Maximum likelihood principle and its application in soil liquefaction assessment - Risk and Reliability in Geotechnical Engineering - page 193

Table 4.4 Calibrated coefficients and model probabilities of the liquefaction models analyzed

Calibrated coefficients Model

probability
posterior

Model
designation Equation for calculating P L b 0 b 1 b 2

1

M 1 (Logistic) P 21.931 −0.123 6.181 0.17

=

{ }

L

1 exp b q b ln( CSR )

+ − + +

0 1 tNcs 1 , 2

M 2 (Probit) P b bq b SR ln( ) 12.939 −0.072 3.635 0.33

= Φ + +

L 0 1 t Ncs 1 , 2

{ }

( )

M 3 (Log-log) P exp xp b bq b SR ln( ) 14.854 −0.080 4.019 0.16

= − −+ +

L 0 1 t Ncs 1 , 2

{ }

M 4 (C-log-log) P 1 exp xp b bq b SR ln( ) 16.148 −0.093 4.721 0.34 =− − + +

L 0 1 t Ncs 1 , 2

CSR , is used because it has been shown that the variable CSR is lognormally distributed
(e.g., Hwang et al. 2005). Based on the generalized linear models in Table 4.3, the four lique-
faction probability models shown in Table 4.4 ( denoted as M 1 , M 2 , M 3 , and M 4 , respectively)
are examined in the subsections that follow.

4.3.2 Calibration database
The database adopted for calibrating the generalized linear models consists of 152 cases
taken from Robertson (2009) and an additional 13 cases taken from Moss et al. (2011). The
152 cases were derived by Robertson (2009) through a rescreening of the cases previously
compiled and evaluated by Moss et al. (2006), which yielded 116 liquefied cases and 36 non-
liquefied cases. The other 13 cases (9 liquefied cases and 4 nonliquefied cases) in the adopted
database are taken from Moss et al. (2011). Thus, the new calibration database consists of
a total of 165 cases (125 liquefied cases and 40 nonliquefied cases). The cases included in
this database are listed in Table 4.5. Interested readers are referred to Ku et al. (2012) for
further details.

4.3.3 evaluation of sampling bias
As noticed in Cetin et al. (2002), sampling bias may exist in the database that is used for cal-
ibrating models that are in turn used to calculate liquefaction probabilities. Evaluating the
effect of sampling bias in model calibration requires the knowledge of Q p , which in practice
is very difficult to obtain. Cetin et al. (2002) assessed the value of Q p through a survey of
expert opinions. Although the exact value of Q p used in Cetin et al. (2002) was not directly
reported, it can be back-calculated as follows: let Q s0 , w L 0 , and w NL 0 denote Q s , w L , and
w NL of the database used in Cetin et al. (2002). As there are 112 liquefied and 89 nonlique-
fied cases in the calibration database used in that database (Cetin et al. 2002), Q s 0 = 0. 557.
The adjusting weights used in Cetin et al. (2002) satisfied the following relationship: w NL 0 /
w L 0 = 1.5. Substituting Equations 4.16 and 4.17 and Q s0 = 0.557 into this relationship yields
Q p = 0.456. Although we adopt Q p = 0.456 in this chapter, it is understood that the true
value of Q p is very difficult to estimate correctly. Further discussions of this issue can be
found in Juang et al. (2009) and Zhang et al. (2013).
The calibration database used in this example consists of 125 liquefied cases and 40 non-
liquefied cases. Thus, Q s = 125/165 = 0.758. Based on Equations 4.16 and 4.17 , the adjusting
weights are determined to be w L = 0.601 and w NL = 2 . 247.