**3. Performance evaluation**

In order to evaluate the performance of two different optimization algorithms involving comparison of solutions obtained, it is vital to first ensure that only one solution exists. In other words, a unique solution has to be guaranteed. This is possible only under the following idealized assumptions [33]:


3. The unknown parameter is piecewise constant.

The first assumption is valid for cases where grid size and time step used in the numerical solution tends to zero. However since the groundwater simulation models used in this study have been proven to be stable and convergent, this assumption approximately holds. The second assumption however, cannot hold in real life scenarios. Hence it becomes necessary to use synthetically generated observation values initially which can be considered free of measurement errors. The third condition is implemented by assuming that the unknown fluxes are constant in every stress period. In such conditions it approximately resembles a well posed problem. Therefore these evaluations are initially carried out for synthetic data (simulated data) with known parameter values. There is another related issue of unique solutions. Whenever numerical simulation and optimization models are used, the convergence of the solutions may be another issue related to unique solutions. These issues are discussed in [4]. In this study the use of synthetic observation data, with known hydro-geologic parameter values reduces the ill-posed nature of the problem. The uniqueness of the solution cannot be guaranteed. However, sufficient iterations were allowed to ensure convergence to the optimal solution. Performance of the source identification methodology is evaluated using synthetic data from a three dimensional aquifer study area. The synthetic contaminant concentration data is obtained by solving the numerical flow and transport simulation models.

#### **3.1. Simulating errors in concentration measurement data**

Once the global optimal solution has been obtained for the idealistic assumption, the performance evaluation of developed methodology can take into account the effects of contaminant concentration measurement errors as well as uncertainty associated with the determination of hydro-geological parameters. To test the performance for realistic scenarios, concentration measurement errors are incorporated by introducing varied amounts of synthetically generated statistical noise in the simulated concentration values. The perturbed simulated concentrations represents erroneous measurements and is defined as follows:

$$\mathsf{C}\_{\text{pert}} = \mathsf{C}\_{\text{ns}} + \mathsf{S}\_{\text{ud}} \times \mathfrak{a} \times \mathsf{C}\_{\text{ns}} \tag{5}$$

Network Design Problems 9

Groundwater Contamination Source Characterization Problem and Monitoring Network Design Problems

Application of Simulated Annealing in Water Resources Management: Optimal Solution of

competing linked simulation-optimization approaches to produce accurate source histories, the errors in estimating source fluxes accurately is also used as a performance criterion. Normalized absolute error of estimation (NAEE) is used as the measure of errors in estimation

of the sources. It can be represented as:

S = Number of Sources = 2 in this case.

**4. Discussion of solution results**

developed methodology and improving it further.

consisting of three unconfined layers as shown in Figure 2.

Where,

 *q j i* 

 *q j i* 

**4.1. Study area**

*NAEE*(%) =

NAEE = Normalized Absolute Error of Estimation

N= number of transport stress periods = 5 in this case.

∑*S <sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>N</sup> j*=1 *q j i est* − *q j i act* 

*act* = Actual source flux for source number i in stress period j

*est* = Estimated source flux for source number i in stress period j

∑*S <sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>N</sup> j*=1 *q j i act*

The developed methodology was applied to a hypothetical illustrative study area with synthetically generated concentration measurements over space and time. Advantage of using a hypothetical study area lies in the fact that unknown data errors do not distort the performance evaluation of the methodology. This helps in understanding the drawbacks of

The hypothetical study area is a heterogeneous aquifer measuring 2100m x 1500m x 30m and

The East and west boundaries are constant head boundaries, whereas north and south boundaries are no flow boundaries. There are two sources (S1 and S2) of contamination. S1 is located in the top layer and S2 in middle layer. Five monitoring location (M1 through M5) are located in the first layer as shown in Figure 3. A grid size of 30m x 30m x 10m is used for finite difference based numerical calculation of groundwater flow and transport equations. Transport time step used for MT3DMS is 36.5 days. Other model parameters are listed in Table 1. Only a conservative contaminant is considered. There are two point sources of contaminants. One in the top layer and another one in the middle layer. A time horizon of 16 years is considered. Entire time horizon is divided into 5 different stress periods. The first four stress periods are each 1.5 years long and the final stress period is of 10 years duration. Sources are assumed to be active only in the first four stress periods or in the initial 6 years. Original source fluxes are presented in Table 2. It is assumed that groundwater contamination is detected at five different locations in the study area at the

× 100 (6)

165

Where,

*Cpert*= Perturbed Concentration values

*Cns*= Simulated Concentration

*Sud*= a uniform random number between -1 and +1

a = a fraction between 0 and 1.0.

### **3.2. Performance evaluation criteria**

The execution time of the algorithms is compared based on convergence curves which represent the value of objective function achieved versus time. To compare the ability of competing linked simulation-optimization approaches to produce accurate source histories, the errors in estimating source fluxes accurately is also used as a performance criterion. Normalized absolute error of estimation (NAEE) is used as the measure of errors in estimation of the sources. It can be represented as:

$$NAEE(\%) = \frac{\sum\_{i=1}^{S} \sum\_{j=1}^{N} \left| \left( q\_i^j \right)\_{est} - \left( q\_i^j \right)\_{act} \right|}{\sum\_{i=1}^{S} \sum\_{j=1}^{N} \left( q\_i^j \right)\_{act}} \times 100 \tag{6}$$

Where,

8 Will-be-set-by-IN-TECH

The first assumption is valid for cases where grid size and time step used in the numerical solution tends to zero. However since the groundwater simulation models used in this study have been proven to be stable and convergent, this assumption approximately holds. The second assumption however, cannot hold in real life scenarios. Hence it becomes necessary to use synthetically generated observation values initially which can be considered free of measurement errors. The third condition is implemented by assuming that the unknown fluxes are constant in every stress period. In such conditions it approximately resembles a well posed problem. Therefore these evaluations are initially carried out for synthetic data (simulated data) with known parameter values. There is another related issue of unique solutions. Whenever numerical simulation and optimization models are used, the convergence of the solutions may be another issue related to unique solutions. These issues are discussed in [4]. In this study the use of synthetic observation data, with known hydro-geologic parameter values reduces the ill-posed nature of the problem. The uniqueness of the solution cannot be guaranteed. However, sufficient iterations were allowed to ensure convergence to the optimal solution. Performance of the source identification methodology is evaluated using synthetic data from a three dimensional aquifer study area. The synthetic contaminant concentration data is obtained by solving the numerical flow and transport

Once the global optimal solution has been obtained for the idealistic assumption, the performance evaluation of developed methodology can take into account the effects of contaminant concentration measurement errors as well as uncertainty associated with the determination of hydro-geological parameters. To test the performance for realistic scenarios, concentration measurement errors are incorporated by introducing varied amounts of synthetically generated statistical noise in the simulated concentration values. The perturbed simulated concentrations represents erroneous measurements and is defined as follows:

The execution time of the algorithms is compared based on convergence curves which represent the value of objective function achieved versus time. To compare the ability of

*Cpert* = *Cns* + *Sud* × *a* × *Cns* (5)

3. The unknown parameter is piecewise constant.

**3.1. Simulating errors in concentration measurement data**

simulation models.

Where,

*Cpert*= Perturbed Concentration values

**3.2. Performance evaluation criteria**

*Sud*= a uniform random number between -1 and +1

*Cns*= Simulated Concentration

a = a fraction between 0 and 1.0.

NAEE = Normalized Absolute Error of Estimation

S = Number of Sources = 2 in this case.

N= number of transport stress periods = 5 in this case.

 *q j i act* = Actual source flux for source number i in stress period j *q j i est* = Estimated source flux for source number i in stress period j
