Understanding 3D analytic signal amplitude

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 ϕ x and the vertical derivative ϕ z are a Hilbert transform pair.

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