GG 652: Homework 7:
Geomagnetic Field and Magnetic Anomalies
Reading: Blakeley Ch.8, Due: Friday1March
1) Do one of the following
1a) After Ex. 8.1. Use Table 8.2 to describe the centered dipole of the Earth in 1965. That is, determine the components and magnitude of m of the dipole. By how many degrees did the dipole rotate between 1965 and 1990?
1b) Problem 3 on p. 181
2) Prove that the spectral power Rn(with r≠ a) is proportional to (a/r)2n+4. Then derive the equation on p. 176 and explain why this equation can be used to estimate the radial distance to the sources of the Earth’s field from the trends shown in Fig. 86. By the way, the 1st sentence of p. 176 is better written “Then to transform Rn based at the surface of the earth into a spectrum that would be determined at some new radius r, we would simply have to multiply Rnat the surface by the factor (a/r)2n+4, or add (2n+4)log(a/r) to the log of Rnat the surface.”Estimate the distance to the source of the field for n<14 and n>14.
3) Building upon your work in HW4, this problem asks you to compute the total field anomaly (T, see p. 178-179) that would be measured over a buried, uniformlymagnetized sphere. Use the Matlab script “dipole.m” as I have set up for HW4 (see course website). The sphere has a radius of a = 2 km and is at a depthd = 3 kmbelow the surface. The sphere’s magnetization M is induced by the Earth’s regional field F and thus M is parallel to F. The magnitude of magnetization is |M| = 30 A/m. Approximate the Earth’s regional field as a geocentric dipole in which magnetic north pole coincides with geographic north pole, so the direction of F (and M) will only depend on latitude
Plot the northward and downward component of the anomalous field (i.e., Fx and Fz) as well as T along a survey profile going south-to-north and passing directly over the center of the sphere in the following situations:
(a) You are at the magnetic equator ( = 0° N) and again the sphere is magnetized by the Earth’s regional field at the equator. Sketch as described below (**) and explain.
(b) You are at the north magnetic pole ( = 90° N). Sketch as described below (**) and explain.
(c) You are at latitude = 30N. Sketch as described below (**) and explain.
**For each case above, make a sketch of a vertical cross-section along your survey profile that shows the sphere, arrows of the magnetic field along the surface of the earth, and a vector denoting the direction of F. Use this sketch to explain why your plots make sense.