A new method for gravity terrain corrections..pdf

8/9/2019 A new method for gravity terrain corrections..pdf

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A NEW

METHO

FOR GR VITY TERR IN CORRECTIONS

by

Woong Bong Chang

B.Sc. Eng.)

A

thesis

submitted to

the

Faculty of Graduate

Studies

and Research in partial fulfillment

of the

requirements for

the

degree

of

Master of Engineering

Department of Mining Engineering

and

Applied

Geophysics

McGill University

Montreal June

197



ACKNOWLEGmEN l

l wish

to express

my heartfe1t

gratitude

to

Professor

W

M

Telford the Director of this thesis for

his

encour.agement

and kind

guidance.

And

l a1so

thank Mr l i

F. King

Mr.

C. K

Park Mr

C R. S.

Haslam

Mr.

S. Vonpaisal in this deputment for a great deal

of

he1p

in preparation

of

the work •



CHAPl ER

Page

IV

v

3.4 Use of Graticules

• • • • • •

•

• • • • • • • •

•

•

PRACTICAL APPLICATION OF

GR TICULES

35

37

4.1 Terrain Corrections

in

Glendyer Brook Grid 37

4.2

Accuracy

of Results

CONCLUSIONS

• • • • •

•

• • • • • • • • • • • • •

• •

• •

•

45

48

BIBLIOGRAPHY

. . . . . . • . . • • . . • . • • • . · · · · •

5

APPENDICES

Appendix A Computer Program for the Calculation

of Radii

for Graticules

• •

53

Appendix B Radii for

Construction of Graticules 55

Appendix C Diagram of Graticules

in pocket)



LIST

OF

FIGURES

Figure

Page

1.

Gravitational Effect of the

Mass Element

2 Bouguer and

Terrain

Correction • • • • • • • • • • • • • • 8

3 Gra:vitational Attraction of a

Cylindrical

Segment

13

4. Gravitational Attraction of

a Prism

19

5 a) b). Construction of Zones

21

6

a) b). Hammer Terrain Correction Chart 22

7 Construction of Angles for

Inner

Zone 27

8.

9

10.

11.

Construction of Angles

for

Outer

Zone

Construction of Graticules

for

the First Inner Zone

Construction of Graticules for

the

Second Inner Zone

Construction

of Graticules

for

the First

Outer Zone

• • • • • •

• • • • • •

28

29

3

31

12 Construction of Graticules for

the

Scond Outer Zone

32

13 a) b). Method of

Drawing Mean Topographie

Profile

33

14 a) b).

Calculation

of the Gravity

Effect J4

15 Interpolation Graphs for the

Fust

Outer Zone 36

16

Interpolation Graphs for

the

Second Outer

Zone

36

17 Glendyer Brook Area

•

•

• • • • • • • •

•

• •

•

• • • • • • •

18

Selection of

Computing

Points

for

the Fust

Outer

Zone

• • • • •

19

a) b) c).

Interpolation

Graphs

for the

First

Outer

Zone

in

Glendyer Brook Grid

20 Interpolation Graphs for

the

Second Outer Zone in Glendyer

Brook Grld • • • • • • • •

38

9

40

43



Figure

Page

A-1.

Graticule

for

Zone

o -

1

•

•

• •

•

• •

•

•

•

•

•

•

in pockek

A-2.

Graticule for

Zone

1 - 2

•

•

•

•

•

•

•

• •

• •

• •

(

A-3.

Graticule for

Zone

2 - 4

•

•

• •

•

•

·

•

•

• • •

•

(

A-4.

Graticule for

Zone

o -

4

•

•

•

•

•

• • •

•

•

•

•

•

(

A-5·

Graticule for

Zone

4 - 8

·

•

•

• •

•

•

•

•

•

•

•

•

(

A-6.

Graticule for Zone

8 - 12

•

•

•

•

• • •

• •

•

•

• •

(

A-7.

Graticule for

Zone

12 - 2

•

•

• •

·

•

•

•

•

•

•

• •

(

A-8.

Graticule

for

Zone

2000

-

3

•

•

•

•

•

• • •

•

•

·

• •

(

A-9.

Graticule for

Zone o - 3000

• •

• •

•

•

•

•

•

•

•

•

•

(

A-10.

Graticule for Zone

3 - 7

•

•

•

•

•

•

• • •

•

•

•

•

(

A-11.

Graticule

for Zone

7000 - 11

·

•

·

· ·

•

•

•

·

•

(

• •

A-12.

Graticule for

Zone

11000 - 15000

•

•

•

• •

·

•

·

•

·

• •

(

A-13.

Graticule for Zone

5000 - 5000

•

·

•

•

• •

•

•

•

•

• •

(

A-14.

Graticule

for

Zone

5

-

15

·

•

· ·

• • • • • •

•

•

(

A-15.

Graticule for Zone

1 5000 - 25

·

·

•

•

•

•

•

•

•

(

A-16.

Graticule for Zone

25 - 35

•

•

• • •

·

•

·

• •

•

•

(

A-17.

Graticule

for Zone 35000 - 45000

·

• •

·

•

·

• •

· ·

•

•

(

A-18.

Graticule for Zone 45000 - 55

•

• •

·

•

•

•

•

• • •

·

(

A-19.

Graticule for Zone

55000 -

•

·

•

·

•

·

•

·

• • •

•

(



1

CHAPTER l

INTRODU TION

1 •

1.

General

n conducting gravity surveys

n areas

of rugged topography i t

is

necessary to

calculate the

effect of local

terrain

about each station

and apply the

results as

terrain

corrections

to each observed gravity

value.

A number of methods for terrain corrections have been presented

both for gravity

prospecting

and geodetic

regional

work.

In large

regional

and geodetic surveys Hayford-Bowie charts are often

used while

Hammer

charts are suitable

for detailed gravity work. However a high speed

digita

computer i

available

is very convenient for much

of

the calculation n

large surveys.

t

is

needless to say that

the

conventional zone

chart terrain

correction methods are time-consuming and tedious.

Since

an electronic

computer

is

not normally

available

in

the field office

a

simple

speedy

an4

reasonably accurate desk-calculator

method is

required

for terrain

corrections. With the Hammer chart method a terrain correction

often

requires one-half to

one hour

per station.

t

is difficult

to

estimate

the correct average

elevation in

each compartment and

the

probable

error

of the observed gravity values

is

of the order of 1/10 milligal even

assUllling that adequate topographie maps are available and the correct

average

elevation

in

each compartment is used •



The present work

was therefore

done with

the

purpose of improving

the speed and accuracy of

the terrain

correction method for gravity

prospecting.

1.2.

The

Principles of Gravit

y

Work

and

Terrain

Corrections

a)

Gravitational Field

The

strength

of

the

gravitational field

of

a body obeys

the

inverse

square

Law and is proportional to i ts

density.

The density ranges with the

mean

value

shown shown

in brackets)

for

various

rock

types

and

minerals

arel

Material

Density

Rock

Density

3

}DJcm

gmJcm

sandstone

1 61 - 2.76 2.32)

petroleum

0.6 - 0.9

shale

1.77 - 2.45 2.42)

graphite

1.9 - 2.3

limestone

1.93 - 2.90 2.54)

diamond

3.5 - 3.6

acidic igneous

chalcopyrite

4.1 - 4.3

rock

2.30 -

3 11

2.61)

Magnetite

4.9 - 5.2

dolomite

2 36 - 2.90 2.70)

cuprite

5.7 - 6.0

metamorphic

smaltite

6.4 - 6.6

rock

2 40

- 3.10 2.74)

galena

7.4 - 7.6

basic igneous uraninite

8.0 - 9.7

rock

2.09 - 3.17 2.79)

gold

15.6 - 19.4

Thus

for

the purpose of the

study

of

the

distribution of

rocks,

minerals

and

general

geological structure

of the earth, gravit

Y methods measure

the

variation

in

magnitude of

the

vertical

gravitational field,

due

to local



variations in

density.

According

to

Newton s aw of

unlversal

gravltation,

two mass points

attract

each other with a force proportional to

the

product of their masses

and

inversely proportional to the

square of

the distance

between them.

f

the magnitude of the force of attractlon

ls

denoted by F, the masses of two

bodies by m

1

and m

2

, and the distance between them by r ,

1.1)

where G ls the

proportionality constant, called the gravitational constant.

In the cgs system the value of G is

6.664

x 10-

8

•

The force

acting

on

a unit

aSS at the general point

P,

distance r

from mass

mi

(Fig. 1), is

defined as

the gravitational

field

of the particle

z

p

y

Fig. 1.

Gravitational Effect of the

Hass Element •



mi' which is written

1-:::

mt

-

F \ r _ - G ---- r

r

3

4

1.2)

Since the gr.avitational field is conservative this force May be

found by differentiation of a

gravitational

scalar potent1a1 function, U

Jf

cr) = :

u

u r)

(1.3)

f

P

is

on

the earth's surface,

the

gravitational

attraction there

is

denoted

bf

the

symbol g which

is

written

(1.4)

g

5

the negative field

intensity

and is

called

the

gravitational

field

or

the

gra itational acceleration. The unit

of g

is

the gal

.

1cm/sec

2

),

and the

direction

5

definition

everywhere vertical and downwards.

n

observed

gravit

y

anomaly which

is

really

the

change

n

the

earth

,s

gravitational

acceleration caused by

local

bodies

of

anomalous density

is

defined by the relation

where gobs

is

the observed gravity

value a t

a station and go is the

theoretical gravit

Y

value

at

the

same

station

or

usually

sorne

relative

value

a t

a

base station.

Because of the magnitude involved the gravity

anomaly

L:

g

is

measured

in milligals

1

milligal

e 10-3 cm/sec

2



5

(b)

Reduction of Gravity Data

The small variations in g recorded by the

gravit

y meter, when

measuring

at

stations on the surface, generally

include

several

effects

unrelated to geology, which must be removed from

the

data. These are the

latitude,

free-air,

Bouguer and

terrain corrections.

The reduction

of

gravit

y data is weIl discussed Qy Grant and

West

1965) and Dobrin

1962).

The latitude correction accounts

for

the variation of gravitational

acceleration of the earth with latitude and

is given

by:

Agrat

e

1,)07 sin

2r milligals/mile N-S

1.6)

where r

is the geocentric latitude. Since

the tendency is

for

g to

increase from

the equator to the poles, the latitude correction is

added

to the readings as one moves toward the

equator.

At

0

latitude the varia

t ion

is

about

0.1

milligals for

each

400

f t

of

displacement

in the

north

south direction.

The Free-air correction is based on the assumption

that

no masses

other than air

exist

between

the

station and

some

datum

plane.

I t is derived

from

equation 1.2) considering the earth s

mass

to

be

concentrated a t

i ts

center. The value of the elevation correction is

d

gFA

=

0.09406

h

mllligals

(1.7)

where h is the elevation difference between the

statiOll

and datum in f t

For

stations

above

the

datum

plane

these corrections are added;

for those

below. they are subtracted•



6

The Bouguer correction accounts for the attraction of the

mass

of

the

material

between the station and datum plane (Fig. 2).

t

is

derived

from

the

formula

for

the

attraction

of

an

1nfinite slab

having a

thickness

h and a

density cr. The

Bouguer

correction is

given

by

gB =

~ G O h

- 0.01277 cr h milligals

(1.8)

f r is taken as 2.67 gm cm

3

for an average of crustal rocks, this

correction

amounts

to 0.034

milligals per foot. This

correction

is

opposite

to

the

free-air correction.

The resultant of the two

corrections

(free-air and Bouguer) is

simply

0.06

milligals

per

foot, when

r

=

2.67

gm cm

3

•

Although

this is

a convenient number, the assumed value of r - 2.67 is not necessarily

correct

in Many cases.

The

terrain

correction

accounts for

the effect

of

aIl

mater1a1

above

and/or

lack of material below the

gravit

y station in i ts vicinity.

That is to

say,

i t corrects

for

local

terrain

irregularities in the form

of hills, valleys

and,

in some cases, known

subsurface

features

such

as

mine workings, caves

etc.

Obviously

the

Bouguer

correction over-estimates

the

gravitational

attraction of the actual mass below

the

station unless

the

surface

is fIat and

doesn t

remove

the effect

of

the maSS

above

the

station,

which

tends

to

reduce

the

observed

gravit

y

value).

This

correc-

tion

ls always

positive.

The

terrain correction

is given

by

the following integral

c Sr c: G J . r r) cos 9

v r

2

1. 9)



7

1n

wh ch

the volume v 1s conta1ned between the land surface and the Datum

plane as shown in F1g. 2. Since topography 1s irregular and can not be

expressed

mathemat1ca11y,

1t

1s

diff1cult

to

make

terra1n

correct10ns.

A method

of

numer1ca1

1ntegrat10n of

equat10n (1.9) 1s needed

for

th1s

purpose.

After app1ying a11 the correct10ns ment10ned above,

the grav1ty

anoma y is obtained. This is ca11ed the Bouguer anoma y which is given

by

the

fo110wing expressions

Applications

of

the

gravity method

in

the search for sma11, local

anoma11es

require

precise data.

To

meet this need terrain

corrections

must be made in areas of rugged topography. However, they are unnecessary

un1ess

the heights

(or

depths)

of

the

topographic

features

exceed one

twentieth

of

their

distance

from

the

station.

1.3.

Previous

Work

on Terrain Corrections

As ment10ned in

section

1.1, two conventiona1 template methods for

terrain corrections often used

for gravit

y surveys in Canada and the

United States. The Hammer and Hayford-Bowie charts are both based on the

block cy1inder type

of

terrain approximation.

Hayford-Bowie 1912)

charts

were deve10ped

in

conjunction with

the

United States Coast and Geodetic Survey s investigation

of

geodetic

prob1ems in 1912 and were later modified by

Bowie

1917). lnasmuch

as the

distances

between

stations

were n

the

order of hundreds of

miles, these



•

Hill

Station

Fig 2 ouguer and Terrain Correction

•

Surface

ouguer plane

Datum plane



9

charts

were designed. to

coyer

wide

area

and

large ranges

in

elevation.

A terrain

correction

method with precision for

gravit

Y prospecting

was presented by

S.

Hammer 1939)e The calculations were

based

upon

the

well-known formula for the gravitational attraction

of

a vert ical cylinder

at a

point

on

the axis

and in

the

plane

of

one end of

the cylinder.

The

gravitY effect of a sectorial column of such a cylinder is

given

by.

1.11)

where

e

is

the

sector

angle

in

radian,

R

1

and

R

2

are

the

1nner and

outer

rad

and h is

the

height

of the

cy1inder. The

ca1culations

were car.ried out y

solving this

equation

for

h in terme

of the radii

and

an adopted unit

gravitationa1 at traction for one compartment.

To

obtain the most nearly

square compartments the ratio of the outer

and

inner radii

(i .e .

the

radial

extent) of

a zone was related to

the

width of

the

compartments in

that

zone

y

the

condition

R2/Rl -

(n

+

fT)

/

(n

-

fT)

where n

is

the

number

of

compartments

in

the zone. The areas

of the

various compartments

in

the

tables

are maximum

(i . e. ,

the total

nUlJlber

of

compartments

i8

a minimum)

consistent with practica1 accuracy in

the

determinations of

the

mean ele-

vat

on

of the terrain in the comp:Lrtments. The tables comprise 12 zones,

Bit

to

Mit,

which in

turn are subdivided into

a

total of 132

oomIBrtments.

The sa l les t zone

3

has an 1nner radius of

6.56

f t . and the largest zone

Mhas an outer radius of 13.5 miles 71,996 f t . . Zone A, the area within

two metera of

the

station,

is

not

given

because on1y very extreme terrain

conditions will give appreciablo effect within

the

sa l1 area and in such

cases

the accuracy

would be

poor.

This is equivalent to saying that a



10

gravit

Y reading taken et

the

ed.ge of a steep

cl iff

is of

l i t t le

signif1cance

Curvature

of

the earth

is ignored

as i ts

effect on

the

tabulated. elevat10ns

1s

practically negligible with1n the area covered. by the

tables.

Gravity

effects

are

tabulated

to

0.001

and

0.01

mil11ga1s. This method permit:::

the

contribution of

terrain

to

the

gravity

value

measured.

t the

station to be

determined

quantitative1y

to a relative accuracy of 0.1 m111iga1s.

C. H.

Sandberg 1958) presented

terra1n

correction tables for an

inc1ined plane. This approximation seems

to

be reasonab1e for an

area

which can be approximated. by such a

two

dimensiona1

feature

near

a

gravit

y

station.

John Bible 1962) modified.

the

Hammer

terrain

correction

t ~ l e

for

use in oi1 exploration by expanding the table

to

accommodate the areas of

rugged topography. The table was

simp1ified

somewhat

to

faci1itate use and

improve

the

accuracy by reducing the

limitation

of certain

estimates.

The zones and

their

radii

remain

the

same

as

proposed. by

Hammer

but

number

of compartments in certain zones is reduced

or increased.

For corrections

1arger than 0.01 mi11iga1s

the

table was graduated in

steps

of 0.01 mi11iga1

A simp1ified gravity terrain correction method was described by H. A.

Winkler 1962).

Corrections

are separated into near and far terrain

effects.

The method for

far

terrain

consists

simp1y

of

computing corrections

by

one

of the standard

techniques at

wide1y spaced

points

and

contouring

the resu1ts

on the basis of topographie

contours.

Near

terrain

effects must be ca1culate

separately.

The method

is

on1y s11ght1y less

accurate

than

standard graphlca

techniques

and

saves

computlng time •



11

Recently computer

oriented

techniques have been

described

by

M H.

P. Bott

1958),

M Talwani and M Ewing 1960), M F. Kane 1962), B.

K a r l e ~

1963), and

D.

Nagy

1966).

Methods programmed

for

digital computers depend

essentially

upon fit t ing mathematical surfaces to

point

values

of the eleva-

tion.

In

the method described by M Talwani and M EWing an expression is

used

for the

gravitY anomaly at an

external point

caused

by

a

h o r i ~ o n t l

lamina with the boundary of an irregular polygone For two-dimensional

features

t

is

simpler

to use the method given y Talwani,

et

al. 1959)

to determine the

terrain

correction. In the methods presented by Bott and

Kane

the

terrain correction for

the inner ~ o n e about 2 km x 2 km) must be

calculated by

conventional methods, and

the correction for the outer ~ o n e

is done

using

approximate expressions with computer. Karlemo, on the

other

hand, includes the

inner

zone

as ~ e l l

D. Nagy 1966) developed a formula

for the gravitational attraction of a prism with sides

parallel to

the

rectangular coordinate axes,

and used

this

formula

to calculate t l ~ i n

corrections.

Another procedure

uses

graticules in which

the

attraction of each

element is

the

same,

regardless of distance or angle.

They may be prepared

for use in plan view with contour lines or in vertical section with

terrain profiles .

Methods

of this

type, due to

K.

Jung and Haalck, are

described by Heiland 1940)

•



12

CHA PTE R II

TTR CTION OF THREE

DIMENSION L MASSES

2.1. Introduction

Numerous

methods have been developed

for calculating the gravitationa

attraction

of simple shapes such as the sphere,

cylinder ellipsoid

and prism

Only

the

most elementary of

these

can

he sol

ved

analytical1y;

genera11y

the

resu1ts are approximationsobtained by numerica1 integration.

In

the

fol1owing

we

will

derive

a c10sed expression

for

ca1cu1ating

gravitational

attraction of

a

cylindrica1 shell

using

cy1indrical coordinate

This solution

is the basis

for the

construction

of graticules to

be used for

terrain corrections as

described

in the

fo1lowing

chapter.

In

order to

estab1ish

the val

d

ty

of the

solution t

is applied

to a prism and the

resu1ts

are

compared with

those

obtained

by two other

workers.

2.2. The Derivation of the

Formula

for the Gravitational Attraction

of

a

Cylindrical Shell

s

indicated

in

Fig. 3,

the mass of a

cylindrica1

segment

is

dm ...

a

de d R dy where R

is the radius

from

point

0 and y the 1ength from

the station

P

to the point

0 of

the masse

The

gravitationa1

attraction

of

such an e1ement

at

station

P

with

distance

1 is

given bYI

G

dm

L:i

g

1

G

Ga tana d de

dy

2.1)



...

-

- .

.:a,

,

y

- - ,

.

,

Fig

3. Gravitational

Attraction

of a Cyl ndrical Segment.

3



14

2

since l _ Y seca, R - Y

tana

and dR . Y sec a

da,

where a

is the

angle

between

OP

and

FQ.

Then,

the

vertical component of gravity bounded by the angles a

1

and

a

2

and

by the angles

e

1

and 9

2

and by

the

distances

Y1

and

Y2'

can be

obtaine

by integrating Ag

1

sina and

sine

over

the

volume,

i .e .

Ag . Go' r

e2

sine de r

Y2

Ja

2

tana sina da dy

e

1

j

Y

1

2.2)

J

JYJa

2

. Go'

2

sine

de 2 2 sin da dy

e Y a cos

1 1 1

2.3)

Carrying out the integration with respect to a and without substituting

the

limits, one obtains

l Sin

2

a

da

cosa

2.4)

The integration of equation 2.4) with respect to y

may

be obtained by

substituting

s1nCl .,.

R/h

2

+ R

2

and tan (T + T) _ y + R + }y2 + R

2

)/

y-R

+

Jy

+

R2)

I

2

JIl

dy

--J

2:

H2

dy +

J

n (

Y

+

R

+ Ji +

R2)

dy

-

JIn

y - H + Jy + R2) dy

2.5)



In

the

second of these, we let

du ca

Ju

2

-

2Ru

dy

u - R

15

and integrating with respect to u and

transforming

back to the

original

variables, we get

1

22

Y ln y R Jy2

R2

-

n y

J 2 Ri

R Jy R

2

- -Z- - 2

2.6)

In the

third

integral

of

equation

2.5), we

let

2

RV

dV.. V R dy

and carrying out

the integration

with

respect to

V and

transforming

back

to

the

original

variables, we

have

With

the

substitution of equation 2.6) and 2.7) into 2.5),

the

integral 2.5)

becomes

l

n{y - R

~

dy

= Y ln Y v R Jy2

R2

, _ R

Y - R

J 2 R2

2.7)

2.8)



16

Applying

the

limits of

integration, the

second term

n

equation

(2.8)

drops out.

e

therefore obtain the

following

general

expression

for the

vert ical

component of the gravitational attraction of a cylindrical shell whose axis

is

paral lel to the horizontal surface,

2.9)

When we

apply

the

limits

of integration,

equation 2.9) becomes,

[

Y2 R

2

J Y ~

R ~

g = Go cose

1

-

cos9

2

) i Y2 ln J

2

y

1 - R

1

Y1 R

2

Y2 - R

1

+

~ - ~ - - R - î ] r

Yl + R

2

+ yî +

- ln 2 - Yl

lln J

2 2

y - R +.1: 2 R

1

Y1 R

2

Y1 R

2

Yl R

1

~ }

- ln

2

(2.10)

Yl - R

1

+ Yl + R

1

Designating

the terms

within

the

brackets by

Ti T

2

,

T)

and T4 one obtains

the followinga

(2.11)



•

f

we

consider the

special case

of R

1

0,

we

have

n the case that R

1

and y1 are both equal to

zero,

we obtain,

Ag os a cos

l

- cos e 2) Y

T

l

17

2.12)

2.13)

As mentioned

in section

2.1

equations 2.11) 2.12) or 2.13)

are used to develop graticules

suitable

for

terrain corrections.

These

are described

in

detail in chapter III .

2.3. Comparison

with

Other Methods

Although

the

formulae in section

2.2

were

derived specifically for

terrain corrections, clearly they can

be

used generally to tletermine

attraction

of three

dimensional mass which can

be

approximated by

the

cylindrical shell.

In

order

to

test the accuracy of the method we will

compare the results

obtained

for a prism using

three different

approaches,

including

the

above.

The attraction

of

a prism can

be

approximated by

that

of an annular

ring with the sarne

height

as the prism,

multiplied

by the ratio of the area

of a horizontal

section of the

prism to that of a horizontal section

of

the rlng as descrlbed by M

F.

Kane 1962). The gravity effect

on

the

axis

of

the

annular

ring

ls

easlly

obtained

as the difference of

two

cylin-

ders.

The formula is .

2.14)



18

where C is the length of a

horizontal

side of

the

prisme

Another formula used is a closed expression for

the

attraction of

a prism, derived by

D

Nagy 1966). However

the

formula has long terms

consisting of logarithmic and arcsin functions.

In

order to get the correct

value for the gravity effect, the prism should be subdivided

into

a number

of horizontal sections

parallel

to the

surface

and the effect of each sectio

summed

up.

Failure to

subdivide

the

prism

May

result

in

errors

in

the gravi

effect

by as much s a factor of two. This is critical for prisms

close to

the

station.

t

is

not

necessary

to

subdivide

prisms whlch

are

remote from

the

station.

Table compares

the

values

obtained

by

these two

methods

with

that developed in sec. 2.2, for

the

attraction of the prism shown in Fig. 4.

TABLE 1

Comparison of Graticules with Prisms and Cylindrical

Approximation of Prisms in milligals).

Elevation of Graticules Prisms Cylindrical

Station

Appr.

of

Prisms

ft . )

Chang) Nagy) Kane)

2,000

.017

.014

.017

3,000

.039

.039 .039

4,000

.069

.070

.068

5,000 .104 .110 .106

6,000

.150

.157

.152

7,000 .206

.209

.206

8,000

.270 .269

.267



•

-

~ - - - - - - - - ~ - - - - - - - - ~

h

~ ~ - - - - - - - - + - - - - - - - - - - - -

station

Distance in

f t

Fig

4 Gravitational Attraction

of a Prism

9



20

CHA PTE R III

GRATICULE

TERRAIN

CORRECTION METHO

3.1.

Introduction

To

make

terrain

corrections

using topographie

profiles, the

area

surrounding the station

is divided

into a number of rectangular zones as

shown

in Fig. 5 Graticules are superimposed over the

profiles

of the

terrain

of

each zone. Hence by counting the number of elements enclosed

by

the

terrain

profile

and

the

graticule

horizon,

the

terrain

effect

is

evaluated in each of

these

zones, and the

sum

gives the

total

terrain

correction.

Graticules

are

drawn

using

equations 2.11), 2.12) or 2.13),

whichever

is

appropriate.

To save computing time and for convenience) the terrain corrections

are made by

calculating

the effect

of the

inner zone, the i rs t outer zone,

and the second

outer

zone separately.

3.2.

Zone

Construction

In

order to make use of graticules, i t is necessary to divide the

terrain surrounding the

station into

several zones of a

certain

width. The

topographic profile of each zone

is

drawn and

then the

graticule correspond

ng

to that

zone

is

superimposed upon

the

terrain

profile.

The

Hammer

chart

shown in Fig.

6 is

included

in

order to compare the dimensions and

shapes of zones with graticules.

The choice of width of zones is important, because i t determines both



5000

1::3

8 )00

35000

45ODO

55DOO

G5 J

o

400

800

1200

2000

3000

a) Outer Zone

o _

3000

=t7 i.

88 6

(b) Inner,Zone

Igg-?

:

2 -?

Fig. 5. Construction of Zones.

I rçw

7000

3Çl.AJ

0

G5

S5OO0

45iXX>

35000

25 CO

15000

5000

21

Scala 1

50 000

•

Station

3000

2000

1200

OO

400

o

Scala 1 n 2000'



a) Zones G h r ~ u h M

Scale 1 50 000

• station

(b)

Zones through

F

Scale 1 - 2000'

Fig.

6.

Hammer Terrain

Correction Chart •



23

the accuracy and s peed of

the

terrain corrections. Thus

the

zones near the

station are

inevitab1y

sma11, whi1e

on

the

other

hand

these

remote from the

station can be very

large.

The present system covers an area 130,000 ft . x 130,000 ft . with the

station

at

the center. he area ls

divided

into the

f lrst and second lnner

zones of areas 400

f t

x 400 f t and

6,000

f t x 6,000 f t

respectlve1y

and

the fl rst and second outer zones having areas of 30,000 f t x 30,000 ft . and

130,000

f t

x 130 000 f t respectlve1y. These zones are subdivided lnto a

number

of

rectangular

strips

or

subzones :

the

f1rst

1nner zone has

three

subzones, the second inner zone has ten subzones, the f irst outer zone has

elght subzones and

the

second

outer

zone has

ten

subzones.

he rectangular

subzones have a definite lnner and outer distance from the center e.g. 200

f t

and

400 f t respective1y

for zone 200-400).

Bear10g

ln mind that

the 0 110e goes through

the station

and

that

the rectangles are

symmetrica1

on

elther

side of the

110e Fig.

5),

the

zones

are

deslgnated

as

fol10ws:

Inner zone

f irst lnner zone : 0-100

100-200

200-400 f t

second

inner zone:

0-400, 400-800, 800-1200

Outer zone

fi rst outer

zone

1200-2000, 2000-3000 f t

0-3000, )000-7000,

7000-11000, 11000-15000 ft .



25

T BLE 2

Cosine

and

Tangent used

for

Construction

of Graticules.

Degrees

Minutes

Cosine

Tangent

Degrees

Minutes

Cosine

Tangent

0

1.

0 0 19

57

940

3630

0

8

99980

0198

21

J4

930

3953

1

17

99975

0224

22

20

925

4108

1

37

99960

0282

2

4

920

4258

1

59

99940

0346

24

30

910 4557

2

17

99920

0399

25

50

900

4841

2

34

9990

0448

31

47

850

6196

3

37

9980

0635

6

52

800

7499

5

8

9960

0898

41

25

750

8822

5

44

9950

1004

45

34

700

1 020

6

17

9940

1101

49

27

650

1 169

7

15 9920

1272

53

8

600

1 334

8

6

990

1423

56

8

550

1 519

8

53

988

1563

60

500

1 732

9

J6

986

1691

63

15

450

1 984

9 56

985

1753

66

25

400

2 291

10

16

984

1811

69

31 350 2 677

10

53

982

1923

72

33

300

3 181

11

29

980

2032

75

31

250

3 871



26

TABLE

2 Contlnued)

Degrees

Minutes

Cosine Tangent Degrees

Minutes Cosine

Tangent

12 5

.975 .2278

78

28

.200 4.901

14

4

.970

.2506

81

22

.150

6.586

15

12

.965

.2717

84

6

.100

9.960

16

16

.960

.2918

87

7

.050

19.855

17

15

.955

.3105

9

0 0

18 12

.950

.3288

Fig. 10,

Fig.

11,

Fig.

12

respectively with

the

same saales mentioned above.

The numbers on the radii in Fig. 9 to Fig.

12

indicate the gravi-

tational attraction

of t

cylinders

in millig.als. The

effect of one element

of the inner zone ls 0.001 milligals for the upper part of the graticules

nd

0.005 milligals for the lower part of

the

gr.aticules,

as

dlstinguished

by the heavy

line. For

the

outer

zone

these value

are

0.002

milligals and

0.01 mll1iga1s respectlvely.

d) Method of usins Graticules

The

graticules

are superimposed on the

terrain

profiles of

the corres-

ponding zone with

the

origin

at

the

sarne

elevation as

the station for whlch

the

terrain correction

is being

calculated: The

number

of

elements between

the horizon and the

terrain

profile ls counted.

The gravlty

effect

of

the

terra n in each zone is then determined.

As an example

the Mean

topographie

profile

for zone

200-400

is drawn

in Fig. 13 a) b) and the calculation of the gravity effect of zone 200-400



0.05 10 15 .20.25 .30 .35 .40

·45

.50 .55

.65

.70

.75

Flg.

8.

Construction

of Angle for Outer Zone.

. .9998

9 9 ~

:.99

.999

·996

.992

.990

.988

.986

984

.982

.98

.85

.97

.96



..

075

0004

Fig 9. Construction

of

Craticules

for

the First Inrler

Zone.



O

•

i



•

0004

.

05

50

Fig 11. Construction of Graticules

for

the

First

Outer Zone.

•



0004

05

50

Fig onstruction of Graticules forthe Second Outer

one

2 50



•

..

50

400

350

1

o

o

\J

o

o

\J

l

~ ~ n Topoera{ hlc Proflle for 7,( Ine 200 - 400

~ t t i o n

o

b) Plan View of Topography of Zone 200 -

400

392

o Station

Fig. 1) a) b).

Hethod

of

Drawing ean Topogra];ilic Profile.

•

Distance

in f t



•

Station

elevat10n

plane

~ e n

to pogra [ilic -

profile for

zone

200 400

ZON

Fig 14 calculation of

the Gravit

Effect.

•

200 - 400

station



35

is shown

in Fig.

14. There are

24t

elements with an

effect of

0.001

milligals

which

l ie

between

the

horizon and

the terrain profile,

and

î

element with an

effect

of

0.005

miIligals.

The

total

effect

of the

zone

is therefore

0.0270

milligals.

3.4.

Use

of Graticules

The terrain corrections for

the

inner

zone must be

calculated

separately at each

station, but

the

sarne

terrain profile of each rectagular

zone

rnay be

used

for aIl the

stations

on the same line.

For

the f i rs t outer

zone a

simplified terrain correction

method

presented

by

H A

Winkler (1962)

may

be used.

The

rnethod consists

simply

of

computing

terrain

corrections at

widely spaced

points

and

then

contouring the

results

on

the œse of

topographie contours.

n this

case

several points

may

be chosen and

corrections

are

calculated

at

these points

for several hypothetical elevations at

convenient intervals

so that the

~ i s t i n actual elevations in the vi

c1n

t of

each

point

are

bracketed.

The computed

values are then plotted against elevation.

With

this graph

corrections

May be

lnterpolated

between the cornputed

points for

aIl

existing

alevation

changes. This

is

sean rom

Fig.

15 a)

b).

n the

second

outer

zone

the

effect

of

terrain

sc&rcely

varies

with

station

location, since the

grid

ls

very

small

compu-ed

to

the area

itself.

However,

the

corrections

do

vary

somewhat

with

station

elevatlon.

Once the

effect of terrain

beyond

the

30,000 f t x 30,000 f t at the

center of the

grid

has been

calculated,

a

graph may be d ra wn of

computed

value of

terrain

correction vs. elevation. The terrain effect of the

whole grid

May then

be

determined by

using

a curve

of

the type shown

in Fig.

16.



•

t

(a)

Topogra}ily Section

(b) Iso-Correction

Diagram.

___________

•

230

....... , . -

2 _

.

.229

. 220 - -:218

•

t

-

~ 2 1 3

.2 /7

_J .21;1

210

.207

,

.207

"

.208

- .210

.2/3

.

.218

_ 220

.

.2/9

fi

-;-:-223

.

232

230

_ . 2 4 0

-

. .237

0 .239

"

.247

--- ;- .255

Coaputing elevation Iso-correction value (Jlilligals

)

Fig. 5

(a)

(b). Interpolation

Gra Ïls for

the

Fus t

Outer

Zone.

Blevation f t . )

Fig.

16. Interpolation

Graphs

for

the

Second

Outer

Zone.



J

The

area

considered on the map.

Scale 1 • 50 000

Fig 17

Glendyer

Brook Area



•

14

ooW

12 ooW

8 ooW

4 ooW

12 ooN

r

o

)c

0

0 0

0

)

a

0

0

0

0

0

0

0

0

0

0

8 ooN

r

0

0

0

0

0

0

0

0

0

)

0

0

0

0

e

0

0

0

4

ooN

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

+00

L

o

x

0 0 0 0

x

0 0

Fig 18

Selection

of

Coaputing

Points for

the Outer Zone

0

)

0

0

0

C

0

0

o

)

B.L.

0

0

0

0

0

0

0

•

o

station

cOllputing point

for

the f1rst outer zone

G Comput1ng point for

the

second outer zone



•

600

1

500

•

-

::=

400

-

œ

Q

r 4

300

•

(a) Line 0 +00

230

220

.210

Iso-correction value ( m i l l s ~ )

200

/90

200

.210

.220

:

230

.240

2S0

1

1

t t

1

•

•

•

16 ooW

14 ooW

12 ooW

10

ooW

8 ooW

6

ooW

4 ooW

2 ooW

B.L

Fig. 19 (a) (b) (c). Interpolation Gra}i1s for the First OUter Zone in Glendyer Brook Grid.



•

b) Line

6 +ooN

600

230

220

- - - - - - - - - - - - - - : I ~ S - o _ - c - o - r r - - e c - t ~ 1 ~ 0 - n - v a ~ 1 ~ u - e ~ m - 1 ~ 1 : 1 ~ 1 - g ~ 1 ~ s ~ ) ~ - - - - - - - - - - - - - -

2 /0

.....

•

s 400

1 11

300

-

=

16 +00\1

14

00\01

12 +ooV

10 00\01

8

+00\1

6 +00\1

•

240

210

220

230

4 +oow

----

+ooW

B.L.



•

•

c) Line

12

ooN

600 r-

-----

2 / 0500 200

-

/ 70

· /80

.. -

Iso-correctlon

value mU11gals)

s::

_ 190

_ ~ _

i 400 r= 200

G

2 /0

220

230

)00 1-

_1

1

1

1 1

1

1

1

1

16 OOW

14 ooW

12

ooW

10 ooW

8 ooW

6

ooW

4

ooW

2

ooW

B.L



240

220

a

200

s

180

160

140

120

200

3

400

500

Elevation

f t . )

Fig 20

Interpolation Graph

for the Second

Outer

Zone

n

Glendyer Brook

Grid

600



•

•

T BLE 3

Terrain Corrections in using Graticules in Glendyer Brook in milligals).

Line

10 00

N

Station

Elevation

Zone Zone Zone

Zone

Zone Zone Zone Zone

Inner

The

fi rst

The

second

Total

1

of Station

o

100

200

0

400

800 1200

2000

Zone

outer Zone

outer Zone

Correc

ft.)

100 200

400

400 800 1200

2000

3600

tion

B.L.

360

.040 .022

.007

.065

.083

.036

.023

.021

.297

.205

.130 .632

2 00

330

.054 .023

.020

.069

.102 .048

.045

.030

.393

.219

.120

.732

4 00

380

.046 .018

.020 .022

.054

.024 .014

.019

.217 .219 .136 .572

6 00

400 .021 .009 .017

.020

.057

.024 .018

.015 .191 .195

.142

.528

8

00

393

.021 .009 .013 .013

.079

.029

.021 .017

.202

.196

.140

.538

10

00

385

.038 .010 .019 .013 .093 .035 .019 .017 .244 .198 .138 .580

12 00

378

.038 .008

.028

.019 .129

.036

.029 .019 .297

.198

.136

.631

14

0 0

383

.038 .009

.018 .019

.105

.044

.034

.017

.281

.197

.137

.615

16

00

370

.036 .011

.021

.022 .112

.039

.037 .015

.292

.210

.135

.637

18 0 0

W

J40

.066 .017 .019 .015

.114

.046

.049

.030

.358

.222

.123

.703



45

six hours,

ten to

thirteen

minutes were

required

to calculate the terrain

corrections of the 10ner zone

for

each station. This resulted

in

about

twelve

ho

urs

more

since

there were

seventy stations.

Therefore

the

whole

operation for seventy stations required about twenty eight hours. I f the

station

spac10g

was

100 f t the

terrain

corrections

for

280

stations

would

consume about

72

yours since the time

for

profiling the

terrain is

not

proportional

to

the number of

stations.

The

locations, elevations, and computed

terrain

corrections

for

these

stations

are

l isted in

Table

4. An

example

of the

calculation

is

shown in Table

3. .

4.2 Accuracy of Results

The

corrections

for

twenty

of the

above

stations

on Line 0 +

00

and Line 10

+ooN

in

varying topographical conditions have been calculated

both by graticules and by the Hammer charts. The corrections

using

the

Hammer

chart required

about

15

hours roughly 45 minutes for each

stat ion),

compared

to about

8 hours

using graticules.

These corrections

are

compared

in Table 5. The

largest discrepancy

between the two methode is 0.09

milligals

and the average

difference ie

0.037

milligals.



46

T BLE 4

Terrain Corrections using Graticules in

Glendyer

Brook

in

milligals).

The figures in parenthesis show the elevation in ft.)

station

0+00 2 +00 4 +00

6 +00 8 +00 10 +00

12

+ooN.

B.L.

)64)

361)

355)

)46)

357)

)60)

365)

.65

.65 .66

.73

.64

.63

.57

2 +00

356)

351)

)43)

335)

335) 330)

324)

.72

.76

.73

.84

.78

.73

.68

4 +00

373)

390)

396)

396)

lW2)

380)

)48)

.80

.79

.74

.70

.64

.57

.53

6 +00

4i.8)

468)

450)

429) 412)

400)

)67)

.79

.81

.70

.63

.56

.53

.50

8 +00

495)

503)

l ~ 6 9

)

437)

411)

393)

367)

.86

.88

.72

.62

.58

.54

.54

10

+00

535)

510)

481)

443) 412)

385)

357)

.89 .76

.73

.67

.60

.58

.56

12 +00

543)

509)

493) 452)

l ~ 0 6

378) )48)

.89

.73

.84

.68

.64

.63

.54

14

+00

539)

509)

484)

440)

402)

383)

350)

.86

.75

.80

.78 .64

.62

.56

16 +00

535)

484)

466)

425)

385)

370)

)46)

.91

.79

.81

.80

.63

.64

.56

18

+00

506) 462)

443)

389)

366)

340)

322)

.85

.83

.85

.81

.70

.70

.58



7



48

CHAPTER V

CONCLUSIONS

Graticules were constructed for the purpose of increasing the speed

and accuracy and

relieving the

tedlum in making terrain corrections.

Equation

2.11) is

a

closed expression suitable for

making gratlcules

for the

gravitational attraction of a cyllndrical shell. I t

is

believed that

the

equation 2.11) can be employed in the analysis

of

gravitational problem

of

three-dimensional

masses

as

weIl

as in

making

terrain

corrections.

The

use of the formula

ln

computer programs for

terrain corrections wouId

have

the

advantage

of

simpler terms

in the

formula

than

those

of

previous methods

As

has been described above

graticules

for terrain correction have

been applied in Glendyer Brook

area

and the results compared wlth those

obtained

from the Hammer

charts.

The

terrain

corrections were

made

faster

wlth

graticules

than

the

Hammer

charts.

The corrections

for

one station

using graticules

required about 24 minutes compared to about

45

minutes using

the Hammer charts. The average difference between the two methods

was

0.037

milligals and the

largest

was 0.09 milligals. This large

discrepancy ls

probably due to one or more of the following reasons. The corrections

for

zones

D

E and F of

the Hammer charts

were calculated

as

0.53 . of a total of

0.80

milligals.

This

was

due

to

relatively

large difference

in

helght

of

those compartments and the

station

elevatlon.

The

estimation

of

the

average

elevatlon

in

each compartment of those zones

was

difficult and a small

difference

in

helght

would have a

large effect on the

corrections.

e.g.

height

of

47 f t

for

a compartment of zone D gives 0.02 milligals,

while



5

should he done carefully

since i t gives

a large

terrain effect in

a

small

distance.

The speed

of

operations increases with the number

of

stations

or

the density of stations.

With an

increase in the density of stations the

corrections by graticule would be done

considerably faster

than the conven-

tional

Methode

About 28 hours

are required

to

complete

the terrain correc-

iOns

for

seventy stations roughly 24 minutes

for

each

station

with a

grid

interval

of

2

f t .

If the

station

spacing

is

100

f t .

the terrain

correc-

t ions

for

28

stations are

calculated.

in

about 72 hours roughly 15 minutes

for

each

station . In

both cases

of

the above a

terrain correction using

the Hammer charts

requirea

O to 6

minutes

per

station,

i.e.

a

minimum

time

of J5

and 140 hours.

In

the graticule method once a

terrain profile

is

drawn

for

a given

area, the

corrections for

an

additional number

of

stations

can be

made

without drawing any

more profiles. I t i8

also

clear that the

values

obtained

by

graticules

and

by

conventional methods

are in

good

agreement •



51

BIBLIOGRAPHY

1.

Bowie

W. Inyestigation of Gravit

y and

Isostasyl U.

S. Coast and

Geod.

Survey Spec. Pub.

40 1917.

2. Bible

J. L.

Terrain Correction

Tables

for

GravitYI Geophysics

V. 27 p.

715-718

1962.

J.

Bott M

P. H.

The Use of Electronic Digital

Computers

for the

Evaluation of Gravimetrie

Terrain Corrections

1

Geo

phys.

Prosp.

V. 7

p.

45-54. 1959.

4. Dobrin M

B. Introduction to

Geophysical

Prospectingl McGraw

Hill Book Co. Inc.

Toronto Second

Edition 1960.

5.

Grant F. S. and West

G. F. Interpretation

Theory

in

Applied

Geophysicsi McGraw-Hi11 Book Co.

Ine.

Toronto 1965.

6. Hammer S.

Terrain

Corrections for

Gravimeter Stationsl

Geo

physics v. 4 p. 184-194.

1939.

7.

Hayford J. F. and Bowie W. The Effect of Topographie and

Isostatic

compensation

upon

the

Intensity

of GravitYI

U. S. Coast and

Geod.

Survey Spec. Pub.

10 1912.

8. Heiland C.

A.

Geophysical

Explorationl

New York

Prentice

Hall

Inc.

1940.

9. Kane

M

F.

A Comprehensive System

of Terrain

Corrections

using

a

Digital

Computera Geophysics

V. 27

p. 455-462

1962•



53

•

Appendix A - Computer Program for

the Calulation

of

nadi

for

Graticules

IV

G ~ E V f . L

C

1, UD

t4A IN

DATE =

7012 -+

0001

0002

0003

0004

0005

0006

0007

0008

ooet

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

O O ~ 3

0024

0025

0026

0027

0028

0029

0030

0031

0032

0033

0034

0035

C

C

C

C

C

C

C

C

C

C

TERRAIN COPRECTICNS FOR MAKING GRATICULES IN

CYLfNORICAL COORDINAlE SYSTEM.

CCCRDJNAlES

AFf

GIVEt.;

IN

FEET.

YI,Y?

APf

H C R I l C ~ T

01STMJCES

F R O ~

THE

STATICN.

R2

IS

RADIUS

OF

THE

CYLp·J

THE VERTICAL COMPONENT

OF G R ~ V I T A T I 0 N A L

A T T P A C T I O

OF l l t CYLINOEP. rs C ~ . L C l J L A T E D IN UNITS OF J/IOO MILLIG

FOR DENSITY 2.67 G P / C M ~ * 3 .

D1

IAf

NSIC N y 1

l 4)

,Y 1 2 ( 4 )

DIMENSION RH2(4CGO) ,CEL G(4000 '

c o r ~ r . 1 0 N Yl,Y2

00 40

1=1,4000

40 RH2( 1

)=0.

REAO(5,9,FND=999)

N

DO

15 1<= l , N

1 5

P

EpD 5 ,

l)

Y

1

1 K ) ,

y 1

2 K )

WP

I l E ( é 7)

DO J

7 t<= 1, N

1 7

v IR

1TE ( (;,

6)

Y1 K , ,

YI

2 )

DO 90 K= 1, N

W ~ I T E é , 3 1

Yl1(K),VI2(K)

Yl=VIl( <)

Y2=YI2(K)

R2=1.

OEL

G{

l)=f)EL

GA(R2)

RH 2 ( 1 ) =1.

DO

41 1=2,4000

RH2(Il=P.H2(I-l)+4.

R 2=P. H 2 1 )

CEL

G(

1'=OEL G[) R2)

t..l Rtl2CI,=P2

DO 50 I=1,10CO

50 wrUTf(l.,,4)

RH?'(I),OEL

G(I),RHZ(I+IOC:0),CEl G(l+lOI)(')

,

lRH2( r +

2rOC'),

DE L G( 1+2000) ,RH2( 1 +3000) ,DEL G( If- 30.00)

90 CONT

1

;U[

CALL EXIT

9 F o r p ~ t \ i t I

5)

l FrHH-1

AT

(Z Fl 0 • 1 ,

7 F 0 RIv

AT (1

H l l l 7 X,

'Y l

, l 7

X,

• Y2

• )

6 F 1 R ~ ~ A T { l H

1 0 X f l C . l l C X F l O . l l

3 fORMAT(lHl,2X,2F20.1l

4 FORMAT{}H

, 5 X , 8 F l ~ . 2 )

999 STOP

END

TOTA L MEt·j CRV

RE

QU

1 P E EN TS C081B0 SV TES



FORTRAN

0002

0003

0004

0005

0006

0001

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

C020

0021

0022

0023

0024

0025

0026

0021

002B

0029

0030

0031

0032

0033

0034

0035

0036

0037

38

0039

0 0 4 ~

Appendix A (Cbntinued)

1 V G

LEV

ELI,

~ 1 O [ 1

4

FUNCTICN

DEL GB(R2

C o r j ~ O N Yl,Y2

DEL

GP..=O.

CONS=O.5428

Y12=Yl ;; *2

y

2 2

=

y

2

)).:r)

2

R22=R2*i,.:2

01 2

=

S 0 P T (

Y

1 2 +R 2 2 )

022=SQRT Y22+P.22)

QP

1=Q22+Y2

QP3=Q12+Yl

R ~ 1 = Q P 1 + R 2

RN2=QP1-R2

RN5=QP3+R2

RN6=QP3-R2

CM=P.Nl/RN2

RATIOl=R2/QPl

DELGB

IF

( R

T

101 -1 .

3

0 l ,7:;

0 , 7

SC

30

1

1

F ( CM) 701 , 1

f)

1 ,

101

101

V 1 =A

lOG ( C'.t

)

IF YI 333 ,138 ,333

333

RATI03=P2/0P3

S ~ = R N 5 / P . N f

IF RATIC3-1.)303,755,755

303 I F S M ) 7 C ~ , 7 0 3 , l C 3

103

V3=AlO ; S ~ 1 )

400 OEl

GB=CCNS*(Y2*Vl-Yl*V3)

500

DEL

GB=ABS(OEL GB)

RETURN

701

DEL

G8=11111.1

GO

TO

500

703

DEL G8=33333.3

GO

TO s o

750

DEL

GB=75000.0

GO TO 5Ct

755 DEL

GB=75555.5

GO TO

500

788 V3=O.

GO TO

~ c o

END

TOTAL M E ~ R Y REQUIPEMENTS

on030E BYTES

D T

E

= 10124



55

Appendix

Radii

for Construction

of

Graticules

TABLE

A 1

Construction

of

Radi1

for Graticules of the

First

Inner

Zone.

Ag

Zone Zone

Zone

Zone

Zone

Zone

o 1

100 200 200 400

o

100 100 200 200 400

mllligals

f t f t

t

mllligals f t

f t

f t

•

4

18

28

.25

8

.005

43

68

.30

58 292 367

.01

55

87

.35

69

.02

73

112

.40

81

373

438

.03

85

13

.45

93

.04

96

145

.50

1 5

466

.05

9.2

1 6

158

.60 1;4

.06

115

17

.70

168

.07

123

181

.80 2 7

.08 131

192

.90

253

.09

139

2 2

1.00

3 8

.10 18.5

147

212

1.10

373

.15

28

183

255

1.20

451

.20

J8

219

294



TABLE

A 2

Construction

of

Radii

for the

Graticules

of

the

Second

Inner

Zone.

Zone Zone

Zone Zone

Zone

A g 0 400

400 800 800 1200

1200 2000 2000 3000

mill1gals)

f t . )

f t . ) f t . )

f t . ) :ft.)

.0004 6

80 100

145

.005

107

187 234

J43

.01

136

238

296

435

.02

174

J03 375 551

.03

200

350

432

633

.04

222

389

478

700

.05

242

422

517

757

.06

259

452

556

808

.07

275

479

584

854

.08

290

504

613

897

.09

303 527

640

936

.10

316

550

660

972

.15

373

647

776

1129

.20

423

732

868

1260

.30

510

879

1024

1478

.40

588

1011

1158

1663

.50 662

1136

1279 1829

.60

7J4

1257

1392

1983



57

T BLE

A 2 Continued)

Zone

Zone

Zone

Zone

Zone

g

0-400

400

-

OO BOO 1200 1200 2000 2000 3000

aUl1ga.l.s

)

ft.) ft.)

ft.)

ft.)

ft.)

.70

B05

1376

1499

2127

.80

876

1495

1602

2265

.90

~

1615

1702

2398

1.00

1020 1737

1801

2527

1.10

1094

1861

1899

2654

1.20

1169

198B

1994

2778

1.)0

1247

2119 20B9 2901

1.40

1327

2252 2176

3022

1.50

1409

2391

22BO

1.60

1493 2533

2375

1.70

1581

2681

2470

1.80

1672

2834

2566

1.90

396

1767

2993 2663

2.00

423

1865

3157

2761

2.10

450

1967

2860

2.20

478 2072

2959

2.)0

508 2183

3060

2.40

538

2297

3161

2 50

569

2417

2.60

602

2541



TABLE

Â-2

Continued)

A g

Zone Zone

Zone

Zone Zone

(m1l11gals) 0 -

4 4

-

8

800-1200

1200-2000 2000- 000

ft.) ft.) (:rt. )

ft.) ft.)

2.70

636

2671

2.80

671

28 9

2.90

7 8

2949

3.00

746

3 96

3.10

786

3.20

828

3 30

871

3.40

916

3.50

963

3.60

1 12

3.70

1 63

3.80

1116

3.90 1172

4.00

1231

4.10

1291

4.20

1355

4 30

1422

4.40

1491

4.50

1564

4.60

1639



59

T BLE A 2 (Continued)

Zone

Zone

Zone

Zone

Zone

Ag 0 -

400 400

-

800

800-1200 1200-2000 2000- 000

(m11ligals) ft.) ft.) ft.) ft.) ft.)

4.70

1720

4.80

1802

4.90

1890

5.00

1981

5.10 2076

5.20 2176

5.30

2279

5.40

2389

5.50

2503

5.60

2623

5.70 2747

5.80

2878

5.90

3015

6.00

3156



60

TAl LE

A 3

Construction

of Radii

for

the Graticules of the

First Outer Zone.

Zone

Zone

Zone Zone

Ag

o -

3000

3000 -

7000

7000 - 11000

11

000 - 15000

milligals)

ft . )

ft.

ft.

ft.

.0004

165

325

490

.01

498

973

14;0

.02

630

·1229 1805

.04

797

15.54

2285

.05

860

1677

2465

.06

916

1785

2622

.08

1011

1970

2895

.10

1093

2128

3123

.20

1395

2711

3981

.40

1795

~ 7 7

5105

.60

2091

4042

5929

.80

2339

4510

6613

1.00

2557

4921

7213

1.20

2755

5293

1.40

2938

56 6

1.60

3110

1.80

3274

2.00

~ 3

6554

9590



61

T BLE A 3 Continued)

Zone Zone Zone Zone

Ag

o

3000

3000

-

7000

7000 11000

11000 15000

mllligals)

ft.) ft.) ft. ) ft.)

2.20

3582

2.40

3729

2.60

3870

2.80

4009

3.00

4145

7875

11509

3.20 4278

3.40

4409

3.60

4538

3.80

4665

4.00

4792

9063

1J233

4.20

4916

4.40

5040

4.60

5161

4.80

5283

5.00

54 4

10185 14861

5·20

5524

5.40

5643

6.00

6000

11273

7.00

6587

12346

8.00

7175

13418

9.00

7767

14498



63

TiBLE A 3 Cont1nued)

Zone

Zone

Zone

Zone

Ag

o -

3000 3000 - 7000 7000 11000 11000-15000

aUl1ga1s)

ft.

)

ft .)

ft.)

ft .

)

JO. 00

9229

31.00

9843

32.00

10493

33.00

11184

34.00

11917

35.00

12694

)6.00

13520

37.00

14396

)8.00

15327



64

T BLE A 4

Construction

of

Radii

for

the

Graticules

of

the

Second Outer Zone.

a) Zone

5000

-

5000

g Radius Radius Radius

milligals)

ft .

mill1gals) ft . milligals)

ft .)

98.00

14796

126.00

25227

154.00

42521

100.00

15382

128.00

26192

156.00

44128

102.00

15988

130.00

27192 156.00

45765

103.00

16300

132.00

28230

160.00

47523

104.00

16616

133.00

28762

162.00

49316

106.00

17267 134.00

29306

163.00

50238

108.00

17941

1 36.00

30421

164.00

51177

110.00

18639

138.00

31580

166.00

53106

112.00

19364

140.00

32778

168.00

55109

113.00

19735

142.00

34021

170.00

57186

114.00 20114

143.00

34661

172.00

59340

116.00

20891

144.00

35313

173.00

60446

118.00

21698

146.00

36651

174.00

61574

120.00

22532

148.00

38039

176.00

63893

122.00

23397

150.00

39478

178.00

66298

123.00

23844

152.00

40972

124.00

24296

153.00

41740



T BLE

A-4 Continued)

b) Zone 5000-15000

Ag

Radius

.Ag

Radius

Ag

Radius

milligals)

ft.)

milllgals)

ft. )

mill1gals) ft.)

22.00

14809 45.00

26915

68.00

4 64e

23.00

15280

46.00

27524 69.00 44528

24.00

15754

47.00 28243

70.00

45423

25.00 16231

48.00

28762

71.00

46333

26.00

16712 49.00

29409

72.00 47260

27.00

17197

50.00

30057 73.00

48201

28.00

17607 51.00

30713

74.00

49159

29.00

18180

52.00

31 80

75.00

50133

30.00

18679 53.00

32059

76.00

51125

31.00

19182

54.00

32748 77.00

52133

32.00

19693

55.00

3J446

78.00

53160

33.00

20207 ,56.00

34157

79.00

54205

34.00

20728

57.00

34879

80.00

55267

35.00

21255

58.00

35613

81.00

.56350

)6.00

21788

59.00

36358

82.00

57449

37.00

22326

60.00

37117

83.00

58569

)8.00

22875

61.00

37866

84.00

59708

39.00

23429

62.00

8669

85.00 60870

40.00

23990

63.00

39464

86.00

62051

41.00

24558

64.00

40274

87.00

63253

42.00

25135

65.00

41096 88.00

64476

43.00

25720

66.00

41933

89.00

65721

44.00

26312

67.00

42783



66

T BLE A-4 Continued.)

c) Zone

15000-25000 Zone

55000-65000

[ .g

15000-25000

25000-35000

35000-45000

45000-55000

55000-65000

milligals)

ft. )

ft. )

f t . )

ft.)

:ft. )

.0004

545

850

1150 1400

1700

.01

1577

2417

3256

4065

4905

.02

1986

3049 4105

5158

6177

.04

2512 3858 5185

6497 7823

.05

2705

4293

5591

7015

8437

.06

288tJ

4421

5947

7460

8967

.08

3173

4873

6552

8218

9889

.10

3423

5257

7052

8865

10667

.20

4339

6664

8955

11234

13509

.40

5521

8473

11385

14284

17172

.50

5972

9162

12313

15448

18571

.60

6372

9776

13135

16475

19809

.80

7066

10835

14555

18257

21952

1.00

7664

11749

15781

19793

23797

1.20

8205

1.40

8682

1.60

9130

1.80

9550

2.00

9945

15225

20441

25633

30814

2.20

10321



T BLE

A 4

Continued)

c)

Zone

15000-2.5000

Zone 55000-65000

.Ag

15000-25000

25000-35000 35000-45000 45000-55000

55000-65

milligals)

ft . )

ft . )

f t . )

ft . )

ft . )

2.40 10680

2.50

10854

16605

22290

27951

33597

2.60 11024

2.80

11357

3.00

11676

17854

23963

30045

36113

4.00

13147

20088

26953

33788

40609

5.00

14468

22088

29629

37138

4463.3

6.00

15688

23935

32097

40230

48345

7.00

16839

25673

4421

43138

51838

8.00

17936

27331

6641

45913

55171

9.00

18996

28930

38779

48590

58382

10.00

20027

30487

408,56

51189

615Qt1

11.00

21034

32007

42886

53729

64554

12.00

22024

33500

44880

56223

67.548

13.00

23001

34971

~ 8 6

58683

14.00

23968

6429

48793

61117

15·00

24928

37875

50723

63532

16.00

25882

39313

52643

65935

17.00 26835 40747

54558

18.00

27785

42178

56470



68

TABLE

A q. Continued.)

c) Zone 15000-25000

Zone 55000-65000

Ag

15000-25000

25000-35000

35000-1.1.5000

45000-55000

55000-65000

m1l11gals)

ft.)

ft.

)

ft.) ft.)

ft.)

19.00

28745

43611

58381

20.00

29688

1.1.5044

60297

21.00

30644

46483

62217

22.00

31603

1 1 7927

64-145

23.00

32567

49380 66083

24.00

33537

50839

25.00

34-513

52309

26.00

35496

53790

27.00

36488

55283

28.00

37488 56789

29.00

38498

58309

30.00

39518

59844-

31.00

4054 8 61395

32.00

41590

62964

33.00

4264-2

64-550

34-.00

43708

66154

35.00

44786

36.00

45877

37.00

46983

38.00

48103



Fi

g

A 4

o-

\



4



i

g.

A 5

4



800



12



Fig. A 7

12



Fig.

A 8

2



A g

A 10

300



~ - - - - - - , - - - - - - - _ - - - - r - - - - _ . - . -

1



g A 11

7

A 12

1100



~

5

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