Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 42

always <1. A bounded random variable is nonnormal. This source of nonnormality can
affect the probability of realizing small values significantly (behavior of lower probabil-
ity tail). For example, if s u is normally distributed with a mean = 100 kPa and a standard
deviation = 40 kPa, the probability of realizing values lower than 25 kPa is 0.03. If s u is
log-normally distributed with the same mean and standard deviation, the probability is
0.0003—two orders of magnitude lower than the corresponding probability under the nor-
mal assumption. The moral of the story here is that it is important to include a lower bound
in a probability model when it exists. It is not recommended to retain the normal assump-
tion out of theoretical simplicity or expediency. A less clear-cut situation arises when one
appeals to engineering judgment or experience to impose a lower bound. For example, one
can venture to stake a lower bound of 10 kPa for s u . The presence of very soft clay (classi-
fied by Terzaghi et al. [1996] as s u < 12.5 kPa) is easy to detect, because it cannot be easily
retained in a conventional sampler. Nonetheless, there is a possibility for small lenses of very
soft clay to exist, because site investigation is typically too limited to eliminate this possibil-
ity with 100% confidence. One can appeal to engineering judgment say based on extensive
knowledge of sites with similar geology elsewhere, but the reason for adopting s u > 10 kPa is
admittedly weaker than s u > 0 kPa.
It is useful to briefly review the concepts of censored and truncated data that are distinct
from the concept of bounded nonnormal random variables. The SPT blow count value
(N-value) lies between 1 and 100. The lower bound is the minimum blow count while the
upper bound is imposed to avoid damage to the sampler. When an SPT-N value of 100 is
recorded, we know that the actual value is at least 100. This type of “censored data” is
commonly produced by field tests, because there are physical limits to the strength that
can be measured without damaging or without exceeding the sensitivity of the test equip-
ment. Another example is a proof load test that is meant to assure an engineer that the
capacity of a pile, soil nail, or ground anchor is larger than a certain value. This type of
“censored data” appears naturally when the intention is to avoid testing a component to
ultimate failure.
In an SPT record, we know the number of data points with a value of 100. There are
examples where the number of data points exceeding a threshold is unknown. For example,
a contractor provides a record of pile head settlement under working load. The reported
settlements are <25 mm. The contractor did not record the number of piles with pile head
settlement exceeding 25 mm, because these piles are deemed defective and new piles are
automatically driven nearby to compensate for these defective piles. We do not know the
number of piles settling more than 25 mm under working load. The SPT record is called
right-censored data. The pile settlement record is called right-truncated data.
It is clear from the above brief digression that nonnormality is only one aspect of statisti-
cal characterization of actual data. We focus on characterizing histograms using nonnormal
probability models below. Many nonnormal models can be found in standard statistical
texts, but only the Johnson system of distributions is reviewed below. The reason is that
the Johnson system of distributions can be transformed into a standard normal distribution
using closed-form equations. This computational advantage is quite significant, because the
only practical method to model multivariate non-normal data is to transform each compo-
nent individually into standard normal data and to link these transformed standard normal
components using a multivariate normal distribution. We have highlighted the usefulness
of multivariate models compared to univariate models. We have noted that a multivariate
model is a more natural fit to geotechnical engineering data, because a number of laboratory
and field tests are typically conducted in a site investigation program. With these observa-
tions in mind, it is obvious that fitting each parameter to a probability distribution is only an
intermediate goal and cannot be conducted without regard to the more stringent theoretical