Methods and guidelines for effective model calibration

Solving the Normal Equations
Using double precision as suggested by Stewart (1972, p. 226-227), equation (4) has been 
solved accurately and efficiently in many applications using Cholesky LDL
T
decomposition (Den-
nis and Schnabel, 1983, p. 50-51). Exceptions were plagued by strong correlations between param-
eters or insensitive parameters, and were resolved by reparameterization. Dennis and Schnabel 
(1983, p. 221) and Seber and Wild (1989, p. 621) suggest that solving the alternative formulation 
X d = (
y - y
’
)
using QR or singular-value decomposition (Dennis and Schnabel, 1983, p. 49-51; 
Seber and Wild, 1989, p. 680-681; Press and others, 1989, p. 52-63) is more stable, but it is unclear 
whether or not they used the scaling and Marquardt parameter which adds stability to equation 4. 
Press and others (1989, p. 515-520) suggest using singular-value decomposition for linear regres-
sion, but use Gauss-Jordon elimination to solve a variation of equation 4 that includes similar scal-
ing and implementation of the Marquardt parameter for nonlinear regression. Considering the 
success experienced using Cholesky decomposition, Cholesky decomposition is used in UCODE 
and MODFLOWP.
References
Cooley, R.L., 1983a, Incorporation of prior information on parameters into nonlinear regression 
groundwater flow models, 2, Applications: Water Resources Research, v. 19, no. 3, p. 662-
676.
Cooley, R.L., 1993, Regression modeling of ground-water flow, Supplement 1 -- Modifications to 
the computer code for nonlinear regression solution of steady-state ground-water flow prob-
lems: U.S Geological Survey Techniques of Water Resources Investigations, book 3, chapt. 
B4, supplement 1, 8p.
Cooley, R.L. and Hill, M.C., 1992, A comparison of three Newton-like nonlinear least-squares 
methods for estimating parameters of ground-water flow models: in Russel, T.F., Ewing, R.E., 
Brebbia, C.A., Gray, W.G., and Pinder, G.F., eds., Proceeding, Computational methods in wa-
ter resources IX: Denver, CO, v. 1, Numerical methods in Water Resources, ***publisher, p. 
379-386.
Dennis, J.E. Gay, D.M. and Welsch, R.E., 1981, An adaptive nonlinear least-squares algorithm: 
ACM Transactions on Mathematical Software, v. 7, no. 3, p. 348-368.
Dennis, J.E., and Schnabel, R.B., 1983, Numerical methods for unconstrained optimization and 
nonlinear equations: Englewood Cliffs, New Jersey, Prentice-Hall, 378 p.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., 1992, Numerical recipes: 
Cambridge, Great Britain, Press Syndicate of the University of Cambridge, 2nd Edition, 963 p. 
Seber, G.A.F., and C.J. Wild 1989, Nonlinear Regression, John Wiley & Sons, NY, 768 p.
Stewart, G.W., 1972, Introduction to matrix computations: New York, Academic Press, 423 p.

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