# Spatial Autocorrelation Incorporated Machine Learning Model for Geotechnical Subsurface Modeling
## 1 Introduction  
In a geotechnical project, an accurate prediction of geotechnical information is important. It could minimize the adjustment of the initial construction plan during the construction process. This results in better efficiency during construction as the designed schedule, methodology, and cost proceed as planned. In situ tests such as the standard penetration test (SPT) and laboratory soil tests are commonly performed to investigate soil properties at a project site [1, 2, 3, 4]. When a site investigation is performed in a limitednumber, estimation of soil information at unexplored points is necessary [5]. However, soil properties vary from place to place due to the complexity of soil formation and geotechnical behavior and due to the spatial variability of the soil [6; 7; 8]. Spatial variability is an important feature of geotechnical design, as it influences the variety of soil behavior at the project site [9; 10; 11; 12]. Empirical techniques that combine engineering judgment and straightforward data analysis in the estimation of geotechnical properties at unobserved site location are frequently employed. Methods such as interpolation based on global mean value and inverse distance squared are relatively rough and inconsistent [13]. Therefore, methods that can predict spatially varied soil properties well are of great importance. Machine learning has been proven to be an appropriate method for predicting geotechnical properties and soil type, and can be considered as an alternative to conventional approaches. Studies utilizing machine learning in predicting soil classification [14; 15] and geotechnical properties [5; 7; 14; 16; 17] exist in the literature. They specifically make use of related geotechnical properties with the target attribute for its prediction, without considering the location of samples in two- or three-dimensional space as input data for machine learning. Recently, there has been increasing interest related to the application of machine learning for geotechnical properties prediction and subsurface modeling by considering the location of the samples as input. Several studies that involve spatial coordinate as input for machine learning modeling have been conducted by several researchers [5; 18; 19]. However, these studies do not consider spatial correlation as an important aspect in spatial modeling because the data for examined locations are all in X-Y-Z coordinate form, which is based on the Cartesian coordinate system. An improved spatial modeling method that considers spatial correlation was proposed by [PERSON] et al. [20]. The method was then applied for geotechnical subsurface modeling by [PERSON] et al. [13]. Their study introduced a spatial modeling framework that provides additional attributes called Euclidean distance field (EDF), which quantify the spatial relationships among training samples, allowing spatial correlations to be reflected in the prediction even when the dataset size is limited. The EDF as in [PERSON] et al. [20] and [PERSON] et al. [13] act as additional machine learning features by integrating several EDF: (1) X and Y coordinates (Euclidean distance fields to the edges of a rectangle around the sample set); (2) Euclidean distances to the corners of a rectangle around the sample set; and (3) the distances to the center location of the sample set. The approach by [PERSON] et al. [13] only considers spatial correlation by incorporating the EDF feature set to the machine learning model, without considering the interrelation between soil properties. The study by [PERSON] et al. [20] combined both of these important information. However, the proposed EDF feature set does not consider the direct spatial relationship between the observed data and query points by not including the measured distance between them in the EDF feature set. In addition, the study by [PERSON] et al. [20] only explains the spatial modeling in two-dimensional space.