Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 54

Table 1.14 Types and parameters for the underlying distributions of (Y 1 , Y 2 , Y 3 )

Distribution parameters

Random variable Soil parameter Distribution type a X b X a Y b Y

Y 1 LI SU 1.434 − 1.068 0.629 0.358

Y 2 SU 4.495 1.199

ln( σ′ p / P a ) − 9.572 − 4.481

Y 3 ln( S t ) SU 2.453 1.888 0.343

− 2.233

random seed initialized by randn('state', 13). The underlying marginal distributions for (Y 1 ,
Y 2 , Y 3 ) are Johnson SU, and the underlying C matrix is

1 0 57 059
0571005
059005

− . .

C = − . . (1.101)

. . 1

The statistics of these underlying distributions are given in Table 1.14 . (Y 1 , Y 2 , Y 3 ) are
realistic clay parameters: Y 1 represents LI, Y 2 represents the logarithm of the normalized
preconsolidation stress ln (

σ p P / (P a is one atmosphere pressure), and Y 3 represents the
logarithm of sensitivity ln( S t ). The scatters and trends in Figure 1.25 are similar to those
observed in the Clay/10/7490 database (Ching and Phoon 2014a). Hence, although the data
are simulated, they are realistic rather than mathematically contrived with no relation to
geotechnical engineering data. Ching et al. (2014a) called data simulated from a realistic
geotechnical engineering context as “virtual site” data. It is significant that nonlinear cor-
relation trends are observed among the simulated data points. Namely, (Y 1 , Y 2 , Y 3 ) are mul-
tivariate non-normal. In fact, (Y 1 , Y 2 , Y 3 ) are individually non-normal. Figure 1.26 shows
the histograms of the simulated (Y 1 , Y 2 , Y 3 ) data.
Given the simulated (Y 1 , Y 2 , Y 3 ) data, the procedure described in Section 1.5.3 is used
to identify the types and parameters for the Johnson distribution. The identified types
and parameters are given in the parentheses in Table 1.15 . This table should be compared
to Table 1.14 . Note that Table 1.14 is the exact solution while Table 1.15 is the estimated

′ )

500 250 250

400 200 200

300 150 150

200 100 100

100 50 50

0 0 0

0 2 4 -2 0 2 4 0 2 4 6

Y 1 = LI

Y 2 = ln(σ′ p / P a ) Y 3 = ln( S t )

Figure 1.26 Histograms of the simulated non-normal (Y 1 , Y 2 , Y 3 ) data. The solid lines are the fitted Johnson
distributions.