Bayesian analysis for learning and updating geotechnical parameters and models with measurements - Risk and Reliability in Geotechnical Engineering - page 226

standard deviation was assumed as fixed σ φ = 3° and only the mean friction angle μ φ was
learned. For the reliability analysis, that is, for prediction, μ φ was taken as the representative
characteristic. As was discussed, this approach is a nonconservative approximation, which
is exact for a soil with correlation length zero. In real applications, this approximation must
be addressed by correcting for the effect of a larger correlation length (Griffiths and Fenton
2001). Optimally, however, an explicit random field modeling approach is selected.
The second modeling approach for spatially variable soil properties is to model them at
each location explicitly. This is the random field approach , where a property X is modeled
through a random variable X ( z ) at each location z (Rackwitz 2000; Baecher and Christian
2008). To numerically represent such random fields, it is necessary to discretize them, for
example, by the Karhunen-Loève expansion (e.g., Betz et al. 2014b). The random field is
then described by a discrete set of random variables, which are part of X ; the number of
random variables in this set can be considerable.
If the soil is modeled through random fields, that is, by explicitly modeling the probabilistic
spatial variation of the soil, it is necessary to define the correlation structure. This a priori cor-
relation model should be selected carefully, as it can have a significant influence on the results
of the Bayesian analysis: Shorter correlation lengths (i.e., larger spatial fluctuations) signify that
the effect of a measurement beyond its immediate vicinity is limited, and longer correlation
lengths (smaller spatial fluctuations) lead to measurements having a more global effect, that is,
the impact of the measurement becomes larger. Note that a model without random field implic-
itly assumes full correlation among the soil properties in the areas represented by the same
random variable. The reader is referred to Rackwitz (2000) for more information on modeling
soil properties with random fields. Once a prior random field is established, the Bayesian analy-
sis follows the general procedure presented in this chapter, with the parameters describing the
random field being part of X . This is independent of the way the random field is discretized.

5.3.2 Computing the reliability and risk based on the prior model
Once the prior probabilistic model of X is defined, it is possible to compute the a priori reli-
ability of the geotechnical construction. In the general case, the failure event (both ultimate
and serviceability) is described through LSFs g ( X ). The probability of failure is

Pr() Pr(( ) F = g X ≤ 0 ), (5.12)

which is computed from

x

Pr() F = f

X xx

() . d

(5.13)

g () ≤ 0

The corresponding reliability index is

Φ 1 [Pr( )].

(5.14)

β=− F

Φ −1 is the inverse of the standard normal CDF.
Pr( F ) and β can be computed by approximating the integral in Equation 5.13 using the
classical methods of structural reliability, such as first- and second-order reliability methods
(FORM/SORMs) or sampling-based methods including crude Monte Carlo, subset simula-
tion, and importance sampling. The application of these methods to geotechnical problems
is presented in detail in Phoon (2008).
It is noted that when weakly informative priors are selected for some of the random vari-
ables in X , it is to be expected that the a priori reliability is low. In these situations, it is