Metrological characterization and measurement model for ELF air-to-undersea fields via compact analytical VED formulas

Accurate measurement of Extremely Low Frequency (ELF) electromagnetic (EM) fields in seawater is essential for oceanographic sensing, geophysical monitoring, and the calibration of undersea EM sensors. However, the lack of real-time, traceable measurement models limits the precision of in-situ sensor calibration, as exact Sommerfeld integrals (SIs) evaluations are computationally prohibitive for embedded systems. In this paper, we derive a compact analytical measurement model for the ELF field components in seawater generated by a vertical electric dipole (VED) reference source. The model is constructed via asymptotic decomposition and trigonometric-series expansion of SIs, explicitly linking instrument readings to the EM field intensity without restriction on observation depth. A comprehensive uncertainty framework is established to quantify the model’s systematic bias and sensitivity to environmental parameters. Comparisons with numerical integration standards confirm that the proposed measurement model achieves high accuracy with negligible computational cost. The resulting formulas provide a theoretical basis for metrological characterization in underwater sensing networks, enabling real-time uncertainty evaluation and dynamic sensor correction. • Compact analytical measurement model for ELF air–to–undersea fields. • Explicit observation equation links voltage to the measurand. • GUM-compliant framework quantifies measurement uncertainty. • Enables real-time in-situ calibration for civilian oceanographic sensors.