Differential Reduction to Magnetic Pole (DRTP)

Use the Differential Reduction to Magnetic Pole option to apply a differential reduction to magnetic pole filter.

Differential Reduction to Magnetic Pole dialog options

Input grid This entry will only appear if an input grid has not already been supplied through the previous dialog. The input grid must carry projection information. Given the projection and the date of the survey, the inclination and declination grids coinciding with the input data will be calculated.
Date (YYYY/MM/DD) Enter the date when the initial survey was conducted so that the corresponding inclination and declination grids are calculated. This is an optional field when the inclination and declination grids exist (see Application Notes).

Amplitude correction inclination

The Amplitude correction inclination Iais a factor that prevents the discontinuity in low magnetic latitudes. The default is 20.0º in the Northern Hemisphere and –20.0º in the Southern Hemisphere (See Application Notes).

Application Notes

Given a georeferenced magnetic input grid and the date of the survey, the Inclination and Declination grids coinciding with the input grid are calculated and used by the DRTP filter. These grids are respectively named _Inc_inp.grd & _Dec_inp.grd, where inp is the name of the input grid. These grids are saved in the working directory. As a result in subsequent invocations of the DRTP filter, if they are present, recalculating them is bypassed. To force the calculation of the Inclination and Declination grids, delete _Inc_inp.grd & _Dec_inp.grd, from your directory and enter the survey date when defining the filter.

If the grid does not have a coordinate system assigned, you will be prompted to define it. The IGRF date and parameters, once defined, are saved in the grid.

If the input grid approaches or crosses the magnetic equator (inclination=0), pay special attention to 3 factors:

  1. To attenuate the potential instability, manifested in the output grid as strong regional highs, increase the Amplitude correction inclination. This parameter helps control the discontinuity at the expense of the amplitude. On output, the grid is well-behaved but the amplitudes of the anomalies at inclinations less than this angle will be attenuated.
  2. Strong anomalies along the edge of the grid, especially near the magnetic equator would be amplified. If you notice such artifacts, perform the grid preparation step separately and ensure that edge anomalies are rapidly attenuated in the grid filled area.
  3. The default amplitude correction of 20.0º may not seem adequate. In order to yield a well behaved output grid you may have to increase the amplitude correction angle as much as 45.0 º. The well behaved nature of the output signal is obtained at the expense of the signal magnitude being attenuated. Such output processed grids should not be subjected to numeric modeling as the outcome will be unreliable.

Formulation

The Reduction to Magnetic Pole in the Fourier domain assumes a constant inclination and declination and is denoted by the equation below:

Where:

Tp is the pole reduced field of the total magnetic anomaly T

F{} designates Fourier Transform

ω is the normalized vector (u2+v2)½ in the wavenumber domain

G = (iu, iv, -ω)

B is the unit vector B along the core geomagnetic field

M is the unit vector in the direction of the magnetization

The reduction to the pole operator makes 2 assumptions:

  1. M=B

  2. The geomagnetic field is a constant across the grid

The Differential Reduction to Magnetic Pole assumes that B and by extension M vary across the survey area. We can thus formulate that the geomagnetic field B can be the sum of a mean value B0 at the centre of the grid and a deviation from the mean δB.

B = B0 + δB

M = M0 + δM

δB & δM are a function of space.

The equation then can be expanded as (see references):

The first term is equivalent to the pole reduction for a constant inclination and declination. The two additional terms account for the perturbation from the mean.

Where:

M0 is set to B0

U is the gradient of the magnetic potential

m0 is vertically integrated magnetization

B0 has components Bx, By, Bz where:

Bx = Sin D Cos I

By = Cos D Cos I

Bz = Sin I

I is the Inclination of the geomagnetic field

D is the Declination of the geomagnetic field

The above equation is solved iteratively in the wavenumber domain using a perturbation method (see Arkani-Hamed 1988 and 2007).

At low magnetic latitudes, in order to stabilize the pole reduction component of the equation (see RTP operator), the B0 & M0 components in the equation denominator are replaced with B'0, M'0 corresponding to the field calculated using an amplitude control inclination Ia > I:

 

k

 Wavenumber domain increment, used to depict a radially symmetrical variable.

where:

np is the number of points

cs is the cell size

u

X component in the wavenumber domain. k = 2π ( i μ+j ν )

v

Y component in the wavenumber domain.

 

r

Radial component in the wavenumber domain.

also 2πk

θ

 Polar component in the wavenumber domain.

See Also:

References

  • [1] Arkani-Hamed L, "Differential reduction to the pole or regional magnetic anomalies", Geophysics, vol. 53, no. 12 (1988), pp. 1529, 1600.
  • [2] Arkani-Hamed L, "Differential reduction to the pole: Revisited", Geophysics, vol. 72, no. 1 (2007), pp. 13, 20.
  • [3]Swain C, "Reduction to the pole of regional magnetic data with variable field direction and its stabilisation at the low inclinations", Exploration Geophysics, vol. 31(2000), pp. 78-83.