Anisotropic Multi-Dimensional Grid-Point Scaling: A Method for Information-Preserving Upscaling of Building-Based Three-Dimensional Urban Geo-Data Cubes

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.