# Estimating CPT Parameters at Unsampled Locations Based on Kriging Interpolation Method
## 2 Materials and Method
### CPT Data Preprocessing  
The first step is to remove outliers from the data. Outliers refer to those abnormal points whose values are obviously different from the surrounding sampled points, which is mainly caused by measurement errors or procedural errors such as rod adding [7]. Outliers do not reflect the actual CPT characteristics of the sampled points but affect the estimation accuracy, so they need to be removed first. This process is accomplished by a Gaussian filter, which replaces the value of the abnormal point with the weighted average of 20 samples around the point. The example of outliers and the smoothed curve is shown in Figure 4.  
Figure 3: The \(q_{\rm c}\) and \(f_{\rm s}\) profiles of the six CPT soundings.  
Figure 2: The locations of the sampled CPT boreholes and the pile.  
The second step is to shift the \(f_{\rm s}\) data to the right place. [PERSON] [15] emphasized that incorrect analysis will be made unless a depth correction, or shift, is applied to the \(f_{\rm s}\) measurements. This correction is required since the cone tip resistance, and sleeve friction load cells are physically separated by a given distance as shown in Figure 1, and hence measurements of \(q_{\rm c}\) do not refer to the same soil as that at which measurements of \(f_{\rm s}\) are taken. [PERSON] [15] recommended that this shift distance should be equal to the distance between the base of the cone and the mid-height of the sleeve, which, for standard electric cone penetrometers, is approximately 75 mm. [PERSON] et al. [16] argued that the shift distance, also termed the \"friction-bearing offset\", is 100 mm and is dependent on the type of soil being penetrated. Actually, when the cone penetrometer is advanced into the subsoil, it will cause a zone of soil to fail and deform plastically, as shown in Figure 5. The \(q_{\rm c}\) and \(f_{\rm s}\) are not point values but spatially averaged values of the failure zone [17], and the shift distance would be different for different cases. [PERSON] [7] recommended cross-correlation function (CCF) to determine the shift distance because of its superiority in demonstrating the cross-correlation between \(q_{\rm c}\) and \(f_{\rm s}\).  
Figure 4: Outlier of raw CPT data.  
Figure 5: Plastic failure zone caused by the cone penetrometer.  
In this study, we adopted the CCF to determine the final shift distance. Taking the \(q_{\rm c}\) and \(f_{\rm s}\) data of borehole F7 as an example, the CCF analysis result is shown in Figure 6. It can be seen clearly from Figure 6 that the maximum cross-correlation coefficient occurs at a spacing of 115 mm, which implies that the optimal shift distance is 115 mm, which is higher than the actual physical spacing of 75 mm. The same process is applied to the other five boreholes, and the final shift distance results are shown in Table 1. Based on the result listed in Table 1, the \(f_{\rm s}\) data of the 6 boreholes will be shifted upward, resulting in a decrease in the maximum available drilling depth of each borehole. The smallest depth is borehole I5, of which the available depth of \(f_{\rm s}\) is reduced to 4775 mm so that in the interpolation part, the maximum interpolation depth is adjusted to 4775 mm. Meanwhile, in order to solve the problem of missing data at some depth in some boreholes, we use linear interpolation to fill in the missing values, and the details are shown in Appendix A.  
After the above preprocessing steps, we divided the soil into a total of 956 sections from the ground down every 5 mm until the depth of 4775 mm, then on each section, we estimated the CPT parameter at the unsampled location using the Kriging method.