Inaccuracies Associated with the Current Method

for Estimating Random Measurement Errors

by

M. B. Jaksa, P. I. Brooker and W. S. Kaggwa

Department of Civil and Environmental Engineering

University of Adelaide

Research Report No. R 122

December, 1994

i

ABSTRACT

This report examines a commonly used procedure, proposed by Baecher (1982),for separating the random measurement error associated with a particular testfrom the inherent spatial variability of the geological material. It is shown thatthe method, while well-founded, requires a number of factors to be investigated,before conclusions can be made regarding the random measurement error of aparticular test. Two case studies are presented, and the sensitivity of the resultsis tested with regard to these factors.

ACKNOWLEDGMENTS

The authors wish to thank the Agency of Road Transport, and in particular,R. Washyn, R. Herraman and I. Forrester, for the use of the Agency’s drillingrigs and technical staff, for without whose generosity the research carried out atthe South Parklands site could not have taken place. In addition, thecooperation of the City of Adelaide, and in particular, A. A. Taylor andM. Underhill, for providing access to this site. The authors also wish togratefully acknowledge Australian National, and in particular P. Gaskill, fortheir generosity and assistance, for allowing access to the Keswick site.

Furthermore, the authors would like to thank the technical staff of theDepartment of Civil and Environmental Engineering, University of Adelaide:T. Sawosko for his significant contribution throughout the field testing;C. Haese for the design and coordination of the drilling apparatusmodifications; L. Collins and R. Kelman for the fabrication of the drillingapparatus, and; B. Lucas for the design and construction of the data acquisitionsystem. Thanks are due also to final year civil engineering students, D. vanHolst Pellekaan and J. Cathro, for their assistance in performing the horizontalcone penetration testing.

ii

CONTENTS

ABSTRACT ...................................................................................................i

ACKNOWLEDGMENTS..............................................................................i

CONTENTS ..................................................................................................ii

1. INTRODUCTION .......................................................................................1

2. TECHNIQUES USED TO DESCRIBE UNCERTAINTY .....................2

2.1 Time Series Analysis.............................................................................3

2.2 Geostatistics ..........................................................................................5

3. ESTIMATION OF MEASUREMENT ERRORS....................................8

4. INADEQUACIES OF PRESENT METHOD.........................................10

4.1 Nugget Effect ......................................................................................11

4.2 Sample Spacing ...................................................................................11

4.3 Stationarity of Data .............................................................................12

5. CASE STUDIES ........................................................................................13

5.1 Vertical Spatial Variability - South Parklands Site ............................13

5.2 Horizontal Spatial Variability - Keswick Site ....................................19

6. CONCLUSIONS........................................................................................ 24

7. REFERENCES ..........................................................................................25

1

1. INTRODUCTION

All engineering design incorporates uncertainty in one form or another. In fact,the overall, or total, uncertainty associated with any particular design mayincorporate one or more of the following:

• uncertainties due to variabilities of material properties;• inconsistencies associated with the magnitude and distribution of design

loads;• uncertainties associated with the measurement and conversion of design

parameters;• inaccuracies that arise from the models which are used to predict the

performance of the design;• anomalies that occur as the result of construction variabilities; and• gross errors and omissions.

Several authors have proposed probabilistic models for incorporating suchuncertainties in the design process (Baecher, 1986; Orchant et al., 1988; Kay,1990; Kay et al., 1991). In essence, each of these models proposes that the totaldesign uncertainty is the sum of each of the individual contributinguncertainties. This report focuses on one aspect of the overall designuncertainty, that is, the uncertainty associated with the in situ test measurementof geotechnical materials.

There are two primary sources of uncertainty which contribute to the variabilityof in situ test measurements, namely, spatial variability, and measurementerror. Spatial variability is the natural variation that soils and rock exhibit,from one location to another, as a result of the myriad and complex processeswhich form them, and to which they have subsequently been subjected.Measurement error, on the other hand, is the error associated with themeasurement process itself, which is explained in greater detail below. Thesetwo sources of uncertainty are related by the following relationship:

mv = v + ε (1)

where: mv measurement of parameter v;v true value of the parameter; andε measurement error.

Orchant et al. (1988) proposed the following model for describing the totaluncertainty, or variance, σT

2, associated with measurement:

22/

2mvsT σ+σ=σ (2)

2

where:22

/22

ropem σ+σ+σ=σ (3)

and:σs v/

2 variance due to spatial variability;

2mσ variance due to measurement error;

2eσ variance due to equipment effects;

2/ opσ variance due to procedure and operator effects;

and,2rσ variance due to random measurement errors.

Uncertainties associated with equipment effects can occur as a result ofelectrical drift, non-linearities, and out-of-calibration errors related to theelectrical transducers and mechanical devices, which each have different levelsof reliability. Procedure and operator effects cause variabilities inmeasurements as a result of inadequate, or limited, testing standards, non-compliance with these standards, as well as uncertainties as a result of differentoperators. Both of these effects, equipment, and operator and procedure, aresystematic, or bias (Lumb, 1974), errors which consistently under-estimate, orover-estimate, a measured parameter. Random measurement errors, on the otherhand, are the variation between measurements that cannot be directly attributedto the inherent variability (spatial variability) of the material, equipment, oroperator and procedural effects.

While the spatial variability of the material, and the random measurementerrors, are both random, or scatter, components of the total measurement error,it is difficult to separate and quantify each. This report investigates the currentmethod proposed by Baecher (1982), and subsequently applied by severalauthors, to separate the random measurement error from the spatial variabilityof the soil. However, before discussing this method, it is necessary to treat thetechniques used to describe these uncertainties.

2. TECHNIQUES USED TO DESCRIBE UNCERTAINTY

To date, the analyses of random measurement errors and the spatial variabilityof geotechnical materials, has centred on two mathematical techniques, namely,time series analysis, and geostatistics. In order to provide a background to thediscussion that follows in later sections, these two techniques are treated briefly,below.

3

2.1 Time Series Analysis

A time series is a chronological sequence of observations of a particularvariable, usually, but not necessarily, at constant time intervals. When appliedto the study of the spatial variability of geotechnical materials and randommeasurement errors, the time domain is replaced by the space, or distance,domain. In all other respects, the analysis procedures and theory are identical.Unlike classical statistics, time series analysis incorporates the observedbehaviour that values at adjacent locations are more related, than those atdistant locations.

For time series analyses to be carried out the data must be stationary, that is, theprobabilistic laws which govern the series must be independent of the locationof the samples. Data are stationary if:

• the mean is constant with distance, that is, no trend, or drift, exists in thedata;

• the variance is constant with distance, that is, the data are homoscedastic;• there are no seasonal variations; and,• there are no irregular fluctuations.

An important implication of the assumption of stationarity is that the statisticalproperties of the time series are unaffected by a shift of the spatial origin. Twoessential statistical properties used in time series analysis are theautocovariance, ck, and the autocorrelation, ρk , at lag, k, which are defined as:

( ) ( )( )[ ] ( ) 2EE,Cov XXXXXXXXXc kiikiikiik −=−−== +++ (4)

and,

ρkkc

c=

0(5)

where: Xi is the value of property, X, at location, i;X is the mean of the property, X;E[..] is the expected value;c0 is the autocovariance at lag 0; and,ck = c-k and ρk = ρ-k .

The autocorrelation, ρk , measures the correlation between any two time seriesobservations separated by a lag of k units.

It is not possible to know ck nor ρk with any certainty, but only to estimate themfrom samples obtained from a population, say X1 , X2 , ... , XN . As a result, the

4

sample autocovariance at lag k, ck* , and sample autocorrelation at lag k, rk , are

given by:

( )( )∑−

=+ −−=

kN

ikiik XXXX

Nc

1

* 1(6)

and,

( )( )

( )∑

∑

=

−

=+

−

−−== N

ii

kN

ikii

kk

XX

XXXX

c

cr

1

2

1*0

*

(7)

where: X is the average of the observations X1 , X2 , ... , XN ;and, 0 ≤ k < N.

The sample autocovariance function (ACVF), or autocovariogram, is the plotof ck

* for lags k = 0, 1, 2, ... . The sample autocorrelation function (ACF), orcorrelogram, is the graph of rk for lags k = 0, 1, 2, ... K, where K is themaximum number of lags that rk should not be calculated beyond. While thesample autocorrelation function can be evaluated for all lags up to N, it is notadvisable, since, as k tends toward N, the number of pairs reduces, and as aconsequence, the reliability of the estimate rk of the true autocorrelationfunction, ρk , also decreases. Most authors suggest that K N= 4, (Box andJenkins, 1970; Chatfield, 1975; Anderson, 1976).

As one might expect, the accuracy of the sample autocorrelation function isdirectly related to the number of observations in the time series, N. Littleguidance is given to the minimum number of observations, though Box andJenkins (1970) and Anderson (1976) recommended that N be greater than 50.With particular reference to the spatial variability of soils, Lumb (1975)suggested that, for a full three-dimensional analysis, the minimum number oftest results needed to give reasonably precise estimates is of the order of 104,which is prohibitively large, even for a special research project. On the otherhand, Lumb (1975) recommended that the best that can be achieved in practiceis to study the one-dimensional variability, either vertically or laterally, using Nof the order of 20 to 100.

The ACF, and to a lesser extent the ACVF, are used widely throughout timeseries analysis literature, and they enable the characteristics of the time series tobe determined. For example, an ACF exhibiting slowly decaying values of rk

with increasing k, suggests long term dependence, whereas rapidly decayingvalues of rk suggest short term dependence (Chatfield, 1975; Hyndman, 1990).

5

A purely random time series, or white noise, is characterised by an ACF withthe following properties:

≠=

=ρ0for 0

0for 1

k

kk (8)

2.2 Geostatistics

The mathematical technique, which is now universally known as geostatistics,was developed to assist in the estimation of changes in ore grade within a mine,and is largely a result of the work of D. G. Krige and G. Matheron (1965).Since its development in the 1960’s, geostatistics has been applied to manydisciplines including: groundwater hydrology and hydrogeology; surfacehydrology; earthquake engineering and seismology; pollution control;geochemical exploration; and geotechnical engineering. In fact, geostatisticscan be applied to any natural phenomena that are spatially or temporallyassociated (Journel and Huijbregts, 1978; Hohn, 1988).

Geostatistics is based on regionalised variables which have properties that arepartly random and partly spatial, and which have continuity from point to point.The changes in these variables, however, are so complex that they cannot bedescribed by a tractable deterministic function (Davis, 1986). This is in contrastto the classical approach which treats samples as independent realisations of arandom function.

One of the basic statistical measures of geostatistics is the semivariogram,which is used to quantify the degree of spatial dependence between samplesalong a specific orientation, and so presents the degree of continuity of theproperty in question. The semivariogram, γh , is defined by Equation (9).

( )[ ]2E2

1ihih XX −=γ + (9)

where: h the displacement between the data pairs.

If the regionalised variable is stationary and normalised to have a mean of zeroand a variance of 1.0, the semivariogram is the mirror image of theautocorrelation function.

Even though a regionalised variable is spatially continuous, it is not possible toknow its value at all locations. Instead its values, like the ACF, can only be

6

determined from samples taken from a population. Thus, in practice, thesemivariogram must be estimated from the available data, and is generallydetermined by the relationship shown in Equation (10).

( )∑=

+ −=γh

ii

N

ixhx

hh

YYN 1

2*

2

1(10)

where: γ h* the experimental semivariogram, that is, one based on the

sampled data set;

Yxithe value of the property, Y, at location, xi ; and,

Nh the number of data pairs separated by the displacement, h.

The accuracy of γ h* is directly related to two parameters: the number of data

pairs, Nh ; and the lag distance, h (Brooker, 1991). In general, as the distancebetween data pairs increases, Nh decreases, consequently reducing the accuracyof the experimental semivariogram. As a result, γ h

* is usually determined up to

half of the total sampled extent (Journel and Huijbregts, 1978; Clark, 1979;Brooker, 1989). For example, if an electrical cone penetrometer (CPT)sounding was performed to a depth of 5,000 mm, the semivariogram would becalculated for values of h from 0 to 2,500 mm. The minimum number of pairsneeded for a reliable estimate of γ h

* is between 30 and 50 (Journel and

Huijbregts, 1978; Brooker, 1989), with some authors suggesting as many as 400to 500 (Clark, 1980).

The strength of geostatistics is that it provides, through the semivariogram, aframework for the estimation of variables. In fact, it can be shown thatgeostatistics provides the best, linear, unbiased estimator (BLUE) (Journel andHuijbregts, 1978; Clark, 1979; Rendu, 1981). Whilst the experimentalsemivariogram is known only at discrete points, the estimation procedure,known as kriging requires the semivariogram values be known for all h. Thus,it is necessary to model the experimental semivariogram, γ h

* , as a continuous

function, γh . The most common model used in the literature is the sphericalmodel, shown in Figure 1, which is characterised by three parameters:

C0 is defined as the nugget effect and arises from the regionalisedvariable being so erratic over a short distance that thesemivariogram goes from zero to the level of the nugget in adistance less than the sampling interval. The nugget effect willbe discussed in greater detail in §4.1;

7

Displacement, h

Range, aC0

C

Figure 1. Example of a semivariogram.

C + C0 is known as the sill which measures half the maximum, onaverage, squared difference between data pairs; and,

a is defined as the range of influence, and is the distance at whichsamples become independent of one another. Data pairsseparated by distances up to a are correlated, but not beyond.

Clark (1979) described the process of fitting a model to an experimentalsemivariogram as essentially a trial-and-error approach, usually achieved byeye. Brooker (1991) suggested the following technique as a first approximationin finding the appropriate parameters for a spherical model:

• The experimental semivariogram and variance of the data are plotted;• The value of the sill, C + C0 , is approximately equal to the variance of the

data;• A line is drawn with the slope of the semivariogram near the origin which

intersects the sill at two thirds the range, a;• This line intersects the ordinate at the value of the nugget effect, C0 .

In addition, Brooker (1991) stated that the accuracy of the modelling processdepends on both the number of pairs in the calculation of the experimentalsemivariogram and the lag distance at which it is evaluated. Journel andHuijbregts (1978) suggested that automatic fitting of models to experimental

8

semivariograms, such as least squares methods, should be avoided. This isbecause each estimator point, γ h

* , of an experimental semivariogram is subject

to an estimation error and fluctuation which is related to the number of datapairs associated with that point. Since the number of pairs varies for each point,so too does the estimation error. The authors recommended that the weightingapplied to each estimated point, γ h

* , should come from a critical appraisal of the

data, and from practical experience.

Similar to the autocorrelation function of random variables, the semivariogramrequires stationarity, that is, the semivariogram depends only on the separationdistance and not on the locality of the data pairs. The regionalised variable canbe regarded as consisting of two components: the residual and the drift. If adrift, or trend, exists in the data, which leads to non-stationarity, it must first beremoved. It has been shown by Davis (1986), that if the drift is subtracted fromthe regionalised variable, the residuals will themselves be a regionalisedvariable and will have local mean values of zero. In other words, the residualswill be stationary and the semivariogram can be evaluated.

3. ESTIMATION OF MEASUREMENT ERRORS

Baecher (1982) proposed a useful method for separating the scatter observed ingeotechnical data into its two component sources: the spatial variability of thematerial, and the random measurement error associated with the test itself.Baecher (1982) suggested, as did Lumb (1974) before him, that the spatialvariation of some parameter, vx , at a location x within a soil mass, may betreated as a combination of a deterministic component, as well as a stochastic,or random, component, as shown in the following equation.

vx = tx + ξx (11)where:

tx is the trend component at location x, usuallydetermined by least squares regression; and,

ξx is the random perturbation off the trend at x.

In addition, Baecher (1982) suggested that the random measurement error ispresumed to be independent from one test to another, to have zero mean, and tohave constant variance. As a consequence, the measurement of vx, mvx

, may be

expressed as:

m tv x x xx= + +ξ ζ (12)

where:ζx is the random measurement error at x.

9

After some algebraic manipulation, Baecher (1982, 1986) stated that theautocovariance, ck mv( )

, of the measurement, mv , at lag, k, may be shown to

equal:

c c ck k kmv v( ) ( ) ( )= + ζ (13)

where:ck v( )

is the autocovariance function of v at lag, k; and

ck( )ζ is the autocovariance function of the random

measurement error, ζ, at lag, k.

A similar relationship can be derived that incorporates the ACF rather than theautocovariance function (ACVF). Baecher (1982, 1986) proposed that, sinceck( )ζ is equal to the variance of ζ at k = 0, and ck( )ζ is equal to zero at k ≠ 0, as

described by Equation (8), the random measurement error may be determined byextrapolating the observed ACVF, or ACF, back to the origin, as shown inFigures 2 and 3.

Figure 2. Procedure for estimating the random measurementcomponent from the ACVF. (After Baecher, 1982).

10

Figure 3. Procedure for estimating the random measurementcomponent from the ACF. (After Baecher, 1986).

Baecher (1986) stated that, by using this method, typical in situ measurementsof soils have been found to contribute random measurement errors anywherebetween 0 and 70% of the data scatter. Several other authors (Tang, 1984, Wuand El-Jandali, 1985; Filippas et al., 1988; Orchant et al., 1988; Spry et al.,1988; Kay, 1990; Kay et al., 1991; DeGroot and Baecher, 1993) have used thismethod, or results based on it, to postulate various aspects relating togeotechnical uncertainty and reliability. As will be shown in the next section,while the method proposed by Baecher (1982, 1986) appears to be well-founded, a number of factors need consideration before conclusions can bemade regarding the level of random measurement error associated with aparticular test.

4. INADEQUACIES OF PRESENT METHOD

While Baecher’s approach focuses on the tools associated with time seriesanalyses, treated in §2.1, three important factors from the study of geostatistics,have highlighted inadequacies with the current method. These factors, thenugget effect, the spacing between samples, and the stationarity of the data,greatly influence the random measurement error obtained by the procedureproposed by Baecher (1982), and are each discussed below.

11

4.1 Nugget Effect

It has long been appreciated, in the study of geostatistics, that many ore bodiesexhibit erratic behaviour at lags close to zero. This erratic behaviour, known asthe nugget, C0 , and described briefly in §2.2, manifests itself as an apparentnon-zero value of the semivariogram at zero lag. The nugget effect is thecombination of three separate phenomena (Rendu, 1981):

1. microstructures within the geological material - which have been observedfrom the study of the spatial variability of mineral concentrations of coresamples. Two adjacent cores will exhibit a nugget effect when one of themcontains a nugget and the other does not (Journel and Huijbregts, 1978).Several researchers have stated that soils also exhibit this behaviour (Liand White, 1987; Soulié et al., 1990; Jaksa et al., 1993);

2. sampling, or statistical, errors - as will be detailed in §4.2 below, C0depends greatly on the spacing between individual samples; and,

3. measurement errors - if it were possible to obtain a repeat measurement atprecisely the same location, the observations would differ by an amountdirectly dependent on the measurement technique. Measurement errors arealso manifested by a non-zero value of the semivariogram at zero lag.

Baecher’s procedure, in essence, attributes the nugget effect solely tomeasurement error, but, as has been shown above, the nugget effect is alsomade up of microstructure variabilities and sampling errors, which must beaccounted for before conclusions can be made regarding the extent of randommeasurement error associated with a particular test.

At this point, it is necessary to define a new parameter, the ACF nugget, R0 ,which is the difference between unity and the value of the autocorrelationcoefficient at lag zero, r0 , obtained by extrapolating the sample ACF back tolag zero, as shown in Equation (14). The ACF nugget, like the nugget fromgeostatistics, accounts for the micro-variability of the geological material,sampling errors, and random measurement errors, but is determined from thesample ACF rather than from the semivariogram.

R0 = 1 - r0 (14)

4.2 Sample Spacing

As mentioned in the previous section, another important factor, again which haslong been established in the field of geostatistics, is the effect of the samplespacing on the observed nugget (Brooker, 1977; Journel and Huijbregts, 1978;

12

Clark, 1979; Clark, 1980; de Marsily, 1982). In fact, the nugget effect that isobtained from the experimental semivariogram, depends greatly on the physicaldistance between the individual samples that form the data set. As the samplingdistance decreases, it is possible to obtain a better estimate of C0 . However,while one is able to reduce the sampling interval to a very small distance, thecost of the exploration programme increases dramatically. As a result, it isoften unreasonable, and in fact unnecessary, to reduce the sampling spacingbelow some nominal minimum value. Unfortunately, this minimum samplingdistance is dependent on the geological material being examined, and cannot beknown prior to investigation. Common practice is to begin sampling with arelatively coarse grid, and then to infill with a repeatedly finer grid, until thesample spacing no longer influences the resultant experimental semivariogram.

In §5, two case studies will be used to demonstrate the effect of sample spacingon the observed nugget. Each of these case studies is based on data obtained atan extremely close sample spacing of 5 mm.

4.3 Stationarity of Data

As defined previously in §2.1, the theory of both time series analysis andgeostatistics, assume that the data are stationary. The ACF, ACVF, and thesemivariogram, are each dependent on the stationarity of the data set, and as aresult, so too is the nugget effect, and hence the random measurement error,obtained from each.

In both time series analysis and geostatistics, it is common practice to transforma non-stationary data set to a stationary one by removing a low-orderpolynomial trend, usually no higher than a quadratic (Journel and Huijbregts,1978), which is usually estimated by means of the method of ordinary leastsquares (OLS) (Lumb, 1974; Brockwell and Davis, 1987). Li (1991) correctlyasserted that OLS assumes that the data are random and uncorrelated, which isinconsistent with spatial variability analyses which, having removed some trenddetermined by OLS, subsequently examine the correlation structure of theresiduals. Li (1991) suggested that a technique based on generalised leastsquares (GLS) be used as an alternative to OLS, and the more complex methodssuggested by Matheron (1973) and Delfiner (1976). Kulatilake (1991) statedthat, while in general agreement with Li (1991), the GLS technique hasdrawbacks when applied in a practical sense. Furthermore, Ripley (1981) foundthat the trend produced by GLS varied only slightly from that produced by OLS.

Regardless of which method is used to evaluate the trend component within anon-stationary data set, the nugget effect is significantly influenced by the

13

stationarity of the data. The following section examines the influence of datastationarity and sample spacing on the nugget effect by means of two casestudies.

5. CASE STUDIES

As part of a research programme currently being undertaken at the University ofAdelaide which focuses on quantifying the spatial variability of soils, a numberof electrical cone penetration tests (CPT) have been performed in a relativelyhomogeneous, stiff, over-consolidated clay known as Keswick Clay. This clayhas been the focus of research because of its local significance, as many ofAdelaide’s high rise buildings are founded on it, and because of its internationalsignificance, since its geotechnical properties are remarkably similar to those ofthe well-documented London Clay (Cox, 1970).

In order to demonstrate the effect of sample spacing and data stationarity on theobserved nugget, and consequently, the random measurement error, two casestudies are presented, the first dealing with vertical spatial variability, and thesecond dealing with horizontal spatial variability. In each case, the CPT wasused to obtain the data. The electric cone penetrometer is a standard 60°, 10cm2 base area type which conforms to the relevant standards which includeISOPT-1 (De Beer et al., 1988), ASTM D3441 (American Society for Testingand Materials, 1986) and AS 1289.F5.1 (Standards Association of Australia,1977). The data were obtained using a micro-computer based data acquisitionsystem which allows CPT measurements to be recorded at 5 mm increments,and stored on disk for subsequent analyses. The data acquisition system istreated in detail by Jaksa and Kaggwa (1994). In order to maintain consistency,and thus to eliminate as many errors as possible, the same cone penetrometer,data acquisition system, and operators were used in both case studies.

The data from each case study were examined using a PC program, SemiAuto,developed by the first author, which is a Windows based application thatenables both time series, and geostatistical, analyses to be performed. Theresults are detailed below.

5.1 Vertical Spatial Variability - South Parklands Site

A large number of vertical CPTs were drilled within a section of the SouthParklands, which lies in the central business district of the city of Adelaide,South Australia. The CPTs, 223 in all, and each drilled to an approximate depthof 5 metres, were arranged in: (i) a 5 × 5 metre grid; (ii) a ‘cross’ formation witheach CPT being spaced at 1 metre centres; and (iii) 50 CPTs which were spaced

14

at 0.5 metre intervals along a line, as shown in Figure 4. Details of the testingprogramme at the South Parklands site, and results of preliminary spatialvariability analyses, are given by Jaksa et al. (1993).

5 m

50 m

50 m

5 m

1 m

1 m

0 1 2 3 4 5 6 7 8 9 10A

B

C

D

E

F

G

H

I

J

K

2.5 m

CD1

CD50

0.5 m

Legend

Continuous core sample

3.3

2.4

1.6

2.3

1.1

2.2

2.2

2.3

1.0 Depth to top of Keswick Clay

Unsuccessful soundings

Successful soundings

Triaxial samples

Figure 4. Layout of vertical CPTs at the South Parklands site.

In order to investigate the sensitivity of the vertical spatial variability of soils,with respect to the factors described in §4, a typical CPT sounding, C8, will beanalysed. Since a continuous core sample was taken adjacent to C8, as shown

15

in Figure 4, it was determined by inspection of the core, that the surface of theKeswick Clay lies 1.1 metres below the ground surface. Eliminating the upper1.1 metres of measurements, a plot of the cone tip resistance, qc , obtained fromthe CPT, for the Keswick Clay at location C8, is shown in Figure 5. A linear,and a quadratic, trend are also shown in Figure 5, and were fitted to the data bymeans of OLS regression.

[ [[[[[[[[[[[[[[[[[[ [ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[

5500

5000

4500

4000

3500

3000

2500

2000

1500

10000 0.5 1 1.5 2 2.5

Cone Tip Resistance, qc (MPa)

Linear Trendr2 = 0.088

Quadratic Trendr2 = 0.719

Figure 5. Measured cone tip resistance, qc , within Keswick Clayfor sounding C8.

The residuals of the data in Figure 5, were obtained by removing the quadratictrend, the results of which are given in Figure 6, and were assumed to bestationary. The sample ACF was obtained by substituting the data shown inFigure 6 into Equation (7), the result of which, is shown in Figure 7.

At lag k = 1, the sample autocorrelation coefficient, r1 , for the residuals of qcfor sounding C8, was found to be equal to 0.90. Extrapolating the sample ACFback to k = 0, in accordance with Baecher’s procedure, yields r0 = 0.90,implying a calculated ACF nugget of 10%.

16

[ [[[[[[[[[[[[ [[[[[[ [ [[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[ [[[[[[[[[ [[[[[[[[[[ [[[[[[[[[[[[

[[[[[[[[[[[[ [[[[ [[[[ [[[[[ [[[[[[[[[[[[ [[[[[ [[[ [[[[[[[[ [[[[[[[[ [[[[[[[[[[[ [[[ [[[[[[[[[[[[[[[ [[[[[[[[[[[ [[[[[[ [[[[[ [[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[ [[[[[[[[[ [[[[[[[[[[[[[[[[

[ [[[[[[ [[[[[[[[[[[[ [ [[[ [[ [[[ [[[[[ [[[[[ [[[[[[[ [[[[[ [[[[[[[[ [[[[[[[[[[ [[[[[[[[[[[[[[ [[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[ [[ [[[[[ [[[[[[[[[[[ [[[[[[[[[[

[[[[[ [[[[[[[[[[[[[[[[[[ [[[[[[[[[ [[[[[ [[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[ [[[[[[[[[[[[ [[[ [[[[[[[[[[[[[[[[[[ [[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[ [[[[[[[[[[[[ [[[ [[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[ [[[[[[ [[[[ [[[[[[[ [[[[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[ [[[[[[[ [[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Cone Tip Resistance, qc (MPa)

Figure 6. Residuals of qc , for sounding C8, after removingthe quadratic trend.

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000

Distance (mm)

Figure 7. Sample ACF of residuals of qc , for sounding C8.

In order to test the sensitivity of the nugget with regard to the stationarity of thedata, the measurements of qc , shown in Figure 5, are examined, firstly, with no

17

trend removal, and secondly, with only a linear trend removed. The sampleACFs obtained for each of these two cases, are shown in Figure 8.

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000

Distance (mm)

(a)

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000

Distance (mm)

(b)

Figure 8. Sample ACFs after: (a) no trend removal,and (b) a linear trend removal.

18

For no trend removal, r1 = 0.59, which yields, r0 = 0.59; and for a linear trendremoval, r1 = 0.94, yielding, r0 = 0.94. The results of these analyses aresummarised in Table 1.

TABLE 1. SUMMARY OF DATA STATIONARITY ANALYSES(VERTICAL SPATIAL VARIABILITY).

Trend Removed from Data r1 r0 ACF Nugget, R0

None 0.59 0.59 41%Linear 0.94 0.94 6%

Quadratic 0.90 0.90 10%

As shown by the results in Table 1, the ACF nugget determined using Baecher’sapproach, varies substantially, from 6% to 41%, and as a result, depends greatlyon the stationarity of the data.

In order to test the sensitivity of the nugget, with respect to sample spacing, theoriginal data set of qc measurements was modified to provide sets of data atdifferent sample spacings. These measurements, shown in Figure 5, weresampled at 5 mm intervals from a depth of 1100 mm to 5055 mm below ground.Data sets at different sample spacings were obtained by simply removingintervening rows of data. For example, to obtain a data set with qcmeasurements at 10 mm spacings, every second row was removed. Thisprovided two data sets of measurements spaced at 10 mm intervals, one from1100 mm to 5050 mm, and the other from 1105 mm to 5055 mm. This processof removing intervening rows was used to provide several data sets of qcmeasurements at spacings of 10, 20, 50, 100, and 200 mm. By removing thequadratic trend from each of these data sets, the residuals were obtained, andsubstituted into Equation (7), to determine the sample ACFs. Two such sampleACFs are shown in Figure 9.

Again using the procedure proposed by Baecher (1982), the ACF nugget can beevaluated by extrapolating the sample ACF back to lag, k = 0. The results foreach of the data sets are summarised in Table 2.

As is indicated by the results shown in Table 2, the calculated ACF nuggetobtained from vertical spatial variability analyses, and determined using theprocedure proposed by Baecher (1982), is significantly dependent on thespacing of the samples, and can vary between 3% and 62%.

19

[[

[[[[

[[

[[[

[[[[

[[

[[

[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000

Distance (mm)

Fitted Curve

(a)

B

B

B

B

B

J

J

J

J

J

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000

Distance (mm)

0.98

0.38

J

Fitted CurvesB

(b)

Figure 9. Sample ACFs for: (a) 50 mm spaced data set,and (b) 200 mm spaced data set.

5.2 Horizontal Spatial Variability - Keswick Site

The lateral spatial variability of the Keswick Clay was studied by Jaksa et al.(1994), by driving an electrical cone penetrometer horizontally into anembankment, as shown in Figure 10. The embankment is situated at theAustralian National railway yards at Keswick, which is located adjacent to thecentral business district of the city of Adelaide.

20

The CPT was carried out to a total horizontal penetration distance of 7.62metres. Removing the first two metres of data, which may be affected byweathering and movement adjacent to the face of the embankment, yields the qcmeasurements shown in Figure 11. The horizontal cone penetration testingcarried out at the Keswick site, as well as results of spatial variability analyses,are treated in detail by Jaksa et al. (1994).

TABLE 2. SUMMARY OF SAMPLE SPACING ANALYSES(VERTICAL SPATIAL VARIABILITY).

Sample Vertical Spatial VariabilitySpacing (mm) r1 r0 ACF Nugget, R0

5 0.90(1) 0.90 10%10 0.84, 0.86(2) 0.89, 0.90 10 - 11%20 0.76 to 0.81(4) 0.93 to 0.95 5 to 7%50 0.48 to 0.64(5) 0.76 to 0.97 3 to 24%100 0.26 to 0.49(5) 0.40 to 0.82 18 to 60%200 -0.03* to 0.29(5) 0.38 to 0.97 3 to 62%

(n) : Separate data sets examined.* : Not possible to sensibly extrapolate R0 when r1 < 0.

40°

Scale

0 1 2 3 4 5(metres)

Track7.62

Extent of horizontal CPT

Trailer mountedhorizontal CPT

3.65

Figure 10. Cross-section of embankment at Keswick site.

Again, the residuals of the data in Figure 11, were obtained by removing thequadratic trend obtained by the method of OLS, and the sample ACF wasobtained by substituting the residuals into Equation (7), the result of which, isshown in Figure 12.

21

[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0

0.5

1

1.5

2

2.5

3

3.5

1000 2000 3000 4000 5000 6000 7000 8000

Horizontal Penetration Distance (mm)

Quadratic Trend

Figure 11. Measured cone tip resistance, qc , within Keswick Clayat the Keswick site.

At lag k = 1, the sample autocorrelation coefficient, r1 , for the residuals of qcfor the horizontal CPT was found to equal 0.97. Extrapolating the sample ACFback to k = 0, in accordance with Baecher’s procedure, yields r0 = 0.97,implying a calculated ACF nugget of 3%.

In order to test the sensitivity of the nugget with regard to the stationarity of thedata, the measurements of qc , shown in Figure 11, are examined, firstly, with notrend removal, and secondly, with a linear trend removed. The results of theseanalyses are shown in Table 3.

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000 1500

Distance (mm)

Figure 12. Sample ACF of residuals of qc , for horizontal CPT.

22

TABLE 3. SUMMARY OF DATA STATIONARITY ANALYSES(HORIZONTAL SPATIAL VARIABILITY).

Trend Removed from Data r1 r0 ACF Nugget, R0

None 0.44 0.44 56%Linear 0.97 0.97 3%

Quadratic 0.97 0.97 3%

As shown by the results in Table 3, the value of R0 determined using Baecher’sapproach, varies substantially, from 3% to 56%, and again shows that thenugget depends greatly on the stationarity of the data. It should be noted thatthere is no difference in the value of R0 obtained by removing the linear trend,as compared to that obtained by removing the quadratic trend.

Again, in order to test the sensitivity of the calculated ACF nugget, with respectto sample spacing, the original horizontal CPT data, which were sampled at 5mm intervals, were modified to provide data sets with spacings of 10, 20, 50,100, and 200 mm, between adjacent measurements of qc , in the same way as forthe vertical spatial variability case, described previously. By removing thequadratic trend from each of these data sets by the method of OLS, the residualswere obtained, and the sample ACFs determined. Two such sample ACFs areshown in Figure 13.

Again, using Baecher’s procedure, the ACF nugget is determined byextrapolating the sample ACF back to lag, k = 0. The results of a number of thehorizontal CPT data sets are summarised in Table 4.

TABLE 4. SUMMARY OF SAMPLE SPACING ANALYSES(HORIZONTAL SPATIAL VARIABILITY).

Sample Horizontal Spatial VariabilitySpacing (mm) r1 r0 ACF Nugget, R0

5 0.97(1) 0.97 3%10 0.95, 0.95(2) 0.97, 0.97 3%20 0.88 to 0.90(4) 0.95 5%50 0.62 to 0.64(5) 0.90 to 0.92 8 to 10%100 0.20 to 0.29(5) 0.50 to 0.67 33 to 50%200 -0.28* to 0.20(5) ? to 0.82 18% to ?

(n) : Separate data sets examined.* : Not possible to sensibly extrapolate R0 when r1 < 0.? : Unknown value of R0 since r1 < 0.

23

[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[[[[

[[[[[[[[[[[[[[[[[[[[[[

[

[

[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000 1500

Distance (mm)

Fitted Curve

(a)

[

[[[[

[

[[[

[[

[[

[

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

500 1000 1500

Distance (mm)

(b)

Figure 13. Sample ACFs for: (a) 20 mm spaced data set,and (b) 100 mm spaced data set.

As is indicated by the results shown in Table 4, the calculated ACF nugget,obtained from horizontal spatial variability analyses, and determined usingBaecher’s method, varies significantly, from 3% to 50%, and again indicatesthat the ACF nugget depends greatly on the sample spacing of the data.Furthermore, for a spacing of 200 mm, 3 of the 5 data sets examined yieldedvalues of r1 less than zero, making it impossible to extrapolate a positive valueof R0 , as indicated in Table 4.

24

6. CONCLUSIONS

This paper has examined the method proposed by Baecher (1982), and adoptedby several researchers since, for separating the spatial variability component ofthe geotechnical material from the random measurement error of the test. It hasbeen demonstrated, using both vertical and horizontal CPT data measured atclose intervals of 5 mm, that conclusions made regarding the randommeasurement error associated with a particular test, depend greatly on: (i) themicro-variability of the geological material; (ii) the spacing of the samples inthe data set; and (iii) the stationarity of the data. In fact, the ACF nugget, R0 ,(the difference between unity and the value of the sample autocorrelationfunction extrapolated back to zero, r0) is a combination of random measurementerror, small-scale variability of the soil, sampling errors, and non-stationarityerrors. It is not solely random measurement errors associated with the particulartest, as several authors have incorrectly assumed.

A series of sample ACFs were obtained by examining the vertical andhorizontal non-stationary CPT data sets, and by removing a linear and quadratictrend, and comparing these with no trend removal. These analyses have shownthat the calculated ACF nugget can vary between 3% and 56% depending onwhich, if any, trend is removed. These results confirm that the ACF nugget issignificantly dependent on data stationarity.

By varying only the sample spacing of a data set, in increments from 5 mm upto200 mm, it has been shown that the calculated ACF nugget can vary between3% and 62%, for vertical spatial variability, and between 3% and 50% forhorizontal spatial variability. As almost all of the information publishedregarding the horizontal spatial variability of soils is based on ACFs derivedfrom samples taken at spacings well in excess of 200 mm, one must question thevalidity of these conclusions.

As a result of the data and analyses presented in this paper, it is likely that therandom measurement error associated with the cone penetration test is less thanor equal to 3%, since this figure accounts for both the random measurementerror and the small-scale variability of the soil, and perhaps, to some extent,non-stationarity errors.

25

7. REFERENCES

Anderson, O. D. (1976). Time Series Analysis and Forecasting: The Box-Jenkins Approach, Butterworths, London, 182 p.

American Society for Testing and Materials (1986). Standard Method forDeep, Quasi-Static, Cone and Friction-Cone Penetration Tests of Soil (D3441).Annual Book of Standards, Vol. 04.08, ASTM, Philadelphia, pp. 552-559.

Baecher, G. B. (1982). Simplified Geotechnical Data Analysis. Proc. of theNATO Advanced Study Institute on Reliability Theory & its Appl’n in Struct’l &Soil Mechanics, Bornholm, Denmark, Martinus Nijhoff (Publ. 1983), pp. 257-277.

Baecher, G. B. (1986). Geotechnical Error Analysis. Transportation ResearchRecord, No. 1105, pp. 23-31.

Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis Forecastingand Control, Holden-Day, San Fransisco, 553 p.

Brockwell, P. J. and Davis, R. A. (1987). Time Series: Theory and Methods,Springer-Verlag, New York, 519 p.

Brooker, P. I. (1977). Robustness of Geostatistical Calculations: A CaseStudy. Proc. Australasian Inst. Min. Metall., No. 264, pp. 61-68.

Brooker, P. I. (1989). Basic Geostatistical Concepts. In Workshop Notes ofAust. Workshop on Geostatistics in Water Resources, Vol. 1, Centre forGroundwater Studies, Adelaide, November, 63 p.

Brooker, P. I. (1991). A Geostatistical Primer, World Scientific, Singapore,95 p.

Chatfield, C. (1975). The Analysis of Time Series: Theory and Practice,Chapman and Hall, London, 263 p.

Clark, I. (1979). Practical Geostatistics, Applied Science Publishers, London,129 p.

Clark, I. (1980). The Semivariogram. Chapters 2 and 3 of Geostatistics,McGraw-Hill Inc., New York, pp. 17-40.

26

Cox, J. B. (1970). A Review of the Geotechnical Characteristics of the Soils inthe Adelaide City Area. Symp. on Soils and Earth Structures in Arid Climates,Adelaide, Inst. Eng., Aust. and Aust. Geomech. Soc., May, 1970, pp. 72-86.

Davis, J. C. (1986). Statistics and Data Analysis in Geology, 2nd ed., JohnWiley and Sons, New York, 646 p.

De Beer, E. E., Goelen, E., Heynen, W. J. and Joustra, K. (1988). ConePenetration Test (CPT): International Reference Test Procedure. In PenetrationTesting, Proc. of the First Int. Symposium on Penetration Testing (ISOPT-1),de Ruiter, J. (ed.), Orlando, Florida, A. A. Balkema, Rotterdam, pp. 27-51.

DeGroot, D. J. and Baecher, G. B. (1993). Estimating Autocovariance of In-Situ Soil Properties. J. Geotech. Eng’g Div., ASCE, Vol. 119, No. GT1, pp.147-166.

Delfiner, P. (1976). Linear Estimation of Non Stationary Spatial Phenomena.In Advanced Geostatistics in the Mining Industry, Guarascio, M. et al. (eds.),D. Reidel Publishing Co., Dordrecht, pp. 49-68.

De Marsily, G. (1982). Spatial Variability of Properties in Porous Media: AStochastic Approach. Proc. of the NATO Advanced Study Institute onMechanics of Fluids in Porous Media, Bear, J. and Corapcioglu, M. Y. (eds.),Newark, Delaware, Martinus Nijhoff (Publ. 1984), pp. 719-769.

Filippas, O. B., Kulhawy, F. H. and Grigoriu, M. D. (1988). Evaluation ofUncertainties in the In-Situ Measurement of Soil Properties. Report EL-5507,Vol. 3, Electric Power Research Institute, Palo Alto.

Hohn, M. E. (1988). Geostatistics and Petroleum Geology, Van NostrandReinhold, New York, 264 p.

Hyndman, R. J. (1990). PEST - A Program for Time Series Analysis,Statistical Consulting Centre, University of Melbourne, 53 p.

Jaksa, M. B., Kaggwa, W. S. and Brooker, P. I. (1993). GeostatisticalModelling of the Undrained Shear Strength of a Stiff, Overconsolidated, Clay.Proc. of Conf. of Probabilistic Methods in Geotechnical Engineering, Canberra,A. A. Balkema, Rotterdam, pp. 185-194.

Jaksa, M. B. and Kaggwa, W. S. (1994). A Micro-Computer Based DataAcquisition System for the Cone Penetration Test. Research Report No. R 116,Dept. Civil & Environmental Eng’g, University of Adelaide, 31 p.

27

Jaksa, M. B., Brooker, P. I., Kaggwa, W. S., van Holst Pellekaan, P. D. A.and Cathro, J. L. (1994). Modelling the Lateral Spatial Variation of theUndrained Shear Strength of a Stiff, Overconsolidated Clay Using anHorizontal Cone Penetration Test. Research Report No. R 117, Dept. Civil &Environmental Eng’g, University of Adelaide, 34 p.

Journel, A. G. and Huijbregts, Ch. J. (1978). Mining Geostatistics,Academic Press, London, 600 p.

Kay, J. N. (1990). Approximate Framework for Probabilistic Evaluation ofSoil Properties. Proc. Uni. of Adelaide Special Symp. on the Occasion ofGeorge Sved’s 80th Birthday, Adelaide, Sth. Aust., pp. 184-197.

Kay, J. N., Kulhawy, F. H. and Grigoriu, M. D. (1991). Assessment ofUncertainties in Geotechnical Design Parameters. Proc. 6th Int. Conf. Statisticsand Probability in Soil and Struct. Eng., Mexico, pp. 683-692.

Kulatilake, P. H. S. W. (1989). Probabilistic Potentiometric Surface Mapping.J. Geotech. Eng’g., ASCE, Vol. 115, No. 11, pp. 1569-1587.

Li, K. S. and White, W. (1987). Probabilistic Characterization of Soil Profiles.Research Report, Dept. Civil Eng’g, Australian Defence Force Academy,Canberra, Australia.

Li, K. S. (1991). Discussion on “Probabilistic Potentiometric SurfaceMapping.” J. Geotech. Eng’g., ASCE, Vol. 117, No. 9, pp. 1457-1458.

Lumb, P. (1974). Application of Statistics in Soil Mechanics. In SoilMechanics - New Horizons, Chapter 3, Lee, I. K. (ed.), American Elsevier, NewYork, pp. 44-111.

Lumb, P. (1975). Spatial Variability of Soil Properties. Proc. 2nd Int. Conf.on Applications of Statistics and Probability in Soil and Struct. Eng’g, Auchen,pp. 397-421.

Matheron, G. (1965). Les Variables Regionalisees et leur Estimation. Massonet Cie, Paris, 212 p.

Matheron, G. (1973). The Intrinsic Random Functions and Their Applications.Advances in Applied. Probability, Vol. 5, pp. 439-468.

Orchant, C. J., Kulhawy, F. H. and Trautmann, C. H. (1988). CriticalEvaluation of In-Situ Test Methods and their Variability. Report EL-5507, Vol.2, Electric Power Research Institute, Palo Alto.

28

Rendu, J.-M. (1981). An Introduction to Geostatistical Methods of MineralExploration, 2nd ed., South African Inst. Mining and Metallurgy,Johannesburg,84 p.

Ripley, B. D. (1981). Spatial Statistics, John Wiley and Sons, New York, 252p.

Soulié, M., Montes, P. and Silvestri, V. (1990). Modelling Spatial Variabilityof Soil Parameters. Canadian Geotech. J., Vol. 27, No. 5, pp. 617-630.

Spry, M. J., Kulhawy, F. H. and Grigoriu, M. D. (1988). A Probability-Based Geotechnical Site Characterization Strategy for Transmission LineStructures. Report EL-5507, Vol. 1, Electric Power Research Institute, PaloAlto.

Standards Association of Australia (1977). Determination of the Static ConePenetration Resistance of a Soil - Field Test Using a Cone or a Friction-ConePenetrometer. In Methods for Testing Soils for Engineering Purposes, AS 1289,Sydney.

Tang, W. H. (1984). Principles of Probabilistic Characterization of SoilProperties. Probabilistic Characterization of Soil Properties: Bridge BetweenTheory and Practice, ASCE, Bowles, D. S. and Ko, H. Y. (eds.), pp. 74-89.

Wu, T. H. and El-Jandali, A. (1985). Use of Time Series in GeotechnicalData Analysis, Geotech. Testing Journal, GTJODJ, Vol. 8, No. 4, pp. 151-158.