Environmental Engineering Reference
In-Depth Information
5.7 ConCluDIng reMarkS
Bayesian analysis and updating of geotechnical models is a flexible and consistent frame-
work for combining information from different sources in a single model and prediction
tool. Its application in research is quickly growing, and geotechnical engineers in practice
start to understand its relevance and potential, as they realize that purely deterministic
models do not facilitate including measurements and observations in a consistent man-
ner. There is also a growing acceptance that increasing the accuracy of deterministic
models is not sufficient to explain the behavior of an inherently uncertain and random
material such as soil.
As demonstrated in this chapter, the application of Bayesian updating is not very difficult
from a mathematical or even computational point of view. Nevertheless, Bayesian updating
is not easy and requires training and experience. The difficulties in Bayesian updating lie in
the probabilistic modeling of (a) the randomness and uncertainty in the material and loading
parameters, (b) the measurements, and (c) the model errors. A particular difficulty in geo-
technical engineering, upon which we have touched occasionally, is the need to represent the
spatially variable soil properties. Ideally, a random field modeling is applied, but for practi-
cal purposes (to reduce the computational effort), it is often necessary to represent the soil
by representative values, possibly in combination with correction factors. While this reduces
the computational efforts, it does not simplify the probabilistic modeling, even though this
is— unfortunately—believed by many. If the variability of the soil is not represented consis-
tently, results are obtained that are misleading and wrong. Multiple criticisms that were pre-
viously made of probabilistic analysis, in general, and Bayesian analysis, in particular, stem
from problems that originate in incorrect probabilistic modeling. It has to be understood by
geotechnical engineers that a basic introduction to statistics is not sufficient for carrying out
probabilistic and reliability analysis of geotechnical systems.
The computational aspects of Bayesian updating are more straightforward and hence
less critical than the modeling aspects. As we have shown in this chapter, once the deter-
ministic and probabilistic models and likelihood functions are defined, computations can
be performed relatively easily, both conceptually and implementation wise. For updating
individual parameters, it is sufficient to numerically solve integrals or use analytical solu-
tions through conjugate priors. For updating of the models or their parameters by means of
a probabilistic inverse analysis, the algorithms presented in Section 5.4 are readily available.
For example, the simple rejection-sampling approach can be implemented with just a few
lines of code. If more efficient algorithms are needed, the BUS approach only necessitates
structural reliability algorithms, which are available in many free and commercial codes.
While we have focused on Bayesian updating of mechanical (geotechnical) models,
Bayesian updating is also a viable tool for other probabilistic models in the context of geo-
technical construction. In particular, probabilistic models of construction performance and
risk can be updated with observations made during the construction progress. As an exam-
ple, Špačková et al. (2013) and Špačková and Straub (2013) show how observed progress
in tunnel construction performance can be used to update the prediction of future perfor-
mance and the total construction time through Bayesian analysis.
aCknoWleDgMent
We acknowledge insightful discussions we had with Dr. Olga Špačková on the applica-
tion and the interpretation of the Bayesian updating results as well as her input on the
illustrations.
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