Graphical Terrane Correction for Gravity Gradient Page: 4
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4 GRAPHICAL TERRANE CORRECTION FOR GRAVITY GRADIENT
where K is the gravity constant, the density of the block is taken
as 1, and x1, y, z1, x,, and z2 are the coordinates of the corners of the
block. For these calculations
w = 5, x2= 5 X 2 , and y= 1 (5+5 X Y ) tan 221/2.
By putting z1 successively equal to 0, 0.1, 0.2, etc., and DbAz equal
successively to 0.1, 0.2, 0.3 E, etc., z2 is left the only unknown in (4),
and by solving for z2 a pile of blocks are obtained which differ succes-
sively in their effect on the gradient by 0.1 E.
Similar calculations were made later with Nikiforov's formula
for a curvilinear prism bounded by radii and by cylindrical surfaces
concentric with the z (vertical) axis; that is,
x = K loge (sin a- sin a2) + p2--1- (5)
where K is the gravity constant, the density of the block is taken as
1, a1 and a2 are the azimuths of the bounding vertical radial planes,
p, and p2 are the radii of the bounding cylindrical surfaces, and
z and z2 are the upper and lower bounding planes. The results of
the calculations by the two methods agreed within the allowable
By the law of similar bodies similarly placed in projection of each
other, which holds for the gravity gradient and differential curvature,
the pile of blocks in the first subzone is expanded successively into
the successive subzones; that is, all dimensions in the first subzone
multiplied by -2 give the corresponding dimensions for the second
subzone, multiplied by -/4 give the dimensions of the third subzone,
and multiplied by -/ give the dimensions of the fourth subzone.
All dimensions of the first zone, multiplied by two, give the second
zone; by four, the third zone; ;---- -; and by 2", the dimensions
of the (n+1) zone. A representation of the 5 to 10 meter zone of
the north octant with its four piles of 0.1 E blocks is given in
The graphic working charts of the method (figs. 2 and 3) repre-
sent a vertical section along the axis of an octant. The radial limits
of each zone, the 5, 10, 20,------ 640 meter vertical lines, are spaced
logarithmically; that is, with a constant interval. The distances
within each zone-that is, from 5 to 10, from 10 to 20, and from
5 X 2n-1 to 5 x 2n meters--however, are divided linearly; and the posi-
tion horizontally in the table of any point between 5 and 10, 10 and
20, 20 and 40, ------ or between 5X 2r-1 and 5 X2 meters distance
from the origin along the radial axis of the octant can be found by
simple linear interpolation between the respective 5 and 10, 10 and
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Barton, Donald C. Graphical Terrane Correction for Gravity Gradient, report, 1929; [Washington D.C.]. (digital.library.unt.edu/ark:/67531/metadc66454/m1/6/: accessed March 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.