Mathematical Modeling, Sensitivity Analysis, and Comparative Evaluation of Height Systems in Engineering, Geodetic, and Cartographic Applications: A Romania-Oriented Computational Study

Height systems form a mathematical interface between physical geodesy, engineering surveying, and digital cartography. Although satellite positioning efficiently provides ellipsoidal heights, practical infrastructure, mapping, hydrological, and monitoring tasks require gravity-related heights that are compatible with national vertical datums. This paper develops a denominator-based framework in which dynamic, orthometric, and normal heights are interpreted as metric realizations of a common geopotential number. Starting from the line-integral definition of geopotential, the principal height formulae are derived; first-order sensitivities to geoid undulation, height anomaly, and mean gravity are established; and uncertainty propagation is analyzed. A Romania-oriented computational experiment, explicitly defined as a representative model-behavior study rather than an official national adjustment, uses lowland, plateau, and mountain-influenced settings consistent with the Constanta and Black Sea 1975 normal-height context. The results show that modeled normal-orthometric separations remain below 3 mm in representative low-relief locations but increase to 17.1 mm in Suceava, 29.5 mm in Cluj-Napoca, and 85.6 mm in the mountain-influenced Brasov case. The dynamic-normal differences remain small at low elevations but become systematic where the normal-gravity denominator departs from the selected reference value. The Monte Carlo experiment indicates standard uncertainties of approximately 4.2–4.4 cm for normal heights when a 1.5 cm ellipsoidal-height uncertainty and a 4.0 cm quasi-geoid uncertainty are assumed. A single-point covariance example gives 31.7 mm for normal-height conversion and 36.9 mm for orthometric-height conversion under the stated correlation assumptions. The transformation-surface comparison further shows that a quadratic local model reduces leave-one-out cross-validation error from 16.73 mm to 9.95 mm relative to a planar model in the synthetic Romania-oriented scenario. The study concludes that the height-system label must be treated as part of the mathematical model and metadata, and it proposes a geopotential-centered computational pathway for survey adjustment, uncertainty control, and metadata-safe geospatial export.