A robust method for calculating the vertical derivative of potential fields based on Hilbert transforms

Calculation of high-order vertical derivatives represents a fundamental challenge in gravity and magnetic data processing, with critical applications in potential field separation, continuation analysis, and geological boundary identification. Conventional methods for obtaining these derivatives often face stability limitations, particularly when computing higher-order derivatives. The Integrated Second Vertical Derivative (ISVD) algorithm emerged as an innovative solution by synergizing spatial and frequency domain approaches, demonstrating improved computational stability for derivative calculations. Building upon these theoretical foundations, this study introduces a novel methodology termed the Recurrence Formula of Vertical Derivative (RFVD). Our approach leverages the Hilbert transform properties in the frequency domain combined with finite difference approximations for horizontal derivatives in the spatial domain, establishing a recursive framework for vertical derivative computation. The derivation begins with the fundamental frequency-domain relationships governing potential field transformations, systematically developing a recursive operator that enables sequential calculation of arbitrary-order vertical derivatives. Testing on synthetic examples and real data demonstrated the RFVD method's superior accuracy and noise sensitivity for 1st- and 2nd-order derivatives. For higher-order derivatives (3rd and 4th), the method shows promise in noise-free conditions but requires cautious application in noisy environments.