Outline
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Gravity above a thin sheet
Equivalent stratum for buried sources (Green’s equivalent layer)
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For gravity
For magnetic field
The uniqueness theorem for potential fields
Upward continuation of the field (integral form)
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Effects of limited survey area
Gravity above a thin sheet
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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:
div  g   4 G 
The total flux through the surface
equals: 4 Gs A
By symmetry, the fluxes through
the lower and upper surfaces are
equal. Each of them also equals:  gA
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Therefore, the gravity above a thin sheet is:
g  2 Gs
Gravity of a line (pipe, cylinder) source
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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:
g  2 rL  4 Gg L

Therefore, the gravity at distance r from a line source:
2Gg
g
r
Note that it decreases as 1/r
Uniqueness theorem
For a potential field ( 2U  0 ), if the normal gradient
( U n ) is known at a closed surface, then the field inside this
surface is uniquely defined (formula (*) below)
2
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Proof: consider integral   U1  U 2   dV over the volume
V
(see pdf notes)
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This means that the field outside the surface can be replaced by an
mass (charge) density at the surface (Green’s equivalent layer):
s 

1 U
2 G n
…and therefore the field inside the surface equals:
Gs
1
U  P   
dS 
r
2
S
1 U
S r n dS
(*)
Effect of survey edge on upward
continuation and modeling
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Consider a half-plane of surface density s 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:


2Gs dx
h
x
1 1
g  x, h   
cos   2Gs  dx

g

arctan
sheet 

2
2
r
2

h



 x  x  h
0
0
where gsheet  2 Gs  g  x   
is the gravity of an infinite sheet
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Thus, the effects of truncation
propagate far into the region x > 0:
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For example, at distance x = 5h
from the edge, the error of gravity
modeled from a limited region is
about 7% (dashed red lines):