An optimal procedure for identifying parameter structure and application to a confined aquifer

Abstract

This study applies an optimal procedure to identify the spatial distribution of groundwater hydraulic conductivity for a confined aquifer in north Taiwan. The parameter structure is determined by the number of zones, zonation pattern, and an uniform hydraulic conductivity associated with each zone. The proposed optimal procedure uses the Voronoi diagram in describing zonation and applies simulated annealing algorithm to optimize its pattern and associated hydraulic conductivity. Three criteria are defined to stop the searching process, including the residual error, the parameter uncertainty, and the structure error. Observation hydraulic heads in years 2000 and 2001 and hydraulic conductivity value from pumping tests are used. The results show that the parameter structure with five zones conforms to the three criteria and, thus, is recommended for future groundwater simulation for the study site. Different heuristic algorithms may also play the role of simulated annealing to optimize the parameter structure. However, which optimization algorithm is more efficient is not discussed and requires further study.

Appendices

Appendix

Covariance matrix for parameter uncertainty

A covariance matrix as given by Yeh and Yoon (1981) is used quantify the parameter uncertainty. The equation is listed as follows:

$$ {\text{Cov}}_n = \frac{1} {{N_h \times T - n}}\sum\limits_{\mu = 1}^{N_h } {\sum\limits_{t = 1}^T {\left( {\frac{{h_\mu (\Omega _n^* ,\,t) - h_\mu ^o (t)}} {{h_\mu (\Omega _{_n }^* ,\,t)}}} \right)^2 } } \left[ {J^{\text{T}} J} \right]^{ - 1} $$

where J is the Jacobian matrix of hydraulic head with respect to K.

Structure error

The three components of structure error are calculated as follows:

$$ d_{_{n,n - 1} }^{{\text{obs}}} = \sum\limits_{g = 1}^{N_g } {\sum\limits_{t = 1}^T {\left| {\frac{{h_g \left( {\Omega _n^* ,t} \right) - h_g \left( {\Omega _{n - 1}^* ,t} \right)}} {{h_g \left( {\Omega _n^* ,t} \right)}}} \right|} } $$

$$ d_{^{n,n - 1} }^{{\text{par}}} = \sum\limits_{g = 1}^{N_g } {\left| {\frac{{K_g \left( {\Omega _n^* } \right) - K_g (\Omega _{n - 1}^* )}} {{K_g \left( {\Omega _n^* } \right)}}} \right|} $$

$$ d_{^{n,n - 1} }^{{\text{pre}}} = \sum\limits_{g = 1}^{N_g } {\sum\limits_{t = 1}^T {\left| {\frac{{h_g \left( {\Omega _n^* ,t} \right) - h_g \left( {\Omega _{n - 1}^* ,t} \right)}} {{h_g \left( {\Omega _n^* ,t} \right)}}} \right|} } $$

where N_g is the number of grids and T is the simulation or prediction periods.