## 2 Research Method
### Improved IDW Interpolation Based on 3D Search Neighborhoods  
#### 2.3.1 3D Search Neighborhoods  
Research on topology, represented by proximity, connectivity, and inclusivity, is becoming more important and more expanded as GIS is implemented from a data visualization tool to an analysis and decision-making support system [34]. 9 IM (9 Intersection Model) is a model that determines the topology between binary objects (A-B) and was developed based on [PERSON]'s mathematical framework [35]. In addition, 9 IM determines the eight spatial relationships (\"Disjoint\", \"Contains\", \"Within\", \"Equals\", \"Touches\", \"Overlaps\", \"Covers\", and \"Covered by\") through a 3 \(\times\) 3 relationship matrix of three elements (Interior, Exterior, Boundary) in binary objects.  
Since it was proposed, the 9 IM framework has expanded into 3 DE-9 IM [36], DE-9 IM [37], and 25 IM [38] among others, and DE-9 IM was adopted as ISO 19125-1 Geospatial International Standard Specification. The realization of a topology depends largely on a spatial data model. In 3D GIS, the standard data model on geometry adopted CityGML [39], and the standard data model on topology adopted IndoorGML [40], but they only deal with cases under traditional urban environments. Therefore, they are limited in being applied to other nonurban environments, such as subsurface, atmosphere, ocean, etc.  
In 3D CAD systems, including GIS, B-rep (Boundary representation) is the most widely used computer graphic processing technique. B-rep also combines geometric primitives, such as node-0D, edge-1D, and face-2D to construct and express 3D objects, but does not store the topology of said object explicitly. Regardless, the reason why most 3D GIS implement B-rep is that it can immediately extract the topology between objects [41]. In this context, [PERSON] and [PERSON] proposed a method for the realization of integrating geometry and topology by implementing the 9-IM model in B-rep [42]. Therefore, this study proposed a 3D search neighborhood based on B-rep for developing a point interpolation in 3D space.  
In this study, to search all directional neighborhoods, use the \"Within3D\" relationship with the observation data, which is included in the spheroid, generated from the unsampled location. Under the 3 DE-9 IM by [PERSON] et al., the \"Within3D\" relationship between 3D spatial objects is defined as follows.  
a. Within3D(b) (dim(a) \(\leq\) dim(b))\(\wedge\)  
Here, the search distance of the neighborhoods is the explicit radius of the spheroid. However, since the search distance in 3D space is formed not only in a horizontal but in a vertical direction, the spheroid is generated using two parameters in the minor and major axes (Figure 4a). Moreover, in order to enhance the performance of the 3D search neighborhoods technique suggested from the study, the neighborhoods search method using bounding box is also developed, which rapidly distinguishes neighbors within the search distance (Figure 4b). Through Figures 3 and 4, it is able to compare how different the neighborhood is determined between the existing approach and the proposed 3D search neighborhood method.