Uzielli, M., Vannucchi, G. & Phoon, K. K. (2005). Ge´otechnique 55, No. 1, 3–20
3
Random field characterisation of stress-normalised cone penetration testing parameters
M . U Z I E L L I , * G . VA N N U C C H I * a n d K . K . P H O O N †
Random field modelling of soil variability allows signifi- cant statistical results to be inferred from field data;
moreover, it provides a consistent framework for incor- porating such variability in reliability-based design. Cone penetration testing (CPT) is increasingly appreciated because of its near-continuity and repeatability. Stress- normalised CPT parameters are included in widely used engineering procedures. Nonetheless, the results of varia- bility analyses for these parameters are surprisingly limited. This paper attempts to characterise normalised cone tip resistance (qc1N) and friction ratio (FR) rigor- ously using a finite-scale weakly stationary random field model. It must be emphasised that inherent soil variabil- ity so determined strictly refers to the variability of the mechanical response of soils to cone penetration. The variability of soil response potentially depends on the failure mode (shear for sleeve friction or bearing for tip resistance) and most probably on the volume of soil influenced (averaging effect). To investigate spatial varia- bility, 70 physically homogeneous CPT profiles were first identified from 304 soundings (subdivided into five regio- nal sites) and subsequently assessed for weak stationarity using the modified Bartlett test. Only 40 qc1N profiles and 25 FR profiles were deemed sufficiently homogeneous from both physical and statistical considerations for the scales of fluctuations to be valid and for estimation of the coefficient of variation of inherent soil variability. The majority of the acceptable profiles were found in sandy soils. The remaining profiles are in fine-grained soils, with a few in intermediate soils. Trends in the estimated random field parameters indicate that qc1N is more strongly autocorrelated than FR, probably because qc1N is influenced by a larger volume of soil around the cone tip, and that the mechanical response of cohesionless soils to cone penetration is significantly more variable and erra- tic than that of cohesive soils. Comparison with literature data indicates that normalisation leads to a decrease in the scale of fluctuation for cone tip resistance and a reduction in the coefficient of variation. A tentative explanation is that normalisation tends to minimise sys- tematic in situ effects that are explainable by physical causes.
KEYWORDS: in situ testing; numerical modelling and analysis;
site investigation; soil classification; statistical analysis; theor- etical analysis
La mode´lisation ale´atoire sur le terrain de la variabilite´
d’un sol permet de de´duire des re´sultats statistiques sig- nificatifs d’apre`s les donne´es de terrain ; de plus, elle donne un cadre de travail constant permettant d’incorporer une telle variabilite´ dans une conception base´e sur la fiabilite´.
L’essai de pe´ne´tration de coˆne (CPT) est de plus en plus appre´cie´ en raison de sa quasi-continuite´ et reproductibi- lite´. Les parame`tres CPT normalise´s du point de vue de la contrainte sont inclus dans des proce´dures d’inge´nierie tre`s utilise´es. Ne´anmoins, les re´sultats des analyses de variabi- lite´ pour ces parame`tres sont e´tonnamment limite´s. Cet expose´ essaie de caracte´riser rigoureusement la re´sistance de la pointe de coˆne normalise´e (qc1N) et le taux de friction (FR) en utilisant un mode`le de champs ale´atoire faiblement stationnaire a` e´chelle finie. Il faut souligner que la variabi- lite´ inhe´rente du sol ainsi de´termine´ se rapporte stricte- ment a` la variabilite´ de la re´ponse me´canique des sols a` la pe´ne´tration des coˆnes. La variabilite´ de la re´ponse du sol de´pend potentiellement du mode de de´faillance (cisaille- ment pour friction de manche ou frottement pour la re´sistance de pointe) et plus probablement du volume de sol influence´ (effet de moyenne). Pour e´tudier la variabilite´
spatiale, 70 profiles CPT physiquement homoge`nes ont e´te´
d’abord identifie´s parmi 304 sondages (subdivise´s en cinq sites re´gionaux) et ensuite e´tudie´s pour trouver la station- narite´ faible en utilisant l’essai de Bartlett modifie´. Seule- ment 40 profiles qc1N et 25 profiles FR ont e´te´ juge´s suffisamment homoge`nes du point de vue physique et statistique pour que les e´chelles de fluctuations soient valables et pour estimer le coefficient de variation de la variabilite´ inhe´rente du sol. La majorite´ des profils accep- tables ont e´te´ trouve´s dans des sols sableux. Les profils restants sont dans des sols a` grains fins et quelques-uns dans des sols interme´diaires. Les tendances des parame`tres de champs ale´atoire estime´s indiquent que qc1N est plus fortement auto-corre´le´ que FR, probablement parce que qc1Nest influence´ par un plus gros volume de sol autour de la pointe du coˆne et que la re´ponse me´canique des sols non cohe´sifs a` la pe´ne´tration de coˆne est bien plus variable et erratique que celle des cols cohe´sifs. La comparaison avec les donne´es publie´es indique que la normalisation me`ne a`
une diminution de l’e´chelle de fluctuation pour la re´sis- tance a` la pointe du coˆne et a` une re´duction du coefficient de variation. Nous tentons de l’expliquer par le fait que la normalisation tend a` minimiser les effets syste´matiques in- situ qui peuvent eˆtre explique´s par des causes physiques.
INTRODUCTION
The importance of variability analysis is increasingly recog- nised in geotechnical engineering as reliability-based meth-
ods of varying degrees of sophistication are gradually forming the basis for the calibration of new design codes.
Notable examples include the Canadian Highway Bridge Design Code (Green & Becker, 2001) and the AASHTO LRFD Bridge Design Specifications for Substructure Design (Withiam, 2003). Kulhawy & Phoon (2002) provided a detailed account of the development of geotechnical design codes in recent years. Several models for geotechnical variability have been proposed in the literature (e.g. Baecher, 1986; Orchant et al., 1988; Phoon & Kulhawy, 1999a, 1999b). The aims of these models are (a) to identify the Manuscript received 23 April 2004; revised manuscript accepted 30
September 2004.
Discussion on this paper closes on 1 August 2005, for further details see p. ii.
* Department of Civil Engineering, University of Florence, Italy.
† Department of Civil Engineering, National University of Singa- pore.
factors that contribute to overall variability, and (b) to evaluate the magnitude of each source of uncertainty. Varia- bility models, though presented using different terminolo- gies, basically identify inherent soil variability, measurement error and transformation uncertainty as the primary sources.
Inherent soil variability results primarily from natural geolo- gic processes that form and continuously modify the soil mass in situ. Measurement error arises from equipment, procedural-operator, and random testing effects. Equipment effects result from inaccuracies in the measuring devices and variations in equipment geometries and systems employed during testing. Procedural-operator effects originate from limitations in test standards and the way they are followed.
Random testing error refers to (a) the remaining scatter in the test results that is not assignable to specific testing parameters and is not caused by inherent soil variability, and (b) statistical uncertainty or sampling error that result from limited amounts of information. Transformation uncertainty is introduced when field or laboratory measurements are transformed into design soil properties using theoretical, semi-empirical or empirical models. These components are assumed to be uncorrelated. Such a hypothesis is important, as it justifies separate treatment of soil variability and test uncertainty (e.g. Agterberg, 1970; Orchant et al., 1988).
This paper focuses on the inherent variability of stress- normalised cone penetration test (CPT) measurements. In general, CPT measurements are ideal for assessing inherent soil variability because a large volume of near-continuous data can be collected in a cost-effective way, the test has good repeatability, the equipment is highly standardised, and the procedure is well defined and almost independent of the operator. Numerous researchers, such as Alonso & Krizek (1975), Tang (1979), Nadim (1986), Campanella et al. (1987), Wu et al. (1987), Reyna & Chameau (1991), Kulhawy et al.
(1992), Fenton (1999), Phoon et al. (2003) and Elkateb et al.
(2003a, 2003b), have assessed inherent soil variability using the CPT. However, results pertaining to stress-normalised CPT parameters are surprisingly limited. It is known that in-situ stress states such as confining stress and stress history influ- ence CPT data quite significantly, and fairly reliable methods for stress normalisation have been proposed. Normalised CPT parameters are included in several widely used correlations for estimation of engineering parameters (in-situ stress state, stress history, strength, compressibility) and liquefaction susceptibil- ity evaluation. Moreover, normalised cone tip resistance and sleeve friction are key parameters in CPT-based soil classifica- tion systems (e.g. Robertson, 1990). Hence it may be possible to relate inherent soil variability to soil type more directly. It must be emphasised that inherent soil variability so determined strictly refers to the variability of the mechanical response of soils to cone penetration. The variability of soil response potentially depends on the failure mode (shear for sleeve friction or bearing for tip resistance) and most probably on the volume of soil influenced (averaging effect). A systematic comparison of normalised cone tip resistance and sleeve fric- tion is undertaken in this study in terms of stationarity (or statistical homogeneity) assessment, scale of fluctuation, and coefficient of variation. Results reported in this study are based on 70 physically homogeneous CPT profiles identified from 304 soundings at various Turkish and North American sites.
Soil types found at these sites cover zones 3 to 6 in the Robertson (1990) soil classification chart: that is, clay, silty clay, clayey silt, sandy silt, silty sand, to clean sand.
DATABASE
Normalised CPT parameters
Various techniques for normalising CPT measurements for vertical stress are available. The normalised, dimensionless
cone penetration resistance (qc1N) proposed by Robertson &
Wride (1998) is adopted in this study:
qc1N¼ qc
Pa2
CQ (1)
where qc is the measured cone tip penetration resistance;
CQ ¼ Pð a=ó 9v0Þn is a correction for overburden stress; the variable stress exponent n takes values of 0.50, 1.00 and 0.70 for cohesionless, cohesive and intermediate soils re- spectively; ó 9v0 is the effective vertical stress; Pa is a reference pressure in the same units as ó9v0 (i.e. Pa ¼ 100 kPa if ó 9v0 is in kPa); and Pa2 is a reference pressure in the same units as qc (i.e. Pa2 ¼ 0.1 MPa if qc is in MPa).
An upper bound of CQ ¼ 1.7 is recommended for data at shallow depths (Youd et al., 2001). The normalised friction ratio is given by (Wroth, 1984):
FR¼ 100 fs
qcóv0
(2)
where fs is the measured sleeve friction and óv0 is the total vertical stress (qc, fs and óv0 in the same units). Owing to the nature of soil formation and deposition processes, the vertical and horizontal correlation structures in soil proper- ties are generally anisotropic, with greater variability in the vertical direction. Here, inherent variability of qc1N and FR
is addressed only in the vertical direction: thus the results of the present study do not apply to the horizontal direction.
Identification of homogeneous soil units
The identification of homogeneous soil units is an impor- tant prerequisite for variability analyses, as the correlation structure of soil properties is potentially dependent on soil type (in terms of composition and behaviour). Also, the assumption of statistical homogeneity—necessary for the application of statistical techniques—is not likely to apply unless the investigated volume of soil is fairly uniform in composition or behaviour (Fenton, 1999). As CPT provides direct information regarding the mechanical response of soil to penetration, homogeneity could also be assessed in a more fundamental way in relation to soil behaviour, rather than simply based on composition.
For the present study, preliminary selection of CPT data was performed among a large number of soundings con- ducted at Turkish and North American sites. Five regional sites were defined on the basis of geographical location and source database:
(a) greater Oakland, CA, area (Alameda and Oakland counties) (hereinafter OAK)
(b) palaeoliquefaction sites in mid-America earthquake regions (hereinafter MAE) (Collierville, TN; Dudley, MO; Marked Tree, AR; Memphis, TN; Mud Island, TN; Opelika, AL; St Louis, MO; Wilson, AR; Wolf River, TN; Wyatt, MO)
(c) Texas A&M University site (TXS)
(d) liquefaction sites in the Adapazari area, Turkey (ADP) (e) Treasure Island, in the San Francisco Bay area (TSI).
Data from OAK were collected as part of the USGS Earth- quake Hazards Program, and are available at http://quake.
wr.usgs.gov/prepare/cpt/; MAE data were collected by the Georgia Tech In-Situ Testing Group (http://www.ce.gatech.
edu/geosys/Faculty/Mayne/Research/index.html); TXS and TSI soundings were conducted at Texas A&M and Treasure Island NGES sites respectively (http://www.unh.edu/nges/);
and ADP data are available on the PEER website (http://
peer.berkeley.edu/turkey/adapazari/). The above data were screened based on the following criteria:
(a) The measurement interval is less than 0.05 m.
(b) Reliable measurement of groundwater level is reported.
(c) The length of the sounding is at least 10 m.
The first criterion ensures that there is sufficient resolution for accurate variability assessment (e.g. Jaksa et al., 1997;
Cafaro & Cherubini, 2002). The second is a necessary condition for the calculation of qc1N and FR. The third ensures that the record length is long enough for stationarity assessment (a stationary process is theoretically infinite in length) and the sample size is large enough for meaningful statistical treatment. A total of 304 soundings were consid- ered adequate for analysis based on the above selection criteria.
Homogeneous soil units (HSU) were identified from the selected soundings using a statistical moving window proce- dure proposed by Uzielli (2004). The procedure involved converting the data to normalised form as given by equations (1) and (2) and applying a moving window on profiles of qc1N, FR and the soil behaviour classification index, Ic (e.g.
Robertson & Wride, 1998), for statistical identification. The soil behaviour classification index maps the usual two-dimen- sional CPT-based soil behaviour classification zones onto a one-dimensional scale, and is calculated from qc1Nand FR:
Ic¼ [(3:47 log qc1N)2þ (1:22 þ log FR)2]0:5 (3) The scale is shown in Fig. 1. Each moving window is made up of two semi-windows of equal height above and below a centre point. At each centre point (identified by its depth zc), the mean, standard deviation and coefficient of variation (COV) were calculated for data lying in the interval zc Wd/2 < z < zc+ Wd/2, corresponding to the upper and lower limits of the moving window of height Wd ¼ 1.50 m. The height of the moving window was set on the basis of past research (e.g. Campanella et al., 1987; Wickremesinghe, 1989; Lunne et al., 1997) and calibration of results with available borehole logs (Uzielli, 2004).
HSUs are essentially identified by delineating soundings
into sections where COV(log qc1N) and COV(Ic) are less than 0.1 over at least 4.50 m. Harr (1987) proposed a value of 0.1 for COV as the upper limit for ‘low dispersion’ in soil properties. Two additional conditions were imposed to ensure that there are sufficient data points for meaningful statistical analysis and log(qc1N) and Ic do not exhibit gradual changes with depth. A total of 70 HSUs were identified from the 304 selected soundings. Details of the CPT database and the identification procedure are given elsewhere (Uzielli, 2004).
To provide a concise overview of the 70 HSUs, the mean values of qc1N and FR and their corresponding standard deviations are plotted on the Robertson (1990) chart as points and error bars respectively (Fig. 1). The ability of the moving window procedure to identify homogeneous soil units was assessed by observing that the mean value of the coefficients of variation of Ic in the HSUs was 0.02, and that values in no case exceeded 0.10. Fig. 1 shows that the identified HSUs cover a wide range of soil types (zones 3 to 6). It may also be observed in Fig. 1 that the standard deviation of qc1N and FR increases approximately from category 3 to 6, i.e. from clay to sand, indicating that CPT profiles are more variable in cohesionless soils. This may also be seen in Fig. 2, in which typical profiles of qc1N and FR are plotted against depth for a cohesive soil (CHS) hsu003, classified in zone 3, and a cohesionless soil (CHL) hsu010, classified in zone 6. Fig. 3 illustrates the subdivision of HSUs by regional site in the Robertson chart. As shown in Table 1, the mean values of qc1N and FR of the 13 HSUs from OAK sites fall in zones 3, 4, 5 and 6; the 27 HSUs from MAE in zones 4 and 6; the 6 HSUs from TXS in zones 4 and 5; the 18 HSUs from ADP in zones 3 and 6;
and the 6 HSUs from TSI in zone 3 only. Hence it is possible to study trends based on geographic location and soil type.
RANDOM FIELD
Inherent soil variability may be modelled using finite-scale stochastic processes, which assume limited spatial correla- tion, or by fractal processes, which admit significant linger-
0·1 1 10
FR: % 1000
100
10
1 qc1N
Ic 5 1·31
Ic 5 2·05
Ic 5 2·60
Ic 5 2·95
Ic 5 3·60 Normally consolidated
7
6
5
4
1
3
2 8
9
after Robertson (1990)
Soil behaviour type classification
Soilbehaviour zone
Ic range Soil behaviour type
1 2 3 4 5 6 7 8 9
Sensitive, fine grained Organic soils - peats Clay, silty clay
Silt mixtures - clayey silt to silty clay Sand mixtures - silty sand to sandy silt Clean sand, silty sand
Gravelly sand to sand Very stiff sand to clayey sand Very stiff, fine grained Non appl.
Ic . 3·60 2·95 , Ic , 3·60 2·60 , Ic , 2·95 2·05 , Ic , 2·60 1·31 , Ic , 2·05
Ic , 1·31 Non appl.
Non appl.
Fig. 1. Mean values and standard deviations of homogeneous soil units as viewed within the context of Robertson’s (1990) soil classification chart, with superimposed boundaries of soil behaviour type index, Ic
RANDOM FIELD CHARACTERISATION OF STRESS-NORMALISED CPT PARAMETERS 5
ing correlation over large distances. CPT measurements are not truly point measurements, but are representative para- meters of the extent of the zone in which soil failure occurs due to penetration; hence small-scale variations may not be detectable. Moreover, previous research (e.g. Fenton, 1999;
Phoon et al., 2003) suggests that it is difficult to distinguish between finite-scale and fractal models over a finite sampling domain. Thus a finite-scale approach is possibly more suita- ble.
In this study, inherent soil variability is modelled as a zero-mean weakly stationary random field with finite-scale correlation structure (Vanmarcke, 1983). This random field is added to a trend function that is assumed to be determi- nistic in nature. It should be emphasised that soil properties are not ‘random’ in the sense that they are intrinsically unknown, but are modelled as ‘random’ because it is practically impossible to obtain measurements at all points.
The mathematical and statistical techniques commonly used to treat stochastic processes are then useful to describe spatial variations in a parsimonious and concise way (Baecher & Christian, 2003).
Decomposition
The ‘real’ value (i.e. neglecting measurement error) of a geotechnical property [î(z)] may be decomposed into a smoothly varying trend function [t(z)] and a fluctuating component [w(z)] representing the inherent soil variability:
7
6 5
8
9 CHL
CHS 1000
100
10
1 qc1N
0·1 1 10
(a)
1000
100
10
1
CHL
CHS 20 15
10 5
0
zt: m (b) 0
5
10
15
20 zt: m
0·1 1 10
FR: % (c)
CHL (hsu010) Mean (qc1N) 5 101·15 COV (qc1N) 5 0·37 Mean (FR) 5 0·58%
COV (FR) 5 0·25
CHS (hsu003) Mean (qc1N) 5 7·93 COV (qc1N) 5 0·18 Mean (FR) 5 1·96%
COV (FR) 5 0·35 1
4
3 2
CHL CHS
qc1N
FR: %
Fig. 2. Representative examples of cohesionless (CHL) and cohesive (CHS) behaviour HSUs: (a) plot on Robertson’s (1990) chart; plots of (b) qc1N and (c) FR against relative depth, zt (depth from top of HSU) with linear trends
7
6
5
4
3 1
8
9
2 OAKMAE
TXS ADP TSI Ic 5 1·31
Ic 5 2·60
Ic 5 3·60 Ic 5 2·95
I 5c 2·05
Normally consolidated
0·1 1 10
Mean (FR): % 1000
100
10
1 Mean (qc1N)
Fig. 3. Categorisation of HSUs by regional site (after Robertson, 1990)
Table 1. Number of best-fit ACMs for HSUs categorised by soil behaviour zone and regional site (qc1N and FR)
Reg. Soil qc1N FR
site zone
SNX CSX SMK SQX Total
STAT
NST NAPP Total
(reg. site)
SNX CSX SMK SQX Total
STAT
NST NAPP Total
(reg. site)
OAK 3 0 0 0 0 0 1 1 2 0 0 0 0 0 1 1 2
4 1 0 1 0 2 1 1 4 0 1 0 0 1 1 2 4
5 0 0 0 0 0 3 1 4 0 1 0 0 1 2 1 4
6 2 0 1 0 3 0 0 3 0 0 1 0 1 1 1 3
Total OAK 3 0 2 0 5 5 3 13 0 2 1 0 3 5 5 13
MAE 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 2 0 2 1 1 4 1 0 0 0 1 2 1 4
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 3 6 10 19 2 2 23 2 2 7 0 11 8 4 23
Total MAE 0 3 8 10 21 3 3 27 3 2 7 0 12 10 5 27
TXS 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 1 1 0 2 3 0 2 0 0 2 1 0 3
5 0 0 1 0 1 2 0 3 0 0 0 0 0 1 2 3
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Total TXS 0 0 1 1 2 2 2 6 0 2 0 0 2 2 2 6
ADP 3 0 2 2 1 5 6 4 15 1 1 1 0 3 10 2 15
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 1 2 0 3 0 0 3 0 1 1 0 2 1 0 3
Total ADP 0 3 4 1 8 6 4 18 1 2 2 0 5 11 2 18
TSI 3 1 2 1 0 4 0 2 6 2 0 1 0 3 1 2 6
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Total TSI 1 2 1 0 4 0 2 6 2 0 1 0 3 1 2 6
40 16 14 70 25 29 16 70
RANDOMFIELDCHARACTERISATIONOFSTRESS-NORMALISEDCPTPARAMETERS7
î zð Þ ¼ t zð Þ þ w zð Þ (4) where z is the depth coordinate. The importance of the decomposition procedure has been recognised by many researchers; inappropriate removal of the deterministic trend component from a measurement profile would result in a biased assessment of correlation. The choice of the trend to be removed is a complex task as it affects the correlation structure and the value of the statistical parameters describ- ing the random field (coefficient of variation and scale of fluctuation). In addition, trend removal should at least result in stationary residuals. Given the above considerations, it is important to limit trend functions to those that are consistent with local geology and well-established principles of soil behaviour in geotechnics. In the present study, the following criteria were taken into account in choosing the method of trend removal:
(a) the existence of a physical motivation (e.g. Akkaya &
Vanmarcke, 2003; Baecher & Christian, 2003)
(b) compatibility with available models for the estimation of random field parameters
(c) compatibility with the adopted stationarity assessment criterion.
Trend removal by linear least-squares regression analysis was found to meet the above requirements. Linear trend removal has been used in several variability studies (e.g.
Campanella et al., 1987; Popescu et al., 1998; Elkateb et al., 2003a, 2003b; Phoon et al., 2003; Uzielli, 2004), though stress normalisation studies have shown that the variation of vertical trends may take other forms, especially for cohe- sionless soils. Moreover, while such an approach may not be fully consistent owing to the presence of correlated resi- duals, and, consequently, could lead to unconservative and biased estimates (e.g. Agterberg, 1970; Campanella et al., 1987; Baecher, 1999; Fenton, 1999), its use seems appro-
priate here, mainly for two reasons. First, normalisation by vertical effective stress should account for systematic physi- cal effects on the soil profiles (e.g. overburden stress and stress history). Thus profiles of vertical stress-normalised variables of homogeneous soil units should no longer display significant trends with depth beyond a simple first-order linear function. Second, it should be emphasised here that the more important goal is not to violate weak stationarity in the resulting residuals or fluctuations, regardless of the method of trend removal. Even though regression is not strictly applicable to correlated residuals, it is possible to assess the impact of this inconsistency independently using the modified Bartlett test described below (Phoon et al., 2003). Fig. 4 shows an example HSU identified using the statistical procedure described above. The performance of the identification procedure may be appreciated through visual examination of the aggregated data points in Fig.
4(a). Profiles of qc1N, FR and Ic in the HSU and the linear trends identified by regression are shown in Figs 4(b), 4(c) and 4(d) respectively.
Estimation of the scale of fluctuation
In finite-scale models, the scale of fluctuation (ä) is a concise indicator of the spatial extent of a strongly corre- lated domain. There are various curve-fitting and statistical techniques available in the geotechnical literature for the estimation of the autocorrelation model and scale of fluctua- tion (e.g. DeGroot & Baecher, 1993, Fenton, 1999). A simple but robust approach is to estimate the sample auto- correlation function (ACF) using the method of moments, fit a plausible theoretical autocorrelation model (ACM) to ACF, and evaluate the scale of fluctuation based on the model parameter in the ACM.
The jth coefficient of the sample autocorrelation function
7
6
5
4
3
2 9 8
1 Ic 5 1·31
Ic 5 2·60
Ic 5 2·95
Ic 5 3·60 Ic 5 2·05
1000
100
10
1 qc1N
0·1 1 10
FR: % 1 2 3 4
Ic 0
2 4 6 8 10 12 14 16 zt: m
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14
1 10 100 1000 160·1 1 10
qc1N FR: %
HSU width: 16·025 m measurement interval: 0·025 m 642 data sets in HSU
(a)
Mean (Ic) 5 1·67 COV(Ic) 5 0·07
(b)
Mean (qc1N) 5 151·59 COV(qc1N) 5 0·27
(c)
Mean (FR) 5 0·68%
COV(FR) 5 0·19 (d) Fig. 4. Example of identified HSU: (a) plot of complete HSU data points in Robertson’s (1990) chart; plots of (b) Ic, (c) qc1N
and (d) FR profiles against relative depth, zt (depth from top of HSU) with superimposed linear trends
(ACF) of the fluctuating component wi ¼ w(zi) (which is a zero-mean stochastic process) is given by
R^ Rð Þ ¼ôj
X
nd j
i¼1
wi wiþ j
X
nd j
i¼1
w2i
(5)
The ACF was calculated for separation distances ôj ¼ j˜z corresponding to j¼ 1, 2, . . ., nd/4, as suggested by Box &
Jenkins (1970), where nd is the number of data points in a given profile and ˜z is the sampling interval. Various kinds of ACM have been employed in the geotechnical literature to fit the ACF (e.g. Spry et al., 1988; DeGroot & Baecher, 1993; Jaksa, 1995; Lacasse & Nadim, 1996; Fenton, 1999;
Phoon et al., 2003). Spry et al. (1988) opined that there is no physical basis to prefer one ACM over another. In this study, it was found that the following four ACMs were sufficient to fit sample ACFs derived from CPT data:
(a) single exponential (SNX) (b) cosine exponential (CSX) (c) second-order Markov (SMK) (d) squared exponential (SQX).
The analytical expressions of the four ACMs and the formulae relating the scales of fluctuation to the model parameters are shown in Table 2.
To increase the reliability of the estimated ä, Uzielli (2004) fitted the ACMs only to the initial part of the sample ACF with coefficients exceeding Bartlett’s limits:
rB¼1:96 ffiffiffiffiffind
p (6)
This guideline has been used by Spry et al. (1988) and is motivated by the well-accepted fact that the estimated auto- correlation coefficients become less reliable with increasing lags, and are deemed not significantly different from zero inside the range rB (e.g. Priestley, 1981; Brockwell &
Davis, 1991; Fenton, 1999). The coefficient of determination, R2, was recorded for each ACM fit. Only ACMs producing R2 . 0.9, with at least four initial autocorrelation coeffi- cients greater than rB to ensure the significance of the fit, were accepted. The procedure outlined above was applied to qc1N and FR profiles for all HSUs. Examples of best-fit for SNX, CSX, SMK and SQX ACMs to sample ACFs from qc1N profiles are shown in Figs 5(a), 5(b), 5(c) and 5(d) respectively. It can be seen that the curves from the four ACMs display distinct shapes at low lags: hence, if the fit of an ACM to a sample ACF is performed with emphasis on low separation distances, the correlation between measure- ments in the volume of soil nearest to the cone (i.e. most directly affected by penetration) assumes a relevant role in random field modelling as defined herein.
Modified Bartlett test
As stated previously, weak stationarity is an important requisite for random field characterisation of soil properties.
Weak stationarity cannot be verified in a strict sense over a finite length because longer scale fluctuation can be mistakenly identified as a non-stationary component
Table 2. Autocorrelation models and relations between scale of fluctuation and characteristic model parameters
Autocorrelation model Equation Scale of fluctuation
SNX R(ô) ¼ exp(kSNXjôj) ä ¼ 2=kSNX
CSX R(ô) ¼ exp(kCSXjôj)cos(kCSXô) ä ¼ 1=kCSX
SMK R(ô) ¼ (1 þ kSMKjôj)exp(kSMKjôj) ä ¼ 4=kSMK
SQX R(ô) ¼ exp[(kSQXô)2] ä ¼ ffiffiffiffiffiffi
pð=
kSQX
sample SNX CSX SMK SQX
R2(SNX) 5 0·98 R2(CSX) 5 0·99 R2(SMK) 5 0·98 R2(SQX) 5 0·95
Measurement interval in HSU: Äz 5 0·012 m rB 5 0·12 value reached at lag 13 (ô 5 0·14 m)
72 coefficients in ACF 13 coefficients used in fit
0 0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8
ô: m (a) 1·0
0·5 0 20·5 21·0
ÂR(ô)
sample SNX CSX SMK SQX
R2(SNX) 5 0·96 R2(CSX) 5 1·00 R2(SMK) 5 0·99 R2(SQX) 5 0·97
Measurement interval in HSU: Äz 5 0·025 m rB 5 0·14 value reached at lag 27 (ô 5 0·65 m)
49 coefficients in ACF 27 coefficients used in fit
0 0·2 0·4 0·6 0·8
ô: m (b) 1·0
0·5 0 20·5 21·0
ÂR(ô)
sample SNX CSX SMK SQX
R2(SNX) 5 0·97 R2(CSX) 5 0·99 R2(SMK) 5 1·00 R2(SQX) 5 0·99
Measurement interval in HSU: Äz 5 0·020 m rB 5 0·13 value reached at lag 20 (ô 5 0·38 m)
54 coefficients in ACF 20 coefficients used in fit
0 0·2 0·4 0·6 0·8
ô: m (c) 1·0
0·5 0 20·5 21·0
ÂR(ô)
sample SNX CSX SMK SQX
R2(SNX) 5 0·94 R2(CSX) 5 0·97 R2(SMK) 5 0·99 R2(SQX) 5 1·00
Measurement interval in HSU: Äz 5 0·025 m rB 5 0·11 value reached at lag 15 (ô 5 0·35 m)
74 coefficients in ACF 15 coefficients used in fit
0 0·2 0·4 0·6 0·8 ô: m
(d) 1·0
0·5 0 20·5 21·0
ÂR(ô)
1 1 1·2 1·4 1·6 1·8
1 1·2
Fig. 5. Example of best-fit case for: (a) SNX (hsu068); (b) CSX (hsu027); (c) SMK (hsu062); (d) SQX (hsu019) ACMs
RANDOM FIELD CHARACTERISATION OF STRESS-NORMALISED CPT PARAMETERS 9
(e.g. Agterberg, 1970; Baecher, 1999; Fenton, 1999; Phoon et al., 2003). Thus only local weak stationarity can be identified. A profile is considered to be weakly stationary if (a) the mean is constant over a given spatial direction, and (b) the autocovariance is only a function of the separation distanceôjbetween observations.
Numerous researchers (e.g. Watson, 1967; Agterberg, 1970; Fenton, 1999; Phoon et al., 2003) have noted that the application of classical statistical tests (e.g. statistical runs test, Spearman’s rank coefficient and Kendall’s tau test) to correlated data may result in biased assessments. The mod- ified Bartlett statistic procedure (MBS) was proposed by Phoon et al. (2003) with the aim of providing a more rational basis for rejecting the null hypothesis of stationarity in the correlated case. MBS has been shown to be more discriminating than other traditional classical tests (Uzielli et al., 2004), as it (a) incorporates the correlation structure in the underlying data, and (b) includes all the key assumptions in geostatistical analysis (stationarity, choice of trend func- tion, and autocorrelation model). This generality is achieved by exploiting powerful digital simulation techniques for random fields. The MBS procedure neglects measurement and random testing errors: this assumption is acceptable for the CPT, which has been shown to be largely operator- independent and to have very low random measurement errors (e.g. Campanella et al., 1987; Kulhawy & Trautmann, 1996; Jaksa et al., 1997).
In the MBS procedure, Bartlett statistics (basically the ratio of variances in two contiguous segments) are computed by applying a moving window method to fluctuating profile.
The test statistic is the peak value of the Bartlett statistic profile (Bmax), and it is compared with a critical value (Bcrit) derived using simulation. The null hypothesis of stationarity in the variance is rejected at the customary 5% level of significance if Bmax. Bcrit. This study basically follows the MBS procedure described by Phoon et al. (2003), with additional steps introduced (fitting ACM to the initial part of ACF and rejecting ACM producing R2 , 0.9) to achieve more robust estimates for the scale of fluctuation. The procedure as revised by Uzielli (2004) will be referred to as MBSR hereon.
For data sets satisfying the condition of weak stationarity, the dimensionless coefficient of variation of inherent varia- bility (ç) is obtained by normalising the standard deviation with respect to the value of the linear trend function at the midpoint of the homogeneous soil unit under investigation (tM) (Phoon & Kulhawy, 1999a):
ç ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
nd 1 Xnd
i¼1
wi
½ 2 s
tM
(7)
STATIONARITY OF CPT SOUNDINGS
While HSUs are defined in terms of homogeneity of both qc1Nand FR (corresponding to tight clusters in the Robertson (1990) chart), it should be emphasised that weak stationarity (or non-stationarity) of one parameter does not necessarily imply the same assessment for the other. Cone tip resistance and sleeve friction are profoundly different measurements, as they are related to distinct aspects of soil behaviour (bearing capacity and friction), and they are influenced by different volumes of penetrated soil (tip resistance being influenced by soils within a few diameters around the tip, and sleeve friction being affected only by soil adjacent to it).
Three outcomes are possible for each profile of qc1N or FR:
(a) MBSR is not applicable in its entirety, for one or more of the following reasons: (i) R2 does not exceed 0.9 for any of the four ACMs shown in Table 2; (ii) one or more of the dimensionless profile factors of the MBS procedure (k, I1, I2), which depend on the scale of fluctuation (ä), sample spacing (˜z), and sample size (nd), fall outside the ranges established by Phoon et al.
(2003).
(b) MBSR could be applied, but Bmax> Bcrit: thus the profile is classified as non-stationary at 5% significance level.
(c) The MBSR procedure could be applied in its entirety, and at least one ACM satisfies the goodness-of-fit and stationarity criteria at the 5% level. If more than one ACM satisfy the above conditions, the scale of fluctuation resulting from the ACM with the maximum R2 is adopted.
Only results from case (c) are analysed further and discussed in detail below. Note that 40 and 25 values of ä were accepted for qc1N and FR respectively, out of values from 70 possible HSUs. Out of the 40 MBSR-stationary qc1Nprofiles, five belonged to OAK sites, 21 to MAE, two to TXS, eight to ADP, and four to TSI (Table 1). Out of the 25 MBSR- stationary FR profiles, three belonged to OAK sites, 12 to MAE, two to TXS, five to ADP, and three to TSI.
Assessment of weak stationarity
Figures 6(a) and 6(b) show the best-fit ACM for each HSU (if MBSR-stationary) for qc1N and FR respectively.
Table 1 summarises the distribution of HSUs according to the best-fit ACM (SNX, CSX, SMK or SQX) if they are stationary. The numbers of cases that are non-stationary or not amenable to MBSR are also noted. In addition, station- ary (STAT), non-stationary (NST) and non-applicable cases (NAPP) are further categorised according to the soil zone in which the mean values of FR and qc1N in the HSU are located in the Robertson (1990) chart.
It is evident from Table 1 that FR is not appropriate for stationary assessment as the rate of rejection of the station- ary hypothesis is too high (25 stationary versus 29 non- stationary). Although physical homogeneity does not neces- sarily correspond to statistical homogeneity (or stationarity), one does expect a stronger correlation such as that produced by qc1N (40 stationary versus 16 non-stationary). It is well known that sleeve friction is much less reliable than tip resistance. DeJong and Frost (2002) demonstrated that more accurate soil characterisation could be achieved by measur- ing sleeve friction at a larger distance behind the tip (.
0.35 m). It is quite surprising that most of the non-stationary HSUs fall in zone 3, which corresponds to clays and silty clays. A closer examination reveals that such evidence may be due to the fact that although CPT profiles in sands are more variable than those in clays, the latter occasionally show large jumps embedded in predominantly smaller fluc- tuations (see Fig. 2), and some of these spikes result in very large—though isolated—peaks in Bartlett statistics. Non- applicability of the MBSR procedure resulted in most cases from insufficient reliability of ACM fitting: out of the 14 NAPP HSUs for qc1N, 12 were not amenable to MBSR owing to insufficiently high R2, while only two did not meet the range of dimensionless MBS factors. The corresponding breakdown for the 16 NAPP FR HSUs is 15 and one respectively.
For qc1N, the SMK and SQX models are more common, particularly in sandy soils (zone 6). Incidentally, SMK and SQX are the ACMs that provide higher autocorrelation coef- ficients in the initial part of the ACF (i.e. those correspond-
ing to smaller separation distances). For FR, the SQX model does not provide the best fit in any soils. These observations are compatible with the fact that the cone tip resistance is influenced by a larger volume of soil, and hence induces larger correlations at short lags. Interestingly, all qc1N and FR profiles with mean values falling within the normally consolidated area of zones 3 and 4 were not amenable to MBSR or were deemed non-stationary (NAPP or NST in Figs 6(a) and 6(b)). More data are needed to clarify the potential role of in-situ soil state and stress history in soil variability.
SECOND-MOMENT STATISTICS OF CPT SOUNDINGS A random field may be described concisely in the second- moment sense by the scale of fluctuation and the coefficient of variation. A careful study of these important statistics is presented below. First, it will be verified that the statistics are not biased by the number of CPT measurement points
within each HSU, the CPT measurement interval, and the best–fit autocorrelation model (ACM). Second, the statistics computed from qc1N and FR are studied separately by plotting them against mean qc1N and mean FR respectively.
The effect of soil type is then interpreted using the Robertson (1990) chart, which considers mean qc1N and mean FR jointly. Finally, if sufficient data are available, site- specific effects not explainable by mean qc1N, mean FR, and soil type are highlighted.
Vertical scale of fluctuation
Possible bias produced by sample size (i.e. the number of measurements in an HSU) and spacing (i.e. CPT measure- ment interval) was first investigated by plotting the estimates for qc1N (Fig. 7(a)) and FR (Fig. 7(b)) against sample size, with data points categorised by measurement interval. No significant trends are visible. Hence the estimates of correla- tion distance for MBSR-stationary HSUs are considered to be unbiased in terms of sample size and spacing.
The scales of fluctuation (ä) for each MBSR-stationary profile are plotted against mean qc1N (Fig. 8(a)) and mean FR (Fig. 8(b)) according to the best-fit ACM. The scales of fluctuation computed by one ACM do not rank consistently above or below the others. This may not come as a surprise given that ä is defined to make different ACMs comparable (Vanmarcke, 1983). Table 3 shows second-moment statistics (mean and coefficient of variation) for the scale of fluctua- tion of qc1N and FR categorised by best-fit ACM and soil type (i.e. location of the mean values of qc1N and FR in the Robertson (1990) chart). On average, the scale of fluctuation of qc1N is higher than the scale of fluctuation of FR (Table 3). The former lies between 0.13 and 1.11 m, with only two values below 0.2 m and one value above 1.0 m, whereas the latter lies between 0.12 and 0.6 m. The overall mean values are 0.70 m for ä(qc1N) and 0.39 m for ä(FR), with lower dispersion for qc1N (COV¼ 0.28) than for FR (COV¼ 0.40).
The stronger correlation structure of qc1N may find a physi- cal basis in the fact that sleeve friction measurements and, consequently, FR are more erratic than cone tip resistance measurements. This is to be expected, given that qc1N is influenced by a volume of soil around the cone tip that is larger than the sampling interval. Hence a few continuous
SNX CSX SMK SQX NST NAPP
0·1 1 10
Mean FR: % (a) 1000
100
10
1 Mean qc1N
7
6
5
4
1 3
2 8 Ic 5 1·31 9
Ic 5 2·05
Ic 5 2·60 Ic 5 2·95
Ic 5 3·60
SNX CSX SMK SQX NST NAPP
0·1 1 10
Mean FR: % (b) 1000
100
10
1 Mean qc1N
7
6
5
4
1 3
2 8 Ic 5 1·31 9
Ic 5 2·05
Ic 5 2·60 Ic 5 2·95
Ic 5 3·60 Norm
ally consolidated Normally consolidated
Fig. 6. Results of MBSR stationarity assessment for (a) qc1Nand (b) FR profiles (after Robertson, 1990)
Äz 5 12 mm Äz 5 16 mm Äz 5 20 mm Äz 5 25 mm Äz 5 50 mm
0 250 500 750 1000
nd (a) 1·2
0·8 0·4 0 ä(qc1N): m
Äz 5 12 mm Äz 5 16 mm Äz 5 20 mm Äz 5 25 mm Äz 5 50 mm
250 500 750 1000
nd (b) 1·2
0·8 0·4 0 ä(FR): m
Fig. 7. Plots of data numerosity of HSUs against scale of fluctuation for: (a) qc1N; (b) FR
RANDOM FIELD CHARACTERISATION OF STRESS-NORMALISED CPT PARAMETERS 11
values of qc1N are basically affected by almost the same volume of soil as the cone penetrates. In addition, more reliable sleeve friction measurements are apparently possible if they are obtained further behind the tip (DeJong & Frost, 2002). This suggests that present sleeve friction measure- ments from standard cones possibly contain more noise, which further weakens the correlation.
Figure 9 is similar to Fig. 8, except that the data points are differentiated according to the regional site. Note that different regional sites contain different proportions of dif- ferent soil types, as shown in Table 4. For example, almost all MAE HSUs are soil type 6, all TSI HSUs are soil type 3, and OAK HSUs cover soil types 4 and 6. Before attribut- ing any trends in Fig. 9 to site-specific effects, it is probably more reasonable to see whether these trends are explainable by differences in soil types or, more specifically, by differ- ences in the mean qc1Nand FR.
It is quite evident from Fig. 9(a) that the lower and upper bounds of ä(qc1N) generally increase with increasing mean qc1Nwhen all the regional sites are considered together. This is also shown in the last row of Table 4. The mean scale of fluctuation for qc1N also exhibits this increase, but not as clearly given the large scatter. This trend can also be discerned within a regional site, particularly for data from ADP and MAE (Table 4).
The cluster of four TSI data points with qc1N , 10 are classified as soil type 3 and probably belong to the Young Bay Mud layer (Fig. 9(a)). The range of ä(qc1N) (0.13–
0.23 m) for this cluster lies entirely below those of the other regional sites. For instance, the scales of fluctuation for ADP HSUs belonging to soil type 3 range between 0.28 and 0.64 m (Table 4). The range of ä(qc1N) values at TSI was computed using different best-fit ACMs (1 SNX, 2 CSX and 1 SMK) as shown in Table 1. Hence these low values are not caused by the choice of a particular ACM. This effect is also exhibited by ä(FR) at the same regional site, as the range 0.13–0.28 m is lower than 0.26–0.45 m for HSUs consisting of soil type 3 at ADP. Hence it would appear that a site-specific effect is present at TSI. It is also possible to compare results for soil type 6 from OAK, MAE, and ADP.
Unlike the case for TSI, there is no strong evidence to suggest that a site-specific effect is present here, given the large scatter and unequal sample sizes at the three regional sites.
The sample sizes for ä(FR) are smaller, and the trends in Fig. 9(b) should be interpreted with this caveat in mind.
Overall, the lower and upper bounds appear to decrease
slightly with increasing mean FR. Note that soil type 3 has high FR but low qc1N, whereas the reverse is true for soil type 6 in the Robertson (1990) chart (Fig. 6). Hence the above observation actually supports the opposing trend ob- served for ä(qc1N) data, which essentially indicates that cohesionless soils are somewhat more correlated than cohe- sive soils. The negative trend for ä(FR) is, however, much weaker than the increasing trend for ä(qc1N). This is also compatible with the results of past research (e.g. Teh &
Houlsby, 1991), which indicate that the extent of the failure zone increases with increasing shear strength and stiffness:
thus the influence zone affecting cone tip resistance is larger in sand, whereas sleeve friction is affected only by the adjacent soil regardless of soil type. A possible site-specific effect can be seen in Fig. 9(b): the MAE soil type 6 data points plot above the ADP data points of the same soil type in the mean FR range between 0.6% and 0.9%. However, it should be noted that there are 11 data points for MAE and only two data points for ADP (Table 4). The single OAK soil type 6 data point agrees with the MAE cluster, although the mean FR is about 2%.
It is interesting to examine the effect of normalisation on the scale of fluctuation through comparison with results from the literature (e.g. Appendix A in Phoon et al. 1995). Phoon
& Kulhawy (1999a) observed a range of 0.1–2.2 m with an average value of 0.9 m for the non-normalised cone tip resistance (qc) from seven studies covering both sands and clays. Cafaro & Cherubini (2002) estimated the average values of the scale of fluctuation of linearly detrended qc
data of two Italian clays as 0.40 and 0.57 m. Elkateb et al.
(2003b) estimated scales of fluctuation of qc ranging from 0.37 to 0.80 m for four soil layers classifiable as types 5 and 6 in the Robertson (1990) chart. For strict comparison with other literature data, the type of trend removed should be specified, as the removal of higher-order polynomial trends results in a decrease in the scale of fluctuation (e.g. Jaksa et al., 1997; Phoon et al., 2003). In addition, it is known that estimates of scales of fluctuation are usually not very precise (besides trend removed, they depend on how ACF is eval- uated, choice of ACM, fitting criteria, etc.). Nevertheless, a broad comparison seems to indicate that the scale of fluctua- tion for qc1N is comparable to or shorter than that for qc. This observation is compatible with the fact that normal- isation tends to minimise systematic physical in-situ effects (depositional processes and confining pressure) that may introduce subtle trends, and hence longer correlation lengths in the data. Literature results for normalised and non-nor-
SNX CSX SMK SQX 1·2
0·8 0·6 0·4 0·2 0 1·0
ä(qc1N): m
1 10 100 1000
Mean (qc1N) (a)
SNX CSX SMK SQX 1·2
0·8 0·6 0·4 0·2 0 1·0
ä(FR): m
0·1 1 10
Mean (FR): % (b)
Fig. 8. Scale of fluctuation against mean value in HSU for MBSR-stationary HSUs, categorised by best-fit ACM: (a) qc1N
profiles; (b) FR profiles
