In a situation where cities are increasingly being developed vertically and complexly, a novel approach for analyzing vertical urban functional patterns is proposed. For this purpose, a multi-dimensional GDC (Geo Data Cube) consisting of spatial and temporal data x, y, z, t, and f dimensions containing layer information was created. At this time, the size of the GDC cell (interval in x, y, z dimensions) is calculated by cell point data using the three-dimensional (3D) Moran’s I index value calculated with the 3D Diversity Factor (DF) based on information entropy proposed to reduce the uncertainty of information for each cell. In other words, the cell with the smallest index value was chosen to minimize the influence of Modifiable Areal Unit Problem (MAUP) that occurs when mapping. The 3D land use index (3D LUI) is calculated as a linearly weighted sum of the spatial accessibility of uses between cells (3D KDF) and the enrichment of uses (3D EF), taking into account the first law of geography. Finally, the 3D LUI value for each use was calculated for each cell of the GDC, and the use with the highest value was determined as the urban function of the cell. As a result of applying this to Seocho-gu, Seoul, Republic of Korea (ROK) in June 2024 and visually evaluating it using the street view provided by Kakao Map, it was confirmed that commercial and residential functions were vertically separated in buildings with residential–commercial complexes or shops on the ground floor. It was also confirmed that such characteristics did not appear in the two-dimensional (2D) urban functional patter analysis.

Urban function labels in multi-dimensional geo-data cubes (GDCs) assign a single dominant function to each voxel, discarding within-voxel composition. This study extends multi-dimensional grid-point (MDGP) scaling from isotropic two-dimensional rasters to anisotropic three-dimensional urban voxels built from building-register floor-level use records. The extension separates planar and vertical aggregation factors, represents voxel composition with gross floor area (GFA), and computes GFA-weighted information retention. The procedure converts each floor-level record into a three-dimensional point, allocates it to an anisotropic voxel, sums GFA by functional category to obtain a continuous composition vector, assigns each voxel to the simplex grid point that maximizes the Czekanowski retention, and then applies a representativeness threshold and adaptive partitioning before the cell-level and global retention are computed. The method was applied to 2,737,362 floor-level use records from 552,560 buildings across Seoul. In a controlled comparison on uniform grids before vocabulary pruning, the three-part mixed label improved information retention over GFA-based majority assignment by 11.63, 16.47, and 20.17 percentage points at 125 m × 8 m, 250 m × 16 m, and 500 m × 32 m, respectively. At 250 m × 16 m, the final adaptive, thresholded configuration retained 89.37% of the GFA composition on the refined grid, against 88.85% for the same label on the uniform grid before partitioning and pruning, and 72.38% for the single-label baseline on that uniform grid. Gains were stable across 30 random grid origins. A cell-level analysis of officially assessed unit building prices showed higher complexity-adjusted association for mixed labels: the complexity-adjusted ω² rose from 0.105 to 0.254 and the out-of-sample R² from 0.443 to 0.501 under spatial-block cross-validation, with the root-mean-square error of the log price falling from 0.480 to 0.454. These results show that mixed labels preserve more within-voxel functional composition under the tested Seoul data and aggregation settings; validation in other cities remains necessary.