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

Potential  Potential (gravity, magnetic) – scalar field U such that the vector field strength ( g ) represents its negative gradient: so that the work of g along any contour C connecting x 1 and x 2 is only determined by the end points:

Potential  Note that this means that g is curl-free (curl of a gradient is always zero):  If the divergence of g is also zero (no sources or sinks), we have the Laplace equation:

Conservative (irrotational, curl-free) fields  For a field with zero curl:, there always exists a potential:  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 x 0  x  x 0 equals zero)

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

Source  In the presence of a source (mass density  for gravity), the last two equations become:  The goal of potential-field methods is to determine the source (  ) 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 Total mass

Basic solutions: point source or sphere  Gravity of a point source or sphere:  Gravity within a hollow spherical cavity in a uniform space ?  Gravity within a uniform Earth : const needed to tie with U(r) above

Basic solutions: line (pipe, cylinder) source  Consider a uniform thin rod of linear mass density   Enclose a portion of this rod of length L in a closed cylinder of radius r  The flux of gravity through the cylinder:  Therefore, the gravity at distance r from a line source: Note that it decreases as 1/r This was copied from Lecture #9

Basic solutions: Gravity above a thin sheet  Consider a uniform thin sheet of surface mass density   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:  Therefore, the gravity above a thin sheet is:

Basic solutions: centrifugal force  Centrifugal force:  Field strength:  Potential:  is the colatitude. The force is directed away from the axis of rotation 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, w 1/2 :  For a spherical anomaly:,  For a cylindrical anomaly:, This is also discussed in Lecture #12

Variance  The “variance” (denoted  2 ) 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: where the mean is defined by: (Also note that: )

Standard deviation  We always have a finite number of measurements, and so need to estimate and  2 from them  For N measurements, these estimates are: Arithmetic mean, or “sample mean” s N-1 is the “standard deviation”, is called “sample variance”  Thus, the expected mean absolute error from N measurements is the standard deviation:

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 This is used in lecture and lab notes