Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 3

intuitive, but is purely empirical with no assurance that it will be effective in all cases. Ching
et al. (2010) have demonstrated that even a more sophisticated variance-weighted averaging
approach is less effective than a rigorous Bayesian updating approach. In addition, it is a
good geotechnical practice to cross-validate interpretation of soil properties from different
sources of information, given the significant assumptions and empiricism underlying most
bivariate correlations. In our opinion, combining multivariate information in a rational and
systematic way is a major useful application of probability theory, because it is impossible
even for an expert probabilist to combine multiple sources of uncertain information that are
dependent on each other in some complex way using pure physical intuition or engineering
judgment.
The purpose of constructing multivariate distributions for soil parameters is to emulate
the information content of a real site as realistically as possible. Ching et al. (2014a) called
data simulated with the intent of replicating the multidimensional correlation structure
underlying a basket of laboratory and field tests as “virtual site” data. It is currently not
possible to emulate every aspect of a real site. One hopes to construct a virtual site that
would at least reproduce realistic data within a scope of interest. In this chapter, the scope
is to reproduce the information content arising from a typical basket of laboratory and field
tests conducted in a clay site for the purpose of estimating important design parameters
such as the undrained shear strength and the preconsolidation stress. The critical feature
here is the consistent and realistic coupling of different test data, which are achieved using
a multivariate normal distribution. Data from different tests will be coupled (or correlated
in the context of a multivariate normal distribution), because they measure the same mass
of soil, although they could be possessing different aspects of soil behavior under different
boundary conditions and over different volumes. The current virtual site does not model
spatial variability. This rather limited virtual site model will be expanded to include more
tests and to improve realism (such as spatial variability) in the future. The purpose of devel-
oping a virtual site is not to replace the actual site investigation. It is meant to serve as a tool
for engineers to explore design savings accrued from conducting better and/or more tests in
the context of RBD.
The idea of simulating a “virtual site” is not new. For example, Jaksa et al. (2003, 2005)
and Goldsworthy et al. (2007) used three-dimensional random fields and Monte Carlo sim-
ulation to simulate the spatially variable elastic modulus of a “virtual” site. Each spatially
variable realization constitutes a plausible full-information scenario. Site investigation is
then carried out numerically by sampling the continuous random field at discrete locations.
The site investigation data so obtained constitute the typical partial-information scenario
commonly encountered in practice. The scope of these studies was to quantify the dis-
crepancy between the design based on partial spatial information (information at discrete
points typically measured in a site investigation) and the design based on complete spatial
information (an ideal state where all subsurface information have been mapped/character-
ized as continuous functions of spatial coordinates without measurement errors). The focus
of these studies is to evaluate how increasing the number of measurement points for a single
test will affect design decisions. Our “virtual site” examines the complementary aspect of
how increasing the number of tests at a given depth will affect design decisions. Spatial vari-
ability is not considered in this chapter.
The basic goal of this chapter is to explain how a useful multivariate non-normal probabil-
ity model can be constructed from actual geotechnical data. Standard undergraduate texts
on probability and statistics present univariate probability models, which are not adequate
for geotechnical data. The construction method is explained by building “need-to-know”
theoretical tools incrementally: (1) single normal random variable, (2) bivariate normal vec-
tor as two normal random variables coupled by a correlation coefficient, (3) multivariate