International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Diagnosis of earth-fill dams by synthesized approach of sounding and surface wave method Nishimura, Shin-ichi; Shibata, Toshifumi; Shuku, Takayuki Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Diagnosis of Earth-fill Dams by Synthesized Approach of Sounding and Surface Wave Method Shin-ichi Nishimura Professor, Graduate School of Environmental and Life Science, Okayama Univ.,  Japan Toshifumi Shibata Senior Lecturer, Graduate School of Environmental and Life Science, Okayama Univ.,  Japan Takayuki Shuku Assistant Professor, Graduate School of Environmental and Life Science, Okayama Univ.,  Japan ABSTRACT: The spatial distribution of the strength inside the earth-fill is identified by the sounding tests. In this research, the Swedish Weight Sounding (SWS) is employed, and the spatially high-density test is performed to identify the spatial correlation structure. Furthermore, the synthesized approach of the SWS and Surface Wave Method (SWM), which is one of the geophysical method,  is proposed to compensate the shortage of each approach. Consequently, the correlation structure of an earth-fill could be identified accurately, and the high resolution of the spatial distribution could be visualized based on the survey results.  1. INTRODUCTION There are many earth-fill dams in Japan.  Some of them are getting old and decrepit, and therefore, have weakened.  Making a diagnosis of the dams is important to increasing the lifetime, and an investigation of the strength inside the embankment is required for this task.  In the present research, the spatial distribution of the strength parameters of decrepit earth-fills is discussed, and an identification method for the distribution is proposed.  Although the strength of the earth-fills is generally predicted from the standard penetration test (SPT) N-values, Swedish Wight Sounding tests (SWS) (e.g. JGS, 2004)  are employed in this research as a static sounding method of obtaining the spatial distribution of the N-values. SWS test is advantageous in that they make short interval exams possible, because of their simplicity.   In general, the identification of the spatial correlation of soil parameters is difficult, since the usual sampling intervals are greater than the spatial correlation.  Therefore, sounding tests are convenient for determining the correlation lengths. Tang (1979) determined the spatial correlation of a ground by cone penetration tests (CPT).  Cafaro and Cherubini (1990) also evaluated the spatial correlation with the CPT results.  Uzielli, et al. (2005) considered several types of correlation functions for the CPT results. Nishimura and Shimizu (2008) determined the correlation parameters of N-value at the coastal dyke with the maximum likelihood method. The information for the spatial correlation structures is important  to perform the random field analysis. Fenton and Griffiths (2002), who analyzed the settlement of the footing on the ground, considering the spatial correlation structure of Young's modulus. In addition, Griffiths et al. (2002) calculated the bearing capacity by analyzing the random field of the undrained shear strength using the elasto-plastic finite elements method. Nishimura et al. (2010) applied the random field theory to the elasto-plastic model and, evaluated the risk of the earth-fill dams.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 The spatial distributions of the N-values can be identified based on the sounding tests with high resolution, since point estimations of the N-values are possible with short testing intervals.  However, the predicted N-values are supposed to involve great prediction errors in parts for which no point estimated data are included. To compensate for this weak point, the surface wave method (SWM) (e.g., Hayashi 2004) is employed here, which is one of the geophysical exploration methods.  By using this method, the secondary wave (S-wave) distribution can be easily estimated as an averaged image over the wide area of an earth-fill, although the actual spatial fluctuations of the S-wave velocities are ignored throughout the inversion process.  S-waves have a close correlation with the soil mechanical parameters, and are transformed into N-values in this research. Finally, two kinds of N-value distributions derived from sounding results and SWM are synthesized and then spatially interpolated with the indicator simulation (Deutsch and Journel 1992).  2. STATISTICAL MODEL OF N-VALUES A representative variable for the soil properties, s is defined by Equation (1) as a function of the location X=(x, y, z). Variable s is assumed to be expressed as the sum of the mean value m and the random variable U, which is a normal random variable in this study.  (1) The random variable function, s(X), is discretized spatially into a random vector st=(s1,s2,...,sM), in which sk is a point estimation value at the location X=(xk, yk, zk). The soil parameters, which are obtained from the tests, are defined here as St=(S1,S2,..., SM). Symbol M signifies the number of test points. Vector S is considered as a realization of the random vector st=(s1,s2,...,sM).  If the variables s1, s2,...,sM constitute the M - variate normal distribution, the probability density function of s can then be given by the following equation. fS s( ) = 2π( )−M2 C −12 exp − 12 s−m( )t C−1 s−m( )"#$ %&'  (2) in which mt=(m1,m2,...,mM) is the mean vector of random function st=(s1,s2,...,sM); and it is assumed to be the following the regression function. In this research, a 2-D statistical model is considered, namely, the horizontal coordinate x, which is parallel to the embankment axis, and the vertical coordinate z are introduced here, while the other horizontal coordinate y, which is perpendicular to the embankment axis, is disregarded. The element of the mean vector is described  as:  (3) in which (xk, zk) means the coordinate corresponding to the position of the parameter sk, and a0, a1, a2, a3, a4, and a5 are the regression coefficients. C is the M×M covariance matrix, which is selected from the following four types in this study.   C = Cij!" #$=σ 2 exp − xi − x j( ( lx − zi − zj( ( lz( )          (a)σ 2 exp − xi − x j( )2 lx2 − zi − zj( )2 lz2{ }      (b) σ 2 exp − xi − x j( )2 lx2 + zi − zj( )2 lz2{ }    (c) Neσ 2 exp − xi − x j( ( lx − zi − zj( ( lz( )       (d)i, j =1, 2, ⋅ ⋅ ⋅,M    (4)  in which the symbol [Cij] signifies an i-j component of the covariance matrix, σ is the standard deviation, and lx and lz are the correlation lengths for the x and z directions, respectively. Parameter Ne is related to the nugget effect. The Akaike’s Information Criterion, AIC (Akaike 1974) is defined by Equation (5), considering the logarithmic likelihood. ( ) ( ) ( )XXX Ums +=kkkkkkk zxazaxazaxaam 52423210 +++++=€ Ne=1     i = j( )Ne≤1     i ≠ j( )$ % & ' & 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 AIC = −2 ⋅max ln fS S( ){ }+ 2L = M ln2π+min ln C + S−m( )t C−1 S−m( ){ }+ 2L  (5) in which L is the number of unknown parameters included in Equation (2). By minimizing AIC (MAIC), the regression coefficients of the mean function, the number of regression coefficients, the standard deviation, σ, a type of the covariance function, the nugget effect parameter, and the correlation lengths are determined. Because the correlation lengths of soil parameters are often short compared with the sampling or the testing interval, sometimes the correlation lengths cannot be determined using the aforementioned method. For such cases, the following two-step approach is proposed as a strategy for identifying the spatial correlation structure. First, the mean (trend) function and the variances are determined by MAIC. Subsequently, the covariance Cij is determined from the semi-variogram. The semi-variogram is evaluated in the horizontal and vertical directions as individual functions of the sampling intervals.                        γ x q ⋅ Δx( ) = U xi, zj( )−U xi + q ⋅ Δx, zj( ){ }2i=1Nx−q∑j=1Nz∑ 2Nz Nx − q( )γ z q ⋅ Δz( ) = U xj, zi( )−U xj, zi + q ⋅ Δz( ){ }2i=1Nz−q∑j=1Nx∑ 2Nx Nz − q( )         q =1,2,  (6) where γx, and γz are the semi-variaograms for the  x, and the z axes, respectively, U(x,z) is a measured parameter at the point (x,z) from which the mean value is removed, namely, the value of (s(x, z)−m(x, z))/σ , ∆x and ∆z are sampling intervals, and Nx and Nz are the number of sampling points for the x and the z axes, respectively.  Next, the calculated semi-variograms are approximated by the following theoretical semi-variogram functions, and the correlation lengths are identified. Since an exponential type of function (Equation (4a)) is selected as the best fitting function by MAIC in many cases, it is also employed here.                           (7) In Equation (7), Cox and C0z are the parameters used for the nugget effect for the x and the z directions, respectively, and C1x, and C1z are the parameters used to express the shape of the semi-variogram functions. Finally, the two-dimensional covariance Cij between two points i and j, is given as                            Cij =σ 2C1xC1z exp − xi − x jlx − zi − zjlz!"## $%&&   i ≠ jCij =σ 2                                                  i = j    (8) 3. INDEICATOR SIMURATION METHOD The N-value is estimated based on the two kinds of data. One is the sounding test and the other is the surface wave method, a kind of elastic wave survey method.  Surface waves are strongly related to shear waves, and are easily related to the N-values. These two sets of results are conveniently synthesized with the indicator simulation method, a kind of geostatistical method, which can simultaneously treat hard data (primary data) and soft data (secondary data).  Herein, the SWS results are considered as the hard data, while the surface wave results are the soft data. An indicator value, i, for a parameter, R is expressed by € i u;rk( )=1,  R u( )≤ rk( )0, R u( )> rk( ) # $ % & %         k =1,...,K               (9) € γxxi− xj( )= C0x+ C1x1− exp − xi− xjlx( ){ }  i ≠ jγzzi− zj( )= C0z+ C1z1− exp − zi− zjlz( ){ }     i ≠ jγx0( )= γz0( )= 012th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 in which the vector u = (x, z) means the positions where the data were measured, and the parameter R is given as a function of u. The values of rk (k=1, 2, ..., K)  are K specific values of R, and the threshold value for the binary parameter i. The probability distribution function of the variable R, F is defined in the following. € F u;rkn + n'( )( )= Prob R u( )≤ rkn + n'( ){ }= λ0F rk( )+ λα u;rk( )α=1n∑ i uα;rk( ) + να ' u;rk( )α '=1n'∑ w uα ' ;rk( )  (10) € λ0=1− λα u;rk( )α=1n∑ − να ' u;rk( )α '=1n '∑  where i(uα;rk) means the binary value of the hard data at the point uα, and for the threshold value rk, w(uα′;rk) is the soft data, and n and n′ are the numbers of the hard and the soft data, respectively.  The parameter λ and ν are the weighting parameters corresponding to the arbitrary point um for the  interpolation; They are determined by solving the Equation (11).   (11) in which Cβα, Cβ'α', Cmα, and Cmα' are the covariance matrices between two points, namely, (uβ, uα), (uβ', uα'), (um, uα), and (um, uα'), respectively. The soft data w is derived from the following process based on the indicatior kriging (Deutsch and Journel 1992). 1) The measured data from the surface wave test results are assigned  for the points α's on the space as the input data for the indicator kriging. 2) The probability distribution function F(rk) of the SWS results is assumed for the measured data. 3) Indicator kriging is conducted based on the measured SWM data and the probability distribution function F(rk). 4) The results of the indicator kriging, which are presented by the probability distributions at the n' points, are employed as the soft data in the indicator simulation. Based on the probability distribution function F(u;rk|n+n') updated by the soft data, the random numbers are created from the Equation (12).                      (12) where p is the uniform random number from 0 to 1.0, and l is the iteration number for the Monte Carlo method. Finally, a random number, r(l), is assigned to the N-value. 4. SWS AND SWM RESULTS AND GEOSTATISCAL ANALYSES 4.1. In-situ test results Although high-density sampling is required in order to evaluate the spatial distribution of soil parameters, the amount of data is not sufficient in the general sampling plans.  In such cases, sounding is a convenient way to identify the spatial distribution structure of the soil parameters.  In this research, an embankment at Site A is analyzed, for which SWS tests were conducted at fifteen points, at 5 m intervals, along the embankment axis, as shown in Figure 1.  Additional tests were conducted between x=18 m and x= 24 m with 2 m interval to identify the lateral correlation length. The soil profile for the embankment is categorized as intermediate soil, and consists of the decomposed granite. Generally, the strength parameters are assumed based on standard penetration tests (SPT) with the use of empirical relationships.  In this research, however, Swedish weight sounding (SWS) tests, which are simpler than SPT, are employed instead of SPT.  Inada (1960) derived the relationship between the results of SPT and € λβ u( )β =1n∑ Cβα + νβ ' u( )β '=1n '∑ Cβ 'α = Cmα ,      α =1,....,nλβ u( )β =1n∑ Cβα ' + νβ ' u( )β '=1n'∑ Cβ 'α ' = Cmα ' ,    α '=1,....,n'€ rl( )u( )= F−1u;pl( )n + n'( )( ) Figure 1. Plan view of embankment and testing interval. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 SWS.  Equation (13) shows the relationship for sandy grounds, while Figure 2 shows the relationship between SWS and SPT N-values.  N = 0.002WSW + 0.067NSW                     (13) in which NSWS is the N-value derived from SWS, NSW is the number of half rations and WSW is the total weight of the loads (N).  Based on this data, the variability of the relationship is evaluated in this study, and the coefficient of variation is determined as 0.354.  The determined σ-limits are also shown in Figure 2 with broken lines.  Considering the variability of the relationship, the SPT N-value, NSPT is modeled by                        (14) in which εr is an N(0,1) random variable (Nishimura et al. 2014). In Figure 3, the distribution of the N-values predicted by the surface wave method (SWM) is exhibited.  The figure shows the averaged image throughout the inversion. Equation (15) is employed to transform the measured shear wave, Vs, to the N-value (Imai, et al. 1975).   VS = 97.0N 0.314                                       (15) Surface waves are closely correlated to shear waves Vs, which in turn have a strong correlation to the elastic modulus and the N-values.  In this research, the surface wave was measured as 70 m along the embankment axis at 2 m intervals.  4.2. Statistical mode The mean function and the covariance function of the SWS N-value, NSWS, are determined with MAIC, and the mean is exhibited in Figure 4.  The mean and the covariance functions given by Equations (3) and (4) were examined, and the optimum functions are determined as Equations (16) and (17).  The horizontal correlation length lx is identified as being approximately 10 m , and the vertical one lz, is 2.66 m. Comparing with the published values (Phoon and Kulhawy 1999, Tang 1979, DeGroot and Beacher 1993, Nishimura et al. 2010), the horizontal one is reasonable, and vertical one  is rather large. The horizontal length, however, is almost three times of the vertical one, and the values could be acceptable in the fact that the horizontal length is much greater than the vertical one. m = 2.52− 0.0279x − 0.226z+0.0003x2 + 0.0465z2 + 0.0038xz            (16) ( ) SWSrSPT NN ε354.01+= Figure 2. Relationship between SWS results and SPT N-values (Nishimura et al. 2014).  Figure 3. N-value distribution by surface wave method.  Figure 4. Distributions of SWS N-value. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 Cij!" #$= Neσ 2 exp(−Δxi / lx −Δzi / lz )Ne =1 (i = j)Ne = 0.73 (i ≠ j)()*σ =1.08, lx = 9.88m, lz = 2.66m         (17) To check the correlation structures, the semi-variograms for the horizontal and vertical directions are calculated. Figures 5(a) and (b) show the semi-variograms for the lateral and vertical directions, respectively. The semi-variogram values of Δx = 2, 5, and  10m, Δz =0.25, 0.5, 0.75, 1.0, 1.25 m are employed to identify the approximate functions of Equation (7) for the lateral and vertical directions. Since the values of Nx and Nz in Equation (6) are large within the short intervals of Δx and Δz, the accuracy of the semi-variogyam values are supposed to be high.  The result is exhibited as follows. Cx1 ⋅Cz1 = 0.45, lx = 27.1m, lz = 2.06m  The value of the nugget effect for the lateral direction, 0.4 seems very large. The reason is supposed are as follows. To identify the lateral correlation, the semi-variogram values are calculated from the data obtaind in the same depth. Since the variability of the measured data along the depth of each test point, is great as sown in Figure 4, the calculated lateral correlation can be easily affected by the variability, and in results, the lateral semi-variogram can include uncertainty as the nugget effect.  The lateral correlation length is identified as almost three times of the  MAIC. While the vertical length is determined as the value similar to that of the MAIC. There is the tendency generally that the variogram exhibits the relatively longer correlation length, comparing with the MAIC, since the correlation lengths lx and lz are identified separately along the single coordinate of x or z, in the calculation of the variogram. While in the MAIC, the multi-dimensional normal distribution is assumed, the correlation structures of obtained data in all test points must be identified simultaneously.  4.3.   Synthesis of SWS and SWM results The SWS are the actual destruction tests, and they can estimate accurate N-values with high resolution as point estimated values.  The accuracy of the measured values at the measuring points is good, while at the mid-points between two measuring points, the accuracy is inferior to that at the measuring points.  While the SWM is convenient for obtaining the averaged profile, the local resolution is not good.  If the results of the two methods are synthesized, the shortcoming of one method can be compensated by the other method.  In this research, the results for SWS and SWM are considered as the hard and the soft data, respectively, and then the two sets of results are synthesized with the indicator simulation method.  (a) Horizontal  (b) Vertical Figure 5. Semi-variograms and approximate functions. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 As for the mean and the covariance functions, Equations (16) and (17) are employed for the embankment. In the Monte Carlo simulation, random numbers for NSWS are generated through Equation (12).  Then, random numbers for NSPT are created  by considering error factor εR in Equation (14).  The spatial statistical values for NSPT are discussed below. Figures 6 and 7  present the analytical cases without and with the soft data, respectively. Figures (a), (b), and (c) correspond to the mean, the standard deviation, and the probability that  N- value falls blow 2, respectively. According to Figures 6(a), around depth z = 3-4 m, x =30-40 m, the lowest value is detected. Corresponding to Figure 6(a), the lowest value of probability is obtained at the same location in Figure 6(c). In Figure 7(a), the left part of the embankment shows lower value, compared with Figure 6(a), and corresponding to this result, the probability is relatively high in the left side as depicted in Figure 7(c). The deeper part of the embankment,  z =7-9 m, the high average of N-value and smaller probability are exhibited in Figures 7(a) and (c) due to the fact the SWM results are affected by the base ground. In comparison between Figure 6(b) and Figure 7(b), (a) Mean (N-Value) (b) Standard deviation (N-Value) (c) Probability (N <2) Figure 6. Statistical values of N-value by indicator simulation without soft data (a) Mean (N-Value) (b) Standard deviation (N-Value) (c) Probability (N <2) Figure 7. Statistical values of N-value by indicator simulation with soft data 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 the standard deviation in the latter case is smaller the former case, and it is understood that the uncertainty  of the N-values can be reduced with the soft data.  According to the comparison between Figures 6(c) and 7(c), safety side results are obtained with the synthesis   with the results of the SWS and SWM in the studied case, since the greater probability is obtained inside the embankment in Figure 7. 5. CONCLUSIONS (1) With minimum lateral interval of SWS of 2 m, the spatial correlation structures of N-values inside the embankment could be evaluated accurately. (2) Correlation structures were obtained by two approaches, MAIC and semi-variogram, and the difference of two results was acceptable. (3) The spatial distribution of the probability that the N-value is lower than the threshold value has been calculated with the indicator simulation.  The spatially distribution of the probability can be used for the health monitoring of the inside of an embankment. 6. ACKNOWLEDGEMENTS This work was partly supported by JSPS KAKENHI Grant Number Number 25292143. 7. REFERENCES Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. on Automatic Control, AC-19(6), 716-723. Cafaro, F. and Cherubini, C. (2002). Large sample spacing in evaluation of vertical strength variability of clayey soil, Journal of Geotechnical and Geoenvironmental Engineering, 128(7): 558-568. DeGroot, D. J. and Beacher, G. B. (1993): Estimating autocovariance of in-situ soft properties, Journal of the geotechnical engineering, ASCE, 119(1), 147-166. Deutsch, C. V. and Journel, A. G. (1992). Geostatistical Software Library and User’s Guide, Oxford University Press. Fenton, G. A. and Griffiths, D. V. (2002). Probabilistic foundation settlement on spatial random soil, Journal of Geotechnical and Geoenvironmental Engineering, 128(5),  381-391. Griffiths, D. V. ,  Fenton, G. A.  and  Manoharan N. (2002). Bearing capacity of rough rigid strip footing on cohesive soil: probabilistic study, Journal of Geotechnical and Geoenvironmental Engineering, 128(9),743-755. Hayashi, K. 2004. Estimation of near-surface shear wave velocity model using surface -waves, Journal of JSNDI, 53(5), :254-259.  Imai, T, , Fumoto, H. and Yokota, K. 1975. Velocity of elastic wave and mechanical properties in Japanese grounds, Proc. of 4th Symp. of Earrhquake Eng: 89-96 (in Japanese). Inada, M. (1960). Usage of Swedish weight sounding results, Geotechnical Engineering Magazine, 8(1), 13-18 (in Japanese). Japanese Geotechnical Society. (2004). Japanese standards for geotechnical and geoenvironmental investigation methods – standards and explanations-, Tokyo: JGS (in Japanese) Nishimura, S., Murakami, A. and Matsuura, K. (2010). Reliability-based design of earth-fill dams based on the spatial distribution of strength parameters, Georisk, 4(3): 140-147. Nishimura, S. and Shimizu, H., (2008). Reliability-based design of ground improvement for liquefaction mitigation, Structural Safety, 30: 200-216. Nishimura, S., Shuku, T. and Shibata, T. (2014): Reliability-based design of earth-fill dams to mitigate damage due to severe earthquakes, Vulnerability, Uncertainty and Risk, Proc. of 2th ICVRAM - 6th ISUMA, 2350-2359.  Phoon, K-K. and Kulhawy F.H. (1999). Evaluation of geotechnical property variability, Can. Geotech. J., 36, 625-639. Soulie, P., Montes, P. and Silvestri, V., (1990). Modelling spatial variability of soil parameters, Canadian Geotechnical Journal,  27: 617-630. Tang, W. H. (1979). Probabilistic evaluation penetration resistances, Journal of the geotechnical engineering, ASCE, 105(GT10): 1173-1191. Uzielli, M., Vannucchi and Phoon, K. K., (2005). Random field characterization of stress-normalized cone penetration testing parameters, Geotechnique, 55(1): 3-20. 

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