INTRODUCTION

In this class and the next we will look at the gravitational effect, "signature" or "anomaly" over some simple geometrical shapes. In deriving these formulas we’ll assume a flat earth, a uniform density background in which the bodies are embedded and no isostatic effects. Given these conditions the formulae can be used to forward model or to invert simple Bouguer anomalies.

We seek formulae for the vertical component of the Newtonian gravitational attraction at ground level caused by a geometrical shape ("body") of uniform density embedded in a uniform density infinite half-space. In general we break the body into tiny parts, use Newton’s law to find the attraction of each and employ simple trigonometry to get the vertical components. Finally we add up ("integrate") the effects of all the individual pieces. In general we must use a triple integral but for some simple shapes we can get away with single or double integrals or, in case of a sphere, no integration at all

GRAVITATIONAL EFFECT OF A SPHERE

We can use a sphere as an approximation for many geological bodies such as magma chambers, some plutons, caverns, salt domes, massive sulfide ore bodies and any more or less "blob-shaped" masses or cavities.

It can be shown that the gravitational effect of a sphere of uniform density is exactly the same as the effect of a mathematical "mass point" of the same mass. The same rule is true for a sphere made of shells each of uniform density. Thus we can apply Newton’s Law directly with no integration at all. In this formula, as in all the others, we’ll use "density contrast" (difference between density of the body and that of the uniform half-space). Thus the formula expresses the anomalous effect of the body compared to a uniform medium.

The anomaly is circular in map view with maximum value directly over the sphere. The maximum slope is located where x=z/2.

Examination of the formula reveals that we have the very same anomaly as long as the product of density times radius^3 is the same and all other parameters are unchanged. In other words there’s an equivalence problem similar to that in resistivity sounding. Of course there is a geologic limit to density (uranium?) and a geometrical limit to the radius (must be less than depth of center of sphere). Only the mass of the sphere and its location matter.

Distributed lens-shaped bodies at shallower depths can have anomalies identical to that of a deeper sphere. Working with data we can determine the anomalous mass uniquely (Gauss’s Law) but not the density distribution.

We can use a similar mass-point formula for the effect of a cube as long as its dimensions are small compared to the depth of burial. By breaking it up into tiny cubes we can approximate the gravitational effect of any three-dimensional body no matter how complex its shape. In practice we would use a computer to carry out the calculations.

INFINITE HORIZONTAL CYLINDER

This shape can be used to approximate the gravitational effect of long, narrow bodies such as caverns, tunnels, anticlines, valley fills and so on as observed on a profile cutting straight across the long axis of the body.

In this case we slice the cylinder into coin-shaped disks. We then treat each disk as a point mass and sum the effects of all. This means we do one integration with location along the cylinder as the independent variable. To get a simple formula we take the integral from minus infinity to plus infinity.

Although this procedure seems a little bogus it turns out that our formula is exact. As long as our profile cuts across the center of the body and the length of the body is more than ten times the width, then the infinite cylinder formula is a good approximation. We can, of course, substitute finite limits for the integration to get a better approximation in any specific situation (2 1/2 D model).

As with the sphere there are nonuniqueness problems. In particular we can only determine the mass per unit length uniquely, not the radius and density separately. In addition there is an infinite number of shallower mass distributions that give the same anomaly.

As long as the sides are very small compared to the depth, we can derive a similar formula for an infinite, horizontal square prism. These can then be used to represent a long, horizontal two-dimensional body of any shape.

THIN VERTICAL CYLINDER

This shape could represent a volcanic neck, a salt dome or any other narrow vertical body.

By slicing the body into horizontal disks, treating them as point masses and integrating over the vertical coordinate we derive a simple formula for the gravitational effect. Unlike our formula for the horizontal cylinder, this formula is only an approximation. Thus it should be used only if the radius is small compared to the burial depth of the top of the cylinder.

Note that we cannot separate the radius from the density in this case also.

As long as we are far from the body we can use a similar formula for a vertical square prism. Some programs for terrain corrections in the complete Bouguer anomaly use these prisms to build up a representation on topography around a gravity station.

USE OF SIMPLE GEOMETRICAL BODY FORMULAE

We can use these formulae to approximate the effects of geological bodies. The formulas are useful for quick forward modeling prior to running a survey. They can indicate how precise our measurements must be to detect and describe bodies of interest. They can be used to tell us what station interval to use. They can be used as building blocks in computer programs to give the effect of arbitrarily complex shapes.

We can also use these formulae to invert field data using least-squares parameter inversion, genetic algorithms, simulated annealing and all the other methods discussed with respect to resistivity soundings.