Understanding 3D analytic signal amplitude

Abstract The concept of the analytic signal goes back at least to Ville (1948). The analytic signal a(x) of function f(x) is a complex quantity defined as where H[f(x)] represents the Hilbert transform of f(x). Nabighian (1972, 1974) applies the analytic signal concept to potential-field data in two dimensions. For a potential field ϕ(x) measured along the x-axis at a constant observation height z and generated by a 2D source aligned parallel to the y-axis, the horizontal derivative ϕx and the vertical derivative ϕz are a Hilbert transform pair.