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Can we estimate total magnetization directions from aeromagnetic data using Helbig’s formulas?

Can we estimate total magnetization directions from aeromagnetic data using Helbig’s formulas?. Jeffrey D. Phillips U.S. Geological Survey. IUGG, Sapporo, Japan, July 2003. Why estimate total magnetization directions?. Useful for modeling the sources of magnetic anomalies

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Can we estimate total magnetization directions from aeromagnetic data using Helbig’s formulas?

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  1. Can we estimate total magnetization directions from aeromagnetic data using Helbig’s formulas? Jeffrey D. Phillips U.S. Geological Survey IUGG, Sapporo, Japan, July 2003

  2. Why estimate total magnetization directions? • Useful for modeling the sources of magnetic anomalies • Useful for filtering operations such as reduction-to-the-pole • May indicate relative geologic ages or tectonic histories of the sources

  3. Previous Work • Helbig(Zeitschrift für Geophysik, 1963) - developed the integral equations for estimating the magnetization vector from magnetic component data. • Schmidt and Clark(Preview, 1997; Exploration Geophysics, 1998) – advocated applying the method to modern total field data using Fourier method component transformations, and gave some Australian examples.

  4. Theory – Helbig's Integral Equations(Zeitschrift für Geophysik, 1963) 1.Given magnetic field components X, Y, Z, the following infinite integrals should vanish:

  5. Theory – Helbig's Integral Equations 2. The following infinite integrals should yield the average magnetization components Mx, My, Mz:

  6. Theory – Helbig's Integral Equations It follows that the average magnetic moment |M|, inclination iM, and declination dM of the total magnetization are given by:

  7. Implementation 1. Gridded total-field aeromagnetic data are converted to magnetic component grids using standard Fourier filtering techniques. Total Field ΔX ΔY ΔZ

  8. Implementation 2. The integrals (I6 through I9) are evaluated from the component grids in sliding n x n windows using a 2-D quadrature algorithm based on the trapezoidal rule:

  9. 3. In order to approximately satisfy Helbig's vanishing integrals (I1 through I5), planar surfaces must be removed fromX, Y, and Z within each small data window before integration. Implementation

  10. Implementation 4. The results are grids of magnetic moment, inclination, and declination for each of the chosen window sizes. 5. Two choices: a. Try to estimate the magnetization directly by comparing results from different window sizes – the "closeness" method. b. Seek solutions with a specified magnetization direction – the "alternate" method.

  11. The basic "closeness" method • Evaluate Helbig's integrals using two different window sizes. • Find the grid nodes where the angular difference between the two solutions is small. • These are the best solutions.

  12. Four Dipole Example – 13 x 13 moment inclination declination

  13. Four Dipole Example – 19 x 19 moment inclination declination

  14. Four Dipole Example Angular Difference Reciprocal Angular Difference

  15. Four Dipole Example Smallest Angular Differences Largest Moments Correct Answer

  16. The extended "closeness" method • Generate solutions for many different pairs of window sizes using the basic closeness method. • Look for clustering in the results. • A cluster indicates repeated solutions in many different pairs of window sizes.

  17. The extended "closeness" method o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 3 5 7 9 11 13 15 17 19 21 23 25 12 window sizes = 66 basic solutions

  18. Four Dipole Example # solutions averaged results

  19. Black Hills Norite – South Australia Aeromagnetic Map Helbig Solution

  20. Black Hills Norite - Comparison

  21. Elongated prisms – extended closeness method results

  22. Pipeline over fault – closeness results Total Field Result

  23. The closeness method • Allows the total magnetization to be estimated from Helbig's integrals without any a priori knowledge. • But, it requires semi-isolated, compact source bodies without significant horizontal elongation (e.g. dipoles, spheres, vertical cylinders).

  24. The alternate method • Where the closeness method fails, the Helbig integral results can still be used to search for sources having a specified total magnetization direction, such as: • The inducing field direction • A measured total magnetization direction • An assumed direction based on geologic age

  25. Elongated prisms - solutions parallel to the inducing field

  26. Black Hills Norite – solutions parallel to the measured total magnetization

  27. Albuquerque basin - volcanic center Aeromagnetic Map Red=Reversed Blue=Normal

  28. Conclusions • The Helbig integral equations can be evaluated in small data windows on magnetic component grids derived from total field data. • The results can always be used to search for magnetic sources having a specified total magnetization direction. • For the special case of isolated, compact sources, the closeness method can be used to locate the sources and estimate their total magnetization directions directly.

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