Outline Uses of Gravity and Magnetic exploration Concept of Potential Field

Conservative Curl-free (irrotational)

Key equations and theorems Laplace and Poisson’s equations Gauss’ theorem

Basic solutions Point source and sphere Solid sphere Line source Inertial (centrifugal) force and potential Characteristic widths of anomalies

Gravity and magnetic modeling Ranges and errors of values Error analysis: Variance and Standard deviation

Why doing Gravity and Magnetic prospecting? Specific physical properties

Density, total mass, magnetization, shape (important for mining)

Basin shape (for oil/gas) Signal at a hierarchical range of spatial scales

(called Integrated in the notes) At a single point, the effects of large (regional) as

well as small (local) structures recorded Thus, “zero-frequency” is present in the data (unlike

in seismology) This allows reconnaissance of large areas by using

wide station spacing Inexpensive (practical with 1-3 person crews)

Why doing not only Gravity and Magnetic prospecting? Much poorer spatial resolution compared to

seismology Resolution quickly decreases with depth

The horizontal size of an anomaly is approx. This is about the distance at which the anomalies

can be separated laterally Uncertainty of depth estimates

For example, we will see that the source of any gravity anomaly can in principle be located right at the observation surface

1/2 0.65w h

Potential Potential (gravity, magnetic) – scalar field U such that the

vector field strength (g) represents its negative gradient:

UU

g

x

r

so that the work of g along any contour C connecting x1 and x2 is only determined by the end points:

1 2

C

d U U g x x x

Potential Note that this means that g is curl-free (curl of a

gradient is always zero):

0 curl g g

If the divergence of g is also zero (no sources or sinks), we have the Laplace equation:

div 0 g g

2 0U

Conservative (irrotational, curl-free) fields For a field with zero curl: , there

always exists a potential: 0 g

0

C

U U d x x g x

Such fields are called conservative (conserving the energy)

By Stokes’ theorem, the contour integral between x0 and x does not depend on the shape of the contour (integral over the loop x0 x x0 equals zero)

Conservative fields

Thus, a field with can always be presented as a gradient of a scalar potential:

UU

g

x

r

0 g

Source

In the presence of a source (mass density r for gravity), the last two equations become:

div 4 G g g

2 4U G

The goal of potential-field methods is to determine the source (r) by using readings of g at different directions at a distance

Poisson’s equation

Gauss’ theorem From the divergence theorem, the flux of g through

a closed surface equals the volume integral of divg Therefore (Gauss’s theorem for gravity):

Total outward flux of g

closed surface volume

4 4ds G dV GM g nÒ

Total mass

Basic solutions: point source or sphere

Gravity of a point source or sphere:

2 3ˆ

M MG Gr r

g r r MU G

r

Gravity within a hollow spherical cavity in a uniform space ?

Gravity within a uniform Earth :

232

MU G r const

R

3 3ˆ

Mr MG GR R

g r r

r R

const needed to

tie with U(r) above

Basic solutions: line (pipe, cylinder) source Consider a uniform thin rod of linear mass density g Enclose a portion of this rod of length L in a closed

cylinder of radius r The flux of gravity through the cylinder:

2 4g rL G L Therefore, the gravity at distance r from a line source:

2

2ˆ 2G

Gr r

r

g r Note that it decreases as 1/r

This was copied from Lecture #9

This was copied from Lecture #9

Basic solutions: Gravity above a thin sheet Consider a uniform thin sheet of surface mass density s Enclose a portion of the thin sheet of area A in a closed

surface From the equations for divergence of the gravity field:

The total flux through the surface equals:

By symmetry, the fluxes through the lower and upper surfaces are equal. Each of them also equals:

div 4 G g

4 G A

gA Therefore, the gravity above a thin sheet is:

2g G

Basic solutions: centrifugal force

Centrifugal force: Field strength:

Potential:

2 sincg r q is the colatitude.

The force is directed away from the axis of rotation

221sin

2cU r const The potential is similarly

cylindrically-symmetric and decreases away from the axis of rotation

Widths of anomalies

The depths h to the sources of anomalies are often estimated from the widths of the anomalies at half-peak magnitude, w1/2:

For a spherical anomaly: ,

For a cylindrical anomaly: ,

1 2 0.65

hw

1 20.65h w

1 2 2w h1 20.5h w

This is also discussed in Lecture #12

Variance The “variance” (denoted s2) is the squared mean

statistical error If we have an infinite number of measurements of g,

each occurring with “probability density” p(g), then the variance is the mean squared deviation from the mean:

2 222 g g g g g p g dg

g gp g dgwhere the mean is defined by:

1p g dg (Also note that: )

Standard deviation We always have a finite number of measurements,

and so need to estimate <g> and s2 from them For N measurements, these estimates are:

22 21

1

1

1

N

N i Ni

s g gN

1

1 N

N ii

g g gN

Arithmetic mean,

or “sample mean”

sN-1 is the “standard deviation”,

is called “sample variance” Thus, the expected mean absolute error from N

measurements is the standard deviation:

2

11

1

1

N

N i Ni

s g gN

21Ns

Another estimate of scatter in the data Sometimes you would estimate the scatter in the data by

averaging the squared differences of consecutive observations:

Note that N-1 here is the number of repeated measurements

This is an approximate standard deviation of the drift This formulas is OK to use with drift-corrected data

1 2

11

1 1

N

j jj

N

g g

sN

%

This is used in lecture and lab notes