Environmental Engineering Reference
In-Depth Information
Table 2.9
Comparison of simulation results associated with different copulas for CU dataset
Gaussian
Plackett
Frank
No.16
τ
Between measured
c
and
ϕ
−
0.544
τ
Between simulated
U
1
and
U
2
−
0.546
−
0.551
−
0.550
−
0.555
τ
Between simulated
X
1
and
X
2
−
0.546
−
0.551
−
0.550
−
0.555
ρ
Between measured
c
and
ϕ
−
0.702
ρ
Between simulated
U
1
and
U
2
−
0.745
−
0.741
−
0.754
−
0.757
ρ
Between simulated
X
1
and
X
2
−
0.749
−
0.719
−
0.710
−
0.684
2.5 IMPaCt oF CoPula SeleCtIon on retaInIng
Wall relIabIlItY
A copula-based approach for modeling and simulating the bivariate distribution of
shear strength parameters was presented in the previous sections. It is concluded that
the Gaussian copula may not provide the best fit to the measured dependence structure
between shear strength parameters. However, in geotechnical practice, the Gaussian cop-
ula is often adopted to characterize the dependence structure among multiple geotechnical
parameters in terms of Nataf transformation or the translation approach (Phoon et al.
2010; Dithinde et al. 2011; Ching and Phoon 2012, 2013). In particular,
Chapter 1
also
uses the Gaussian copula for constructing the multivariate distribution of multiple geo-
technical parameters.
There are three practical reasons to stick to the Gaussian copula: (1) Reliability-based
design in geotechnical engineering often involves multiple geotechnical parameters. The
Gaussian copula is one of the relatively few copulas that have practical
n
-dimensional
generalizations; (2) small sample size is a real feature of geotechnical data. On the basis
of the limited data, only marginal distributions and covariance underlying geotechnical
parameters can be determined, which are the only required inputs of the Gaussian copula.
In addition, there is large statistical uncertainty in identifying a best-fit copula using lim-
ited data even for the bivariate case; and (3) complete multivariate data (i.e., multiple geo-
technical parameters for a single depth at a site known simultaneously) are not available
in geotechnical practice. With the incomplete multivariate data (i.e., multiple geotechnical
parameters for a single depth at a site known partially) at hand, the Gaussian copula is
the only copula that can be constructed using bivariate data as demonstrated in Section
1.7 of
Chapter 1
.
Although the application of copulas is associated with the above limitations, there are
still three cases in which the copula approach can be applied: (1) There is one pair or
multiple independent pairs of correlated geotechnical parameters in a specific geotechni-
cal problem such as the shear strength parameters studied in this chapter; (2) the sample
size of the geotechnical data is sufficiently large enough to accurately identify the best-fit
copula; and (3) complete multivariate data are available in geotechnical engineering. In this
case, the multivariate
t
copula as well as the multivariate Gaussian copula can be adopted
to model the multivariate distribution of the multiple geotechnical parameters. When the
above conditions are satisfied, it is desirable to try other copulas (not just the Gaussian
copula) since the copula type will influence the geotechnical reliability significantly (e.g.,
Tang et al. 2013a). With this in mind, this section presents some investigations to show the
effect of copula selection on reliability. The reliability of a retaining wall with overturning
failure mode is studied.
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