Reduction to Pole

Introduced by Baranov (1957) (see also, Baranov and Naudy, 1964), the reduction‐to‐pole transformation of total field magnetic anomalies (see crustal magnetic field) is intended to remove the skewness of the anomalies (see Figure R1). The transformation makes the anomalies overlie the sources, makes it possible to correlate the magnetic anomalies with other types of geophysical anomalies (e.g., gravity) and geological information, and aids their interpretation. In reality, even the amplitude of the anomaly is affected (increased) when sources of induced magnetization are observed at poles in comparison to lower magnetic latitudes because the Earth's field intensity increases from equator to poles; some of the reduction‐to‐pole methods can take this change in amplitude into account (e.g., equivalent source method) while the others typically do not (e.g., rectangular coordinate wavenumber domain methods). The expression of a magnetic anomaly, ΔT, due to a localized spherical source of uniform magnetization is helpful in understanding the transformation

where r is source to observation distance, ΔV _α is anomalous potential due to the uniform anomalous magnetization direction α, ΔJ is the intensity of anomalous magnetization (q.v.), and β is the direction of the Earth's main field (assumed uniform). To derive the anomalous source function (ΔJ/r), one integrates the equation twice, once with respect to β (to find the anomalous potential), and once with respect to α. The magnetization direction of the source is usually not known and, therefore, induced magnetization is assumed, leading to α = β. The reduced‐to‐pole magnetic anomaly (vertical intensity anomaly due to vertical magnetization) can then be computed by twice differentiating this source function in the vertical direction (to find first the potential due to the vertically magnetized source, and then its anomaly in the vertical direction) as

where both ΔZ and ΔT _z represent the vertical intensity magnetic anomaly.

Figure R1

Skewness of a magnetic anomaly due to a uniform arbitrarily magnetized source below Earth's surface in an obliquely oriented Earth's magnetic field (left) and its reduced‐to‐pole expression in the vertical magnetization and vertical field conditions (right).

In practice, these computations are significantly easier to perform in the wavenumber domain, where the process of integration involves division by a factor and differentiation involves multiplication by a factor. Under induced magnetization conditions (i.e., α = β), the reduced‐to‐pole anomaly in the wavenumber domain is given by

where asterisks denote wavenumber domain representation of the respective anomalies, k is the radial wavenumber; in Cartesian coordinates, , where k _x and k _y are the wavenumbers in x and y directions; and , where I and D are the main field inclination and declination, respectively, and the trigonometric quantities represent direction cosines in the north, east, and down directions, respectively. An iterative wavenumber domain method for variable directions of magnetization and Earth's field intensity appropriate for a large region is described by Arkani‐Hamed (1988).

Another customary approach of achieving reduction to pole is through the equivalent source technique (Dampney, 1969; Emilia, 1973; von Frese et al., 1981, 1988; Silva, 1986), where a configuration of equivalent sources is first assumed. Magnetization is generally assumed in the direction of the inducing field, but it can also be different if known for particular sources. Using inverse methods, and taking advantage of Green's principle of the equivalent layer (see Blakely, 1995), one can map magnetization variation for the region where anomalies are available. Using the derived magnetization distribution, it is possible to compute the reduced‐to‐pole anomaly under vertical magnetization and vertical Earth's field conditions.

The equivalent source method is subject to instabilities due to a variety of reasons (e.g., spacing of sources, altitude difference between observations and the sources, low magnetic inclinations, etc.), but most of the instabilities can be reduced or eliminated using damped least‐squares or ridge regression approach (Silva, 1986; von Frese et al., 1988). The wavenumber domain reduction‐to‐pole operations also encounter instabilities in low magnetic latitudes (< 30° inclination, i.e., when the terms involving vertical component of direction cosines are close to zero). Under these circumstances, when the other two direction cosines nearly negate one another, which happens along a line in a k _x –k _y plane due to the shape of the reduction‐to‐pole filter (see, e.g., Blakely, 1995), the quantity B ² is nearly zero and small errors in the anomaly field are significantly enlarged in the reduction‐to‐pole process. Hansen and Pawlowski (1989) describe methods to overcome these artifacts by designing a Wiener filter for this purpose.

Cross‐references

Crustal Magnetic Field Magnetic Anomalies