Methods and guidelines for effective model calibration

References
Brockwell, P.J and Davis, R.A., 1989, Time series, Theory and methods: New York, Springer-Ver-
lag, 519 p.
Carrera, Jesus and Neuman, S.P., 1986, Estimation of aquifer parameters under transient and 
steady-state conditions: Water Resources Research, v.22, no. 2, p. 199-242.

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77
APPENDIX B: CALCULATION DETAILS
Three aspects of the calculations needed for the nonlinear regression methods described in 
this work require more detailed explanation. These include a more detailed description of the ma-
trices and vectors of equations 2, 3, 4, and 26, discussing a possible addition to equation 4a, and 
calculation of the damping parameter and convergence of equation 4.
Vectors and Matrices for Observations and Prior Information
The primary vectors and matrices of concern in nonlinear regression are the measured val-
ues of vector , the simulated values of vector , the sensitivities of matrix X, the weights of ma-
trix 
ω
, the residuals of vector e (equal to 
) and the true errors of vector 
ε
. These vectors and 
matrices, including terms for both the observations and prior information used in the regression, 
are as follows. Except for e and 
ε
, these vectors and matrices are used in equations 2, 3 and 4a of 
this report. The vectors e and 
ε
are included here because they appear frequently in regression lit-
erature. A few common relationships are displayed at the bottom of this section using vector nota-
tion.
y = 
,
X = 
,
ω
=
V is the weighting for the observations; U is the weight matrix for the prior information, and it is 
assumed that the true errors in the observations are independent of the true errors in the prior 
information.
y
y’
y
y’
–
y
1
y
2
y
ND
P
1
---------
P
2
P
NPR






































•
•
•
•
•
•
•
•
x
1 1
,
x
1 2
,
…
x
1 NP
,
x
2 1
,
x
2 2
,
…
x
2 NP
,
x
ND 1
,
a
1 1
,
--------------
x
ND 2
,
a
1 2
,
--------------
…
…
------------
x
ND,NP
a
1 NP
,
-------------------
a
2
1
,
a
2 2
,
…
a
2 NP
,
a
NPR,1
a
NPR,2
…
a
NRP,NP
•
•
•
•
•
V 0
0 U

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78
y’= 
,
e =
, and
ε
=
These definitions are used in the following sections of this appendix.