Engineering Geology 122 (2011) 239–248

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Engineering Geology

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Probabilistic evaluation of spatial distribution of secondary compression by usingkriging estimates of geo-layers

Woojin Lee a, Donghee Kim c, Youngho Chae a, Dongwoo Ryu b,⁎a School of Civil, Environmental and Architectural Engineering, Korea Univ., Seoul, 136-713, Republic of Koreab Korea Institute of Geoscience and Mineral Resources (KIGAM), Daejeon, 305-350, Republic of Koreac Infrastructure Engineering Team, SK E&C, Seoul, 100-130, Republic of Korea

⁎ Corresponding author. Tel.: +82 42 868 3099; fax:E-mail address: dwryu@kigam.re.kr (D. Ryu).

0013-7952/$ – see front matter. Crown Copyright © 20doi:10.1016/j.enggeo.2011.06.008

a b s t r a c t

a r t i c l e i n f o

Article history:Received 12 October 2010Received in revised form 2 May 2011Accepted 11 June 2011Available online 30 July 2011

Keywords:Secondary compressionKrigingCoefficient of variationProbabilistic method

This paper presents a procedure for evaluating the spatial uncertainty in the secondary compression (ss) usinga probabilistic method. In order to evaluate the spatial distribution of ss, the spatial maps of three geo-layers(the thickness and depth of the consolidating layer, the bottom elevation of the reclaimed sandfill) areestimated by using kriging techniques. For all three geo-layers considered in this study, the ordinary kriging isfound to give more reliable estimates than the kriging with a trend and simple kriging. It is observed that thecoefficients of variation (COVs) of Cα/Cc and Cc/(1+e0) have similar influences on the COV of ss. It is alsoshown that theCOVof cvhas less effect on theCOVof ss than theCOVsofCα/Cc andCc/(1+e0) although theCOVof cvis larger than that of Cα/Cc and Cc/(1+e0). The COV of ss evaluated by considering all the COVs of soil properties is0.420, which is 1.4–2.7 times larger than that determined by considering the COV of an individual soil propertyseparately. It is observed that the area exceeding a design criterion increases as the COV of Cα/(1+e0) increases andtheprobabilistic designcriterion(α)decreases. For SongdoNewCity, the area ratiodecreases from0.47 forαvalueof0.05 to 0.04 forα value of 0.45. The design procedure presented in this paper could be used in the decision makingprocess for a geotechnical engineering design.

Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction

Land reclamation on the foreshore of existing coastlines oftenoverlies soft marine clays, which require soil improvement to reduceundesirable long term settlement, i.e., secondary compression (ss). Suchundesirable long term settlement gives rise to a negative skin friction onpiles, cracks on pavements, and differential settlement between roadsand buildings, etc. Therefore, ss needs to be predicted to evaluate itseffect on the structure at every design and construction stage. Themagnitude of ss should be estimated spatially because the properties,thickness and depth of geo-layers vary irregularly at every location.

The concept of kriging was first introduced by Krige (1951), andmathematical expressions were subsequently developed by Matheron(1962, 1963a, 1963b). In geotechnical engineering, kriging has beenused to estimate the spatial distribution of undrained shear strength, SPTN value, liquefaction potential, and high liquefaction probability values,as well as the thickness and depth of geo-layers. Kriging has also beenused to model the spatial variation in the results of cone penetrationtests, to characterize the spatial variability of soil properties, and toevaluate the quality of site investigations, etc. (Christakos, 1985; Soulieet al., 1990; Chiasson et al., 1995; Jaksa et al., 1997; Parsons and Frost,2002;Baise et al., 2006;Mendes andLorandi, 2008;Maracheet al., 2009).

+82 42 861 9720.

11 Published by Elsevier B.V. All rig

The prediction of ss is made with many uncertainties due to theinherent variability of soil properties. Although the determination ofreliable and representative properties for compressible layers is offundamental importance in the estimation of long term settlement,the nature of soil deposits suggests that these properties should bedescribed by probability distributions (Corotis et al., 1975). Aprobabilistic method has been frequently used to evaluate theuncertainty of primary consolidation and secondary compression(Corotis et al., 1975; Freeze, 1977; Athanasiou-Grivas and Harr, 1978;Chang, 1985; Hong and Shang, 1998; Zhou et al., 1999).

This paper presents aprocedure for evaluating the spatial uncertaintyin ss using a probabilistic method. The spatial distributions of geo-layersare estimated by using kriging techniques such as ordinary kriging,krigingwith a trend, and simple kriging. Based on theevaluated statisticsof the soil properties, the mean (μ) and standard deviation (σ) of ss arecalculated by using the first order second moment method. With theseresults, a probabilistic approach is applied to evaluate the uncertainty inss while a deterministic method is used to evaluate the spatialdistribution of μ of ss. Finally, amethodology considering the uncertaintyin ss in the geotechnical engineering design process is presented.

2. Description of study site

Songdo New City has been built on a large scale artificial island of53.4 km2 area. As shown in Fig. 1a, Songdo New City is located in the

hts reserved.

Fig. 1. Study site: (a) location of Songdo New City, and (b) study area (location of Sections 1–4).

Fig. 2. Schematic soil profile of the study site.

240 W. Lee et al. / Engineering Geology 122 (2011) 239–248

western part of Incheon, South Korea (37°21′17″–37°24′33″N,126°36′40″–126°40′57″E). Sections 1–4 of Songdo New City, whichwere reclaimed with dredged silty sands between 1997 and 2003,comprise the study area (Fig. 1b). The subsoil of the study areaconsists of eight distinct units, as shown in Fig. 2: a reclaimed sandfill(NSPTb15), upper soft silty clay (NSPTb6), upper silty sand (NSPTN30),lower medium and stiff silty clay (NSPTN10), lower silty sand(NSPTN40), weathered soil, weathered rock, and bedrock. Silty sandand silt (NSPTb10) are partially observed between the reclaimedsandfill (RSF) and the upper soft silty clay (USSC). NSPT represents theN value of the standard penetration test (SPT). The USSC is a relativelyhomogeneous layer with average values of NSPT and undrained shearstrength of 4 and 30 kPa, respectively. Most of the primaryconsolidation settlement is expected to occur in the USSC.

3. Geostatistical approach

In a geostatistical approach, the spatial variability and its patternare usually expressed by an experimental semivariogram, γ(h), whichis computed as half of the average squared difference between paireddata values separated by a vector h:

γ hð Þ = 12N hð Þ ∑

N hð Þ

α=1z uαð Þ−z uα + hð Þ½ �2 ð1Þ

in which N(h) is the number of data pairs within a given class ofdistance and direction; z(uα) is the value at the start or tail of the pairα at the location uα; and z(uα+h) is the corresponding end or headvalue at a lag of h from the location uα. Experimental values for a finitenumber of separation vectors are obtained using a semivariogram. Atheoretical semivariogrammodelmust be fitted to these experimentalvalues to obtain semivariogram values for any possible separationvector used in the kriging interpolation algorithm or kriging matrices.The spherical, exponential, Gaussian, and power models are the mostfrequently used basic models. The semivariogram stops increasingand fluctuates around a specific value at a specific separation distance.These specific value and separation distance are called the sill andrange, respectively. The range denotes the distance between locationsbeyond which data values appear independent. However, short scalevariability, sampling and measurement errors may cause discontinu-ity at the origin of the variogram. These phenomena are expressed bythe nugget effect, and the ratio of the nugget discontinuity to the sillvalue is referred to as the relative nugget effect (Journel andHuijbregts, 1978; Isaaks and Srivastava, 1989; Cressie, 1991; Deutsch

and Journel, 1992; Goovaerts, 1997; Chiles and Delfiner, 1999;Kanevski, 2008).

The purpose of kriging is to estimate the value of a continuousattribute, z, at any unsampled location, u, using the neighboringz-data {z(uα), α=1,…,n} available over the study area. All krigingestimators are variants of the basic linear regression estimator, Z⁎(u),defined as:

Z� uð Þ−m uð Þ = ∑n uð Þ

α=1λα uð Þ Z uαð Þ−m uαð Þ½ � ð2Þ

in which λα(u) is the weight assigned to z(uα), which is a realizationof the random variable Z(uα); and n(u) is the number of data involvedin the estimation; and m(u) and m(uα) are the expected values ofrandom variables Z(u) and Z(uα). All kriging methods have the sameobjective of minimizing the error variance, σE

2(u), under theconstraint of unbiasedness of the estimator. That is, λα(u) aredetermined by minimizing Eq. (3) under the constraint of Eq. (4)(Deutsch and Journel, 1992; Goovaerts, 1997).

σ2E uð Þ = Var Z� uð Þ−Z uð Þ� � ð3Þ

E Z� uð Þ−Z uð Þ� �= 0 ð4Þ

Fig. 3. Location of 702 borehole data; a test set (closed circles, 121 data) and aprediction set (open circles, 581 data).

Table 1Summary of statistics of the thickness of the USSC, top elevation of the USSC, andbottom elevation of the RSF.

Parameters Thickness ofthe USSC (m)

Top elevationof the USSC (m)

Bottom elevationof the RSF (m)

No. of samples 581 581 581Mean 10.19 −3.14 −0.91Standard deviation 6.79 5.10 2.41Minimum 0 −20.94 −8.40Lower quartiles 5.00 −4.50 −2.44Median 10.00 −1.92 −1.06Upper quartiles 15.10 −0.13 0.49Maximum 26.70 5.40 5.93

241W. Lee et al. / Engineering Geology 122 (2011) 239–248

The kriging estimator changes depending on the model adoptedfor the random function itself. The random function is composed of aresidual component R(u) and a trend component m(u). The type ofkriging can be distinguished according to the model considered forthe trend m(u). In simple kriging (SK), the mean m(u) is consideredto be known and constant throughout the study area. The ordinarykriging (OK) considers local variations in the mean by limiting thedomain of stationarity of the mean to the local neighborhood. As thelocation u for the prediction gets farther away from data locations, theSK estimate becomes closer to the global mean, whereas the OKestimate becomes closer to the local mean. The kriging with a trend(KT) considers that the unknown local mean, m(u′), fluently varieswithin each local neighborhood. The trend component is modeled as alinear combination of functions of coordinates (Goovaerts, 1997).

The reliability of kriging estimates is evaluated by comparingestimated values and known values at the same locations. The meanabsolute error (MAE) and mean squared error (MSE) measure theaccuracy of kriging estimates. The goodness-of-prediction (G-value)measures the effectiveness of estimated results. The MAE, MSE, andG-value are calculated as follows:

MAE =1n

∑n uð Þ

α=1jz uαð Þ−z� uαð Þj� � ð5Þ

MSE =1n

∑n uð Þ

α=1z uαð Þ−z� uαð Þ� �2 ð6Þ

G = 1−∑n uð Þ

α=1z uαð Þ−z� uαð Þ½ �2

∑n uð Þ

α=1z uαð Þ−z½ �2

0BBB@

1CCCA × 100 ð7Þ

in which z*(uα) is an estimate of value z(uα) at location uα, and z isthe sample mean of z-datum value z(uα). Small MAE and MSE valuesrepresent a more accurate prediction, on a point-by-point basis. TheG-value gives an indication of how effective an estimation might be,relative to that derived using a sample average alone. If the G-value is100, the estimation is regarded as being perfect, while a negativeG-valuemeans that the estimation is less reliable thanwhen using thesample average as the estimator (Santra et al., 2008).

4. Probabilistic evaluation of secondary compression

4.1. Evaluation of spatial distribution of geo-layers

4.1.1. Exploratory data analysis and variographyData from 702 boreholes are used to map the thickness of the

USSC, the top elevation of the USSC, and the bottom elevation of theRSF. These borehole data are collected from private companies andpublic administration offices. Fig. 3 presents the location of 702boreholes, and the shaded regions represent the area where relativelysmaller numbers of borehole data are available. To validate thereliability of kriging estimates, data from 702 boreholes were dividedinto a test set (121 test locations) and a prediction set (581 locations).The closed and open circles in Fig. 3 represent the test (validation) andprediction sets, respectively. Table 1 presents a summary of statisticsfor the thickness of the USSC, top elevation of the USSC, and bottomelevation of the RSF. In this study, cell declustering technique is usedto obtain the representative statistics of the thickness and depth of thegeo-layers because data collected in the study area are clusteredpreferentially (Deutsch, 1989; Deutsch and Journel, 1992). Fig. 4presents the symbol plots of the thickness of the USSC, top elevation ofthe USSC, and bottom elevation of the RSF. It can be seen in Fig. 4 thatthe thickness and depth of the geo-layers do not show evident spatialtrend within the study area.

Fig. 5a–c shows experimental semivariograms with theoreticalsemivariogram models taken to determine their respective fitting foreach geo-layer. Omnidirectional semivariograms are computed forthe thickness of the USSC, top elevation of the USSC, and bottomelevation of the RSF because no significant directional trend isobserved. In Fig. 5, the spherical model is found to best representthe spatial variability of all geo-layers and the model parameters(nugget, sill, and range) for each geo-layer are presented. The relativenugget effects of theoretical semivariogram models for the thicknessof the USSC, top elevation of the USSC, and bottom elevation of the RSFare 0.36, 0.19, and 0.55, respectively.

4.1.2. Spatial distribution of geo-layersIn general, most estimation techniques underestimate high values

and overestimate low values. This tendency is referred to as thesmoothing effect. Fig. 6 shows the scatterplots of the true values andkriging estimates for the thickness of the USSC. The slopes of theregression lines of the scatterplots of the true and estimated values byOK, KT, and SK are 0.61, 0.59, and 0.49, respectively. As the smoothingeffects for the OK and KT estimates are smaller than those for the SKestimates, the thickness of the USSC estimated by OK and KT isbelieved to be more reliable than that estimated by SK.

Table 2 shows a summary of scatterplots for three geo-layersestimated by different kriging techniques. It is shown in Table 2 thatOK and KT give better estimated results than SK for all three geo-layers. The smoothing effect for the thickness of the USSC is lesspronounced than that for the top elevation of the USSC and thebottom elevation of the RSF, regardless of the kriging technique.Table 3 shows the statistics for the true and estimated values by OK,

Fig. 5. Experimental semivariogram and adjustment model of samples: (a) thethickness of the USSC, (b) top elevation of the USSC, and (c) bottom elevation of the RSF.

242 W. Lee et al. / Engineering Geology 122 (2011) 239–248

KT and SK for the thickness of the USSC, top elevation of the USSC, andbottom elevation of the RSF, respectively, at 121 test locations. For allthree geo-layers, the σ, IQR (interquartile range), and range ofestimated values appear to be smaller than those of the true valuesdue to the smoothing effect. It is also shown that the SK estimates forall three geo-layers provides a smaller σ, IQR, and range than otherkriging estimates. This indicates that SK gives less reliable estimatedresults than OK and KT.

The evaluation indices (MAE, MSE, and G-value) of the valuesestimated by OK, KT, and SK for the thickness of the USSC, the topelevation of the USSC, and the bottom elevation of the RSF,respectively, are given in Table 4. In all geo-layers, the G-values arefound to be greater than zero, which means that the predictionestimated by kriging is better than that by taking mean values at anyunsampled location. As shown in Table 4, for all geo-layers, the MAEand MSE of the OK estimates are smaller than that of the KT and SKestimates, and the G-value of the OK estimate is larger than those ofthe SK and KT estimates. This means that OK gives more reliableestimates than KT and SK.

Fig. 7a–c shows the spatial maps of the thickness of the USSCestimated by OK, KT, and SK, respectively. As shown in Fig. 7a and b,OK and KT yield the similar estimates. This is because the thickness of

Fig. 4. Symbol plots: (a) the thickness of the USSC, (b) top elevation of the USSC, and(c) bottom elevation of the RSF.

Fig. 6. Scatterplots of the true and estimated values for the thickness of the USSC: (a) OK,(b) KT, and (c) SK.

243W. Lee et al. / Engineering Geology 122 (2011) 239–248

the USSC for Songdo New City does not show evident spatial trendwithin the study area (Fig. 4). Fig. 7b shows that KT gives negativeestimates for the thickness of the USSC at some locations in Sections 2and 4. As it is not possible for the thickness of the USSC to be negative,the KT estimates at some locations in Sections 2 and 4 are not used tocalculate the ss. As shown in Fig. 7, the values estimated by OK, KT andSK for Sections 1 and 3 are similar; whereas, the estimated values forSections 2 and 4 are different somewhat. At some locations inSections 2 and 4, the thickness of the USSC estimated by OK is close tothe local mean which is smaller than 9 m, and the thickness of the

Table 2Slopes of regression lines of scatterplots of true and estimated values by OK, KT, and SK.

Geo-layer OK KT SK

Thickness of the USSC 0.61 0.59 0.49Top elevation of the USSC 0.48 0.46 0.30Bottom elevation of the RSF 0.36 0.39 0.32

USSC estimated by SK is close to the overall mean (10.19 m). As mostof the thicknesses of the USSC in Sections 2 and 4 are smaller than 9 m,as shown in Fig. 4(a), it appears that OK gives more reasonableestimates than SK in these areas. A similar estimation tendencyappears in the estimated results for the top elevation of the USSC andthe bottom elevation of the RSF. It is concluded from the above resultsthat OK is the most satisfactory estimation technique.

In order to estimate the spatial distribution of ss, the spatial mapsfor the thickness and depth of geo-layers are evaluated by OK. Figs. 7a,8, and 9 show the spatial maps of the thickness of the USSC, topelevation of the USSC, and bottom elevation of the RSF estimated byOK using a 50×50 m grid. Fig. 7a shows that the thicknesses of theUSSC in Sections 1 and 3 (northwestern part of the study area) arelarger than those in Sections 2 and 4. Especially, the thickness of theUSSC in Section 1 is the largest in the entire study area. Estimatedresults must be used cautiously for the region where relatively smalldata sets are available, e.g., the southern part of Section 4. Thethickness of the USSC is used directly to calculate the ss. The estimatedthickness and depth of all the geo-layers are used to calculate the in-situ effective stress at the middle of the consolidating layer. Thebottom elevation of the RSF is used to calculate the load incrementdue to the fill.

4.2. Statistics and probability density functions of soil properties

The statistical analysis of soil properties of the USSC is performedto estimate the spatial distribution of μ and σ of ss. The μ, σ, coefficientof variation (COV), and type of probability density function (PDF) forsoil properties are obtained from the statistical analysis of laboratorytest results for the USSC and are summarized in Table 5. The μ of Cα/Cc

for Songdo New City is 0.034, which is within the range of 0.04±0.01for inorganic soft clays suggested by Mesri and Castro (1987). In thisstudy, all the soil properties are checked for the normality using theKolmogorov–Smirnov (K–S) test at a 5% level of significance; theparameters of PDFs are estimated using the method of moments(Benjamin and Cornell, 1970; Ang and Tang, 1975). Logarithmictransformations are conducted for the coefficient of consolidation (cv)to fit to a normal distribution. While cv fits to a lognormal distribution(LN), other soil properties fit to a normal distribution (N) at a 5% levelof significance. The average values of the moist unit weight (γt) andbuoyant unit weight (γb) of the reclaimed sandfill material (siltysand) are 19.0 and 9.2 kN/m3, respectively. The COVs for γt and γb areassumed to be 5 and 7%, respectively. The type of PDFs for γt and γb ofthe reclaimed sandfill material is considered to be a normaldistribution (Phoon and Kulhawy, 1999; Duncan, 2000).

4.3. Spatial distribution of secondary compression

The use of the Cα/Cc concept is an economical, rapid, and powerfultool for the estimation of ss (Mesri and Godlewski, 1977; Mesri andChoi, 1985). Assuming a constant Cα/Cc value during secondarycompression, themagnitude of ss during the time increment from tp totf is computed as follows:

ss =Cα = Ccð Þ × Cc

1 + e0H0 log

tftp

ð8Þ

in which Cα is the secondary compression index; Cc is thecompression index; e0 is the initial void ratio of the USSC; H0 is thethickness of the USSC; and tp is the time at the end of the primaryconsolidation process.

In the absence of pore water pressure measurement data, the tpcan be estimated as the time required for 95% primary consolidationsettlement (Mesri and Choi, 1985). In this study, tp is calculated usingTerzaghi's 1-D consolidation theory (Taylor, 1948). Fig. 10 shows thespatial distribution of tp for the study area. The maximum tp is about

Table 3Statistics of true and estimated values by OK, KT, and SK.

SummaryStatistics

Thickness of USSC (m) Top elevation of the USSC (m) Bottom elevation of the RSF (m)

True OK KT SK True OK KT SK True OK KT SK

μ 13.3 13.3 13.3 13.3 −1.89 −1.76 −1.71 −1.94 −1.15 −0.73 −0.71 −0.75σ 4.7 3.8 3.8 3.3 2.18 1.68 1.69 1.31 1.63 0.99 1.11 0.88Minimum 1.2 3.0 3.7 4.6 −8.7 −6.61 −6.14 −5.35 −4.65 −3.32 −3.83 −3.01IQR 6.0 4.8 5.0 4.8 2.44 2.00 2.26 1.94 1.62 1.12 1.30 1.10Maximum 22.5 20.1 20.0 20.1 5.27 2.11 2.50 1.52 4.1 2.22 2.84 1.82Range 21.3 17.0 16.3 15.6 13.97 8.72 8.63 6.87 8.75 5.55 6.67 4.83

244 W. Lee et al. / Engineering Geology 122 (2011) 239–248

108 months (9 years) in Section 1; the range of tp in Sections 2–4 isapproximately 0–60 months (0–5 years). The tp for the southern partof Section 4 is zero because of zero thickness of the USSC. As shown inFig. 10, the primary consolidation process may take quite a long timein some areas due to the thick consolidating layer with lowpermeability. In such cases, the ground improvement with prefabri-cated vertical drains (PVDs) is required to accelerate the primaryconsolidation process. In Songdo New City, PVDs are used to shortentp for the reclaimed area where primary consolidation is notcompleted within 36 months after the completion of reclamationwork. Assuming that the COV of tp of the PVDs area is the same as thatof the non-PVDs area, the μ and σ of ss are determined using the firstorder second moment method (Benjamin and Cornell, 1970; Ang andTang, 1975) for tf=50 yr, which is the design life of the pavement andpipeline of Songdo New City.

In order to investigate the reliability of the cumulative distributionfunction (CDF) of ss evaluated by using the first order second moment(FOSM) method, the CDFs of ss are determined by the FOSM methodand Monte Carlo Simulation (MCS) for the thicknesses of the USSC of10 m and 15 m, respectively. The PDF of ss is assumed to be a normaldistribution because the PDFs of most of soil properties used toevaluate ss appear to be the normal distribution as shown in Table 5. Itis shown in Fig. 11 that the CDFs of ss determined by FOSM methodandMCS are almost the same. It is, therefore, concluded that the FOSMmethod provides reliable CDFs of ss although the FOSM methodapproximates the μ and σ of ss using a Taylor series expansion. In thisstudy, the probabilistic analysis for ss is performed using the CDFsdetermined by the FOSM method.

Fig. 12 shows a spatial map of the μ of ss, which is computed with μvalues of all soil properties. As shown in Fig. 12, μ values of ss forSections 1 and 3 are larger than those for Sections 2 and 4 due to thelarger thickness of the USSC, and the maximum μ of ss is 115.4 mm inSection 1. In a deterministic method, the spatial map of μ of ss is usedto determine the area where the magnitude of ss exceeds the designcriterion. In this study, the design criterion is an allowable long termsettlement of 100 mm. The ratio of the area where ss exceeds 100 mmto the total area is 0.013, which can be determined from Fig. 12. Mostgeotechnical engineering designs use the deterministic analysis toevaluate the area potentially damaging to structures, pavements andpipelines. Fig. 13 shows the spatial map of the σ of ss, which is

Table 4Reliability analysis results of OK, KT, and SK estimates.

Geo-layer Kriging technique MAE, m MSE, m2 G

Thickness of USSC OK 2.408 9.831 54.5KT 2.447 9.843 54.5SK 2.601 10.917 49.5

Top elevation of the USSC OK 1.209 2.973 36.7KT 1.269 3.195 32.0SK 1.325 3.517 25.1

Bottom elevation of the RSF OK 1.064 1.870 29.0KT 1.102 1.997 24.3SK 1.065 1.879 28.7

estimated by considering the uncertainties of all soil properties. In thisstudy, the effect of variability of soil properties on the spatialdistribution of ss is focused because the variation of soil propertieshas larger effect on ss than the variation of the thickness and depth ofgeo-layers. The probability that ss at a certain location is larger than aspecific value can be computed using spatial distributions of the μ andσ of ss. In this paper, the probability that ss is larger than a specificvalue (s) at a specific time (t) is represented by P(ssNs, t). Fig. 14shows a spatial map of P(ssN100 mm, 50 yrs), i.e., the probability thatss is larger than 100 mm at 50 yrs. The P(ssN100 mm, 50 yrs) forSection 1 ranges from 0.20 to 0.62, and that for Sections 2–4 is 0 to0.40.

5. Application to geotechnical engineering design

The magnitude of ss depends on H0, Cα/Cc, Cc/(1+e0), and cv.Considering Cα/Cc, Cc/(1+e0), and cv as random variables, thesensitivity analysis of ss is performed for a 15 m thick consolidatinglayer. Parameters for the sensitivity analysis of ss are presented inTable 6. Case 1 involves evaluating the uncertainty of ss by consideringuncertainties of all soil properties. Case 2 involves evaluating theinfluence of the COV of cv on the uncertainty of ss. Cases 3 and 4 areused to evaluate the influence of the COVs of Cα/Cc and Cc/(1+e0) onthe uncertainty of ss. The last three columns in Table 6 present the μ, σ,and the COV of ss for each case. In this study, the probability densityfunction of ss is assumed to be a normal distribution. It is observedfrom the last column in Table 6 that the COVs of Cα/Cc and Cc/(1+e0)have similar influences on the COV of ss. It is also shown that the COVof cv has less effect on the COVof ss than the COVs of Cα/Cc and Cc/(1+e0)although the COV of cv used to estimate the tp is larger than that of Cα/Ccand Cc/(1+e0). The COV of ss evaluated by considering all the COVs ofsoil properties is 0.421, which is 1.4–2.7 times larger than the COV of ssdetermined by considering the individual COV of soil property separately.

As mentioned previously, the COVs of Cα/Cc and Cc/(1+e0) have alarger influence on the uncertainty of ss than the COV of cv. Therefore,the influence of the COV of Cα/(1+e0) on the area where ss exceeds adesign criterion (100 mm) at a specific time (50 yr) is evaluated usingthe FOSMmethod and the result is given in Fig. 15. The vertical axis inFig. 15 represents the ratio of the area of P(ssN100 mm, 50 yrs)Nα tothe total area. Various α values between 0.05 and 0.45 are assigned inthe analysis to show the variation of the area with the probabilisticdesign criterion (α). It is observed in Fig. 15 that the area exceedingthe design criterion increases as the COV of Cα/(1+e0) increases andthe α value decreases. As the COV of Cα/(1+e0) for Songdo New Cityis 0.39, area ratios for α values of 0.05–0.45 are in the range of 0.47–0.04. Fig. 16a–c presents the areas of P(ssN100 mm, 50 yr)Nα for αvalues of 0.10, 0.20 and 0.30, respectively.

6. Summary and Conclusions

The magnitude of ss should be evaluated spatially because theproperties, thickness and depth of geo-layers vary at every location. Thespatial variability and distribution of three geo-layers (thickness ofthe USSC, top elevation of the USSC, and bottom elevation of the RSF) are

Fig. 7. Estimated thickness of the USSC: (a) OK, (b) KT, and (c) SK.

Fig. 8. Spatial map of the top elevation of the USSC.

Fig. 9. Spatial map of the bottom elevation of the RSF.

Table 5Statistics and type of PDFs for soil properties.

Soil layer(soil type)

Soilproperties

Units No. ofsamples

μ σ COV PDF

Reclaimedsandfill(RSF)

γt kN/m3 – 19.0 0.95 0.05 Nγb kN/m3 – 9.2 0.64 0.07 N

Upper softsilty clay(USSC)

γt kN/m3 124 18.3 0.54 0.03 Nγb kN/m3 124 8.5 0.54 0.06 NCc/(1+e0)* – 124 0.122 0.036 0.30 NCr/(1+e0)* – 20 0.030 0.007 0.22 Ncv cm2/s 58 6.22×10−3 2.76×10−3 0.44 LNCα/Cc* – 60 0.034 0.0087 0.26 NOCR(G.L.0–−4.0 m)*

– 30 3.1 0.70 0.23 N

*Cc — compression index; Cr — recompression index; e0 — void ratio; Cα — secondarycompression index; and OCR — overconsolidation ratio.

245W. Lee et al. / Engineering Geology 122 (2011) 239–248

evaluated by using semivariograms and kriging techniques. Based on theCα/Cc concept, the spatial maps of the μ and σ of ss are obtained by usingthe statistical parameters of soil properties and the spatial distributions of

geo-layers. With these results, a probabilistic approach is applied toevaluate the spatial uncertainty in ss and an exemplary procedureconsidering the uncertainty of ss in the geotechnical engineering design ispresented.

Fig. 10. Spatial map of tp.

Fig. 11. Comparison of CDFs obtained by FOSM method and MCS.

Fig. 12. Spatial map of μ of ss in 50 yr.

Fig. 13. Spatial map of σ of ss in 50 yr.

Fig. 14. Spatial map of P(ssN100 mm, 50 yr).

246 W. Lee et al. / Engineering Geology 122 (2011) 239–248

Although the σ, IQR, and range of estimated values appear to besmaller than those of true values due to the smoothing effect, thesmoothing effect for the thickness of the USSC is less pronounced thanthat for the top elevation of the USSC and the bottom elevation of theRSF, regardless of the kriging technique. For all three geo-layers, theMAE andMSE of the OK estimates are smaller than those of the KT andSK estimates, and the G-value of the OK estimate is larger than thoseof the SK and KT estimates. The estimated thicknesses of USSC withinthe area containing plenty of databases are similar, regardless ofthe kriging technique, while the estimated values within the area

Table 6Parameters of sensitivity analysis of ss and the uncertainty in ss at 50 yrs.

Case cv, cm2/s Cα/Cc Cc/(1+e0) ss, mm

μ COV μ COV μ COV μ σ COV

1 0.0062 0.44 0.034 0.26 0.122 0.30 76.0 32.0 0.4212 0.0062 0.44 0.034 0 0.122 0 76.0 12.0 0.1583 0.0062 0 0.034 0.26 0.122 0 76.0 19.5 0.2564 0.0062 0 0.034 0 0.122 0.30 76.0 22.4 0.295

Fig. 15. Influence of COV of Cα/(1+e0) on the area where ss exceeds 100 mm at 50 yr.

247W. Lee et al. / Engineering Geology 122 (2011) 239–248

including a small amount of database are heavily dependent on thekriging technique. A similar estimation tendency appears in theestimated results for the top elevation of the USSC and the bottomelevation of the RSF. It is therefore concluded that OK gives morereliable estimates of geo-layers than KT and SK.

It is observed that the COVs of Cα/Cc and Cc/(1+e0) for SongdoNew City, which are the most important properties in thecalculation of ss, have similar influences on the COV of ss. It isalso shown that the COV of cv has less effect on the COV of ss thanthe COVs of Cα/Cc and Cc/(1+e0) although the COV of cv used toestimate the tp is larger than that of Cα/Cc and Cc/(1+e0). The COVof ss evaluated by considering all the COVs of soil properties is0.420, which is 1.4–2.7 times larger than the COVs of ss determinedby considering the COV of the individual soil property separately.The spatial map of P(ssNs, t) is evaluated by using a probabilisticapproach to determine the area where ss exceeds a design criterion(100 mm) at a specific time (50 yr). It is observed that the areaexceeding a design criterion increases as the COV of Cα/(1+e0)increases and the probabilistic design criterion (α) decreases. ForSongdo New City, which has a COV of 0.39 for Cα/(1+e0), arearatios for α values of 0.05–0.45 are found to be in the range of0.47–0.04.

Fig. 16. Area where ss exceeds 100 mm for α value

7. Notation

The following symbols are used in this paper.

α probabilistic design criterioncv coefficient of consolidationCα secondary compression indexCc compression indexe0 initial void ratioCc/(1+e0) primary compression ratioCα/(1+e0) secondary compression ratioγ(h) experimental semivariogramγt moist unit weightγb buoyant unit weightH0 thickness of consolidating layerh separation distance or lagμ meanN(h) number of data pairsλα(u) kriging weight associated to z-datum at location uα

m(u) expected value of random variable Z(u); trend componentmodel in the decomposition Z(u)=R(u)+m(u)

m(uα) expected value of random variable Z(uα)n(u) number of data involved in the estimation at location uP(ssNs, t) probability that ss is larger than a specific value (s) at a specific

time (t)R(u) residual component model in the decomposition Z(u)=R

(u)+m(u)ss secondary compressiontp time at the end of the primary consolidation processtf time at which the secondary compression is computedσ standard deviationσE2(u) kriging error variance

u coordinate vectoruα datum locationZ(uα) random variable at location uα

Z⁎(u) kriging estimatorz(uα) z-datum value at location uα

z*(uα) an estimate of value z(uα) at location uα

z sample mean of z-datum value z(uα)

Acknowledgements

This paper was supported by a POSCO E&C grant, and wassupported by the Basic Research Project of the Korea Institute of

: (a) α=0.10, (b) α=0.20, and (c) α=0.30.

248 W. Lee et al. / Engineering Geology 122 (2011) 239–248

Geoscience and Mineral Resources (KIGAM) (GP2009-019) funded bythe Ministry of Science and Technology of Korea.

References

Ang, A.H.-S., Tang, W.H., 1975. Probability concepts in engineering planning and design.Basic Principles, vol. 1. John Wiley & Sons, New York.

Athanasiou-Grivas, D., Harr, M.E., 1978. Consolidation — a probabilistic approach.Journal of the Engineering Mechanics Division, ASCE 104 (EM3), 681–690.

Baise, L.G., Higgins, R.B., Brankman, C.M., 2006. Liquefaction hazard mapping-statisticaland spatial characterization of susceptible units. Journal of Geotechnical andGeoenvironmental Engineering, ASCE 132 (6), 705–715.

Benjamin, J.R., Cornell, C.A., 1970. Probability, Statistics, and Decision for CivilEngineers. McGraw-Hill Book Company, New York.

Chang, C.S., 1985. Uncertainty of one-dimensional consolidation analysis. Journal ofGeotechnical Engineering, ASCE 111 (12), 1411–1424.

Chiasson, P., Lafleur, J., Soulie, M., Law, K.T., 1995. Characterizing spatial variability of aclay by geostatistics. Canadian Geotechnical Journal 32, 1–10.

Chiles, J.P., Delfiner, P., 1999. Geostatistics, Modeling Spatial Uncertainty. A Wiley-Interscience Publication, New York.

Christakos, G., 1985. Modern statistical analysis and optimal estimation of geotechnicaldata. Engineering Geology 22, 175–200.

Corotis, R.B., Krizek, R.J., El-Moursi, H.H., 1975. Probabilistic approach to prediction ofconsolidation settlement. TRB, Transportation Research Board 548, 47–61.

Cressie, N., 1991. Statistics for Spatial Data. John Wiley & Sons, Inc., New York.Deutsch, C.V., 1989. DECLUS: a Fortran 77 program for determining optimum spatial

declustering weights. Computers and Geosciences 15 (3), 325–332.Deutsch, C.V., Journel, A.G., 1992. GSLIB: Geostatistical software library and user's guide.

Oxford University Press, New York.Duncan, J., 2000. Factors of safety and reliability in geotechnical engineering. Journal of

Geotechnical and Geoenvironmental Engineering, ASCE 126 (4), 307–316.Freeze, R.A., 1977. Probabilistic one-dimensional consolidation. Journal of the

Geotechnical Engineering Division, ASCE 103 (GT7), 725–741.Goovaerts, P., 1997. Geostatistics for Natural Resources Evaluation. Oxford University

Press, New York.Hong, H.P., Shang, J.Q., 1998. Probabilistic analysis of consolidation with prefabricated

vertical drains for soil improvement. Canadian Geotechnical Journal 35, 666–677.Isaaks, E.H., Srivastava, R.M., 1989. Applied Geostatistics. Oxford University Press, New York.Jaksa, M.B., Brooker, P.I., Kaggwa, W.S., 1997. Modeling the spatial variation of the

undrained shear strength of clays soils using geostatistics. Proc., 5th int. Geostat.Congr. Kulwer Academic, Dordrecht, The Netherlands, pp. 1284–1295.

Journel, A.G., Huijbregts, C.J., 1978. Mining Geostatistics. Academic Press, London.Kanevski, M. (Ed.), 2008. Advanced Mapping of Environmental Data: Geostatistics,

Machine Learning and Bayesian Maximum Entropy. iSTE and Wiley, London.Krige, D.G., 1951. A statistical approach to some basic mine valuation problems on the

Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of SouthAfrica 52, 119–139.

Marache, A., Breysse, D., Piette, C., Thierry, P., 2009. Geotechnical modeling at the cityscale using statistical and geostatistical tools: the Pressac case (France).Engineering Geology 107, 67–76.

Matheron, G., 1962. Traite de geostatisque appliquee. Vol. I. Memoires du Bureau deRecherches Geologiques et Minieres, 14. Editions Technip, Paris.

Matheron, G., 1963a. Traite de geostatisque appliqué. Vol. II: Le Krigeage. Memoires duBureau de Recherches Geologiques et Minieres, 24. Editions Bureau de RechercheGeologiques et Minieres, Paris.

Matheron, G., 1963b. Principles of geostatistics. Economic Geology 58, 1246–1266.Mendes, R.M., Lorandi, R., 2008. Analysis of spatial variability of SPT penetration

resistance in collapsible soils considering water table depth. Engineering Geology101, 218–225.

Mesri, G., Castro, A., 1987. Cα/Cc concept and K0 during secondary compression. Journalof Geotechnical Engineering, ASCE 113 (3), 230–247.

Mesri, G., Choi, Y.K., 1985. Settlement analysis of embankments on soft clays. Journal ofGeotechnical Engineering, ASCE 111 (4), 441–464.

Mesri, G., Godlewski, P.M., 1977. Time- and stress-compressibility interrelationship.Journal of Geotechnical Engineering, ASCE 103 (GT5), 417–430.

Parsons, R.L., Frost, J.D., 2002. Evaluating site investigation quality using GIS andgeostatistics. Jounal of Geotechnical and Geoenvironmental Engineering, ASCE 128(6), 451–461.

Phoon, K.K., Kulhawy, F.H., 1999. Characterization of geotechnical variability. CanadianGeotechnical Journal 36 (4), 612–624.

Santra, P., Chopra, U.K., Chakraborty, D., 2008. Spatial variability of soil properties andits application in predicting surface map of hydraulic parameters in agriculturalfarm. Current Science 95 (7), 937–945.

Soulie, M., Montes, P., Silvestri, V., 1990. Modelling spatial variability of soil parameters.Canadian Geotechnical Journal 27 (5), 617–630.

Taylor, D.W., 1948. Fundamentals of Soil Mechanics. John Wiley & Sons, Inc., New York.Zhou, W., Hong, H.P., Shang, J.Q., 1999. Probabilistic design method of prefabricated

vertical drains for soil improvement. Journal of Geotechnical and Geoenviron-mental Engineering, ASCE 125 (8), 659–664.