On the calculation of vertical pseudoderivative from nonharmonic functions in gravimetry and magnetometry

The vertical differentiation of a nonharmonic quantity derived from a potential field - such as the horizontal gradient, analytic signal modulus, or tilt angle - using conventional wavenumber-domain filters designed for harmonic functions does not yield the true vertical derivative (VD) of the original quantity. Instead, it produces a pseudoderivative. However, this pseudoderivative shows characteristics similar to those of the true VD, so distinguishing between the two can be challenging in practice. Pseudoderivatives are commonly used in qualitative interpretation because they perform well in several tasks (e.g., edge analysis). This good performance stems from the fact that pseudoderivatives correspond to the Hilbert transform of the horizontal derivative of the original function. Nevertheless, analytical expressions exist, enabling the computation of true VDs for nonharmonic functions. Our analysis demonstrates that true VDs provide superior delineation of structural boundaries, particularly in geologically complex settings with interfering anomalies. Based on these results, we recommend the use of true VDs when analyzing transformed nonharmonic fields.