# 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.

$2U

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

error.

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

Figure 1.

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 June 26, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.