Fourier domain vertical derivative of the non-potential squared analytical signal of dike and step magnetic anomalies: a case of serendipity

Vertical derivatives of non-potential fields are, intentionally or not, often performed in the Fourier domain producing nonphysical but interpretable results. Using the dike model, we prove that the vertical derivative of the squared Analytic Signal Amplitude calculated in the Fourier domain does not correspond to the true one. We derive an analytical expression for this pseudo-vertical derivative, providing a mathematical meaning for it. One significant difference between the pseudo and true vertical derivative is that the former possesses real roots, while the latter does not. Taking advantage of this attribute, we show using synthetic and field data that the pseudo-vertical derivative can be used for qualitative and quantitative interpretation of magnetic data, despite being nonphysical. As an example of the usefulness of this filter in the qualitative interpretation we convert the image of the pseudo-derivative to a binary image where the anomalies are treated as discrete objects. This allows us to morphologically enhance, disconnect, classify and filter them using tools of shape analysis and mathematical morphology. We also illustrate its usefulness in quantitative interpretation by deriving a formula for estimating the depths of magnetic thin dikes and infinite steps. Our outcomes were also corroborated by outcrops observation found by field surveys.