# Estimating CPT Parameters at Unsampled Locations Based on Kriging Interpolation Method
## 1 Introduction  
The cone penetration test (CPT) has been widely used as a quick and reliable soil exploration test that provides subsurface soil properties [1]. As the electric cone penetrometer (Figure 1) advances into the soil with a constant speed, parameters such as the cone tip resistance (\(q_{\mathrm{c}}\)) and the sleeve friction (\(f_{\mathrm{s}}\)) are simultaneously measured by the load cells on the penetrometer. Compared with other in situ tests, CPT can provide continuous parameter profiles that can yield much more detailed information [2]. As the two major measured parameters \(q_{\mathrm{c}}\) and \(f_{\mathrm{s}}\) vary significantly with the change of the soil layers, CPT shows advantages in geotechnical investigations to classify soil strata, and several methods have been proposed to classify soils upon CPT data [3; 4; 5]. Moreover, CPT can also be used to estimate the strength and deformation characteristics of soils, such as the undrained shear strength of soil \(S_{\mathrm{u}}\)[6]. Generally speaking, CPT is only conducted at necessary locations to obtain the required soil properties, for example, around the underground structure to be constructed. Considering the spatial variability of the soil, the CPT data obtained from adjacent soundings could be quite different, which makes it difficult to estimate the soil properties at the unsampled locations. Therefore, for large areas, how to reasonably estimate the soil properties of the entire area through limited CPT data is of great importance.  
Soil properties are regionalized variables; that is, within a soil layer or rock mass, samples that are close to each other indicate a stronger correlation than distant ones [7]. Because CPT parameters directly depend on the soil properties, they are also autocorrelated. In statistics, there are two ways to consider the correlation of the data sequence with distance: one is random field theory, and the other is geostatistics. Random field theory is often usedto analyze time series. Although it has been successfully applied in analyzing soil spatial variability by different researchers [8; 9], it is mostly applied to one-dimensional situations.  
Geostatistics often refers to the Kriging method, which has been widely used in the field of geographic science. Different from random field theory, the Kriging method can be applied to multi-dimensional analysis since it is an algorithm for spatial modeling and regression interpolation [10]. It has been used by many researchers to analyze the autocorrelation of geotechnical parameters and achieved promising results [11; 12]. However, there are only a few case studies on adopting the Kriging method to predict the soil parameters at the unsampled locations, and the data used in previous studies is often scarce, which is not enough to prove the accuracy and effectiveness of the Kriging method. Currently, there have been several studies using Markov chain and machine learning methods combined with CPT test to predict the soil parameters at the unsampled locations [13; 14]; however, they need to meet some strong prerequisites. For example, the Markov chain method is suitable when the unsampled location and the sampled CPT boreholes are located on the same horizontal line, while the machine learning method needs to obtain the soil layer information in advance and perform extensive training to obtain a better prediction result, which is difficult to be applied into practical engineering. In this study, we used CPT parameters with small sampling intervals as the research objects to assess the performance of the Kriging method in predicting spatial soil parameters, thus avoiding data insufficiency due to excessive sample spacing. The study was further supplemented by axial bearing capacity calculation, which we used to examine the reliability of the interpolation results and the applicability of the Kriging method in solving practical engineering problems.