INTRODUCTION

As with all the other geophysical methods we need formulae for the effects of simple geometrical bodies. We use these to forward model hypothetical situations as part of survey planning. The formulae are also used as part of various computerized inversion methods.

When we derived gravity formulae we had to do a triple integral over all the anomalous mass (all the volume) of the body. In magnetics we have to do a line integral (that is, apply the Biot Savart law) over all the electrical currents flowing on the surface of the body.

After learning how to locate and evaluate these currents in terms of magnetization and background field, we’ll derive some formulae for vertical field anomalies caused by a vertical background field.

MAGNETIZATION AND BOUNDARY CURRENTS

Our basic strategy is to apply the Biot Savart law to the system of boundary currents flowing on the surface of the body. But how do we find the currents?

The basic concept is due to Ampere. He visualized a thin slice of the body taken normal to the external field as being filled with aligned tiny dipoles or current loops (this was long before people had learned about electron spin or even about electrons). Inside the slice any current would be matched by an equal and opposite current and thus having no magnetic effect outside the body. Only around the periphery of the slice would there be a net current to produce a magnetic field.

To find the magnitude of the current we simply calculate the dipole moment of the slice. By the very definition of magnetic moment this equals the area of the slice times the boundary current. By definition of magnetization, the dipole moment equals the magnetization times the area times the thickness of the slice. Equating these two expressions gives the desired result: the current equals the magnetization times the thickness.

In the case of induced magnetization we can replace the magnetization by the equivalent susceptibility. Thus the current equals the susceptibility times Bo times thickness all divided by the permeability of free space.

If there is remanent magnetization (known M) we take slices normal to the direction of the NRM. Some cases have both induced and remanent magnetization so we end up with two sets of slices and two boundary currents. We can calculate the effects of each separately and then just add them up vectorially.

Lets’ look at a variety of simple cases.

VERTICAL FIELD, VERTICAL COMPNENT ANOMALIES

For today’s class we assume that Bo is vertical. This is exactly true at dip poles and not a bad approximation as far south as Durham where the inclination is about 75 degrees. We’ll also assume that we’re using a vertical component magnetometer. These were the common field instrument for ground surveys up through the 1950’s and, at high inclinations, give results roughly the same as do proton-precession (total field) magnetometers. For simplicity we’ll also assume only induced magnetization.

THIN SEMI-INFINITE HORIZONTAL SHEET

First we’ll consider only a "thin" sheet with thickness less than the depth (actually the distance below magnetometer sensor). Here, and in all the other cases, we’ll use "susceptibility contrast" in the formulae. The contrast is simply the difference between the susceptibility of the body and the background material.

Referring to a previous section we see that the only boundary current flows along the straight free edge of the semi-infinite sheet. The other "edges" are infinitely far away so any current there has no effect at the free edge. Thus we are dealing with the magnetic field of an infinitely long straight current, a problem we have already solved (recall lecture on Biot Savart Law). The only extra work is to take the vertical component of the field.

The anomaly is confined to the vicinity of the edge and decays to zero far from the edge. This tells us that the anomaly over a (complete) infinite sheet is zero everywhere, even over the sheet! Note that the anomaly has a positive peak near the edge (and over the sheet) and a corresponding negative one off the sheet. The slope of the anomaly profile is maximum exactly over the edge. The slope is steep for a shallow sheet and gentle for a deep one. The average values of the anomaly for a whole profile is zero; this is a consequence of the dipole nature of the source. Note that we cannot break up the product of susceptibility and thickness.

In most today’s cases we assume a two-dimensional geometry with the body extending from minus infinity to plus infinity in y direction. This assumption is not necessary because we could go back to the original infinite line of current derivation and substitute finite limits of integration (2 1/2 D model). That effort would complicate all the algebra without adding much understanding. Modern computer modeling "packages" generally let you specify finite limits in the y direction.

THIN INFINITE HORIZONTAL STRIP

This situation is easy to deal with. Just as we did with gravity, we can overlap a positive semi-infinite sheet with a negative one to get a positive strip of finite width.

For a narrow strip we see a large positive peak over the body flanked by week but wide negative anomalies. For a wider sheet we see separate anomalies located over each edge (as in the previous case). Note that the anomaly over a very wide sheet is essentially zero except at the edges.

THICK SEMI-INFINITE HORIZONTAL SHEET

In this case we slice the body horizontally into thin sheets. Then , using calculus, we add up (integrate) the effects of all the sheets. We have to be careful to take the vertical components of the individual fields before integrating.

By letting the depth of the base of the sheet become very large we obtain a formula for a vertical contact.

HORIZONTAL RECTANGULAR PRISM

This is perhaps the most useful of all the formulae based on the field of an infinitely long, straight current. To derive it we apply our overlap trick to the previous case. The utility of the formula is that it includes all the previous ones as special cases. It can represent a sill, a thin lava flow, a thick pile of narrow flows, a thin vertical dike, a thick dike and so on. With a set of rectangular prisms we can build up a cross-section of any shape.

SPHERE

To derive a formula for a sphere we use the formulae for the radial and tangential components of a dipole. Then we take the vertical components of each and add them up. To get an expression for dipole moment of the sphere we multiply the magnetization by the volume of the sphere.

The anomaly shows a positive peak over the sphere and weak, broad flanking negatives at greater distances. On a map view the anomaly contours are circles. Note that we cannot break down the product of susceptibility contrast times radius cubed. We can use this formula for any more or less equant "blob".

THIN VERTICAL CYLINDER

In this case we assume that the radius of the cylinder is less than the depth to the top. With this assumption we can treat the cylinder as a stack of disks, treat each disk as a dipole and sum (integrate) the effects of each. As before we take the vertical components of both the radial and tangential field components.

If the top and bottom depths are more or less the same, the resulting anomaly resembles that of a sphere. If the bottom depth is very large the anomaly is just a circular positive peak with almost imperceptible negative flanks. In this case we can only determine susceptibility times radius squared, not both parameters separately.

In the next lecture we’ll look at total field anomalies.