Unlike the 2D case, the total gradient (magnitude of the gradient vector) in 3D is not the envelope of either the vertical or horizontal derivatives of the total-field anomaly over all possible directions of the earth''s field and source magnetization. Instead, the total gradient of the reduced-to-pole data defines the envelope for the vertical derivative while the total gradient of the reduced-to-equator data defines the envelope for the horizontal derivative.
Nabighian (1972) showed that the horizontal and vertical derivatives of the magnetic anomaly produced by a 2D source form a Hilbert transform pair and define an analytic signal.
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2003. Society of Exploration Geophysicists