Technical information

4-9
and the subgrade modulus (
3
E ). In order to derive their numerical values, an objective
(scalar) function describing the agreement between the model and test data was
formulated. First, for each gauge separately, out of the total twelve gauges available, an
error term was defined as follows:
[]
∑
=
−⋅=
N
n
nng
RR
N
ERR
1
2
modelAPT
1
.................................................................... (4.2.6)
in which
N is the number of data points used for the comparison for the
th
g gauge
(i.e.,
25=N ).
APT
R
represents the measured APT response of either stress or strain and
model
R is the corresponding isotropic LET response. Note that
g
ERR has the same units
as
APT
R (or equivalently
model
R ) and is always positive. Next, these individual errors
were combined to formulate a unitless global error term, defined as follows:
∑
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⋅=
G
g
g
g
ERR
ERR
G
ErrorGlobal
1
1
)min(
1
_
..................................................... (4.2.7)
where
G is the total number of gauges considered in the analysis (i.e., 12=G ), and
)min(
g
ERR
represents the lowest achievable error between the model and the test data
for the
th
g gauge. The numerical value of )min(
g
ERR was obtained by employing an
over-fitting technique; i.e., the layer moduli were first manipulated using the
optimization algorithm in an effort to separately minimize each of the individual errors
(equation 4.2.6).
Note that )min(
g
ERR is always greater than zero; even if the model were
perfect, all test data contain some random noise. However, the global error term can, in
principal, equal zero. This situation occurs mathematically when all individual errors
are minimal. Therefore, equation 4.2.7 serves as a weighted average of the individual
errors, making sure that neither of the gauge readings is underweighted or overweighted
in the backcalculation process compared to the others. In order to enable a direct
comparison between the global error for pass #5,000 and pass #80,000, values of