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. An important property of the 2D analytic
signal is that its amplitude is the envelope of
its underlying signal (Kanasewich, 1981) - the horizontal
or vertical derivative in the 2D magnetic problem.
It follows that the magnitude of the gradient of
magnetic data (henceforth referred to as the total
gradient) is equal to the envelope of both the horizontal
and vertical derivatives over all possible inclinations
(this is for 2D only, not 3D!). For processing magnetic
data, the amplitude of the analytic signal in 2D
is remarkable in that it allows one to obtain a
signal that is independent of the source magnetization
direction.
Attempts
have been made to generalize the analytic signal
to 3D (Nabighian, 1984; Craig, 1996). These studies
have employed the concept of the complex field but
have not defined a 3D analytic signal amplitude.
In spite of this, the applied geophysical community
has adopted a simple extension from 2D to 3D following
Nabighian's (1984) generalized Hilbert transform
and its connection to the 2D counterpart. It states
that the total gradient in 3D is also the envelope
of the derivatives of the magnetic anomaly over
all inclinations and declinations (Roest et al.,
1992). This extension has been accepted without
theoretical proof or numerical verification.
We
now understand that such an extension is incorrect.
The total gradient is only the envelope of the vertical
derivative if, and only if, the field has
been reduced to the pole. If the field has not been
reduced to the pole, then the total gradient is
nothing more than the total gradient. It is not
the 3D analytic signal!
