SUMMARY, CONCLUSIONS AND FUTURE SCOPE
CHAPTER 4 ESTIMATION OF THE VERTICAL SCALE OF FLUCTUATION
4.3. ONE-DIMENSIONAL RANDOM FIELD MODELLING OF FIELD SPT-N VALUES In the standard penetration test, a standard split spoon sampler is driven into the soil by dropping
4.3.2. Random Field Model for Residuals
BH
No. Depth (m) Detrending Function Coefficient of Regression, R2
1 0-10.5 y=0.7249x2 −6.1508x+23.286 0.8522
10.5-19.5 y=5.8286x−28.467 0.9928
2 0-19.5 y=0.2724x2−1.9364x+14.336 0.9715
3 0-19.5 y=0.2857x2−2.348x+15.462 0.9896
4 0-19.5 y=0.2375x2 −1.8894x+14.399 0.9877
5 0-19.5 y=0.3699x2−4.0453x+10.664 0.9731
6 0-15 y=0.1801x2−1.2955x+13.783 0.9879
15-19.5 y=0.4444x2−14.667x+143 1
7 0-19.5 No detrending function ----
8 0-19.5 y=0.1162x2−0.6722x+13.083 0.8
9 0-19.5 y=0.0852x2−0.3763x+18.406 0.9582
10 0-12 y=1.1587x+7.9286 0.8602
12-19.5 y=5.2x−1.4 0.9826
11 0-19.5 y=0.2862x2−1.2364x+11.112 0.9728
12 0-13.5 y=0.1438x2 −1.1462x+13.905 0.8504
13.5-19.5 y=5.2x−11.2 0.9909
13 0-19.5 y=0.1166x2 −0.6197x+9.7552 0.9386
14 0-19.5 y=0.2866x2 −1.319x+9.6853 0.9701
coordinate. The stationary residuals are modelled as a random field by an auto-correlation function described by the scale of fluctuation.
The autocorrelation function represents the variation of spatial correlation as a function of spatial separation length between two locations at which data are available. The autocorrelation function is most commonly used for investigation of spatial variation of soil properties in the context of geotechnical engineering. The popularly used approach is to estimate the sample autocorrelation function (ACF) and fit a probable theoretical autocorrelation model (ACM) to the estimated ACF.
As there is no reliable physical resemblance between the type of soil and nature of correlation model, a numerical approach is acceptable (Uzielli et al., 2006). For a given dataset, the choice of the correlation structure can be made depending on the comparative assessment of goodness-of-fit of the empirical ACF of the dataset to one or more theoretical ACMs. This is achieved by optimization of the characteristic parameter of model for each ACM (e.g., by least-squares regression or any other optimization techniques) and the subsequent estimation of the goodness- of-fit parameter for each ACM. The ACM with the maximum fitness can be chosen as best-fit function. Knowing the correlation model, SoF can be easily evaluated from the model parameter of the ACM.
The sample ACF for the residual random field can be estimated as (Fenton and Griffiths, 2008):
1
1 1
2 1
( )( )
( ) 1, 2,...,
( )
n j
i X i j X
i
j n
i X
i
X X
j n
X
− +
= + −
=
− −
= =
−
(4.3)where, j =(j− 1) z is the lag distance, n is the total number of data in set, z is the sampling interval and X is the average of the sample and is estimated as
1
1 n
X i
i
n X
=
=
(4.4)Several theoretical 1-D ACMs are available in the literature. Table 4.2 lists four such correlation models commonly used in geotechnical engineering, namely the Single exponential, Squared exponential, Second-order Markov and Cosine exponential ACMs. The sample auto-correlation function is first estimated for all the borehole profiles and then fitted to the theoretical ACMs. The unknown parameters of the ACMs are estimated from the normalized residuals by minimizing the mean square error between the computed ACF and the theoretical ACM. To illustrate the procedure, Fig. 4.6 shows the autocorrelation function estimated for the BH 2 and the corresponding fitted ACMs. The auto-correlation value is equal to 1.0 (its maximum value) at zero correlation distance, which tends to zero with the increase in separation distance, and can take intermediate negative values. It is well known from the literature that at higher lag distances, the sample auto-correlation function computed from the data becomes noisy (Uzielli et al., 2006). At large lags, there is practically no correlation between two random quantities under consideration.
The presence of such correlations coefficients at large lags obscures the interpretation about the actual decay of correlation function with spatial distance, and hence, considered as noise. Thus, correlation coefficients at large lags are ignored, and only the initial part of the correlation function (defined by the correlation function within the spatial lags until which the correlation coefficients are greater than or just reached zero values) is fitted to the ACMs. The exercise is conducted for all the 14 borehole profiles considered in this study (BH 1-14). Table 4.3 summarizes the best fitted ACMs for all the borehole profiles indicating correlation coefficient value (R2), along with
the least root mean square error (RMSE) and sum of squares of errors (SSE) for the fitted curve.
It is seen that SoF varies from 0.27 m to 0.92 m, with a mean value of 0.6 m and a CoV of 38.75%.
Phoon and Kulhawy (1996) reported a mean SoF of SPT-N data as 2.4 m in sandy soil. The geotechnical site investigation conducted in present study area mostly encountered clayey soil with presence of sand and silt in some locations. Therefore, a comparatively smaller SoF value than in sand is expected. Moreover, the present study found that cosine exponential function is best suited for all the 14 borehole residual profiles. Vanmarcke (1978) also reported the cosine exponential function to be the best fit function for geotechnical properties. Hence, the cosine exponential function can be recommended to be used as characterizing correlation model for SPT-N data.
However, it is to be noted that the distinction between trend in dataset and stochastic variation of residuals is not inherent to the geotechnical property, rather is dependent on the judgment of the modeler. Moreover, the mean SoF so estimated, depending on a particular in-situ test dataset, is not recommended of direct application to other geotechnical sites and properties without proper engineering judgement. However, data reported in present work can be used as preliminary indication of correlation distance, which will enable the sample spacing and correlation model to be chosen for a geotechnical site investigation or simulation based geotechnical reliability analysis with no prior knowledge of the site.
Fig. 4.6 Auto correlation model fitted to the estimated auto correlation function for BH 2
Table 4.2: 1-D correlation functions and their characteristics as typically determined for BH 2 in the present study
ACM Function Model
Parameter
Correlation
length (m) RMSE SSE R2 Single
exponential
y=e−ax a = 1.549 1.29 0.2338 0.3827 0.7208 Squared
exponential
(ax)2
y=e− a = 0.8595 2.06 0.2279 0.3634 0.7348
Second-order
Markov (1 )
y=e−ax +ax a = 2.313 1.73 0.1276 0.3736 0.7274 Cosine
exponential cos(2 )
y=e−ax bx a = 0.2653
b = 0.1214 0.8135 0.1261 0.09546 0.9187
Table 4.3: Summary of the soil variability analyses on the borehole data BH
No.
Fitted ACM Unknown parameter
RMSE SSE R2 SoF
(m)
1 Cosine
exponential
a = 0.637 b = 0.3332
0.1919 0.2209 0.8005 0.27
2 Cosine
exponential
a = 0.2653 b = 0.1214
0.1261 0.09546 0.9187 0.81
3 Cosine
exponential
a=0.4346 b=0.1653
0.1126 0.07601 0.9227 0.69
4 Cosine
exponential
a = 0.2375 b = 0.1572
0.08515 0.07975 0.9361 0.46
5 Cosine
exponential
a = 0.2684 b = 0.216
0.1046 0.1204
0.8993 0.28
6 Cosine
exponential
a = 0.5372 b = 0.1593
0.138 0.2094 0.8146 0.83
7 Cosine
exponential
a = 0.5304 b = 0.1591
0.1659 0.1652 0.8325 0.83
8 Cosine
exponential
a = 0.2487 b = 0.1148
0.1968 0.2325 0.7975 0.85
9 Cosine
exponential
a = 0.7638 b = 0.1656
0.1308 0.1026 0.8917 0.92
10 Cosine
exponential
a = 0.185 b = 0.1775
0.1374 0.1132 0.9089 0.29
11 Cosine
exponential
a = 0.1428
b = 0.101 0.1477 0.131 0.9103 0.68
12 Cosine
exponential
a = 0.1662
b = 0.1363 0.0975 0.05704 0.9601 0.44
13 Cosine
exponential
a = 0.2288
b = 0.1534 0.1273 0.0972 0.9145 0.47
14 Cosine
exponential
a = 0.1602
b = 0.1134 0.08648 0.04488 0.9655 0.6