Inaccuracies Associated with the Current Method for Estimating Random Measurement Errors. M. B. Jaksa, P. I. Brooker and W. S.

Transcription

1 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 December, 994

2 ABSTRACT This report examines a commonly used procedure, proposed by Baecher (98), for separating the random measurement error associated with a particular test from the inherent spatial variability of the geological material. It is shown that the method, while well-founded, requires a number of factors to be investigated, before conclusions can be made regarding the random measurement error of a particular test. Two case studies are presented, and the sensitivity of the results is 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 drilling rigs and technical staff, for without whose generosity the research carried out at the South Parklands site could not have taken place. In addition, the cooperation of the City of Adelaide, and in particular, A. A. Taylor and M. Underhill, for providing access to this site. The authors also wish to gratefully acknowledge Australian National, and in particular P. Gaskill, for their generosity and assistance, for allowing access to the Keswick site. Furthermore, the authors would like to thank the technical staff of the Department 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 apparatus modifications; L. Collins and R. Kelman for the fabrication of the drilling apparatus, and; B. Lucas for the design and construction of the data acquisition system. Thanks are due also to final year civil engineering students, D. van Holst Pellekaan and J. Cathro, for their assistance in performing the horizontal cone penetration testing. i

3 CONTENTS ABSTRACT...i ACKNOWLEDGMENTS...i CONTENTS...ii. INTRODUCTION.... TECHNIQUES USED TO DESCRIBE UNCERTAINTY.... Time Series Analysis...3. Geostatistics ESTIMATION OF MEASUREMENT ERRORS INADEQUACIES OF PRESENT METHOD Nugget Effect Sample Spacing Stationarity of Data CASE STUDIES Vertical Spatial Variability - South Parklands Site Horizontal Spatial Variability - Keswick Site CONCLUSIONS REFERENCES...5 ii

4 . INTRODUCTION All engineering design incorporates uncertainty in one form or another. In fact, the overall, or total, uncertainty associated with any particular design may incorporate 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 such uncertainties in the design process (Baecher, 986; Orchant et al., 988; Kay, 99; Kay et al., 99). In essence, each of these models proposes that the total design uncertainty is the sum of each of the individual contributing uncertainties. This report focuses on one aspect of the overall design uncertainty, that is, the uncertainty associated with the in situ test measurement of geotechnical materials. There are two primary sources of uncertainty which contribute to the variability of in situ test measurements, namely, spatial variability, and measurement error. Spatial variability is the natural variation that soils and rock exhibit, from one location to another, as a result of the myriad and complex processes which form them, and to which they have subsequently been subjected. Measurement error, on the other hand, is the error associated with the measurement process itself, which is explained in greater detail below. These two sources of uncertainty are related by the following relationship: m v = v + ε () where: m v measurement of parameter v; v true value of the parameter; and ε measurement error. Orchant et al. (988) proposed the following model for describing the total uncertainty, or variance, σ T, associated with measurement: T s / v m σ = σ + σ ()

5 where: and: m e p / o r σ = σ + σ + σ (3) σ sv / variance due to spatial variability; σm σe σ p / o σr variance due to measurement error; variance due to equipment effects; variance due to procedure and operator effects; and, variance due to random measurement errors. Uncertainties associated with equipment effects can occur as a result of electrical drift, non-linearities, and out-of-calibration errors related to the electrical transducers and mechanical devices, which each have different levels of reliability. Procedure and operator effects cause variabilities in measurements as a result of inadequate, or limited, testing standards, noncompliance with these standards, as well as uncertainties as a result of different operators. Both of these effects, equipment, and operator and procedure, are systematic, or bias (Lumb, 974), errors which consistently under-estimate, or over-estimate, a measured parameter. Random measurement errors, on the other hand, are the variation between measurements that cannot be directly attributed to the inherent variability (spatial variability) of the material, equipment, or operator and procedural effects. While the spatial variability of the material, and the random measurement errors, are both random, or scatter, components of the total measurement error, it is difficult to separate and quantify each. This report investigates the current method proposed by Baecher (98), and subsequently applied by several authors, to separate the random measurement error from the spatial variability of the soil. However, before discussing this method, it is necessary to treat the techniques used to describe these uncertainties.. TECHNIQUES USED TO DESCRIBE UNCERTAINTY To date, the analyses of random measurement errors and the spatial variability of geotechnical materials, has centred on two mathematical techniques, namely, time series analysis, and geostatistics. In order to provide a background to the discussion that follows in later sections, these two techniques are treated briefly, below.

6 . Time Series Analysis A time series is a chronological sequence of observations of a particular variable, usually, but not necessarily, at constant time intervals. When applied to the study of the spatial variability of geotechnical materials and random measurement 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 observed behaviour that values at adjacent locations are more related, than those at distant locations. For time series analyses to be carried out the data must be stationary, that is, the probabilistic laws which govern the series must be independent of the location of the samples. Data are stationary if: the mean is constant with distance, that is, no trend, or drift, exists in the data; 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 statistical properties of the time series are unaffected by a shift of the spatial origin. Two essential statistical properties used in time series analysis are the autocovariance, c k, and the autocorrelation, ρ k, at lag, k, which are defined as: and, c k ( X, X ) = E ( X X )( X X )] = E( X X ) X = Cov i i+ k i i+ k i i+ k (4) ρ k ck = c where: X i is the value of property, X, at location, i; X is the mean of the property, X; E..] is the expected value; c is the autocovariance at lag ; and, c k = c -k and ρ k = ρ -k. (5) The autocorrelation, ρ k, measures the correlation between any two time series observations separated by a lag of k units. It is not possible to know c k nor ρ k with any certainty, but only to estimate them from samples obtained from a population, say X, X,..., X N. As a result, the 3

7 sample autocovariance at lag k, c k *, and sample autocorrelation at lag k, r k, are given by: and, c * k = c N N k i= N k * k = k * c = i= N r ( X X )( X X ) i i+ k ( X i X )( X i+ k X ) ( X i X ) i= (6) (7) where: X is the average of the observations X, X,..., X N ; and, k < N. The sample autocovariance function (ACVF), or autocovariogram, is the plot of c* k for lags k =,,,.... The sample autocorrelation function (ACF), or correlogram, is the graph of r k for lags k =,,,... K, where K is the maximum number of lags that r k should not be calculated beyond. While the sample autocorrelation function can be evaluated for all lags up to N, it is not advisable, since, as k tends toward N, the number of pairs reduces, and as a consequence, the reliability of the estimate r k of the true autocorrelation function, ρ k, also decreases. Most authors suggest that K = N 4, (Box and Jenkins, 97; Chatfield, 975; Anderson, 976). As one might expect, the accuracy of the sample autocorrelation function is directly related to the number of observations in the time series, N. Little guidance is given to the minimum number of observations, though Box and Jenkins (97) and Anderson (976) recommended that N be greater than 5. With particular reference to the spatial variability of soils, Lumb (975) suggested that, for a full three-dimensional analysis, the minimum number of test results needed to give reasonably precise estimates is of the order of 4, which is prohibitively large, even for a special research project. On the other hand, Lumb (975) recommended that the best that can be achieved in practice is to study the one-dimensional variability, either vertically or laterally, using N of the order of to. The ACF, and to a lesser extent the ACVF, are used widely throughout time series analysis literature, and they enable the characteristics of the time series to be determined. For example, an ACF exhibiting slowly decaying values of r k with increasing k, suggests long term dependence, whereas rapidly decaying values of r k suggest short term dependence (Chatfield, 975; Hyndman, 99). 4

8 A purely random time series, or white noise, is characterised by an ACF with the following properties: for k = ρk = (8) for k. 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 (965). Since its development in the 96 s, geostatistics has been applied to many disciplines including: groundwater hydrology and hydrogeology; surface hydrology; earthquake engineering and seismology; pollution control; geochemical exploration; and geotechnical engineering. In fact, geostatistics can be applied to any natural phenomena that are spatially or temporally associated (Journel and Huijbregts, 978; Hohn, 988). Geostatistics is based on regionalised variables which have properties that are partly random and partly spatial, and which have continuity from point to point. The changes in these variables, however, are so complex that they cannot be described by a tractable deterministic function (Davis, 986). This is in contrast to the classical approach which treats samples as independent realisations of a random function. One of the basic statistical measures of geostatistics is the semivariogram, which is used to quantify the degree of spatial dependence between samples along a specific orientation, and so presents the degree of continuity of the property in question. The semivariogram, γ h, is defined by Equation (9). ( X X ) ] γ h = E (9) i+ h i where: h the displacement between the data pairs. If the regionalised variable is stationary and normalised to have a mean of zero and a variance of., the semivariogram is the mirror image of the autocorrelation function. Even though a regionalised variable is spatially continuous, it is not possible to know its value at all locations. Instead its values, like the ACF, can only be 5

9 determined from samples taken from a population. Thus, in practice, the semivariogram must be estimated from the available data, and is generally determined by the relationship shown in Equation (). γ * h = N Nh i= h ( Yx + ) i h Yx i () where: γ * h the experimental semivariogram, that is, one based on the sampled data set; Y xi the value of the property, Y, at location, x i ; and, N h 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, N h ; and the lag distance, h (Brooker, 99). In general, as the distance between data pairs increases, N h decreases, consequently reducing the accuracy of the experimental semivariogram. As a result, γ h * is usually determined up to half of the total sampled extent (Journel and Huijbregts, 978; Clark, 979; Brooker, 989). For example, if an electrical cone penetrometer (CPT) sounding was performed to a depth of 5, mm, the semivariogram would be calculated for values of h from to,5 mm. The minimum number of pairs needed for a reliable estimate of γ h * is between 3 and 5 (Journel and Huijbregts, 978; Brooker, 989), with some authors suggesting as many as 4 to 5 (Clark, 98). The strength of geostatistics is that it provides, through the semivariogram, a framework for the estimation of variables. In fact, it can be shown that geostatistics provides the best, linear, unbiased estimator (BLUE) (Journel and Huijbregts, 978; Clark, 979; Rendu, 98). Whilst the experimental semivariogram 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, γ h. The most common model used in the literature is the spherical model, shown in Figure, which is characterised by three parameters: C is defined as the nugget effect and arises from the regionalised variable being so erratic over a short distance that the semivariogram goes from zero to the level of the nugget in a distance less than the sampling interval. The nugget effect will be discussed in greater detail in 4.; 6

10 C C Range, a Displacement, h Figure. Example of a semivariogram. C + C a is known as the sill which measures half the maximum, on average, squared difference between data pairs; and, is defined as the range of influence, and is the distance at which samples become independent of one another. Data pairs separated by distances up to a are correlated, but not beyond. Clark (979) described the process of fitting a model to an experimental semivariogram as essentially a trial-and-error approach, usually achieved by eye. Brooker (99) suggested the following technique as a first approximation in finding the appropriate parameters for a spherical model: The experimental semivariogram and variance of the data are plotted; The value of the sill, C + C, 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, C. In addition, Brooker (99) stated that the accuracy of the modelling process depends on both the number of pairs in the calculation of the experimental semivariogram and the lag distance at which it is evaluated. Journel and Huijbregts (978) suggested that automatic fitting of models to experimental 7

11 semivariograms, such as least squares methods, should be avoided. This is because each estimator point, γ h *, of an experimental semivariogram is subject to an estimation error and fluctuation which is related to the number of data pairs associated with that point. Since the number of pairs varies for each point, so too does the estimation error. The authors recommended that the weighting applied 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 semivariogram requires stationarity, that is, the semivariogram depends only on the separation distance and not on the locality of the data pairs. The regionalised variable can be regarded as consisting of two components: the residual and the drift. If a drift, or trend, exists in the data, which leads to non-stationarity, it must first be removed. It has been shown by Davis (986), that if the drift is subtracted from the regionalised variable, the residuals will themselves be a regionalised variable and will have local mean values of zero. In other words, the residuals will be stationary and the semivariogram can be evaluated. 3. ESTIMATION OF MEASUREMENT ERRORS Baecher (98) proposed a useful method for separating the scatter observed in geotechnical data into its two component sources: the spatial variability of the material, and the random measurement error associated with the test itself. Baecher (98) suggested, as did Lumb (974) before him, that the spatial variation of some parameter, v x, at a location x within a soil mass, may be treated as a combination of a deterministic component, as well as a stochastic, or random, component, as shown in the following equation. where: t x v x = t x + ξ x () is the trend component at location x, usually determined by least squares regression; and, ξ x is the random perturbation off the trend at x. In addition, Baecher (98) suggested that the random measurement error is presumed to be independent from one test to another, to have zero mean, and to have constant variance. As a consequence, the measurement of v x, m vx, may be expressed as: where: mv = tx + ξ x x + ζ x () ζ x is the random measurement error at x. 8

12 After some algebraic manipulation, Baecher (98, 986) stated that the autocovariance, c k, of the measurement, m ( m v ) v, at lag, k, may be shown to equal: where: c k v c = c + c (3) ζ k( mv ) k( v) k( ) ( ) is the autocovariance function of v at lag, k; and is the autocovariance function of the random measurement error, ζ, at lag, k. c k( ζ) A similar relationship can be derived that incorporates the ACF rather than the autocovariance function (ACVF). Baecher (98, 986) proposed that, since c k( ζ ) is equal to the variance of ζ at k =, and c k( ζ ) is equal to zero at k, as described by Equation (8), the random measurement error may be determined by extrapolating the observed ACVF, or ACF, back to the origin, as shown in Figures and 3. Figure. Procedure for estimating the random measurement component from the ACVF. (After Baecher, 98). 9

13 Figure 3. Procedure for estimating the random measurement component from the ACF. (After Baecher, 986). Baecher (986) stated that, by using this method, typical in situ measurements of soils have been found to contribute random measurement errors anywhere between and 7% of the data scatter. Several other authors (Tang, 984, Wu and El-Jandali, 985; Filippas et al., 988; Orchant et al., 988; Spry et al., 988; Kay, 99; Kay et al., 99; DeGroot and Baecher, 993) have used this method, or results based on it, to postulate various aspects relating to geotechnical uncertainty and reliability. As will be shown in the next section, while the method proposed by Baecher (98, 986) appears to be wellfounded, a number of factors need consideration before conclusions can be made regarding the level of random measurement error associated with a particular test. 4. INADEQUACIES OF PRESENT METHOD While Baecher s approach focuses on the tools associated with time series analyses, treated in., three important factors from the study of geostatistics, have highlighted inadequacies with the current method. These factors, the nugget effect, the spacing between samples, and the stationarity of the data, greatly influence the random measurement error obtained by the procedure proposed by Baecher (98), and are each discussed below.

14 4. Nugget Effect It has long been appreciated, in the study of geostatistics, that many ore bodies exhibit erratic behaviour at lags close to zero. This erratic behaviour, known as the nugget, C, and described briefly in., manifests itself as an apparent non-zero value of the semivariogram at zero lag. The nugget effect is the combination of three separate phenomena (Rendu, 98):. microstructures within the geological material - which have been observed from the study of the spatial variability of mineral concentrations of core samples. Two adjacent cores will exhibit a nugget effect when one of them contains a nugget and the other does not (Journel and Huijbregts, 978). Several researchers have stated that soils also exhibit this behaviour (Li and White, 987; Soulié et al., 99; Jaksa et al., 993);. sampling, or statistical, errors - as will be detailed in 4. below, C depends greatly on the spacing between individual samples; and, 3. measurement errors - if it were possible to obtain a repeat measurement at precisely the same location, the observations would differ by an amount directly dependent on the measurement technique. Measurement errors are also manifested by a non-zero value of the semivariogram at zero lag. Baecher s procedure, in essence, attributes the nugget effect solely to measurement error, but, as has been shown above, the nugget effect is also made up of microstructure variabilities and sampling errors, which must be accounted for before conclusions can be made regarding the extent of random measurement error associated with a particular test. At this point, it is necessary to define a new parameter, the ACF nugget, R, which is the difference between unity and the value of the autocorrelation coefficient at lag zero, r, obtained by extrapolating the sample ACF back to lag zero, as shown in Equation (4). The ACF nugget, like the nugget from geostatistics, accounts for the micro-variability of the geological material, sampling errors, and random measurement errors, but is determined from the sample ACF rather than from the semivariogram. R = - r (4) 4. Sample Spacing As mentioned in the previous section, another important factor, again which has long been established in the field of geostatistics, is the effect of the sample spacing on the observed nugget (Brooker, 977; Journel and Huijbregts, 978;

15 Clark, 979; Clark, 98; de Marsily, 98). In fact, the nugget effect that is obtained from the experimental semivariogram, depends greatly on the physical distance between the individual samples that form the data set. As the sampling distance decreases, it is possible to obtain a better estimate of C. However, while one is able to reduce the sampling interval to a very small distance, the cost of the exploration programme increases dramatically. As a result, it is often unreasonable, and in fact unnecessary, to reduce the sampling spacing below some nominal minimum value. Unfortunately, this minimum sampling distance is dependent on the geological material being examined, and cannot be known prior to investigation. Common practice is to begin sampling with a relatively coarse grid, and then to infill with a repeatedly finer grid, until the sample spacing no longer influences the resultant experimental semivariogram. In 5, two case studies will be used to demonstrate the effect of sample spacing on the observed nugget. Each of these case studies is based on data obtained at an extremely close sample spacing of 5 mm. 4.3 Stationarity of Data As defined previously in., the theory of both time series analysis and geostatistics, assume that the data are stationary. The ACF, ACVF, and the semivariogram, are each dependent on the stationarity of the data set, and as a result, 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 transform a non-stationary data set to a stationary one by removing a low-order polynomial trend, usually no higher than a quadratic (Journel and Huijbregts, 978), which is usually estimated by means of the method of ordinary least squares (OLS) (Lumb, 974; Brockwell and Davis, 987). Li (99) correctly asserted that OLS assumes that the data are random and uncorrelated, which is inconsistent with spatial variability analyses which, having removed some trend determined by OLS, subsequently examine the correlation structure of the residuals. Li (99) suggested that a technique based on generalised least squares (GLS) be used as an alternative to OLS, and the more complex methods suggested by Matheron (973) and Delfiner (976). Kulatilake (99) stated that, while in general agreement with Li (99), the GLS technique has drawbacks when applied in a practical sense. Furthermore, Ripley (98) found that 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 a non-stationary data set, the nugget effect is significantly influenced by the

16 stationarity of the data. The following section examines the influence of data stationarity and sample spacing on the nugget effect by means of two case studies. 5. CASE STUDIES As part of a research programme currently being undertaken at the University of Adelaide which focuses on quantifying the spatial variability of soils, a number of electrical cone penetration tests (CPT) have been performed in a relatively homogeneous, stiff, over-consolidated clay known as Keswick Clay. This clay has been the focus of research because of its local significance, as many of Adelaide s high rise buildings are founded on it, and because of its international significance, since its geotechnical properties are remarkably similar to those of the well-documented London Clay (Cox, 97). In order to demonstrate the effect of sample spacing and data stationarity on the observed nugget, and consequently, the random measurement error, two case studies are presented, the first dealing with vertical spatial variability, and the second dealing with horizontal spatial variability. In each case, the CPT was used to obtain the data. The electric cone penetrometer is a standard 6, cm base area type which conforms to the relevant standards which include ISOPT- (De Beer et al., 988), ASTM D344 (American Society for Testing and Materials, 986) and AS 89.F5. (Standards Association of Australia, 977). The data were obtained using a micro-computer based data acquisition system which allows CPT measurements to be recorded at 5 mm increments, and stored on disk for subsequent analyses. The data acquisition system is treated in detail by Jaksa and Kaggwa (994). 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 that enables both time series, and geostatistical, analyses to be performed. The results are detailed below. 5. Vertical Spatial Variability - South Parklands Site A large number of vertical CPTs were drilled within a section of the South Parklands, which lies in the central business district of the city of Adelaide, South Australia. The CPTs, 3 in all, and each drilled to an approximate depth of 5 metres, were arranged in: (i) a 5 5 metre grid; (ii) a cross formation with each CPT being spaced at metre centres; and (iii) 5 CPTs which were spaced 3

17 at.5 metre intervals along a line, as shown in Figure 4. Details of the testing programme at the South Parklands site, and results of preliminary spatial variability analyses, are given by Jaksa et al. (993). A B C D E CD.3.5 m 5 m. CD5.5 m F G.6 m 5 m H I J.4. K 3.3 m 5 m. 5 m Legend Successful soundings Unsuccessful soundings Continuous core sample. Depth to top of Keswick Clay 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 be analysed. Since a continuous core sample was taken adjacent to C8, as shown 4

18 in Figure 4, it was determined by inspection of the core, that the surface of the Keswick Clay lies. metres below the ground surface. Eliminating the upper. metres of measurements, a plot of the cone tip resistance, q c, obtained from the 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 by means of OLS regression. Cone Tip Resistance, q c (MPa) Linear Trend r =.88 Quadratic Trend r = Figure 5. Measured cone tip resistance, q c, within Keswick Clay for sounding C8. The residuals of the data in Figure 5, were obtained by removing the quadratic trend, the results of which are given in Figure 6, and were assumed to be stationary. The sample ACF was obtained by substituting the data shown in Figure 6 into Equation (7), the result of which, is shown in Figure 7. At lag k =, the sample autocorrelation coefficient, r, for the residuals of q c for sounding C8, was found to be equal to.9. Extrapolating the sample ACF back to k =, in accordance with Baecher s procedure, yields r =.9, implying a calculated ACF nugget of %. 5

19 Cone Tip Resistance, q c (MPa) Figure 6. Residuals of q c, for sounding C8, after removing the quadratic trend Distance (mm) Figure 7. Sample ACF of residuals of q c, for sounding C8. In order to test the sensitivity of the nugget with regard to the stationarity of the data, the measurements of q c, shown in Figure 5, are examined, firstly, with no 6

20 trend removal, and secondly, with only a linear trend removed. The sample ACFs obtained for each of these two cases, are shown in Figure Distance (mm) (a) Distance (mm) (b) Figure 8. Sample ACFs after: (a) no trend removal, and (b) a linear trend removal. 7

21 For no trend removal, r =.59, which yields, r =.59; and for a linear trend removal, r =.94, yielding, r =.94. The results of these analyses are summarised in Table. TABLE. SUMMARY OF DATA STATIONARITY ANALYSES (VERTICAL SPATIAL VARIABILITY). Trend Removed from Data r r ACF Nugget, R None % Linear % Quadratic.9.9 % As shown by the results in Table, the ACF nugget determined using Baecher s approach, varies substantially, from 6% to 4%, and as a result, depends greatly on the stationarity of the data. In order to test the sensitivity of the nugget, with respect to sample spacing, the original data set of q c measurements was modified to provide sets of data at different sample spacings. These measurements, shown in Figure 5, were sampled at 5 mm intervals from a depth of mm to 555 mm below ground. Data sets at different sample spacings were obtained by simply removing intervening rows of data. For example, to obtain a data set with q c measurements at mm spacings, every second row was removed. This provided two data sets of measurements spaced at mm intervals, one from mm to 55 mm, and the other from 5 mm to 555 mm. This process of removing intervening rows was used to provide several data sets of q c measurements at spacings of,, 5,, and mm. By removing the quadratic trend from each of these data sets, the residuals were obtained, and substituted into Equation (7), to determine the sample ACFs. Two such sample ACFs are shown in Figure 9. Again using the procedure proposed by Baecher (98), the ACF nugget can be evaluated by extrapolating the sample ACF back to lag, k =. The results for each of the data sets are summarised in Table. As is indicated by the results shown in Table, the calculated ACF nugget obtained from vertical spatial variability analyses, and determined using the procedure proposed by Baecher (98), is significantly dependent on the spacing of the samples, and can vary between 3% and 6%. 8

22 Fitted Curve 5 Distance (mm) (a).8.98 Fitted Curves.6.4. B.38 B J J B J J B 5 J B J B Distance (mm) - (b) Figure 9. Sample ACFs for: (a) 5 mm spaced data set, and (b) mm spaced data set. 5. Horizontal Spatial Variability - Keswick Site The lateral spatial variability of the Keswick Clay was studied by Jaksa et al. (994), by driving an electrical cone penetrometer horizontally into an embankment, as shown in Figure. The embankment is situated at the Australian National railway yards at Keswick, which is located adjacent to the central business district of the city of Adelaide. 9

23 The CPT was carried out to a total horizontal penetration distance of 7.6 metres. Removing the first two metres of data, which may be affected by weathering and movement adjacent to the face of the embankment, yields the q c measurements shown in Figure. The horizontal cone penetration testing carried out at the Keswick site, as well as results of spatial variability analyses, are treated in detail by Jaksa et al. (994). TABLE. SUMMARY OF SAMPLE SPACING ANALYSES (VERTICAL SPATIAL VARIABILITY). Sample Vertical Spatial Variability Spacing (mm) r r ACF Nugget, R 5.9 ().9 %.84,.86 ().89,.9 - %.76 to.8 (4).93 to.95 5 to 7% 5.48 to.64 (5).76 to.97 3 to 4%.6 to.49 (5).4 to.8 8 to 6% -.3* to.9 (5).38 to.97 3 to 6% (n) : Separate data sets examined. * : Not possible to sensibly extrapolate R when r <. Trailer mounted horizontal CPT Track 7.6 Extent of horizontal CPT Scale (metres) Figure. Cross-section of embankment at Keswick site. Again, the residuals of the data in Figure, were obtained by removing the quadratic trend obtained by the method of OLS, and the sample ACF was obtained by substituting the residuals into Equation (7), the result of which, is shown in Figure.

24 Quadratic Trend Horizontal Penetration Distance (mm) Figure. Measured cone tip resistance, q c, within Keswick Clay at the Keswick site. At lag k =, the sample autocorrelation coefficient, r, for the residuals of q c for the horizontal CPT was found to equal.97. Extrapolating the sample ACF back to k =, in accordance with Baecher s procedure, yields r =.97, implying a calculated ACF nugget of 3%. In order to test the sensitivity of the nugget with regard to the stationarity of the data, the measurements of q c, shown in Figure, are examined, firstly, with no trend removal, and secondly, with a linear trend removed. The results of these analyses are shown in Table Distance (mm) Figure. Sample ACF of residuals of q c, for horizontal CPT.

25 TABLE 3. SUMMARY OF DATA STATIONARITY ANALYSES (HORIZONTAL SPATIAL VARIABILITY). Trend Removed from Data r r ACF Nugget, R None % Linear % Quadratic % As shown by the results in Table 3, the value of R determined using Baecher s approach, varies substantially, from 3% to 56%, and again shows that the nugget depends greatly on the stationarity of the data. It should be noted that there is no difference in the value of R 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 respect to sample spacing, the original horizontal CPT data, which were sampled at 5 mm intervals, were modified to provide data sets with spacings of,, 5,, and mm, between adjacent measurements of q c, in the same way as for the vertical spatial variability case, described previously. By removing the quadratic trend from each of these data sets by the method of OLS, the residuals were obtained, and the sample ACFs determined. Two such sample ACFs are shown in Figure 3. Again, using Baecher s procedure, the ACF nugget is determined by extrapolating the sample ACF back to lag, k =. The results of a number of the horizontal CPT data sets are summarised in Table 4. TABLE 4. SUMMARY OF SAMPLE SPACING ANALYSES (HORIZONTAL SPATIAL VARIABILITY). Sample Horizontal Spatial Variability Spacing (mm) r r ACF Nugget, R 5.97 ().97 3%.95,.95 ().97,.97 3%.88 to.9 (4).95 5% 5.6 to.64 (5).9 to.9 8 to %. to.9 (5).5 to to 5% -.8* to. (5)? to.8 8% to? (n) : Separate data sets examined. * : Not possible to sensibly extrapolate R when r <.? : Unknown value of R since r <.

26 Fitted Curve 5 5 Distance (mm) (a) Distance (mm) (b) Figure 3. Sample ACFs for: (a) mm spaced data set, and (b) 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 using Baecher s method, varies significantly, from 3% to 5%, and again indicates that the ACF nugget depends greatly on the sample spacing of the data. Furthermore, for a spacing of mm, 3 of the 5 data sets examined yielded values of r less than zero, making it impossible to extrapolate a positive value of R, as indicated in Table 4. 3

27 6. CONCLUSIONS This paper has examined the method proposed by Baecher (98), and adopted by several researchers since, for separating the spatial variability component of the geotechnical material from the random measurement error of the test. It has been demonstrated, using both vertical and horizontal CPT data measured at close intervals of 5 mm, that conclusions made regarding the random measurement error associated with a particular test, depend greatly on: (i) the micro-variability of the geological material; (ii) the spacing of the samples in the data set; and (iii) the stationarity of the data. In fact, the ACF nugget, R, (the difference between unity and the value of the sample autocorrelation function extrapolated back to zero, r ) is a combination of random measurement error, small-scale variability of the soil, sampling errors, and non-stationarity errors. It is not solely random measurement errors associated with the particular test, as several authors have incorrectly assumed. A series of sample ACFs were obtained by examining the vertical and horizontal non-stationary CPT data sets, and by removing a linear and quadratic trend, and comparing these with no trend removal. These analyses have shown that the calculated ACF nugget can vary between 3% and 56% depending on which, if any, trend is removed. These results confirm that the ACF nugget is significantly dependent on data stationarity. By varying only the sample spacing of a data set, in increments from 5 mm up to mm, it has been shown that the calculated ACF nugget can vary between 3% and 6%, for vertical spatial variability, and between 3% and 5% for horizontal spatial variability. As almost all of the information published regarding the horizontal spatial variability of soils is based on ACFs derived from samples taken at spacings well in excess of mm, one must question the validity of these conclusions. As a result of the data and analyses presented in this paper, it is likely that the random measurement error associated with the cone penetration test is less than or equal to 3%, since this figure accounts for both the random measurement error and the small-scale variability of the soil, and perhaps, to some extent, non-stationarity errors. 4

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