Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 70

Hence, the bivariate PDF of (Y 5 , Y 6 ) is

yb
a

− yb

a

−

5 Y 5 6 Y 6

fy (,y) = f x = +× b a sinh

− 1

, x = ba + × si nh −

1

Y 56 X 5 X 5 X 5 6 X 6 X 6

Y 5 Y 6

d
dy

x d

dy

x

×

5
5

×

6
6

(1.121)

Before the conditioning,

0
0

1001
0011

.

= ˆ
Š

'
'

=

µ []

1 [] 11

C (1.122)

.

The updated mean vector and covariance matrix have been computed in Equations 1.115
and 1.116:

− 0 769

0 0189

. 0 359 . 0 . 204

[] 1 [] 1 [] 11 [] 11

µµ = = CC = = (1.123)

update update

. 0 204 . 0 451 .

The unconditioned and conditioned bivariate PDFs for (Y 5 , Y 6 ) are shown in Figure 1.36 .
The cross-symbol in the figure indicates the actual measured values of (Y 5 , Y 6 ). The reduc-
tion in the scatter is even more pronounced in Figure 1.36 than in Figure 1.35 .

(a) 4 (b) 4

3 3

2 2

1 1

0 0

-1 -1

-2 -2

-3 -3

-4 -4

-5 -5

-2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5

Y 5 = ln(σ′ p / P a ) Y 5 = ln(σ′ p / P a )

Figure 1.36 Unconditional and conditional bivariate PDFs of (Y 5 , Y 6 ), based on the information Y 3 = 1.793,
Y 4 = − 0.912 , Y 8 = 0.504, and Y 10 = 1.265; (a) unconditional PDF and (b) conditional PDF. (The
cross indicates actual measured values of (Y 5 , Y 6 ) reported by Rochelle, P.L., Zebdi, M., Leroueil,
S., Tavenas, F., and Virely, D. 1988. Proceedings of the 1st International Symposium on Penetration
Testing , Orlando, 2, 831-841.)