Cross-regional evaluation of thermodynamically consistent fractional-order integration for observation-constrained ocean heat content estimation

Abstract Ocean heat content (OHC) is sensitive to errors in the vertical representation of temperature, particularly in regions characterized by strong thermohaline gradients. This study evaluates an empirical fractional-order integral procedure in six scenarios encompassing the tropical Atlantic, Indian, equatorial Pacific, and South Atlantic oceans. The scenarios compare observation-derived OHC from PIRATA, RAMA, and TAO/TRITON profiles with estimates from ArpHDv2, SODA, EN4, GODAS, and the ORAS5 ensemble using depth-dependent fractional orders selected according to scenario-specific error metrics. In Scenario 1, the mean relative error decreased by between 57.4% and 69.97%, depending on the season, relative to the classical method. Across Scenarios 2 through 5, the selected α values remained close to unity and varied with depth, dataset, regional setting, and vertical sampling configuration. At depths where α different from 1 was selected, it produced a lower absolute value of the corresponding scenario-specific error metric than the classical order α = 1. Under the recommended operational acceptance interval $$\:0.985\le\:{\upalpha\:}<1.015$$ , Scenario 6 defined an applicability boundary: $$\:{\upalpha\:}=1$$ was selected between 5 and 25 m, whereas the minimum-error orders at greater depths were $$\:{\upalpha\:}\ge\:1.015$$ and were therefore classified as diagnostic rather than admissible. Values obtained with α different from 1 represent locally adjusted OHC estimates rather than thermodynamically identical forms of conventional OHC. The results support conditional application after local estimation of α and demonstrate the need for explicit acceptance criteria that can reject unsuitable profile adjustments.