## 3 Research Results
### Improved IDW Interpolation and Mapping  
#### 3.3.1 Development of an Improved IDW Interpolation Algorithm  
In this study, the 3D search neighborhoods require generating and representing B-rep-based precise spheroid. 3D CAD systems can be divided largely into polygonal modeling and Non-Uniform Rational B-splines (NURBS) modeling. Polygonal modeling produces meshes by combining polygons (generally triangles) to express objects, which has the advantage of allowing for fast modeling but is limited in expressing curved surfaces. On the other hand, NURBS modeling is the way of creating curves with the Bezier curve and B-splines and generates a surface from these curves. It is widely used in the fields  
Figure 7: Borehole density in the study area. where precision modeling is required, such as ships, cars, and architecture, among others. Therefore, this section offers the results of the algorithm development using Rhinoceros(r) Grasshopper, which is a NURBS-based 3D CAD system, to realize the proposed IDW interpolation method. The first stage of the algorithm is to adjust the parameters. The developed algorithm is divided into the case in which the search distance is given (fixed distance; FD) and the case in which the number of neighbors is given (number of points; NP). Such a division implements a main role in decision making whether to limit search 'distance' or to limit 'number of neighbors' when determining the neighborhoods. For instance, it is possible to search a neighborhood within a fixed distance with a spheroid-based approach, but it is unable to strictly maintain the number of neighbors. Thus, FD has five parameters, horizontal search distance (VI), vertical search distance (V2), assessment height/depth (V3), elevation exaggeration factor (V4), and power (V5), while NP has seven parameters, the maximum search distance toggle (V1), the maximum horizontal search distance (V2), the maximum vertical search distance (V3), the number of neighbors (V4), assessment height/depth (V5), altitude exaggeration factor (V6), and power (V7). In the second stage, the 3 D grid data by altitude is produced for storing interpolation results by the given FD\(-\)V3, FD\(-\)V4, or NP\(-\)V5, NP\(-\)V6 assessment height/depth parameters. To promote the applicability of the assessment result, the study set the grid origin and size identical to 250 m grid data offered by the National Geographic Information Institute ([[https://map.ngi.go.kr/](https://map.ngi.go.kr/)]([https://map.ngi.go.kr/](https://map.ngi.go.kr/)) (accessed on 22 July 2022)), and the altitude was set to 1 m to 20 m under the subsurface based on the FL assessment standards. The third stage Is to search for neighbors. In FD, the spheroid is formed using FD\(-\)V1 and FD\(-\)V2, and the Within3D relationship to the borehole data is processed. Later, the distance from the unsampled location to the borehole data that has the Within3D relationship is calculated, and the IDW processing is performed using FD\(-\)V5. Figure 8a shows the example of the application of the improved IDW interpolation with 250 m of the horizontal search distance, 2 m of the vertical search distance, 1 m of the assessment depth, 100 m of the elevation exaggeration factor, and 2 power as the parameters. NP uses as the horizontal and vertical search distances the diagonal length of the assessment grid extent and NP\(-\)V3 if NP\(-\)V1 is false, or NP\(-\)V2 and NP\(-\)V3 if it is not. Figure 8: Limiting neighborhoods used for improved IDW: (**a**) Fixed search radius; (**b**) Fixed neighborhood counts. Later, a virtual 3D box is generated in the direction of X-Y-Z axes to the given maximum search distance, and through the Within3D relationship processing, the neighbor is selected. Here, the final neighbors for interpolation are selected for up to NP\(-\)V4 based on the distance in ascending order and using NP\(-\)V5, the IDW process is performed.