monitoring well is installed at (x, y) = (30, 6) near the source of contaminant, which is located at the origin (0, 0).
3.2 Sampling concentration data
Previously, Skaggs and Kabala (1994) used 1-D spatial concentration data to recover the release history. The plumes generated by the source history function, i.e., Eq (3), were given at time T = 300 day and 25 different locations for the distance x = [0.01 m, 25.05 m, 50 m, … , 250 m, 275 m, 300 m] with a 10 m interval for x ranging from 50 to 250 m. Therefore, a total of 25 spatial concentration data were used in the recovery of source release history. After that, most articles in recovering the release history adopts their concentration data set for case studies.
In addition, Woodbury and Ulrych (1996) also used 1-D spatial concentration data sampled at the downstream of source ranging from 5 m to 300 m with a 5 m interval and thus a total of 60 data points were used in the recovery of source release history.
In principle, the sampling data should cover the whole range of plume concentration in order to recover the entire release history.
Consider three source history patterns, namely the exponential function, the triangle function and the step function. The concentration data generated by those release patterns and measured from a monitoring well, where data points are less than those of used in previous studies, are used to recover the source release histories by FSRM.
CHAPTER 4 CASE STUDIES AND RESULTS
4.1 Two-dimensional source recovery
This study is based on the analytical approach to recover the source release history, each case assumes that the aquifer is isotropic and homogeneous; the flow is steady and uniform; the contaminant is conservative, no decay, and no adsorbed on the aquifer. Various aquifer parameters and the source geometry and location are assumed known. The target of this study is to reconstruct the contaminant release history in the groundwater by FSRM. The method has an advantage that it can be used to reconstruct release history with arbitrary pattern, including smooth curve and non smooth curve. However, FSRM has a limitation that the observed concentration data must be uniform time interval. In this study, cubic spline was used to overcome the problem. For the concentration data with uniform time interval, three source patterns are designed to demonstrate the proposed method in solving the source release history recovery problem.
Three cases are designed to demonstrate the application of FSRM to the cases of two-dimensional area source for three different source release patterns. The aquifer is assumed to be homogeneous, isotropic, and of infinite width and the groundwater flow is steady and uniform. The contaminant is conservative. Case 1 attempts to recover the release history for a release pattern expressed in terms of a
combination of exponential functions. Cases 2 and 3 aim to recover the source release history in terms of the triangle and step functions, respectively.
4.2 Scenario 1: Recovering release history with FSRM
4.2.1 Sampling time with a 7 day intervalFigure 1(a) shows the behavior of a “true” source release history generated based on the exponential functions of Eq. (6) with tj = 130, 150, and 190, bj = 5, 10, and 7, and aj = 1, 0.3, and 0.5 and the recovered release histories estimated by FSRM. The value r is chosen to be 3, 4 or 5 to assess the performance of the FSRM in case 1.1. The recovery of the entire release history needs the sampled data covering the full range of plume concentrations in response to the true release history. Note the monitoring well is located at the downstream of the source with a distance of 30m and the average groundwater velocity is 1 m/day. The plume concentration is sampled starting at the time 112 day with a 7 day interval and thus 22 data points are uniformly spaced for an exponential source pattern. For r = 3, the solution of FSRM is divergent as indicated in Fig. 1(a). When r is increased to 4, the recovered release history is in fairly good agreement with the true release history although the peaks of the concentration curve are slightly lower and shifted.
For r = 5, FSRM gives a smoother curve with significant lower concentration in the peaks than the true ones.
Figure 1(b) displays the distribution of the “true” source release history with the triangle function of Eq. (7) with t0 = 100, t1 = 175 and t2 = 250 days for case 1.2.
For triangle function form, 28 measured data points are uniformly sampled from t = 98 to t = 287 days. The recovered release history for a triangle form gives fairly good match with the assumed release history for FSRM with r = 4, 5, and 6.
However, Fig. 1(b) indicates that the shape of the source release history is better recovered for FSRM with r = 5 than that with r = 4 and 6.
The case 1.3 considers that the source release function is a step function, Eq.
(8), with t1 = 130 and t2 = 225 days. Twenty eight data points with a fixed time interval are in the range from t = 119 to t = 273 days. Figure 1(c) shows that when r = 4, an obvious oscillation is observed at the beginning of the step function. The
recovered release history almost has no oscillation throughout the whole step function when r = 5. Although the release times at the beginning and the end of the step function are not recovered exactly, the percent of error of the release period is about 12%. As r is increased to 6, the time shifting is more obvious. Those results imply that FSRM can recover the release history reasonably well with an appropriate value of parameter r.
4.2.2 Sampling time with 1 day and 3 day interval
The use of smaller time interval, i.e., more sampling data, for the observed data
may improve the estimated results of recovering release history. This section intends to investigate the use of of smaller sampling time interval on the estimated results.
For an exponential source release pattern of Eq. (6), the sampling period of the observed data is exactly the same as that used in section 4.2.1, yet, the sampling time intervals are reduced to 1 day and 3 day instead of 7 day. The plume is sampled starting at the time 112 day with 1 day and 3 day intervals and thus resulting 114 data and 48 data points, respectively. For the cases of the time intervals 1 day and 3 day, the solutions of FSRM with r = 5 and r = 3, respectively, are shown in Fig. 2(a). For the case with 1 day time interval, the recovered release history has a very good agreement with the true release history. On the other hand, for the case of 3 day time interval, the solution still matches well with the true one although the first peak of the concentration curve is slightly lower.
For the source release in terms of the triangle function, 158 and 64 measured data points are uniformly sampled from t = 98 days with time intervals 1 day and 3 day, respectively. Fig. 2(b) demonstrates that the FSRM with r = 5 and r = 4 gives fairly good recovered release histories if compared with the true one.
For the step function form, the observed data are sampled from t = 119 with time intervals 1 day and 3 day; and thus the totals of available data are 120 and 52,
respectively. Figure 2(c) shows that when r = 5, the release time is recovered reasonably well, although a small oscillation is observed at the beginning of the step function. The recovered release history exhibits no oscillation throughout the whole step function when r = 4 with 3 day time interval. Obviously, those results indicate that the use of smaller time interval will yield better estimations.
4.3 Scenario 2: Nonuniform sample data and Cubic spline
interpolation
In reality the concentration measurements may not be sampled with a fixed time interval, which restricts the use of FSRM in recovering the source release history. Under this circumstance, the cubic spline can be chosen to interpolate the nonuniform observed concentration data into uniform ones. A set of 25 concentration data produced by the analytical model for the sampling period from 100 day to 300 day with non-uniform time intervals illustrated in Figure 4 is considered. The interpolated concentrations with 7 day interval by the cubic spline are used for FSRM in recovering the release history. Thus, a total number of temporal concentrations, the same as those used in scenario 1, are used to recover the source release history.
Figure 3(a) shows the interpolated data and recovered release history for the case that the source pattern is of exponential function. The result indicates that the
recovered release history with interpolated concentration data by cubic spline is still as good as the one obtained by FSRM with the uniformly spaced data. The recovered history exhibits three peaks clearly with r = 4, though the peaks are slightly lower and the location is lightly shifted. Moreover, the FSRM gives a poor result when r > 4.
In the case of a triangle release function, the recovered history using FSRM with interpolated concentration data is almost identical to that with nonuniform observed concentrations for r = 5 as indicated in Fig. 3(b). Similarly, the recovered history shown in Figure 3(c) for the case of a step release obtained by FSRM with r
= 5 and interpolated data is also close to the one obtained with nonuniform observed data.
4.4 Scenario 3: Measurement errors
One of the major advantages of using FSRM in recovering release history is that the solutions are not affected apparently by the measured error. Three cases with different magnitudes of uncertainty representing possible field measurement errors are considered. The error is added to the concentration data generated by Eq.
(3) with assumed release history and known aquifer configuration. Cases 3.1 to 3.3 consider that the ε in Eq. (16) are 0.01, 0.05, and 0.1, respectively, representing different level of measurement error. Those data with measurement errors are
shown in Fig. 4. Table 1 lists the mean and the standard deviation of the measurement error εδnCext for different error level ε in those three cases. With different values of ε, the recovered histories by FSRM with r = 4 for the exponential release function are shown in Fig. 5 (a), with r = 5 for the triangular function are shown in Fig. 5 (b), and with r = 5 for the step function are shown in Fig. 5 (c).
The results indicate that the recovered release histories are very close to those without the measurement error except that the data with larger uncertainty give small fluctuation in the recovered history.
4.5 Scenario 4: Five other methods for the source recovery
Five methods included the least squares (LS), bounded variables least squares (BVLS), minimum relative entropy (MRE), second-order Tikhonov regularization (TR), and simulated annealing (SA) with the exponential function fitting approach are used to recover the release histories for the triangle and step source history functions in this scenario. Except SA and MRE methods, the other four methods are applied to the case of 1-D solute transport with the observed spatial concentrations sampled from 25 monitoring wells at time T = 300 day where the data points are given by Skaggs and Kabala (1994). For the MRE method, a total of 60 data points given in Woodbury and Ulrych (1996) are used. The computer codes developed in Aster (2005) for LS, BVLS, MRE, and TR are used to recover
the release history for the triangle and step function patterns.
Using the temporal concentration data in scenario 1, SA method with the exponential function fitting approach developed in Chen and Yeh (2006) is applied to recover the release histories. Figures 6 (a) and (b) demonstrates the recovering release histories for the triangle and step functions, respectively, by SA when j is equal to 1, 2, or 3. For the case of triangle function, when j = 1, the solution is the best although the climax of concentration is slightly lower than that of the true one.
However, the recovered history has a long tail at very low concentration region which gives poor prediction at the beginning and end periods of the release history in triangle shape. As j is increased to 2, the recovered history deviates from the true one significantly after time T = 225. For j = 3, a spike appears at time T = 125 which reflects that an extra exponential term used in the fitting model gives a poor result.
The recovering release histories for various j are shown in Fig. 6(b) exhibit sinuous curves with obvious oscillation when j = 2 and 3. These results imply that the exponential function is not suitable to recover the source release pattern in the form of triangle and step functions
The recovered result by the MRE method for the case of a triangle pattern is shown in Fig. 7(a) which indicates that the recovered release history has obvious
The recovered result by the MRE method for the case of a triangle pattern is shown in Fig. 7(a) which indicates that the recovered release history has obvious