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)

Poisson relationship: relating gravity and magnetics

Magnetic potential:

Magnetic field:

The key term, tells us to:

1. Project g in the direction M

2. 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

Horizontal components:

Poisson relationship: relating gravity and magnetics

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

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

The construction is equivalent to a new body, with positive and negative monopoles on the two surfaces.

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

Poisson relationship: examples

Poisson relationship: examples

Bouguer gravity anomaly Total magnetic field anomaly

Psuedo-magnetic field Psuedo-gravity field

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

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.

1. Use the Poisson relationship to transform into pseudo-gravity

2. Use the Poisson relationship again to transform to a new pseudo-magnetic field, with the magnetization vector pointing downward.

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:

Aside: Justification of Laplace’s equation:

Examine the divergence of g (or B) in free space:

If there are no monopoles, this must be zero, thus

0

Since

Then

B or g

N.b: at a monopole this is not zero!

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:

1. Start from a magnetic or gravity map i.e., f(x,y)

2. Measure the two x and y second derivatives everywhere on the map

3. From Laplace’s equation find the z second derivative, everywhere on the map

4. Using the second derivative, extrapolate the current map to a new elevation

Magnetic and gravity field transformations

Upward continuation example

(Crete)

a) Ground magnetic survey

b) Aeromagnetic survey (2000 m)

c) Ground survey after upward

continuation

Upward continuation example

Original gravity/magnetics

After upward continuation

After subtraction of upward continued field

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.

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”

Next lecture: Rock magnetism

All rock magnetism is related to dipole moments at atomic scales

Contributions to magnetization arise from

1. Dipole moments of electron “spin”

2. 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