Outline  Gravity above a thin sheet  Equivalent stratum for buried sources (Green’s equivalent layer)  For gravity  For magnetic field  The uniqueness theorem for potential fields  Upward continuation of the field (integral form)  Effects of limited survey area

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:

Gravity of a 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

Uniqueness theorem  For a potential field ( ), if the normal gradient ( ) is known at a closed surface, then the field inside this surface is uniquely defined (formula (*) below)  Proof: consider integral over the volume (see pdf notes)  This means that the field outside the surface can be replaced by an mass (charge) density at the surface (Green’s equivalent layer):  …and therefore the field inside the surface equals: (*)

Effect of survey edge on upward continuation and modeling  Consider a half-plane of surface density  at the surface, with gravity recorded at height h  Splitting the half-plane into line sources, we can easily show that the gravity, as a function of distance x from the edge, equals:  Thus, the effects of truncation propagate far into the region x > 0:  For example, at distance x = 5h from the edge, the error of gravity modeled from a limited region is about 7% (dashed red lines): where is the gravity of an infinite sheet