etude de la qualite des eaux d'un hydrosysteme fluvial ... - LTHE

eal list variables with the argument either the program variable with time or the programvariable with space coordinate. The standard deviations σ meas,i can be defined individually foreach data point or globally for all data points of each real list variable. The sum χ 2 extendsover all data points of all real list variables specified as fit targets as shown above.Simultaneous comparisons of data for measurements corresponding to different variables,compartments and zones are possible. AQUASIM performs a minimization of the sum ofsquares (χ 2 ) with the constraintsP min,i ≤ P i ≤ P max,iwhere p min,i and p max,i are the minimum and maximum of the constant variable representing p iwhich are defined by modellers.Due to the possible nonlinearity of the model equations and due to the numerical integrationprocedure, the sum χ 2 (P) must be minimized numerically. The user has the choice betweentwo numerical minimization algorithms: The simplex algorithm (Nelder and Mead, 1965) andthe secant algorithm (Ralston and Jennrich, 1978). Both of these techniques are well-suitedfor the minimization of numerically integrated equations, because they avoid the calculationof derivatives of the solutions with respect to the parameters.Depending on the nature of parameter subset to be estimated and our knowledge on theparameter variation range, the simplex or secant technique will be in use. Technically, thesimplex technique may be applied even to a poorly defined parameter estimation process withstarting values of the parameters far from those leading to the minimum of χ 2 . In contrast, thesecant method has more problems with bad starting values and poorly defined minimum of χ 2 ,but it leads to much faster end convergence close to a well-defined minimum. Estimates forthe standard errors of the estimated parameters and for the parameter correlation matrix areonly calculated by the secant method, and only if no parameter estimated is on one of thebounds p min,i or p max,i of the parameter.224