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Site Sampling: Assessing Residual Uncertainty
Échantillonnage du site : évaluation de l'incertitude résiduelle
Fenton G.A.
Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands
Department of Engineering Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada
Hicks M.A.
Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands
ABSTRACT: Geotechnical design is plagued by the uncertainty associated with site characterization. Common questions are “How
many samples should be taken?” and “How do these samples reduce my uncertainty?” Of considerable interest is the question “What
site sampling plan will give the best cost to effectiveness ratio?” This papers looks specifically at the effect of the number of samples
on residual uncertainty. The results can be used to quantitatively select the required number of samples needed to achieve a target
maximum residual uncertainty level. To study this problem, a square domain is selected (the site) and a stationary Gaussian random
field is simulated within the domain (the random soil properties). The random field is sampled at a series of locations and a trend is
estimated from the samples. The trend is then removed from the random field and the residual random field is statistically analyzed to
determine various measures of the effectiveness of the sampling scheme. These measures include: 1) the variance of the residual field
average (i.e. does the estimate represent the average?), 2) the residual standard deviation (i.e. how much residual uncertainty
remains?), and 3) the residual correlation length (i.e. how does trend removal affect the perceived correlation lengths?).
RÉSUMÉ : Le design géotechnique est traditionnellement affecté par des incertitudes associées à la caractérisation du site. Les
questions les plus courantes sont : combien d’échantillons devraient être prélevés ? Comment ces échantillons peuvent réduire mon
incertitude ? Un des intérêts les plus importants vient de cette question. Quel plan d’échantillonnage du site donnera le meilleur
coefficient d’efficacité? Cet article examine spécifiquement l’effet du nombre d’échantillons sur des incertitudes résiduelles. Les
résultats peuvent être utilisés pour quantifier et sélectionner le nombre demandé d’échantillons nécessaires pour atteindre un objectif
d’incertitude maximal avec le niveau résiduel. Pour étudier ce problème, un domaine carré est sélectionné (le site) et un champ
gaussien aléatoire stationnaire est simulé dans le domaine (les propriétés du sol aléatoires). Le champ aléatoire est échantillonné à une
série d’emplacements et une tendance a été estimée à partir de l’échantillon. La tendance retirée du champ aléatoire et le champ
résiduel aléatoire est statistiquement analysées afin de déterminer les mesures diverses de l’efficacité du plan d’échantillonnage. Ces
mesures comprennent : 1) la variance de la moyenne de champ résiduel, c’est à dire comment la tendance estimée représentent la
moyenne réelle sur le terrain ? 2) l’écart type résiduel, c’est-à-dire à quel degré d’incertitude résiduelle demeure, et 3) la valeur
longueur résiduelle de corrélation, c’est-à-dire comment la suppression tendance affecte les longueurs de corrélation ?.
KEYWORDS: geotechnical design, site characterization, residual uncertainty, sampling, required number of samples, sampling plans.
1 INTRODUCTION
Site characterization is clearly an essential component of any
geotechnical design and a great deal of effort has been devoted
over recent decades on how to best perform such a
characterization. How many samples should be taken? How
should these samples be used in the design process?
The ground is one of the most complex of engineering
materials, and yet is the most fundamental, in all senses of the
word. While steel, concrete, and wood, for example, have fairly
well established and relatively small uncertainties, the ground
can vary by orders of magnitude from site to site, and even
within a site.
As a result of the large uncertainty in the ground, all
geotechnical designs must start with a geotechnical
investigation so that the best “nominal” or “characteristic”
ground parameters can be used in the design process.
Traditionally, the intensity of the site investigation has not been
particularly important, so long as a reasonable estimate of the
characteristic design values can be estimated. However, recent
impetus has been towards providing reasonable estimates of the
reliability of designed geotechnical systems. In order to do so
the ground used to provide the geotechnical resistance needs to
be properly evaluated, in both the mean and the covariance.
In this paper, the ability of a soil sampling scheme to predict
the actual mean, variance, and correlation length of the soil at a
site is investigated. A key question is how does the number of
samples affect the accuracy of the estimate? Or, put another
way, how many samples are required to achieve a certain
desired accuracy? The answer is found by considering a square
site and using random field simulation to generate realizations
of the soil properties over the site, sampling each realization,
and then comparing the estimated mean, variance, and
correlation length to the ‘true’ values. The goal here is to
investigate the discrepancies between the estimated statistics
and the true ‘local’ statistics, the latter obtained by sampling the
field at all locations. Note that the ‘local’ statistics will differ
from the population parameters,
(mean),
(standard
deviation), and
(correlation length), which are used by the
random field generator, due to the fact that the local statistics
are derived from a single realization. In detail, the soil is
represented by a stationary Gaussian random field,
x
X
, at
spatial position , which is simulated within the domain and
sampled at
x
s
n
locations. The samples are then used to estimate a
mean trend,
x
ˆ
, which can then be compared to the field
realization to assess its ability to represent the actual mean
trend. Defining the residual to be
ˆ
( )
( )
( )
X X
r
x
x x
(1)
then
ˆ
x
is a good estimate of the mean trend if
is
generally small. If the site is sampled at all locations, then
r
X
x
ˆ
can be taken to be equal to
X
x
, in the event that a pointwise
trend is assumed (as in Kriging), in which case
0
r
X
x
everywhere. Sampling at all locations is the best case since
there is then minimum residual uncertainty (zero in the case of
Kriging).
