Geology 319: Magnetic and Gravity Field Transformations

1. Geol 319: Important Dates Monday, Oct 1st – problem set #3 due (today, by end of the day) Wednesday, Oct 3rd – last magnetics lecture Wednesday afternoon – Midterm review (4:30 pm, M 210) Friday, Oct 5th – Midterm exam Week of Oct 8th – 14th: No lectures (field trip week)

2. Magnetic potential: Magnetic field: • The key term, tells us to: • Project g in the direction M • Take the gradient Note that the component of the gradient in any direction is the rate of change in that direction, so for example the x component of is Poisson relationship: relating gravity and magnetics

3. For non-horizontal components it is less obvious. For example, for the total field anomaly we need (i.e., pointing in the direction of the B field) Poisson relationship: relating gravity and magnetics Horizontal components:

4. For the total field anomaly we need (i.e., pointing in the direction of the B field) • Imagine instead, we move the target a small amount in the B direction, and change the sign of the gravity field • the sum of the two fields is equal to the required derivative Poisson relationship: relating gravity and magnetics

5. Poisson relationship: relating gravity and magnetics The construction is equivalent to a new body, with positive and negative monopoles on the two surfaces.

6. Poisson relationship: relating gravity and magnetics Common applications: • Model the magnetic anomaly from the predicted gravity anomaly • Calculate the “pseudo-gravity” directly from the magnetic field data • Calculate the “pseudo-magnetic field” directly from the gravity field

7. Poisson relationship: examples

8. Poisson relationship: examples Bouguer gravity anomaly Total magnetic field anomaly Psuedo-magnetic field Psuedo-gravity field

9. Magnetic and gravity field transformations • Poisson relationship is an example of a data transformation • Many other gravity and magnetic field transformations are also widely used • The recorded data are used to predict something different about the field • Based on either principles of physics, or on principles of image processing, or both

10. Magnetic and gravity field transformations Reduction to the pole: This corrects for the asymmetry in magnetic field anomalies – peaks and troughs of total field anomaly are not directly above targets. Exception is at the North/South poles – reduction to the pole transforms observations to those that would have been recorded if the inducing field were vertical. • Use the Poisson relationship to transform into pseudo-gravity • Use the Poisson relationship again to transform to a new pseudo-magnetic field, with the magnetization vector pointing downward.

11. Magnetic and gravity field transformations • Upward / downward continuation (used for both gravity and magnetic field data): • Objective is to obtain data from a different elevation from that actually used. • Matching ground data to airborne data • Correcting field data for elevation changes • Removing local anomalies to enhance regional trends, or • Remove regional trends, enhance local anomalies • Upward and downward continuation rely on an understanding of Laplace’s equation:

12. B or g If there are no monopoles, this must be zero, thus 0 Since Then N.b: at a monopole this is not zero! Aside: Justification of Laplace’s equation: Examine the divergence of g (or B) in free space:

13. Magnetic and gravity field transformations Upward / downward continuation (used for both gravity and magnetic field data): • If you measure second derivatives in any two directions, Laplace’s equation tells you the second derivatives in the third direction. • This points to a computer algorithm for upward/downward continuation: • Start from a magnetic or gravity map i.e., f(x,y) • Measure the two x and y second derivatives everywhere on the map • From Laplace’s equation find the z second derivative, everywhere on the map • Using the second derivative, extrapolate the current map to a new elevation

14. Magnetic and gravity field transformations

15. Upward continuation example • (Crete) • Ground magnetic survey • Aeromagnetic survey (2000 m) • Ground survey after upward continuation

16. Upward continuation example Original gravity/magnetics After upward continuation After subtraction of upward continued field

17. Magnetic and gravity field transformations Other transformations Laplace’s equation can be used to create a full, 3-D (“cube”) of data (i.e., that would have been sampled at any optional elevation) This cube of data can be operated on with a wide variety of derivative operators, image processing algorithms, filters, etc.

18. Other transformations Vertical first derivatives of gravity / magnetics Total horizontal derivative of gravity / magnetics Gray scale vertical derivative of gravity, and pseudo-gravity after “lineament detection”

19. Next lecture: Rock magnetism • All rock magnetism is related to dipole moments at atomic scales • Contributions to magnetization arise from • Dipole moments of electron “spin” • Dipole moments of the electron orbital shells • If these dipole moments are organized, the macroscopic crystal will be magnetically susceptible. • There are several varieties of macroscopic magnetization: • Diamagnetism • Paramagnetism • Ferromagnetism • Anti-ferromagnetism • Ferrimagnetism