Fourier domain vertical derivative of the nonpotential squared analytical signal of dike and step magnetic anomalies: A case of serendipity

Abstract

Vertical derivatives of nonpotential 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 pseudovertical derivative, providing a mathematical meaning for it. One significant difference between the pseudo and true vertical derivative is that the former possesses real roots, whereas the latter does not. Taking advantage of this attribute, we find using synthetic and field data that the pseudovertical 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 qualitative interpretation, we convert the image of the pseudoderivative to a binary image where the anomalies are treated as discrete objects. This allows us to morphologically enhance, disconnect, classify, and filter them using the 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 are also corroborated by the observation of outcrops found by field surveys.