Methods and guidelines for effective model calibration

Weighted Observations Versus Weighted Simulated Values and Correlation Coeffi-
cient R
Ideally, simulated values are close to observations, so that weighted simulated values are 
close to weighted observations. When weighted observations are plotted against weighted simulat-
ed values, the hope is that the points fall close to a line with slope equal to 1.0 and an intercept of 
zero. A summary statistic that reflects how well this is accomplished is the correlation coefficient 
between the weighted observations and the weighted simulated values. The correlation coefficient, 
R, is calculated as (Cooley and Naff, 1990, p. l66):
R = 
(18)
where , , and were defined for equation 2. m
y
and m
y
’
are vectors with all ND elements equal 
to:
(19)
(20)
Thus, 
is simply a vector with each component equal to the average of the weighted dependent-
ω
1 2
⁄
y
m
y
–
(
)
T
ω
1 2
⁄
y’ m
y’
–
(
)
ω
1 2
⁄
y
m
y
–
(
)
T
ω
1 2
⁄
y
m
y
–
(
) ω
1 2
⁄
y
m
y’
–
(
)
T
ω
1 2
⁄
y
m
y’
–
(
)
[
]
1 2
⁄
---------------------------------------------------------------------------------------------------------------------------------------------------------------------
y y’
ω
m
y
i
ω
1 2
⁄
y
(
)
q
q
1
=
ND
∑
ND
⁄
=
m
y’
i
ω
1 2
⁄
y’
(
)
q
q
1
=
ND
∑
ND
⁄
=
m
y
i

---

22
variable observations, and 
is an analogous vector using the weighted simulated values. Gen-
erally, R needs to be greater than 0.90. When there is prior information, R also is calculated with
, , and augmented with prior information as in Appendix A, in which case ND+NPR replaces 
ND when calculating 
and 
.