Normatividad internacional

One possible reason for the models inability to find a good combination of realistic inter- action points and interaction rates was the tendency of Levenberg-Marquardt to converge to local minima. To overcome this the Particle Swarm Optimization technique was used. Because Particle Swarm Optimization (PSO) tests many different solutions at a time it is less likely to get stuck in a local minima. Also in order to simplify the model and reduce the total simulation time required the data used in the sloping base model was aggregated up to a 2km x 10km grid cell size.

The PSO routine has several parameters that need to be set before running the model. The parameters used in the PSO routine are as follows:

UseMinElevation If this value is set to yes in the code (Appendix B) then the minimum

elevation in a grid cell is used instead of the average elevation. mode Can use the depth to water mode (DTW) or the interval mode.

maxDTW Maximum depth to water value to use if using the depth to water method of finding points of interaction.

xmin The minimum value of the initial guess (see section 4.3) xmax The maximum value of the initial guess (see section 4.3)

xmaxEnforce The maximum value of extraction allowed before the objective function is penalized

enforceXMAX If this value is set to yes then xmaxEnforce is used if it is set to no then no penalty will be enforced for values larger than xmaxEnforce

xminEnforce The minimum value of extraction allowed before the objective function is penalized

enforceXMIN If this value is set to yes then xminEnforce is used if it is set to no then no penalty will be enforced for values smaller than xminEnforce

popsize The number of particles used

maxIter The maximimum number of iterations before the optimization terminates

Vmax The maximum velocity (see section 4.3)

amax Parameter movement of a particle (see section 4.3)

bmax Parameter movement of a particle (see section 4.3)

penalty Description of penalty functions used if any (see section 4.3)

The PSO routine was used for several model runs. Figures 5.7 and 5.9 show the error and extraction maps of the best results achieved through usage of PSO. Table5.2 shows the parameters used to achieve these results.

Table 5.2: Parameters used to Obtain Best PSO Results

Parameter Name Parameter Value

useMinElevation Yes mode DTW maxDTW 10 xmin 0 xmax 0.0005 xmaxEnforce 2.5 enforceXMAX Yes xminEnforce -10 enforceXMIN Yes popsize 200 maxIter 350 Vmax 0.0001 amax 1.5 bmax 2.5 penalty +50000 for ea. E > xmaxEnforce & +3000 for ea. E not = 0

The model did well in simulating heads in the southern two thirds of the High Plains Aquifer but struggled in Nebraska. This is likely due to the high level of complexity present in this basin. In Nebraska the aquifer transects are quite long west to east compared to other portions of the aquifer. Also there are many rivers and many areas where the water table is near to the land surface. This results in a transect have a very high number of possible points of interaction. This can be seen in the map in figure 5.9 where there is interaction occurring in most cells in Nebraska. While PSO is much better at avoiding local minima than Levenberg-Marquardt it will still struggle when the parameter space becomes too complex. Many of the transects in Nebraska have around 250 possible points of interaction. This means the PSO routine has to search a 250 dimensional parameter space to try and find the best combination of interaction rates. In such a complex parameter space it is probable that the PSO routine is simply unable to perform an effective search. The search can be made more effective by increasing the number of particles and the maximum number of iterations but the search time becomes prohibitive as the population size and maximum number of iterations grows too large.

Greater numbers of particles and increased maximum number of iterations were tried to see if the solution in Nebraska could be improved. Results showed that increasing the population size and maximum number of iteration had a negligible impact on the quality of the results while drastically increasing the simulation time required to achieve those results. Figures 5.10-5.13 show the impact of increased populations on a transect in Nebraska. A maximum iteration of 400 was used for all figures. The results shown were typical for other transects in the area as well, increasing the population size up to 2500 resulted in little to no improvement in the final results.

Figure 5.10: PSO Transect Results with Population of 500

Figure 5.12: PSO Transect Results with Population of 2000

Increasing the maximum number of iterations was also tried. Here the objective function of the best particle and the worst particle in the swarm were plotted to see if an increased number of iterations would improve convergence of the solution. Figures5.14-5.17 show the results. A population size of 350 was used for all figures. This increase also had almost no effect on the final outcome of the transect and the results shown here were typical for other transects tested as well. It can be seen in the objective function plots that the particles converge quite early in the optimization routine. This was true of all the Nebraska transects tested. The failure of these attempts to improve the PSO results in Nebraska point to the likelihood that the parameter space is too complex for the PSO routine to handle with a reasonable population size and maximum number of iterations.

Figure 5.15: PSO Transect Results with Maximum Iterations of 800

5.1.3

Levenberg-Marquardt Results (2km x 10km grid cells)