Sand, Semivariogram, Ordinary Kriging
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Journal of Civil Engineering Research 2014, 4(3A): 36-40 DOI: 10.5923/c.jce.201402.05 Prediction of Sand Thickness Using Ordinary Kriging Ahmad Shukri Yahaya*, Fauziah Ahmad, Lo Yean Tiing School of Civil Engineering, Engineering Campus, Universiti Sains Malaysia, Pulau Pinang, Malaysia Abstract The objective of the research is to determine the characteristics of sand in Batu Ferringhi, Penang. Then an ordinary kriging model was developed which can be used to predict thickness of sand. A set of data from three different sites in Batu Ferringhi were collected from the site investigation report. There are three main stages in predicting sand thickness. The first stage is to determine the location and coordinate of each boreholes from the three different sites in Batu Ferringhi. The second stage is to obtain descriptive statistics of the thickness of sand. The third stage is to model the sand thickness using ordinary kriging and to obtain the best model. The results show that the best model variogram is the exponential function. Five boreholes were then used for prediction using the best ordinary kriging model. It was found that the prediction is quite accurate. Keywords Sand, Semivariogram, Ordinary kriging prediction case study was carried out on clay thickness data 1. Introduction in Kuala Lumpur (Saffur, 2003). This prediction technique is capable in predicting the clay thickness at an unsampled Penang is located at the north-eastern coast and location provided that the data used were collected at constituted by two geographically varying entities which is reasonable distances so that it is correlated, for the technique 2 an island with area of 293 km called Penang Island and a to work. The kriging method was used to estimate the portion of mainland called Seberang Perai having the area of presumed unknown values of sorptivity for a soil under tilled 2 about 755 km . Batu Ferringhi is located in Penang island. and no-tilled condition in some locations in the 0.5 m × 0.5 m The development in Penang Island is very rapid because it is grid distances (Sepaskhah et al., 2004). The result shows that one of the most industrialized area in Malaysia. In Penang, the method is capable to estimate the unknown sorptivity of the most frequent hazard that always occurs is landslide. the soil. Malaysia is a hilly or a mountainous country and the slope This study aims to determine the characteristics of sand failure are a common occurrence. Geologically, this area is thickness in Batu Ferringhi, Penang Island and to predict also underlain by medium to coarse-grained biotite granite sand thickness using ordinary kriging so that engineers can layer with predominant orthoclase and subordinate use this model to predict sand thickness at unsampled microcline (Fauziah Ahmad et al., 2005). Thus it is locations. important to analyze the soil conditions as well as to predict sand thickness in Penang Island. Based on the kriging principle, Ishii and Suzuki (1989) 2. Methodology presented a simple probabilistic model that evaluates an unknown value of ground thickness and estimation of error 2.1. Study Area for the soil properties at unsampled location in the ground. The data was obtained from 22 boreholes which were This model is capable to evaluate the borehole spacing. collected from three sites around the area of Batu Ferringhi However, the model required statistical parameters, namely, in Penang Island. The information was obtained from the a correlation distance and variance of soil properties as Public Works Department, Malaysia from the site important data input. The geotechnical database system for investigation reports. The data that was extracted from the Saga Plain, Japan provided the parameters needed. Finally, reports t are location, description of soils, depth of soil and the exploration spacing for different values of estimation thickness of soil. Batu Ferringhi is located in the northern error was suggested for site investigation. A kriging part of Penang Island as shown in Figure 1. The coordinates of the boreholes are obtained using ArcView GIS software * Corresponding author: [email protected] (Ahmad Shukri Yahaya) (Lee and Wong, 2001) by plotting the boreholes based on the Published online at http://journal.sapub.org/jce contour levels in the report and hence the coordinate for each Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved borehole was obtained. Journal of Civil Engineering Research 2014, 4(3A): 36-40 37 Figure 1. Map of Penang showing the study location (Source: Penang map, 2012) 3 2.2. Descriptive Statistics h h C 1.5 − 0.5 if h ≤ a Descriptive statistics for measures of location such as 0 (1) γ (h) = a a mean, median, minimum and maximum values were obtained. The mean represents the average value of all the C 0 otherwise observations of size n while the median is the middle value in where ( ) is the semivariogram model, is the a set of observations when they are arranged in the order of Euclidean distance between two points, is the range and magnitude. The measures of spread such as standard C is the ℎnugget. ℎ deviation, variance and range were calculated. The standard 0 deviation measures the variability of the data from the mean. The coefficient of variation (COV) was also calculated to 2.3.2. Exponential Model determine the extent of variability of the data on sand The equations for exponential model with a nugget are thickness. The value of COV less than 1 shows low 3 variability in the data while COV greater than 1 indicates h γ (h) = C + (32ha) − high variability in the data. 0 C1 3 2a 2.3. Semivariogram Models (0 ≤ h ≤ a), C1 > 0 (2) For predicting sand thickness five semivariogram models γ (h) = C + C (h ≥ a) (3) were fitted namely Spherical, Exponential, Gaussian, Linear 0 1 and Power functions. These models are discussed in the where C0 + C1 is the sill of the semivariogram, C0 is the following sections. More information about these models nugget effect that represents the micro-scale variation and/or can be found in Kaluzny et al. (1997). The best measurement error, is the Euclidean distance between two semivariogram model was chosen to be used in ordinary points and a is the range. kriging. ℎ 2.3.3. Gaussian Model 2.3.1. Spherical Model The form of the Gaussian model is The spherical model is given in Equation 1. Spherical 2 model has a linear behaviour at small separation distances ( ) = 0 1 exp 2 (4) near the origin but flattens out at larger distances, and ℎ reaches the sill at a . In fitting this model to a sample where C0 is theℎ nugget �effect− that� −represents�� measurement semivariogram it is useful to note that the tangent at the error, is the Euclidean distance between two points and origin reaches the sill at about two thirds of the range. a is the range. Asymptotically, ( ) approaches the sill. ℎ ℎ 38 Ahmad Shukri Yahaya et al.: Prediction of Sand Thickness Using Ordinary Kriging 2.3.4. Linear Model 2.5. Ordinary Kriging The equation for the linear model is The ordinary kriging model (Webster and Oliver, 2000) 0 if = 0 must meet two assumptions that are given as follows: ( ) = (5) 0 + 1 otherwise (1) The first assumption is that the sampling is a partial ℎ ℎ � realization of a random function Z(x), where x where C0 + C1 is the sill of the semivariogram and is denotes spatial location. the Euclidean distance between two points. (2) The second assumption is that the random function ℎ is second order stationary over sampling domain, 2.3.5. Power Model which implies that, [ ( )] = and Cov( , + The equation for the Power model with a nugget is = . where E[.] denotes expected value, m is a constant mean, h is ( ) = 0 (6) ℎ (ℎ) a vectorial distance in the sampling space and Cov [.] is the where C0 is the nugget ℎ effect, ℎ is the Euclidean distance covariance of the random function. between two points and a is the range. For this model the The best semivariogram model was used to obtain the ℎ ordinary kriging model that is given below: parameter a is a dimensionless quantity, C0 has −1 dimensions of the variance and there is no sill for the power Z OK (x0 ) = Z ′L = Z ′V v (9) model. ′ 2.4. Error Measures where Z = [Z(x1 ) Z(x2 ) ... Z(xk ) 0] and Z(xi ) are the random variables of a random function sampled at Goodness-of-fit was used to examine how well a sample of sand thickness data agrees with a given semivariogram sites xi , i =1, 2,…, k and x0 is the estimated location, model. From this measures, the best semivariogram model ′ L = [λ λ ... λ µ] with λ ’s be the optimal that fit the sand thickness data will be chosen. In this research, 1 2 k i error measures such as the mean absolute error (MAE) and weights for the estimator and µ is the Lagrange multiplier, normalized absolute error (MAE) were used. The equations is the covariance matrix. are given by Lu (2002). The mean absolute error (MAE) is: n 3. Results and Discussions PO− ∑ ii (7) The analyses are done using the descriptive statistics to get MAE = i=1 the typical values that represent the study area. Table 1 n below shows the summary of the descriptive statistics for all boreholes in Batu Ferringhi. The normalized absolute error (NAE) is: Table 1. Summary of descriptive statistics for thickness of sand and clay n ∑ Abs() Pii− O Sand Thickness i=1 = (8) Number of boreholes, N 22 NAE n Mean (cm) 39.5968 O ∑ i Std.