I MUELLER, Ivan Iatvan. the GRADIENTS of GRAVITY and THEIR APPLICATIONS in GEODESY* the Ohio State University, Ph.D., 1960 Geology
This dissertation has been microfilmed exactly as received
| Mic 00-4118 X ’ I MUELLER, Ivan Iatvan. THE GRADIENTS OF GRAVITY AND THEIR APPLICATIONS IN GEODESY* The Ohio State University, Ph.D., 1960 Geology
University Microfilms, Inc., Ann Arbor, Michigan THE GRADIENTS CP GRAVITY AND THEIR
APPLICATIONS IN GEODESY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the, Graduate School of The Ohio State University
By
IVAN ISTVAN MUELIER, DIPL. ENG.
******
The Ohio State University i960
Approved by
Adviser Department of Geology (Division of Geodetic Science) TO THE MEMORY OF ROLAND EOTVOS PREFACE
In 1901 Professor Roland Efttvtts, Hungarian physicist, in his opening speech as President of the Hungarian Academy of Sciences,
said that scientists have tried to determine the shape and size of the Earth for centuries but
... good results can be obtained only if we concentrate upon the investigation of the Earth's gravity field, since it was gravity which determined the shape of the oceans and’ the surface of the continents when the Earth was formed....
He mentioned that Geodesy is that science which deals with these problems but
...at the present time (1901) Geodesy cannot determine the detailed shape of the Earth. It cannot answer questions such as: What does the detailed shape of the surface, formed by gravity, look like? What is the shape of the surface of a body of water immediately around us? How large is the curvature of this surface, and in which di rection is this curvature a maximum? In which direction dees gravity change with a maximum rate, and what is the magnitude of that change? All of these questions are yet unanswered. The geodesist is like the farsighted person who can enjoy the sight of the distant blue moun tains but is unable to read a letter bringing good news. This same man, to use another analogy, can determine the curvature of the oceans, but not the curvature of a glass of water.
Etitvtis, in order to bring all these problems closer to solution, had designed an instrument, the "torsion balance," which measures the horizontal gradients of gravity. In 1915, only a few years after the first Instruments were in use, Helmert wrote that Geodesy has 1. two miraculous instruments, the level and the torsion balance. Both are very simple instruments^even so they give crucial informa tion about the shape and structure of the Earth. In spite of this fact, the torsion balance has been used only in commercial geophysi cal explorations and not for geodetic purposes. Even in geophysics it was superseded by the faster and cheaper gravimeters after about thirty years' use. It is interesting to note that with the develop ment of the sensitive gravimeters it has become possible to measure the vertical gradient of gravity with a relatively high accuracy.
The horizontal gradients, together with the vertical gradient, i.e., the torsion balance together with the gravimeter, can give even more geodetic information than described in the Efltvfls quotation above.
Thus we see that the advent of sensitive gravimeters leads to the further development of geodetic applications of the torsion balance.
However, not much further has been done as far as geodetic applica tions are concerned since Efitvfls' death. This can be partly ex plained by the fact that geodesists in the past did not need very detailed information about the gravity field of the Earth or about the shape of the equipotentlal surfaces. At present the situation is different. In the age of space flight and of the developing world geodetic system even the small details of the gravity field may have Increasing significance. Someone might say that we still do not need all of the detailed geodetic data which it is possible to obtain from the gradients of gravity. But with the present rate of development of science the time is near when the best of these data may not be sufficient. When the author first became interested in this topic in 1951*-> his original aim was to emphasize the importance of the horizontal and the vertical gradients of gravity in geodesy. During this work, which continued until the end of 1956 and was started again the be ginning of 1959# the writer realized that no publication was avail able which covered all of the problems of determining and applying the gradients of gravity in geodesy. One purpose of this paper is to report and Introduce these problems with as many details as are needed for practical applications. (For instance, help tables for computations which are out of publication were also included. See
Tables 3> 7 and 8 .) The author collected and used all of the available publications (350, mostly in German, Russian, French,
English and Hungarian languages.) Most of these publications are listed in two sections of the Bibliography. In the first section those publications are given to which reference is made. The second section contains the rest. Note the relatively few publications in
English, especially in connection with the horizontal gradients of gravity. Another purpose of this paper is to fill this gap in the
English coverage of the subject. The third section of the Biblio graphy contains those papers which deal with the geophysical appli cations of the gradients of gravity. Since the topic of this dis sertation is the geodetic applications of the gradients of gravity, the third section is included only for the sake of those who are interested also in the geophysical applications. Vi In preparing this paper several conclusions and suggestions
have been made (see chapters 2*21, 2.22, 2*31? 3*5* 3*6? 4.14, 4.33/
and 4.6.) These together with chapters 3*1* 3»7. 3*8# 3*9* 3«10»
4.4 and 4.5 are the contributions of this dissertation to geodesy.
During the summers from 1954 to 1956 numerous field measurements were carried out. The computations based upon measurements together with other computations have been also included as examples in the proper chapters (see Tables 1, 2, 5> 6 , 10, 14, 15,and l6 .) In addi tion, several formulas have been derived, which are given also in other publications but without derivations (see Formulas 49, 107, 135
136, 181, 184, 193 and 196.)
This paper is dedicated to the memory of Professor Roland
E&tv8s,and therefore the author feels that it is proper to end this preface with another quotation from his work: "The scientist is not a person who knows all about science, but one who forwards it."
Measured against this scale the author considers himself only on the very first rung of the ladder of science. vii
ACKNOWLEDGMENT
The author wishes to acknowledge his indebtedness for the help he received during the preparation of this dissertation, to Pro fessor W. A. Heiskanen, Director of the Institute of Geodesy, Photo- grammetry and Cartography, The Ohio State University, who has been his adviser; to Dr. R. A. Hirvonen, Professor of Geodesy of the Finland
Institute of Technology, to Dr. H. J. Pincus, Professor of Geology of
The Ohio State University, the members of the reading committee; and to Mr. R. J. Feely for his great work in correcting the manuscript.
The first part of this paper in a shorter and somewhat different form was prepared in Hungary, and was circulated among leading Hun garian scientists for criticism. The author is especially grateful to Professor I. Redey, who has been his teacher and adviser for more than five years; to Professor A. Tarczy Hornoch; to Dr. J. Renner; and to Dr. L. Homor&Ly, for their constructive criticisms.
An Etttvfts torsion balance, has been made available for field measurements by Mr. T. Dombay, Director of the Hungarian Roland
EtttviJs Geophysical Institute, whom the writer wishes to thank.
Special thanks are due to the author's wife, Mrs. Marianne
Mueller, for her patience and help during the preparation of this dissertation; to Mrs. Margery Corrigan for her excellent typing; to Mr. Leslie Cunningham for his help with the figures and the repro duction; and to all others who have assisted with this paper. viii
TABLE OF CONTENTS
; Chapter Title Page
1. INTRODUCTION ...... 1
1.1. The Gravitational Force...... 1
1.2. The Potential of the Gravitational Force . . . « ■ . . ' 4
1.3* The Gravity. The Gradients of Gravity...... 7
2. THE DETERMINATION OF THE HORIZONTAL GRADIENTS OF GRAVITY . 13
2.1. The Principle of the EdtvBs Torsion Balance . . . . 13
2.11. The Curvature Variometer...... 13
2.12. The Horizontal Variometer ...... 24
2.121. The Horizontal Variometer with One Swinging System ...... 24
2; 122. The Horizontal Variometer with Two Swinging Systems...... 30
2.13. The Gradiometer...... 36
2.2. The Torsion Balance Apparatus...... 39
2.21. The Different Instruments...... 39
2*22. Field Operations...... 58
2.3. The Reduction and Transformation of the Torsion Balance Measurements ...... 6l
2.31> Definitions...... 6l
2.32. The Normal Gradients...... 64
2.33• The Terrain Effect...... 67
2.331«- Analytical Methods...... 67
2*332. Mechanical M e t h o d s ...... 86 ix TABLE OF CONTENTS
Chapter Title Page
2.34. The Cartographic E f f e c t ...... 91
2.341. Analytical Methods...... * . . 91
2.342. Mechanical Methods...... 94
2.343. Graphical Methods ...... 95
2.35* The Transformation of the Torsion Balance ■ Measurements...... 100
2.4. The Determination of the Instrument Constants. . . . 103
2.41. The Instrument Constants...... 103
2.42. The Determination of the Quantity — ...... 104
2 A 3 . The Determination of the Torque Constant, r • 111
2.431. The Swinging Period Method...... Ill
2.432. The Cavendish Method...... 114
3. THE APPLICATIONS OF THE TORSION BALANCE MEASUREMENTS IN G E O D E S Y ...... 119
3.1. Introduction and the Problem of the Analytical Determination of the Equipotential Surfaces of Gravity...... 119
3*2. The Total Horizontal Gradient...... 124
3.3* The Horizontal Directivity or the Sphericity Coefficient...... 123
3*4. The Radius of Curvature of the Plumb Line...... 135
3»5* The Reduction of Astronomic Coordinates to the Geoid, to the Telluroid, and to the Quasi-geoid. . . 138
3.6. The Densification of a Net of Deflections of the V e r t i c a l ...... 142 X
TABLE OF CONTENTS
Chapter Title Page
3.7* The Determination of the Geoid ...... 154
3.8. The Determination of the Geopotential Value and the Shape of the G e o p s ...... 159
3.9* Correction of Observed Directions for Devia tions (Azimuth Correction) ...... 163
3.10. The Effect of the Inner Circle upon the Gravi metric Determination of the Absolute Deflec tions ...... 165
4. THE DETERMINATION OF THE VERTICAL GRADIENT OF GRAVITY. . 169
4.1. The Determination of the Vertical Gradient by Measurements ...... 169
4.11. Measurements by the Classical Method. . . . 169
4.12. Measurements b y Means of Gravimeters. . . . 171
4.13. Measurements by Means of Special In struments ...... 178
4.131. The Haalck Horizontal Pendulum. . . 178
4.132. Other Instruments ...... 186
4.14. Conclusions...... 187
4.2. The Normal Vertical Gradient ...... 188
4.3. The Computation of the Vertical Gradient Anomaly from Gravity Anomalies ...... 194
4.31. The Principle of the Computation...... 194
4.32. The practical Computation...... 197
4.33. Remarks Regarding the Applications...... 201
4.4. The Computation of the Relative Vertical Gra dient from the Torsion Balance Measurements. . . . 202 xl
TABIE OF CONTENTS
Chapter Title Page
U.5. The Computation of the Regional Vertical Gradient Anomaly from the Deflections of ...... the Vertical ...... 207
k.6, Applications...... 210
Bibliography
1. References ...... 212
2. Other Publications in Connection with the Topic of this Paper...... * ...... 218
3. Publications in Connection with the Geophysical Applications of the Gradients of Gravity ...... 223 xii
U S T OF TABLES
Number Title Page
1 . Computation of the Horizontal' Gradients' from the Measurement Results ...... 3^
2. Comparison between the Different Torsion Balances. . . 57
3 . The Normal Horizontal' Gradients’ in Function* of the ' Latitude...... 69
1+. Coefficients for Terraln-effect Computations . .... 80
5. Leveling Record...... : ...... 82
6 . Computation of the Terrain Effect...... 83
7. Help-table for Terrain-effeet Computation...... 8U
8 . The Numerov Coefficients...... 86,87
9 . Computation of the Terrain Effect with the Numerov Method...... 89
10. Computation of the Cartographic Effect ...... 97*98 K 11. Determination of the Instrument Constant — ...... 110 T 12. Comparison between Astro-geodetic and Torsion Balance Deflections...... 153
13. Comparison between Astro-geodetic and Torsion Balance Deflections ...... 153 l*t-. Formulas for Densification of a Net of Deflections •157,158
15. Vertical Gradient Measurements by Means of Gravi meters...... 176,177
16. Vertical Gradient Determinations by Means of Gravity A n o malies...... 200 1
1. INTRODUCTION
1.1. The Gravitational Force
The gravitational force is the effect produced by an object on
its base in vacuum and in the gravity field of the Earth. This
force is the resultant of all the acting forces in the Earth's po
tential field. The two most important components of the gravitational
force are the attraction of the Earth's body (t ) and the centrifugal
force (C) caused by the rotating Earth.
The law of the attraction component was laid down by Newton.
This law states that the mutual force of attraction T of two bodies
with masses m and m' separated by distance r is:
T ir2 mm' - r2
2 —9 -1 3 ”2 where k is the Newtonian gravitational constant (66.73 10 gr cm sec ).
[HEISKANEN-VENING MEINESZ, 1958].
The centrifugal force at any point of mass m is perpendicular
to the Earth rotation axis and its magnitude is expressed as follows:
2 C = rnpo) where p is the radius of rotation, to is the angular rotation speed of the Earth.
In addition to these T and C components there are many more forces acting in the gravitational potential field of the Earth. The attraction of the sun, the moon and the planets are the most ' prominent among them. In the first approximation we are going to
neglect the effect of all celestial bodies. To justify this step,
note that if the attraction of the Earth at one point on its surface i < is T_ then the maximum attractions of the sun and of the moon at the Cj same point are only
1 10“6 T, S 19.31 E
’m ° S ^ 2 5 10"6 TE and these effects can be observed only by means of the most accurate gravimeters [SOROKIN, 1953].
The positions of the celestial bodies relative to the Earth change with time. There are also constant changes in the distribu tion of the attracting masses both inside and outside of the Earth.
Then, too, the rotation speed of the Earth is not constant. All these facts prove that the gravitational force at one point is not constant but changes with time. However, this change is very small and, therefore, in the second approximation in our investigation, we neglect it.
There are some presently unexplained phenomena which also in fluence the gravitational force. Such a question is, for instance, whether the attraction between a pair of moving bodies ( is the same as the attraction between two motionless bodies. According to Professor Weber the mutual attraction between two moving bodies is
T = -k2 S f (1 ♦ ig c '2 + % rp”) r c c
where c is the propagation speed of the attraction
c' is the relative speed between the two bodies m and m'
p" is the relative acceleration between these masses*
The existence and the numerical value of the attraction-propagation
speed is very uncertain [REDEY, 19^9]•
Similarly, we do not have definite knowledge about the influence
of the temperature, energy radiation and other physical properties of
the masses on the attraction.
Summarizing our analysis we shall assume that:
1. The gravitational force is the resultant of the attraction
of the Earth and the centrifugal force caused by the j Earth's rotation;
2. The gravitational force is independent of the influences of
the celestial bodies;
3* The gravitational force at one point is independent of the
time;
h. The gravitational force is independent of the temperature
or energy radiation of the bodies involved.
5* The gravitational forces between moving bodies and that be
tween motionless bodies are the same.
We should emphasize two properties of the gravitational force which follow from the first assumption above and which have great significance in our investigation:
1. The gravitational force continuously changes its magnitude
and direction in space.
2. The gravitational force is a potential force and, therefore,
it follows the law of persisting energy, i.e., the work
done against it depends only on the height difference of the
starting and end points.
1.2. The Potential of the Gravitational Force
Since the gravitational force is a potential force, there exists a W function, which has the property that its first partial deriva tive taken in any direction is equal to the gravitational force components in the same direction. Let us find this function. In order to make our investigations independent of the mass, we always imagine 1 gram mass at the investigated point. Using a rectangular coordinate system we place the origin at the center of gravity of the
Earth, and we make the z axis coincide with the mean rotation axis of the Earth. The x and y axes are in the plane of the equator. In this system at any point P = (x,y,z), we want to determine the G , G , x y G components of the gravitational force. The attraction at this z point P caused by a mass element of the Earth, dm, at an arbitrary point, A = (£,tj,£), referred to the x, y, z coordinate system is where r is the distance between A and P.
r2 . (I - xf * (T) - y f + « - zf
The components of the attraction in the direction of the coordinate axes can he obtained by simple multiplication by the cosine values:
_ _k2 to |_;jc x r r
dl = -ka « S 2 ^ 2 y r2 r
2 dm t - z = 'k “ 2 r r
The attraction components of the whole Earth at the same point P can be computed by integration:
Tx = -k2 / dm
Ty = -k2 f dm
T = -k2 / dm z J r3
The centrifugal force components obviously are Therefore, the components of the gravitational force at the point P are
Gx * Tx + Cx =» -k2 / — - 3- dm + o>2x r
Gy = Ty + Cy = “k2 f ^ +
G_ = T + C = -k2 f S. I. z dm Z Z Z j r
From these equations it is relatively easy to find the potential func
tion of the gravitational force [HEISKANEN-VENING MEINESZ, 1958]•
It is
It is easy to show that the first partial derivatives of this func
tion, in the x, y, z directions, and the G , G , G components above x y z are, as stated, equal. Naturally, the partial derivative of W in any
s direction is also equal to the G_ component of the gravitational
force
|| = 0 COS (G,s) = Gs
Let us investigate two special cases. In the first case we assume
that the s direction isperpendicular to thegravitational force. In / \ SiW this case cos (G,s) = 0 and^ » 0. After integration we get the following result
W » constant •
Since W is the function of the x, y, z coordinates, the W a constant equation is the equation of a surface on which the potential is constant and which has the property that at any point the gravi tational force is perpendicular to the surface. This type of surface is generally called "equipotential surface” or "potential surface."
Different surfaces correspond to different constants. We can get with a certain constant an equipotential surface which coincides with the surface of the undisturbed oceans. This surface is called the "geold."
In the second case we assume that the s direction is parallel to the gravitational force. In this case cos (G,s) = 1 and Sw ^ = G. This demonstrates that the gravitational force is perpendic ular to the equipotential surface, as stated.
We could see that the W "potential” or "force function" de pends only upon the mass and on the position (x, y, z coordinates) of the investigated point. The unit of the potential is 2 -2 gr cm sec , i.e., it is work. Each equipotential surface is characterized by the work which has to be done by the gravitational force bringing 1 gram mass from the surface to infinity.
1.3. The Gravity. The Gradients of the Gravity
We mark as before the x, y, z components of the G gravitational force with G . G„, G . Then, x y z '
&M r , dw n ■ aw r m 5E ■ Gx ' Sy - V SS ■
_p The units of these force components are gr cm sec . In general, any force can be expressed as the product of an acceleration and a mass. Therefore, the gravitational force also con be expressed as the
product of the gravitational acceleration, g, and the investigated
mass, m.
G o mg gr cm sec-2
In our investigations m = 1 gr } therefore,
-2 G = g gr cm sec
In other words, the numerical value of the gravitational force and
of the acceleration is equal. This fact holds also for the force
components:
Gx ■ "B* * Gy = "«y * \ - "Sz where g , g , g are the gravitational acceleration components in x y z the x, y, z directions.
It follows from (l) that
dW . dW dW 5 c - ^ x 5 ^ " “Sy ' S ’ = “Sz or if m = 1 gr
|W . | a c . & - e (2) Sc “ gx 9 Sy sy ' ^ " gz •••12;
In other words, the numerical values of the partial derivatives of the potential and the gravitational acceleration components are equal.
However, we have to realize that (2) does not hold when units are .2 concerned. The unit of the partial derivatives is gr cm sec and of _p the acceleration components is cm sec . Keeping this in mind we can use the expressions "gravitational force" and "gravitational accelera tion" with common meaning and call them by the shorter name "gravity*11
Let us investigate the change of gravity in a limited field
where linear variation can be assumed. The origin of an arbitrary
rectangular coordinate system is 0. The gQ is the gravity at this
point; 6x q j gyQ » Bzo are its components in the directions of the co
ordinate axes. Close to the point 0, an arbitrary point is located by
its x, y, z coordinates. The g , g , g gravity components at this x y z point can be computed by means of the Taylor series:
dgv dg„ dgv 6k = bx o + (sr}ox + ( sr + z + •••
dgv be bg gy ‘ gyo + X + (5 y ^ 0 y + (5E^ 0 2 + •" — <3>
^gz gz - gzo + (5T)0 x + ( W }o y + Z + ” ■
It follows from (2)
b 2 !,! 3gx aSt ;• S T = ^2 ' S T = 55?’ *" etc* and since the investigated field is limited (3 ) becomes:
fb2^ \ \ gx - gxo = x + y + Z
gy - gyo - x + y + <&>„ 2 - W
gz - gzo ■ & o x + y + (S }o 2
These equations give the change of gravity in the directions of the 10
coordinate axes in the neighborhood of the point 0. The second
partial derivatives of the potential in (4) are the so-called “gra
dients of gravity” or just "gradients." They express the change of
a certain gravity component in a certain direction for a unit dis
tance. The following relations exist between the gradients:
b2^ b2^ _ _ dgy bxby = 5y5x Sy = cbc
b2^ b2^ dsy Sgz ^ = ^ = 5r- = 5 T ...(5)
b2^ b2^ hgz ^Sx S5E = = 5F" = £ “
Another relation is expressed by the Poisson equation
b2^ b2^ , b2^ ,,.2 0 2 ,, x — 5 + — 5 + — 5 = - w k p+ 2m ...(6a) bx b y bz
or, if the density is zero, by the Inplace equation,
b2^ b2^ b2^ . 2 //-. n — o + 5 = 2a> ...(6b) & T dy2 3z2
These general equations are valid— depending on the density--at any point within the gravity field.
Fromequation ( we can see that nine derivatives are needed to determine the change of gravity in a limitedfield where
linear variation can be assumed. Between these nine constants we have the relations expressed by (5 ) which show that three ofthem are equal to three others. Between the remaining sixgradients we have relations expressed by either the Laplace or the Poisson equation therefore, we can state that five constants are necessary to determine the change of gravity in a single point. If we know these five values then the change of gravity in the directions of the coordinate axes,
i.e., the gx -gxo > Sy “ g ^ Sz-gzo values are determined.
Since in this paper we are going to deal frequently with the derivatives of the potential we introduce the following customary symbols:
dW TT . dW __ . dW „ cbc 53 x ' 5y y ' ciz z
^ = W ; ^ = W 4 = ...(7) ax2 ** dy W 5za zz
= y . _ „ . dxSy xy 9 5y§z yz
Up to now we used an arbitrary coordinate system. Let us change this arbitrary system to a coordinate system whose z axis co incides with the local vertical at 0 (positive downwards), with posi tive x and y axes pointing respectively to the North and East. In this system
gxo ^ 9 gyo 0 9 gzo 3 go and,therefore, (4) becomes 12
We want to repeat that the W , W ... W quantities (gradients) are xx xy zz considered as constants near the 0 point and that they express the change of a gravity component in a certain direction. For instance, W xy symbolizes this change of the g component in the y direction and also the change of the g component in the x direction, both for a unit y distance.
The unit of these second partial derivatives can be derived from 2 -2 the unit of the potential, which is gr cm sec • The unit of the first -2 partial derivatives, i.e., of the gravity components is gr cm sec .
Therefore, the unit of the second partial derivatives, i.e., of the - o gradients is gr sec or in a more appreciable form, gr cm sec /cm.
The latter expresses the physical meaning of the gradients; the change of gravity component in a unit distance.
The potential generally is given in ergs, the gravitational force in dynes, the gravitational acceleration and the gravity in gals. .2 "Gal" is named after Galileo Galilei (l gal = 1 cm sec ). The second partial derivatives or the gradients are given in E units (l E = —Q *• p 0.0001 gal/km - 10 cm sec" /cm). These units were named in honor of Etitvfls. 13
2. THE DETERMINATION OF THE HORIZONTAL GRADIENTS OF GRAVITY
i 2*1. The Principle of the Efttvfls Torsion Balance
2.11. THE CURVATURE VARIOMETER
At the end of the last century Roland Etitviis designed and con
structed a very sensitive and rather clever instrument for measuring
of the horizontal gradients of gravity [E&TV&S,l896]. Actually he
designed two types of instruments. With the first one which he
called a "curvature variometer" he was able to measure the quantities
W and W. = W - W . With the second type of instrument, which xy A yy xx was called a "horizontal variometer," he could measure the quantities
W , W , W and WA. In the English literature there is no dis- xy' xz* yz A tinction between these instruments and both of them are called
"torsion balance" or"Etitvfls torsion balance."
We deal first with the curvature variometer. The instrument
is based on the Coulomb torsion balance, which was used to determine o the gravitation constant k [BUCHHOIZ, 1916]. This apparatus has been perfected by Efltvfis for more sensitive measurements. In prin
ciple this consists of a horizontal beam with weights at each end
(swinging system). The beam is suspended at its center from a fixed point by means of a very thin and precisely manufactured fiber.
The direction of the fiber goes through on the center of gravity of
the swinging system. In Figure 1 point A is the suspension point, Figure 1 Figure 2
*z
Figure 3 Figure 4
14 1 5 and Pg are the weights/and point S is the center of gravity of the swinging system. The fiber AS, which is called a "torsion fiber," tends in the direction of gravity at point S. The gravitational forces, G^ and Gg respectively at P^ and P y are not parallel with the gravitational force at S. Let us divide the forces, G^ and Gg, into three rectangular components. The first component should be parallel to the vertical at S. The second component should fall into the PnSP0 axis of the beam and the third component should be perpen- (Figure 2). dicular to the other two/ These last components produce a torque in the plan perpendicular to the torsion fiber. This torque swings,or rather rotates, the beam and weights about the fiber. The rotation theoretically will continue until the torque becomes equal to the in creasing torsion resistance of the fiber. In this position the swinging system will be in equilibrium. Let us express this equili brium in mathematical form. We investigate this problem in a rec tangular coordinate system, whose origin is at S. The z axis coincides with the fiber AS. The axes, x and y, are perpendicular to z and to each other. The directions of the axes x, y in the horizon tal plane are arbitrary (Figure 3). In this arbitrary position the axis of the beam P^P^ should form an angle, a, with the x axis. In this coordinate system x, y and z are the coordinates of an arbitrary mass element of the beam, dm. According to the laws of mechanics we can substitute for the gravitational force at dm three rectangular force components acting at S and three torque moments in the three coordinate planes. Since any force acting at the center of gravity 16
does not cause rotation, we can neglect the force components at S
where beam rotation is concerned. Similarly, the torque moments in
the planes, xz and yz, do not cause rotation in the horizontal plane,
xy. Therefore, the swinging system will rotate only from the in
fluence of torque in the plane xy. When this torque and the torsion
resistance of the fiber become equal the swinging system will be in
equilibrium. We designate the x and y components of the gravitational
force G at dm as G and G . The torque moment, df, about the z axis x y caused by these force components, i.e., the torque in the xy plane
is (Figure 4),
df = (xGy - yGx ) = (xgy - ygx ) dm .... -*.(9)
The torque moment F, caused by the gravitational force components acting on the whole swinging system, can be computed by integration,
'-yg^Xdm______... (10) where the integration should be extended to all mass-elements of the beam and of the weights.
Let us assume now that the variation of gravity is linear within the field of the swinging system. In this case the gravity components g and g at dm, according to the expressions in (8 ) are: x y
g = W x + W y + z °x xx xjr w xz
g„ = W, x + W y + W,„ z y yx yjr yz
The coefficients W , W , ... W relate here to the point S. The xx xy yx x, y,z values are coordinates of the mass element dm. We substitute 17
these g and g values Into expression (10) to get x y
F = / [ (Wyxx + Wyyy + W^zjx - ( y + Wxyy + Wx2z)y] dm
After multiplication separate the terms 2 F = / W x dm - f W xy dm + / W _xz dm - M yx M 77 M ^
" ^ Wxxxy “ f Vxyy2 dm " / Wxzyz dm M M M and since the W , W .... W values are constants at S. this yx' yy xz expression yields
^ xy dm ♦ Wyx/P = w^ j xy dm ♦+ WWyx/P f (x - y )dm + W x z dm - W f yz dm ...(11) M M
The integrals in this formula should be extended to all mass elements of the swinging system.
We introduce now a new rectangular coordinate system, having x, y, z axes. The origin of this system is at the center of gravity,
S. The axis z coincides with the fiber.-AS.^ The axis x coincides with axis P^SPg of the beam, i.e., it should form an angle, a, with the x axis. The y axis should be perpendicular to the x axis in the plane xy. Between the new system (x, y, z) and the former system
(x, y, z) we have the following transformation equations:
x = x cos a - y sin a
y « x sin a + y cos a ...(12)
z a z 18
We substitute these values into expression (11) to get
F = WA/ (x cos a - y sin a)(x sin a + y cos a) dm + n w r — p — — 2 + W J E(x cos a - y sin a) - (x sin a + y cos a) ] dm + xyM
+ W / z(x cos a - y sin a) dm - W / z(x sin a + yZM M
+ y cos a) dm ... (13 )
We have the following trigonometric relations.:
sin a cos a = - sin 2 a a. 2 2 cos a - sin a = cos 2 a
Substituting these expressions into equation (13)> we get after reduction: ■j n n ^ — F = WA ~ sin 2a/ (x - y ) dm + W cos 2af xy dm + A d M A M + W cos 2a f(x - y ) dm - W 2 sin 2 a /xy dm + Xy M Xy M ■f W cos a/ xz dm - W , sin af yz dm - yz m yZ M - W sin af xz” dm - W cos af yz dm . ..(14) XZ M XZ M
Since the swinging system is symmetrical with reference to the planes, xz, yz and xy, three of the integrals in expression (14) are zero:
/ xy dm = 0 ; / xz dm = 0 j / yz dm = 0 ...(15) M M M
With these expressions removed (1^+) becomes
P = WA \ Sln 2(X f y2^ ^ + Wxy cos 2CL f (x2 - y2 ) . ..(16)
The moment of inertia of the swinging system referred to 19
the z axis is in general form,
K = / (x2 + y2 ) dm=/x2dm + /y2dm M MM
y2 Since in practical swinging systems the proportion is less than
-2 * 1/1*000 we can assume that f y dm = 0. In this case the moment of M inertia becomes:
K = / x2 dm M
But since we can add or subtract zero from any quantity without
changing it we may write
/ x2 dm S / x2 dm - / y2 dm £ f (x2 - y2 ) dm M M M M and, therefore, get
K = / (x2 - y2) dm -.(17) M
With this, equation (l6 ) becomes
F = W ^ K sin 2a + W K cos 2a ...(l8) A 2 xy ' '
This torque-moment is equal to the torsion-resistance of the torsion-fiber when the swinging system is in equilibrium. The tor
sion-resistance of the fiber is
fpT ...(19) where cp is the angle of rotation between the torsion-free po
sition and the equilibrium position of the beam and r is the torque constant of the fiber (dyne cm per radian).
At the position of equilibrium equations (18) and (19) are 20 equal, therefore,
rep = WA 5 K sin 2a + Ww K cos 2a La Cm xy or, dividing both sides by t,
WA sin 2a + — W cos 2a (2 0 ) A t xy
The torque constant, r, depends upon the shape, length, diameter and material of the fiber. Therefore, the — quantity is a constant in one particular instrument. The determination of this constant is a laborious procedure which can only be done in a laboratory. We will deal with this problem in Chapter .
The angle, a, is the angle between the axis of the beam in
equilibrium position and an arbitrary direction (the x axis). In order to be able to determine, i.e., to read this angle, the swing ing system together with the torsion fiber are placed in a case. The case can be rotated about a vertical axis rigidly connected to the base of the instrument. This base supports the whole apparatus. On the base there is a horizontal circle with angular divisions. The position of the case, i.e., the angle, a, can be read on this horizon tal circle by means of an index connected to the case. If we know the azimuth of the zero index and the position of the case, the azi muth of the case can be computed. The direction of the zero index is considered to be the +x axis. When the apparatus is set up this direction points North. This can be effected by means of a compass mounted on the base. In the working position the vertical axis, the center of the horizontal circle, and the center of gravity, S, have 21 to “be in a vertical line.
There is another scale mounted on the case andused to deter mine the angle (p. This is the angleof rotation of the swinging sys tem from torsion free position to equilibrium. Since in the gravity field the fiber is always twisted, the torsion free position can not be determined directly. The 5 ^ 8^ .... 8^ directions should correspond to different beam positions in equilibrium and 8q should be the assumed direction of the beam in torsion free position. If we knew this value the angle cp could be computed by i cp = 6 - & o ,..(21)
Let us substitute this value into equation (20)
S - 6 = §- W A sin 2a + - W cos 2a ...(22) o 2t A t xy ' 1
In this equation we have only three unknowns: 8 , W. and W . The O Za Xjr value 8 can be read on the scale mounted on the case. The azimuth a can be read from the base horizontal circle. The letters K and t are instrumental constants. We can determine the three unknowns by reading the & values in three different positions of the case while at the same station. Let us assume that a^, a^ and a^ are the cor responding azimuths of these three positions. At each station we can write three equations:
_ K_ - 6 sin 2a. 1 - “oo - 2T ''aA sln ^1 + §7t wVxy cos 2*1 K - 8 sin 2ag + - W ...(23) 2 0 o 2 2rt A 2 t xy 2
e* _ K_ - 8 sin 2a 0 + ^ W 3 woo “ 22tt "AA ‘ 3 ^ t "xyxy 1 3 22
From these equations the three unknowns can he computed.
The following is the procedure, viz: At each station— after
setting up the instrument oriented towards the North and after level
ling— we turn the case to three different azimuths, cc^, a£g and
in turn. At each azimuth we wait until the swinging system reaches
the equilibrium position, i.e., stops swinging. The values are then
read, one value, &, at each azimuth. Knowing the instrumental con
stants K and t, the torsion free direction 5q , the gradients
and W can be computed by means of equations (23). The computed xy \ gradients refer to the center of gravity of the swinging system, S.
The angle,
cp a 8 - 5 , is observed n (a) indirectly by means of a
mirror mounted on the
beam close to the suspen
sion point. This mirror
swings with the beam;
therefore, the angle n of rotation of the mirror is equal to that n-n of the beam. The mirror
is observed through a -h i , telescope with a verti
cal cross hair. The telescope is mounted on Figure 5 the case. Directly below the telescope a scale is mounted on the case (Figure 5)» The re flected picture of the scale in the mirror is observed through the telescope. Different readings on the scale as read beneath the vertical cross hair correspond to different positions of the mirror and hence, also, of the beam (Figure 5/a)« When the mirror is parallel to the scale the reading should be nQ . After an arbitrary rotation, cp, the corresponding reading should be n« From Figure 5/b it is obvious that n - n tan 2 q> - tan 2 (6 - 5 ) = ------° D where D is the distance between the mirror and the scale expressed in the units of the scale divisions. Since the angle cp is small the equation above yields n - n 2
Substituting this value into expressions (23)s solving them for n - nQ and using the abbreviation,
we obtain the following equations? 2b
In these equations the values, n^, n^, n^, Q^, and ci^ are readings as before but a is an instrument constant. The values nQ ,
W and W are the unknowns to be determined by means of the above A xy equations. The actual sensitivity of the curvature variometer de pends upon the constant, a.
Currently, the curvature variometer has only historical value in so far as practical applications are concerned.
2.12. THE HORIZONTAL VARIOMETER
2.121. The Horizontal Variometer with One Swinging System
Figure 6/a shows the swinging system of the horizontal vario meter [EOTVOS, 1896]. The beam--similar to that of the curvature variometer— is loaded with the weights and Pg. A difference is that the weight, Pg, is suspended from the beam by means of a thin metal fiber of length, h. The masses of the weights are so deter mined as to insure equilibrium in the horizontal position of the beam. In this position the center of gravity of the swinging system,
S, lies below the beam and almost in a line with the torsion fiber.
This type of torsion balance is also manufactured in two other models.
Figure 6/b shows the swinging system of a model with a beam having a
Z shape [ SCHWEYDAR, 1926]. The weights are placed at the upper and lower ends of the beam, i.e., they are rigidly connected to the hori zontal part of the beam by vertical extensions. Figure 6/c shows a model with an inclined swinging system [IMHOF AND GRAF, 193^]* In the Soviet Union a torsion balance was designed with a beam similar to the one in = 0= Figure 6/b but turned by S' h (a) 90° in the vertical plane
[NIKIFOROV, 1927]. Here after we shall call the first model a torsion balance, the second model h 00 a Z-beam torsion balance and the third model an ln- A clined-beam torsion balance.
P All of them belong to the 1 family of horizontal vario (c) meters. At the present time
Germany, the Soviet Union and Hungary are manufac Figure 6 turing these instruments.
The torsion balance is manufactured in Germany and in Hungary. The Z-beam and the inclined- beara torsion balances are manufactured in Germany and the Soviet
Union.
Here we will deal with the principle of the original torsion balance, i.e., of the horizontal variometer only. The principle of the Z-beam and of the inclined-beam torsion balances are similar and the results of the measurements are the same*
The torque acting on the swinging system of the horizontal
variometer is also described by equation (l1*-). In this expression
the gradients WA, W , W _ and W refer to the center of gravity of A xy yz xz • the curvature variometer. This point geometrically coincides with
the suspension point, 0, of the horizontal variometer (Figure 6/a).
The coordinates x, y, z determine the position of a mass element, dm.
The origin of this coordinate system is at 0; the z axis coincides
with the torsion fiber, the x axis coincides with the axis of the
horizontal part of the beam. The y axis is perpendicular to z
and to x.
Since the swinging system is symmetrical about the plane,"xz,
two integrals in egression (l^) are zero;
/ xy dm =* 0 ; /. yz dm = 0 (25) M M With these equation (lk) becomes
F = W. ^ sin 2a / (x2 - y2) dm + W cos 2a J (x2 - y2) dm + A d M xy M
(26)
Inspecting the swinging system from the top, i.e., projecting it
into the plane xy, we can observe that it looks exactly like the pro
jection of the swinging system of the curvature variometer. There fore, the Integrals in expression (26) which contain only x and y coordinates are exactly the same as the integrals of the curvature variometer in expression (16). The situation is different in the plane xz. The curvature variometer does not have a suspended weight; 27.
therefore, it is symmetrical with reference to the horizontal plane,
"xy. For this reason the integrals of the curvature variometer which
contain the products of the coordinates x and z are zero. In the
horizontal variometer the horizontal part of the swinging system is
also symmetrical with reference to the plane,"xy. However the sus
pended weight, Pg, has no equivalent symmetrical situation above the
plane xy. For this reason the integrals /*xz dm in equation (26) M are zero where the horizontal part of the swinging system is con
cerned, but they are not zero when they are extended to include the
suspended weight. It follows that the integral forthe entire
swinging system is
/"xz dm = h t m ...(27) M where m is the mass of the weight, Pg,
h is the z coordinate of the mass,
& is the x coordinate of the mass.
In equation (27) we neglected the influence of the mass of the sus pending metal fiber. Substituting expressions (17) and (27) into equation (26) to get,
F = WA r K sin 2a + W JK cos 2a + W h Z m cos a - V7 h $, m sin a A 2 x r ** ** ...(28)
K is again the moment of inertia of the swinging system referred to the z axis.
As in Chapter 2.11 the torque, F, is equal to the torsion resistance, pr,
F =■ CpT 28
and since cp = 8 - 8q (see equation (2l)), expression (28) yields
8 • 8o ” WA lr Bln 80 + Wxy T 008 801 • Wxz sin “ + Wyz ^ cos “ • • • (2 9)
In this equation K, t, h, I and m are instrument constants; 5 and Of are readings as before. The unknown gradients are:
WA, W , W and W A xy* xz yz
The torsion free direction, 8 , is also unknown. o We introduce the following abbreviations:
DK a _ r ...(30) b = 2DrahiJ T
These values are instrument constants. The actual sensitivity of the horizontal variometer depends upon the constant, a, when and are concerned and upon the constant, b, when the gradients W and XZ W are concerned. Using the same mirror scale device that we intro- yz duced in Chapter 2.11 for reading the equilibrium position of the swinging system, we have the following relation between the linear scale divisions and the rotation angle
8 - 8 - n " "o o 2D
Substituting this and the expressions (30) into equation (29), we obtain
n - n = aW. sin 2a + a2W„. cos 2a - bW sin a + bW cos a o A xy xz yz ...(31;
This equation contains in addition to the instrument constants and 29 the five unknowns only quantities which can he read when making the observation..
If we rotate the case of the instrument into five different azimuths (ct^, o^, CC^, a^, a^) at a single station and if we read the equilibrium positions of the swinging system each time (n^, ng, n^, n^ n<~), we can write five equations® Introducing the abbreviation, v = - nQ, the five equations ares
v = aW. sin 20L + a2W cos 2a, - bW sin a., + bW cos a, 1 A 1 xy 1 xz 1 yz 1
v_ = aWA sin 2o^ + a2W cos 2a_ - bW sin a_ + bW cos a_ 2 A 2 xy 2 xz 2 yz 2
V3 = aWA sin Qoc3 + a2Wxy cos 2a3 ” bWxz sin a3 + bWyz 003 a3 *•••(32)
vk ~ aWA sin ^ 4 + a2Wxy cos ~ bWxz s:I’n ak + bWyz cos a4
v = aWA sin 2aK + a2W cos 2aK - bW „ sin ca + bW cos ol. 5 A 5 xy 5 xz yz
If we use, for example, the azimuthss
0°, 72°, lhb°, 216°, and 288°, equations (32) give the following solutions for the unknowns?
no = 3 ^nl + n2 + n3 + nk +
Wxz = - | [0.2351 (v3 - vh ) + 0.380U (vg - v? )]
V “ ‘ I [0-TS36
Wxy = - | [0.1382 (v3 - vu) + 0.3618 (vg - v5)]
WA - Wyy ’ Wxx =+ I [°,a351 - v? ) - 0.3801. (v3 - v^)] 30
Check: V1 + v2 + v3 + \ + v5 - 0
In certain applications when we do not want the gradient W^,
It is sufficient to observe in four azimuths; If the azimuths are:
0°, 90°, 180° and 270° the unknowns can be computed from the following simple expressions:
no = 5 (ni + n2 + n3 + n^)
" n2 Wxz = b ...(3*0 nl - n3 wyz = b
nx - n2 + n^ - n^ W xy 8a where n^, n2 , n^ and n^ are the readings at the different azimuths,
2„122. The Horizontal Variometer with Two Swinging Systems
E8tv8s, in order to shorten the observation time, later modernized his horizontal variometer by building two swinging sys tems into the case of the apparatus
[EftTVOS, 19063. The first swinging system forms an angle of 180° with the second one (Figure 7)* Since the quan tity, nQ, of the second swinging sys tem is also unknown, the total number of unknowns has increased to six; TO obtain these unknowns we need six Figure 7 31
equations at each station. If we turn the case to three different
azimuths we can write three equations for each swinging system* giving altogether six equations:
v. ** n- - n m aWA sin 2a, + a2W„. cos 2a, - l l o A l xy 1 - bW „ sin a. + bW cos a n xz 1 yz 1
v 2 = n 2 - n o = ®WA sin 2ag + a2Wxy cos 2ag -
- bWxz sin a2 + bW^ cos aQ
v = n_ - n = aWA sin 2a_ + a2W cos 2a_ - 3 3 o A 3 xy 3 - bW sin a0 + bW cos a0 xz 3 yz 3
v' = n' - n 1 = a'WA sin 2a' + a'2W cos 2a' - 1 1 o A 1 xy 1 -b'w sin a 1 + b'W cos a* xz ~ yz 1
v' n' - n' - a'WA sin 2a' + a'2W cos 2a' - 2 2 o A 2 xy 2 - b'w sin a' + b'W cos a' xz 2 yz 2
v' = n' - n' = a'WA sin 2a' + a'2W cos 2ai - 3 3 o A 3 xy 3 - b'W „ sin a' + b'W cos a' xz 3 yz 3 where a^, a^, a^ are the azimuths of the first swinging system,
a|, a^, are the azimuths of the second swinging system,
(a* = a + 180°)
ni* ng, n^ are the readings at the equilibrium positions of the
first swinging system for each azimuth,
n^, n^, n^ are the readings at the equilibrium positions of
the second swinging system, 32
« where a and b are instrument sensitivity constants of the first
system,
a' and b ’ are instrument Sensitivity constants of the second
system.
The four gradients, WA , W„ , W , W , and the readings at the tor- XjT xz yz sion free positions of both swinging systems, nQ and n^, are the un
knowns of these equations.
Generally, the following azimuths are used:
I position : ^ a 0°, = 180°
II position : a2 = 120°, = 300° ••*(36)
III position': a 3 = 2^0°, a' = 60°
With these azimuths the solution of equations (35) is:
WX2 = - A [va . v3 - fr (v'z - v.)]
V = ' B [V2 + V3 ■ b (vS + v3 )]
WA = - C [v2 - v3 + F r (vj, - v')]
2 V - - D [v2 + v3 + b (vS + v3 )]
no = I ^nl + n2 + n3^
no * 3 ^nl + n2 + np where: a* •J3 (&'”*> + a b 1)
...(38) n ______a 1 a'b + ab' These quantities are instrument constants. The values below are the
constants of one particular instrument (Number 36618 torsion balance
of the Roland Etitvfis Geophysical Institute, Budapest) and are included
for informational purposes.
A = 1.1+23 X 10"9
B = 2.1+65 X 10"9
C = 1+.519 X 10”9
D = 7.827 X 10’9
£ r = 0 - 9 8 7 8
£7 = 1.0192
All quantities are in CGS units. Table 1 shows the computation pro cedure based on formulas (37)--on the computation form of the Etttvtts
Institute. The values, (n - nQ), were computed as the mean of four repetitions. In the first column the Arabic numbers symbolize the order of repetition. The Roman numbers indicate the azimuth posi tions according to expressions (36). The results are in "E" units, i.e., 10’9 CGS;.
We might mention that sometimes one measures the gradients in five azimuths with this type of instrument. This is done if we want to verify that both swinging systems give the same answer, i.e., as a V*
APPARATUS! 36618 OBSERVER: X. MUELIER Arsn • Dates Time • GONCZ Started 5n40m PM 4 Aug, 2 , 1954 Point N o . : 271 Removed 6^5 0m AM 0 Posi i Swinging System No. 1 Swinging Systen11 No. 2 Tem pera tions n n-n n* no 0 ~ * L n '-no ture
^-I«. 72.. 9 -90.0 4-II.. 67.5 -71,5 -4.0 -83.O 87.7 +4.7 4-III. 74.0 -70.4 +3*6 -90.0 87<.7 -2.3 i • H U) 70.. 0 -69,9 +0,1 -90.1 88.0 -2.1 3-II., 65.8 -69.6 -3,8 -84.1 88.4 +4.3 3-III. 73.. 0 -69.3 +3,7 -90.9 88.4 -2.5
2-1 ., 69,0 -69.0 0.0 -90.1 88.6• -1.5 2-IIa. 65.. 0 -68.7 -3°7 -84.9 88.7 +3.8 2-IXI. 72.. 2 -68.5 +3.7 -91.-3 89.0 -2.3 l-Io. 68,. 2 -68.3 -0,1 -90.9 89.O -1.9
1—XX o. 64.5 -68.0 -3.5 -84.9 88.6 +3.7 1—XXX a 71.4 -91 ol s I------Swinging System Swinaina System 2 No 1 No. v' V1 V2 Vi V2
V3 T 7 3 «° 1 H H V no *11*0 “ill"”© 4 - n; 4 i - ni 0.0 -3.8 +3-7 -1.8 +4.3 -2.4 a=V v3 -7.5 a - (1-0.0122) b -14.4 -1.423 ! -A b-v'-v- +6.7 c - (1-0.0122) d - l.l -2.465 -B ,-0.1 a + (I+O.OI92) b -0.6 -C C=,V2+V3 -4.519 daV*,+V^ +1*9 c + (1+0 .0192) d +0.9 -7.827 , -D
Observed G radients W*z Wyz WA 2Wxy v. (in Etttvfis Units) +20.5 +2.7 +2.7 -7.0
Table 1 guard against one system's malfunctioning. Should one system fail, the apparatus functions essentially as did the original horizontal variometer described in Chapter 2.121.
When we derived our basic equation (ll) for the torque moment in Chapter 2.11 we assumed that the change of gravity is linear in the field of the swinging system. This assumption was permissible where the curvature variometer was concerned. In this apparatus the weights lie on the same equipotential surface. On such a surface and within a small area the change of gravity is very small and can be assumed to be linear. The situation is different with the horizontal vario meter because the weights lie on two different equipotential sur faces. In the derivations we have assumed that the change of gravity in the direction z, i.e., between these two surfaces, is also linear.
If thisis not the case, then equation (28) should read as follows:
? = W. ^ K sin 2ct + W K cos 2a + W hJbn cos a - A 2 xy yz
- W him sin a + h (W„_ sin a - W,„ cos a) XZ OZ XZ yZ
The last term of this equation considers that the change in the ver tical direction is not linear. This terra is significant if there are significant non-symmetrical masses near the instrument. There fore, if we neglect this term from our equation— as we have done— we have to take care that during the measurement there are no signifi cant non-symmetrical masses near the instrument. We shall deal with this question in Chapter 2.21 in further detail.
The horizontal variometers with two swinging systems are in 3 6
practice the most frequently used torsion balances.
We should mention that there was an experiment in the Soviet
Union with three swinging systems, but it has not been used in prac tice [NUMEROV, 1932].
2.13. THE GRADIOMETER
In 1932 Lancaster-Jones published an article about the prin ciples and use of a third type of variometer [LAICASTER-JONES, 1932].
He called his instrument a "gravity gradiometer." With this appara tus we are able to determine the gradients, Wv„ and Ww .
The gradio meter has a special sort of swinging system (Figure 8 ). 1,2 The beam carries o- Sideview three weights. Two of these, and m^, are in the xy plane; the third one is at +x 120° TopvieWi an elevation, h, above them and is rigidly connected 2 to the Y-shaped * horizontal part of Figure 8 37
the beam. The appendages of Z length holding the weights form an angle of 120° to one another. These appendages meet at the suspen
sion point. In the coordinate system shown (Figure 8 ) the coordinates of the three weights are as follows:
x 1 » -Z cos 60°=* Z% - -Z sin 6o°a Z; a 0
x2 a -Z cos 6o°=* -- Z; y2 » +Z sin 6o°a ^ Z; z2 * 0 •••(39)
X3 a Z ^ = 0 Z^ a h
Let us investigate in the case of the gradiometer the values of the different integrals in the general equation (l^). We had there the following types of integrals:
—2 —2 ) ___ / (x - y ' dm ; / xy dm ; / x z dm ; / yz dm M M M M
Each integral has to be extended to the weights and also to the supporting staffs.
The first integral extended only to the weights yields:
/ (x2 - y2 ) dm a (x^ - + (x| - y|) m + (x® - yi?) m_ m^gin^
Substituting the quantities from expression (39) into the equation above,
/ (J2 - y2) dm - (ii2 - |i2) + (J*2 - Ji2) nu + iStL.
If m^ = nig = m^ this integral is equal to zero. Since any three corresponding mass elements of the appendages could be considered as the masses m^, and in the equation above/.it is obvious that this integral is zero also when these appendages are concerned*
Therefore, we can state that
t * * J (x2 - y2) dm » 0 ..,(40) M
Since the swinging system is symmetrical with reference to the plane,"xz, the second and third integrals are also zero,
f xy dm = / yz dm a 0 . ..(4l) M M The fourth integral, / xz dm, is also zero when the horizontal part of the beam and the weights, ^ and V are concerned. This must be so, because these parts of the swinging system are symmetri cally referred to the plane, xy„ If we neglect the influence of the vertical appendage, this integral is limited to the mass, m^.
xz dm a x^z^ni^ = i5 h m .. . (42)
If we substitute expressions (40), (4l) and (42) into equation
(14) then after the usual reductions we get the equation of the gra- diometer,
n - n * b W cos a - b W sin a .**(43) o yz xz ' '
This equation has three unknowns, n . W and W , which can be de- o yz xz termined by means of observations in three azimuths, n and a are readings; b is the sensitivity constant of the instrument.
\ We should mention that Haalck had also designed a similar vario meter [HAAICK, 1942]. In his apparatus there are two swinging ays- terns perpendicular each to the other. Observations in two azi
muths (0° and 180° for the first swinging system and simultaneously • " * ' * ‘ ^ ...... 90° and 270° for the second one) give four equations. From these the four unknowns, the two gradients and the two zero readings, can be computed (see Chapter if. 131).
2.2. The Torsion Balance Apparatus
2.21. THE DIFFERENT INSTRUMENTS
When we wrote about the principle of the torsion balance we pointed out certain conditions for which great care must be taken during construction and in the practice. Now we will discuss these conditions in more detail.
The heart of the apparatus is the torsion fiber. The goal sought in manufacturing the fiber is to make the angle of rotation a linear function of the torque moment acting in the horizontal plane. This moment is very small and, therefore, the torsion resis tance of the fiber is also very small. This torsion resistance is the product of the angle of rotation, q>, and the torque constant, t , of the fiber. With cp constant, the smaller the torque constant, j the smaller is the torsion resistance. The quantity r appears in - the denominators of the expression (30)(sensitivity constants); therefore, the sensitivity of any variometer is inversely propor tional to T. The torque constant depends upon the material, the 1*0
length of the fitter.and its diameter. Smaller diameter and longer
fiber result in smaller torque constant. It can be shown that
T * f (j")
where |i is the diameter and i is the length of the fiber. EiJtvds,
in his first instrument, [E0TV5S, 1896], used a fiber of platlnum-
iridium alloy. The diameter was 0.01* millimeters and the length
110 centimeters. In modem instruments, in order to decrease the
dimensions of the apparatus, the fiber is shorter; but in order to
have the same sensitivity fiber diameters of 0.01 millimeters are
used. Choice of material is extremely important. To be satisfac
tory, a fiber must not drift and must have small temperature coef
ficients of elasticity and of angular position. These properties de
pend upon the material used. Today Wolfram fibers and Tungsten
fibers as well as a platinum-iridium alloy are used by manufacturers.
The different factories do not publish the exact proportions of
materials used in the different alloys because this is the “Achilles
heel" of the apparatus. With better materials the diameter of the
fiber can be decreased and with this decrease the dimensions of the
whole instrument can be reduced. The stability of the torque con
stant also depends upon the materials used and upon the method of
manufacture. The finished fibers generally are exposed to different
influences and afterward their torque constants are redetermined. We
can expect that r remains constant in a temperature range of about
70° Fahrenheit and in an angle of rotation range of about 20° k l
amplitude. The apparatus 3hould he used only under those same condi
tions which existed when the fiber constants and properties were de
termined* Each manufacturer gives the temperature limits within which
the apparatus should be used. Drift is another problem. The fiber
should be conjpletely free of drift. Because of these severe require
ments only about 15 to 20 per cent of the finished fibers can be used
in the instruments. The post-manufacture preparation of a good fiber
lasts for about one year. The long and laborious preparation of the
torsion fiber is one reason why the torsion balance is an expensive
apparatus.
The second very important part of the instrument is the
swinging system, which consists of the beam and of the weights. From
the foregoing derivations we know that the whole swinging system
should be symmetrical with reference to the xz plane. (See equa
tions (25)). The horizontal part of the swinging system should also
be symmetrical with reference to the plane xy", (see equation (27))*
Another requirement was that the swinging system should have the
smallest possible dimensions in the y direction. (See equation (17))«
To fulfill these conditions is not easy, especially since there are
certain attachments to the swinging system which make the system more
stable and carry the mirror. In the derivations we have assumed
" that the suspension point lies at the center of gravity (See Figures 1,
6/b and 6/c). if this were a fact, the swinging system would not be
stable in the vertical plane. A very small vertical force could
swing it from the vertical equilibrium position. For this reason
the suspension point has been lifted to the end of a small vertical metal staff which is attached to the center of the beam (Figure 9)*
The other task of this staff is to carry the observation mirror. For this reason we also find a staff on the swinging system of the torsion balance, though here it is not necessary for stability purposes since the suspension point is above the center of gravity
(Figure 6/a).
The most dangerous effect inhibiting precise measurements is the air circulation around the swinging system caused by the changing temperature - To prevent this circulation the swinging system is placed in an insulated case. The vail of the case consists of two to four layers of metal. Generally, the outside layer is aluminum while the inside layers are brass. The layers are separated by air spaces. The air circulation between the layers is inhibited by means of small baffles perpendicular to the walls. The device of layers and baffles prevents the air circulation only when the rate of temperature change is small. In order to secure this, all observa tions should be made in tents or protecting sheds (Figure 10). In addition, it is advisable to measure only on cloudy days or at night.
If the sun is shining, a layer of canvas above the protecting shed might prevent fast temperature changes inside. Iri order to check the success of these precautions we also have to observe the tem perature. On the modern torsion balances there are mounted three thermometers, one at the torsion-fiber and one at each weight.
The temperatures both outside and inside the shed should be ob- < Torque fiber
Mirror Scale
7 ------“ Telescope Beam
Figure 9
Figure 10
*6 kk t ■ served as well.
The instrument is very sensitive to non-symmetrical disturbing * 1 t . masses close to the gvingjng system (See the end of Chapter 2 . 1 2 2 ) .
This means that the parts of the instrument should be distributed
evenly around the torsion fiber by the designer. This can be attained
by building the layers of the case in cylindrical form and by using a
cylinder instead of a tripod as the base of the apparatus (Figure 1 7 ) .
For the same reason the location of the station should be in a rela
tively flat area. The ground surrounding the instrument should be
levelled by means of a hoe or some other tool within a circle of 3
meters diameter. The effect of the topography should be computed.
(See Chapter 2.33). The accoutrements of the field party should be
at least 100 meters' distance from the Instrument. The rectangular
protecting sheds (as in Figure 10) are actually not sufficient.
They should be cylindrical.
In Chapter 2.121 we saw that in order to determine the gra
dients of gravity at one station we have to observe the equilibrium position of the swinging system in five azimuths. The time required for the swinging system to settle, down to the equilibrium position depends upon the particular instrument. The relaxation period of the first instruments were about 120 minutes; thus, one set of five azimuth observations lasted for about ten hours. The horizontal variometer with two swinging Bystems reduced this observation time by half. The relaxation period of the modern Instruments varies from twenty to sixty minutes. The torsion balances working with ■ ' ' “ ...... w t to >2-3 E accuracy have a period of twenty minutes, i.e., one set
of observations lasts for about one hour. The most modern torsion ... ^ .. ' ' ' ’ t balance, the 12-5^, has a relaxation period of forty minutes, (two
hours’ observation time), and gives an average accuracy of -0.50 E
[RYBAR, 19571.
We can divide the different torsion balances into two groups,
depending upon whether the reading is obtained by visual methods
or by photographic registration. Another grouping is possible ac
cording to whether the case is rotated to the successive azimuths by
automatic methods or by hand. The first instruments had to be turned by hand and the reading was obtained by means of a telescope as described in Chapter 2. 1 1 . The instruments shown in Figures 16,
17, 18 and 19 belong to this group. The modern instruments are turned by clockwork and the readings are registered by photographic methods. This type of Instrument is shown in Figures 21, 23, 25,
26 and 27.
The photographic registration device in its most simple form consists of an electric bulb, a lens system and a photographic plate.
The plate is moved at a constant speed by clockwork. The light rays of the bulb, after going through a complex diaphragm-lens system, reflect from the mirror of the swinging system on to the moving photographic plate. As the mirror swings the rays trace a path on the plate proportional to the angle of rotation. In order to have a reference line, other rays are projected on to the plate from a motionless mirror. The distance between the reference line and the 1*6 line projected from the moving mirror is at any instant proportional to the angle of rotation. The instrument design generally provides for the inside temperature, the serial number of the instrument and the station designation to be recorded on the plate. The electric bulb is lit either all the time or Just for a few seconds after the swinging system has stopped in equilibrium. In the first case damped sine curves can be seen on the plate with breaks indicating that the instrument was turning into another azimuth (Figure 1 1 ) .
In the second case only dots are recorded, which means the plate was exposed once every twenty to sixty minutes during this period, de pending upon instrument design (Figure 12). In the instrument with two swinging systems both are .registered on the same plate. In order to be able to differentiate between the marks, they usually are given different shapes (Figure 1 3 ) . Naturally, there are many other possi bilities of registration. For instance, the photographic registra tion of the British Oertling [OERTLING, 1935] torsion balance con sists of photographing the scale itself as seen in the telescope,
(Figure ll+).
Both the visual and the photographic methods have advantages and disadvantages. The main advantage of the visual method is that the readings are immediately available to the observer who is then able to compute the results without delay. This is important as a check and also helps to determine the positions of the next stations.
The drawback to this method is that the observer has to remain with the instrument all the time in order to make readings every twenty Figure 11
•H
Figure 12
Zero line 'Beam positions
Temperature
Beam No. 1 IB Beam No. 2
Figure 13
^7 Figure Ik
o TO M <3 19 i9 TO 1 1 10 13 1 00 T II ]!,; ;!ij::1 :■ h ' iHi:|! 1 r j ■ 1if. i‘ 'i ■ p i |if ;ii i i-t !•' ■ Ii:: i!':r •it r . 'I ■j i I 1i:i; ■ ' 3 M1 •I' i; yM ifj1,.j : ii'fi II aM :i 1ti; j: ■ ■ 1 •-'I'i ■ i i' h i I i;! j: ■ t r: : 1 ■: , '' II a ; .1. ► ’1 1 ■ J V iiI ■ ii!1 1 r;jii-i: ■i j 1 I i ; ii <: II j. ■: ;M 1 ■it i > | r'if •1 h ! ■ -i i 1 ! ‘it II !| '1 a I 1 .1 > e ;II i : fi ;j[iit j; .‘r! j ■if 1 ■! :• !i ll ' 1:' '■!ii il: •i, .:ii IIi- O » B *> i9 19 W I9 i) 10 1> 10 ca Jo
Figure 15
1*8 - ' - • ■ minutes to one hour and, therefore, he can not operate several
■balances at different stations simultaneously. Another disadvantage
of making visual readings is that if the observer is slow the attrac
tion of his body can move the swinging system from the equilibrium
position. The influence of an observer can be computed to on ap
proximate value very easily. Let us assume that the mass of the ob
server is seventy kilograms and that he stands fifty centimeters from
one weight and sixty centimeters from the other. The attraction-
difference for this 60 - 50 = 10 centimeter distance is approximately
k2m (-i-- -i=) . 66.73 X 10_9X 70000 <-K - -ts) « 0.6 X 10 COS v * r22 60 50 -8 or for one centimeter distance, 6 X 10” CGS, which is about sixty
times larger than the accuracy of the torsion balance. Due to the
large moment of inertia of the swinging system this effect is criti
cal only when the observer stands next to the instrument for a long
period. If he makes his readings quickly the swinging system will
not have time to move.
The advantages of photographic registration are obvious. The
plate represents a lasting record and eliminates recording blunders.
The observer can work at once with several instruments and there is
no disturbing effect caused by his body. A drawback is that the
plate is developed at the distant headquarters and, therefore, in case
of Instrumental failure a new trip is necessary to repeat the measure ments. If the plate is to be developed in the field, then much additional equipment, such as instruments, chemicals, darkroom, 50 and so forth, is necessary.
To measure the values n on the plate we generally use a reading grid plate placed ovex> the photographic plate (Figure 15)• In the model E-5^ a reticle is photographed on the plate so that the equi librium positions can be read simply by means of a magnifying glass and without measuring (Figure 22). Note that in the case of photo graphic registration the distance, D, in expressions (30) is the length of the light path from the mirror to the plate expressed in units of the divisions of the reading plate (Figure 15) or in ■units of the scale photographed on the plate (Figures 14 and 22).
Figures 16 and 17 show the torsion balances designed by EtitvSs with one and with two swinging systems respectively [E&FVOS, 1906].
Figures 18 and 19 illustrate the large and the small "Suss- instruments" designed by Desidarlue Pekar and manufactured by the
Suss factory in Budapest [FEKETE-MIKOLA., 1918] • These instruments were used exclusively before the first World War throughout the world [BRILLOtflN, 1908; GAVAZZI, 19165 BflCKH, 19175 KLUPATHY, 19065
MIKOLA, 1918]. Figures 20 and 21 show the modern',E6tvtis-Rybar" apparatus with visual reading and with photographic registration respectively, manufactured by the EBtvBs Geophysical Institute of Budapest [R7BAR, 1934].
Figure 23 is the picture of the most modern apparatus,
Model E-5designed by Dr. I. Rybar in 195^ [RYBAR, 19573* Tkis model is an improvement upon the Etitvtis-Rybar model, mentioned above. When designing the instrument Rybar did not aim at the . a 79 a h Figure 1.6 k Figure 17 a a a
a i I 7 u 1J Is' fci fci I&1 Tal tel IsJ W fel W 13 Is I W Is W 13 fel IsJ tel Tal I&1 fci fci u Is' I 1J n £ I I 3. [3 rr^i H rr^i H M M N H
J&SS
Figure 19
51 Figure 21
Figure 20
^
Figure 22 Figure 23
52 ■ ^ ^ _ . 53 reduction of the relaxation time or dimensions. To have done this would mean sacrifice of accuracy as previously explained. His object was :,to construct with the use of all experiences gathered in the course of several decades of laboratory and field work a per fectly reliable torsion balance coining to rest in a position of equi librium which exclusively is Buiting the field of gravity." Thus, he wanted to eliminate the need for repetitions of the readings at each azimuth and to render a single series of readings sufficient.
This naturally reduces the observation time considerably depending upon the number of repetitions customarily used (four) with other instruments.
Figure 2^ shows schematically the optical system of this in strument. G represents the case of one of the swinging systems and
Tg the mirror of that system.The light from the small electric bulb
F, reflected by prism P^, passes through the condensing lens K-^ and illuminates the transparent glass scale S, placed directly below a second condensing lens. Scale S has 0.25 millimeter divisions with every tenth line numbered. The scale lies in the torsion direction of the mirror Tg, i.e., perpendicular to the plane of the drawing. The condensers gather the light from the lens L^. The light reflected by mirror passes through the lens , then is re- fleeted by the nirror Ta and paeses again through lens ^ but in the opposite direction. The lens gives a clear picture of scale S on the metal plate B which has a 3 X ^*5 millimeter rectangular slit having in its center a thin wire perpendicular to the longer sides. 4
Figure 24 Figure 25
54 55
The thin wire of the slit In diaphragm B Is parallel to the di
visions of the scale. The lens Lg follows the diaphragm and pro
duces on the photographic plate A the image present at the slit,
i.e., the picture of scale and reticle (Figure 22). Should the
swinging system with the mirror Tg rotate about the vertical axis
and take another equilibrium position, another scale-division would be projected upon the slit. This scale-division is then read from
the scale image on the photographic plate. The other optical ele
ments in the figure, condenser Kg, diaphragm N, and the lens L^,
serve to project the serial number of the instrument and the station designation upon the plate A.
The Askania Werke in Germany manufactures three modern
instruments. The large torsion balance (Model L-Uo) is shown in
Figure 25 [SCHWEYDAR, 1921]. The Z-beam torsion balance (Model Z-40)
[SCHWEYDAR, 1926; HEILAND, 1926] and the inclined-beam torsion balance (Model S-20) [IMHOF-GRAF, 193*+] are shown in Figures 26 and
27, respectively. The Soviet Union also manufactures models simi lar to the Askania Z-40 and S-20 in the Factory for Geophysical In struments in Leningrad [SOROKIN, 19533*
Table 2 gives the specifications for the most important models. On comparing the different instruments the Etttvtis-Rybar model is outstanding with its small torque constant (t « 0 .03) and its relatively short period. Note also the incredibly thin tor sion fiber (|i o 0.012 millimeters).
The accuracy of model E-5*+ is - 0.255 E for Wxz and W' Figure 27
56 57 t friginal Efltvfis Izdtrds Rybar E-54 L-40 Z-40 S-20 ' Fig. Fig. Fig. Fig. 27 Maes of Weights (m) gr 30(25) 15(15) -(9 ) 42(42) 22(2 2 ) ,40(40) Length of Torsion Fiber cm 56 21 56 30 26 Length of (Beam (2 f) cm \ko 28 20 40 40 20 1 Torsion Con dyne stant (t) cm/rad 0.46 O. 0 3 ' 0.066 0 .5 0.25 1.2 Diameter of Fiber (n) HUH 0 .0 4 0.012 m a§ 0.04 0 .0 3 6 , , 0. 1 4
Moment of gr2 Inertia (k ) cm 21100 1 4 0 0 1900 2 6 7 0 0 19500 8800 .Total ,Welght kg « ■» ^5 44 59 vr 37 Height of C • of G. cm 100 100 90 90 90 Total Height cm 195 mm « 183 120 126 Relaxation Period min | 60 40 40 60 40 20 Accuracy i s - 1 0.5 0.5 0.75 1.5 2 . 5 Distance of Lower Weight cm 66 — 10 56 2 2.5 from Center o i Beam (h)
Table 2 58 # . . . • • , and - 0.72^ E for WA and W . The relaxation period is forty A xy minutes.
It can be seen that modern torsion balances give the gradients + -9 of. gravity with an accuracy of - 0.25 to 3*00 E. r Since 1 E * 10
CGS « 0.1 milligals/kilometers, we can observe a 0.025 to 0.3 milli- gal change of gravity in a distance of 1 kilometer. No modem gravimeter can even approach this accuracy. The accuracy of the
Worden gravimeter is about 0.01 mllllgals ■ (10 ^ CGS). The re cently developed La Coste and Romberg Earth-Tide Gravimeter claims -6 an accuracy somewhat better them 1 microgal * (10* CGS), which is about 1000 times less accurate than the Etitvtie-Rybar or E-5^ torsion balances, and about 100 times less accurate than the As kania model S-20.
2.22. FIELD OPERATIONS
The procedure for making a torsion balance observation with photographic registration is as follows;
The station is selected and its horizontal coordinates de termined. The ground around the station is smoothed within a circle of three meters'diameter. Radial levelling is made within a circle of 100 meter radiuB for topographic corrections (see » Chapter 2.^). The protective shed is erected and the instrument Is set up within ltB base. The apparatus is levelled and then oriented by means of a compass, x axis towards the North. The clock- ...... 59
work which turns the case Into the different azimuths is wound* ' ' ‘ •' ' ~ - t The clockwork control stops are set properly* The swinging system ,...... 4 is uncaged and the photographic plate is inserted* The operator
then leaves the shed removing all packing boxes, trucks, and so forth, to at least 100 meters' distance from the apparatus. He may
then attend to the next station, generally 500 to 1000 meters away,
leaving a watchman on the premises* At the end of the relaxation period as pre-set on the controls, the clockwork makes an electrical
contact which turns on the light for a few seconds* A ray of light reflected from the mirror records the equilibrium position on the plate. The clockwork control next turns off the light and starts the clockwork (spring motor) which slowly turns the case of the instrument into the next azimuth. The apparatus remains still for another relaxation period, after which the procedure is repeated and the instrument similarly turns into the third azimuth. If the ap paratus is left alone it will continue these operations until the clockwork runs down (12 to 15 azimuths, i.e., 3 to 4 sets). This repetitious procedure is not necessary with the model E-5^* When the observer returns to the instrument he takes out the photographic plate,which is developed and read at headquarters. The swinging system is caged and the electrical leads are disconnected* The in strument is then packed in its carrying case and the protecting shed and other accoutrements are loaded on a truck and moved to the next station.
The organization of an observation group depends upon many 6 ° ' 4 economic and technical factors. Generally, a group consists of three or four observation parties. Each party works with two or three in struments at once. The parties have the following personnel:
1 observer
2 assistants
2 or 3 night watchmen.
The number of stations observed per day and per party depends upon the relaxation time of the swinging system and upon the number and type of instruments In the party. If the relaxation period is forty minutes and the observation is carried through in three azi muths with four repetitions at each azimuth, then the necessary ob servation time for one station is:
^ x 3 x a U80 minutes ■ 8 hours. I Therefore, using three instruments simultaneously the party can ob- r serve from seven to nine stations per day (28-36 with the E-5*0»
Each group has a headquarters not too far from the observa tion locations. Here the photographic plates are developed and evaluated. The results are computed by a computer party located at headquarters. 61
2.3* The Reduction and Transformation of the Torsion
Balance Measurements
2.31. DEFINITIONS
The observed gradients (w , W . W. and W ) refer to the ' xz yz A xy equipotential surface of the gravity passing through the center point, 0, of the beam (Figure 6). This surface will be called the geop of the observation point* Any of the gradients may be divided into three parts, namely:
1. The normal value called the normal gradient. This refers to the equi-spheropotential surface through point 0. This surface will be called the spherop of the observation point. The normal gradient is related to the normal gravity field of the Earth.
2. The gradient of the attraction of the topography.
3- The gradient of the attraction of the subterranean mass- anomalies, i.e., the attraction of those masses whose densities do not correspond to the assumed density distribution of the normal-
Earth.
The effect of the topography can be further divided into two parts:
1. The Influence of the topography within a circle of 50 to
100 meter radius. This effect will be called the terrain effect.
2. The influence of the topography outside of the circle mentioned above. This effect is determined by means of maps and, therefore, it will be called the cartographic effect. We introduce the following designations:
1. Topographic value » observed value minus terrain- offeet.
2* Total anomaly = observed value minus normal value.
Topographic anomaly = topographic value minus normal value,
5. Subterranean anomaly = topographic anomaly minus carto
graphic effect.
Let us designate the gradients related to these designations
with the following symbols:
Symbol Example t .t Terrain-effect v Vxz
Cartographic-effect v* V* xz T.t t Topographic-value W xz
Total-anomaly AW AWxz t t Topographic-anomaly AW AWxz
Subterranean-anomaly AW^ AW 11 xz
Normal gradient U Uxz
With the above symbols our previous designation can be written as follows (for W as an example): XZ In Chapter k we will discuss the applications of the gra
dients to Geodesy and to Geophysics. The question is: Which re
duced value of the expression (Mf) should he used in these applica
tions? Only general principles and not strict rules can he given
here for the selection.
In geodetic applications the objects of our Investigations are
always in one way or another related to the geops. If we are in
terested, for instance, in the detailed shape of these surfaces in
the vicinity of the observation points, then the observed values
should be used as they are. If we would like to see how the equipo-
tential surface would look if no topography existed, the topographic values should be used after reduction for the cartographic effects.
These values will give a smoother equipotential surface. They will eliminate the "microrelief" caused by the surrounding topographic masses. If the subject of the investigation is somehow related to the differences between a geop and a spherop then one of the anoma lies should be used, depending upon the problem. It will be seen in
Chapters 2.33 and 2.3^ that the influence of the topography (terrain plus cartographic effects) is computed based only on the visible masses. Wo effort has been taken yet to introduce the "isostatic- terrain" and the "isostatic-cartographic" effects which would con sider also the influence of the invisible compensating masses. The observed values reduced with these corrections would refer to the co-geops, which are the corresponding equipotential surfaces of the co-geolds or quasl-geold, as they are sometimes called.
In geophysical applications the situation seems to be much 6 k simpler. Usually the goal of the investigation is to find mass anomalies relatively close to the surface. It is obvious that if we subtract from the observed value the influence of the topography
(terrain and cartographic effects) and the normal value (which repre sents an Earth of a regular density distribution), then the remain der, the subterrain anomaly will represent the mass anomalies.
In the following chapters we shall introduce the computa tional methods for the different effects mentioned above.
2.32. THE NORMAL GRADIENTS
In the following investigation we shall assume that the change of the U , U _, UA and U gradients of the normal gravity can be xz yz A xy neglected along the vertical, between the spheroid and the spherop of the observation point. The difference between the reference ellipsoid and spheroid is also neglected; therefore, the normal gra dients will be determined on the surface of the reference ellipsoid but they refer also to the spherop of the observation point.
We will employ our usual coordinate system. The +x axis points North, the +y axis points East and the z axis coincides with the tangent to the plumb line at 0, and is positive downwards. In this coordinate system the normal gravity has no component in the Su y direction, i.e., 7^ a ^ a 0 and, therefore, the normal gra dients, U „ and U , are zero, xy yz '
uxy “ S5SF ‘ I ? 0 " 0 uyz = isss?" S£ 0 ■ 0
It will be shown in Chapter 3 that the following relation holds between the gradients, WA and W , and the principal curvatures, fad AJT 1 1 — and — - of the geop, P1 2
(w? + )1/2 = s(t- - 7r) xy “ 'P-l P2
In case of the normal gravity field, since U is zero, this equa- xy tion leads to
where M and N are the radius of curvatures on the reference ellip soid in the planes of the meridian and of the prime vertical, re spectively. The symbol 7 is the normal gravity at the observation i point. It is well known from the geometry of the ellipsoid that
ut a(l - e2 ) (l - e2 sin2 (p)^2
N (1 - e2 sin2 cp)*/2 where a is the semi-major axis, e is the first eccentricity of the ellipsoid and
UA * 7 (^ “ |) * a (®2 cos2 * + I cos2 ‘P +
For all practical purposes the first term of the expression in the parenthesis is sufficient; so
U. - Zf- COB2
The normal gradient U can also be computed very easily. We xz know that
„ a ,au> a?
On the ellipsoid dx = Mdcp, and, therefore,
it 1 & UXB - M 55
The change of the normal gravity 7 on the surface of the ellipsoid is a function of cp and is given by the normal gravity formula,
2 2 7 = 7e (1 + 3 sin cp + € sin 2cp) where 7g is the normal gravity av the equator, p and e are con stants depending upon the parameters of the used ellipsoid. From this equation, and after neglecting the second term, we get
a 27g 3 sin cp cos cp « y& 3 sin 2cp
With this the normal gradient U is XZ 67
Equations (45), (1*6), (47) and (48) give the normal gra
dients of gravity on the surface of any reference ellipsoid, i.e.,
according to our assumption at the beginning of this chapter, on the
spherop through the observation point.
Using the dimensions of the International Ellipsoid and the
International Gravity Formula, equations (47) and (48) become;
- 10.3 cos2 cp » 5«15 (l + cos 2cp) - 5.15 + 5.15 cos 2
Both expressions give the normal gradients in E units. A change
in 7 in equation (47) and in M in equation (48) where both y and
M are taken on the ellipsoid will not affect the tenths decimal place (0.1 E) of the constants in expressions (1*9).
Figures 28 and 29, respectively, show the variation of U„_ XZ and with the latitude. Table 3 gives these values for every
5° of latitude.
2.33. THE TERRAIN EFFECT
2.331. Analytical Methods
In order to determine the terrain effect, the physical surface of the Earth is replaced by mathematically describable surface pieces out to the border of the effect (50 - 100 meters). The main dif ference between the various computational methods is in the selection uxz w
J C t9m ■ SO*
Figure 28 Figure &9
/'•\SOm
rm\iOm
Figure j
68 69
10 20 30 40 9° 0 5 15 25. 35 45
10.3 10.2 9.0 8.4 6.0 UA 9-9 9.5 7.7 6.9 5.15
0.0 1.4 2.8 4.1 7.0 7.6 8.0 8.1 Uxz . 5.2 6.2
9°
4.2 3.4 2.6 1.8 1.2 0.1 0.0 UA 0.7 0.3
8.0 7.6 7.0 6.2 5.2 4.1 2.8 1.4 0.0 Uxz ]
Values in E units
Table 3 7°
of these representative surface elements.
All methods agree that the surface of the Earth within a
circle of three to four meters' diameter around the instrument has
to be nearly level. There should not be any ditch, pit, building
or other irregular object within 50 to 100 meters, the limit of
the terrain effect. \ Starting at the instrument, the area is divided into com partments by concentric circles and radial lines (Figure 30). The
physical surface of the Earth is replaced by a representative sur
face element within each compartment. The comer points of the
compartments are common to both the physical and representative
surfaces. Authors have used for this compartmentation various radii and various sector widths. For instance, E&tvtts [eOTVCS, 1906] used
circles with radii of 1.5* 5> 20, 50, 100 and 1000 meters, and azimuths of 0°, k'f, 90°, 135°1 180°, 225°, 270°, 315° and 3^0°.
These azimuths are designated by numbers from 1 to 8, (Figure 30).
Schweydar [SCHWEYDAR, 192^-1925 AND 1927] divided the area with the same eight azimuths but his radii are different: 1.5, 3, 5,
10, 20, 30, 40, 50, 70 and 100 meters. The radii of Nuroerov
[NUMER0V, 1931] are: 1.2, 2, 3, ^.5, 6, 10, 18 and 50 meters. He uses either eight or sixteen azimuths. In addition to these three most significant methods the formulas of Nikiforov [MrHAJLOV, 1939] and of Haalck [HAAICK, 1930] are often cited in the literature on variometers.
The relative elevations, z, of the Intersection points of the 71 circles and radial lines are determined by means of levelling.
These elevations are referenced to a horizontal plane passing through the observation point. Above this plane z is positive; below, nega tive. As mentioned before, these intersection points on the physical
surface are also considered to be points on the different representa tive surface elements. The different authors have used various assumptions to find the best-fitting surface shapes.
EBtvtts assumed that z changes linearly in the directions of the radial lines and in azimuth,
z a c + aa + bra + d ...(50) where a, b, c and d are constants, and r and a are the polar coordi nates (distance and azimuth) of an intersection point.
The assumption of Schweydar and also of Numerov was that z changes linearly along the radial lines but that it follows a Fourier series in azimuth,
z = a + b sin a + c cos a + d sin 2a + e cos 2a + f r ...( 5 1 ) where a, b, c, d, e and f are constants, and r and a are the polar coordinates of the point.
Nikiforov kept the change of z linear in azimuth but assumed quadratic curves fit the situation along the radial lines.
The method of E&tvfts gives good results if the surface is flat; however, the formulas are rather complicated and, therefore, the computation is laborious. If the topography is irregular, one of the Nikiforov, Schweydar or Numerov methods should be used. For the 72
latter two, tables are available which speed up the computations
considerably. In rugged mountain areas observations are useless be
cause the influence of the surrounding masses completely masks the
true gradient values.
Here we shall introduce the most significant formulas of E&tvtts,
Schweydar and Numerov without derivation and give the basis of the
computations.
We shall use a cylindrical coordinate system wherein the axis
of the cylinder coincides with the tangent to the plumb line at
point 0. The origin of the system is at the observation point. In
this coordinate system the position of a mass element, dm, is given by its horizontal distance r, from the z axis, by the azimuth a
of the radius and by its vertical distance z from the observation point. The relation between these new r, a and z coordinates and
the cartesian coordinates x, y and z is:
x a r cos a
y - r sin a ...(52)
Z a Z x is positive North; y, East; z upwards; and a is measured clockwise from North. The mass element dm is at a distance d from the center point 0 of the beam; thus,
d = [x2 + y2 + (z - h)2]1/2 *[r2 + (z - h)2]1/2 ...(53) where h is the vertical distance of point 0 from the observation point, i.e., it is the z coordinate of the point 0. 73 The attraction potential at point 0 due to the mass element
dm is
, 2 dm dV = k ~ a
From this |V S t a 3 d d 3V .a dm 3d. S a2 ax ^ a2 ay
From expression (53) it follows that
dd x d d y Sdz-h H “ a> 3? * a and SI * ~ d ~
Therefore, 3V 2 am 3x “ 3 X a
^ dV , 2 dm d = - k —5- y 5? d3
The second partial derivatives are:
, 5 1 -,2 x . 5d ,2 dm ,.2x^ . ,2 dm d — 5 = 3 k -T- d m k -r » 3 k dm - k -=■ dx d dx d^ dP 6T
d ^ 5 - 3k2 h dm ^ - k2 32 . 3k£ - ^ am - k2 32 dy d dy d** d d
d = 3 k2 dm — = 3 k2 ^ dm *..(5 *0 dxdy d dy d
d = 3 k2 \ am M . 3 k 2 x(z - h) ^ cixdz d dz d?
d = 3 k2 ^j- dm — * 3k2 ^ - - g - dm dyd» d dz d -
Substituting x, y and d from equations (52) and (53)t and using
the symbol dVA ■ dV - d V , expressions (5^) become: yy xx 2 ^,.2 r cos 2a dVA - -3k — 5------p~g7o am A [r2 + (z - h )2]5'2
1 v2 rg sin 2a . 2 TZTTTTTTA^Ta[r + (z - h )*]• ...(55) -,,.2 r (z - h) cos a dV ■ 3k---- — f --- "p H/p ^ xz [r + (i - h)2r ''2
dV - 3k2 -£ ' h? SiB-q .., ■ [r2 + (t ' h)2]5/2 or since
dm m a r da dr dz
(a is the density of the mass element) equations (55) yield:
3 -.,.2 ^ rJ cos 2a dV^ a -3k a — 5------5“tt7q dr da dz . U )2]5/2 tr2 + (z 3 sin 2a dV - - k2a — dr da dz xy o [r2 + (z - h)2]5/2 ...(56) 2 r (z - h) cos a dVxz a 3k o — 1 7 o PtM dr da dz - h)2]5/2 [r2 + (* 2 r (z - h) sin a yz [r2 + (z - h )2]?/2
In order to get the total terrain effect within the radius of the terrain effect, rQ , we have to integrate expressions (56) in the directions a, r and z from 0 to 2?r, from 0 to rQ and from 75
0 to z, respectively. The terrain effects are:
„t „,_2 ? / ° fz ar3 cos 2a VA * “3k i f f ~ p „ ■ o s/o ^ r 2 [r + (z - h)2]5/2
or3 sin 2a r 2 , [r + (z - h)S ]5/a
+ , Sir r0 z 2 , or (z h) cos a Vxz - 3k2 / / / " '----- ' TriJo da dr dz [r2 + (z - h)2 ]5'2
t ? 2ir r0 z 2 or (z - h) sin a V.„yz » 3k f f f - -g- r/ H v n da dr dz o o o [r + (z - h) ]*
We develop the denominators into a series:
1 1 /t z2 - 2zhv“^ 2 [r2 + h2 -2zh + z2]5/2 [r2 + h2]5/2 r2 + h2
1 5 z2- 2zh v (r2 . h ^ ? / 2 (1 2 7 7 ? ^
Within more or less flat topography z is much smaller than r and, therefore, the second term can be neglected. The new denominator is
[r2 + (z1- h)2)5/2 (r2 A 2 )5'2
With this, expressions (57) after integrating with reference to z become: pjr yi *3 V*.-3k2 / ; °o|r °°BSatodr o o (r + h ) ' ...(58) ™ t n,_2 ^ fr° azr3 sin 2a ... ^ x v * 3k J J 7~2--- 2372 y o o (r^ + h dy f d 76 2 2ir rQ oz(£ - h)r2 cos a da dr V*xz = ...(58) 2 2jr rQ c z ( | - h)r2 .sin a “3k / f da dr yz o o
Up to this point all methods of finding terrain effect are similar.
In order to solve the integrals (58), z has to be expressed as a function of r and a. As we already know, Etttvbs assumed this func tion to be identical to expression (50) when considering that area beyond the inner circle (r a 1.5 meters). He also assumed that inside the inner circle the terrain can be represented by a plane whose equation is
z a x tan € + y tan k where € and k are the tilt angles of the plane in the North and
East directions respectively. Since these angles are very small this equation becomes,
z = xe + yx = r (e cos a + tc sin a) *..(59)
If we substitute equations (50) and (59) into expressions (58), and integrate compartment by compartment after summation we get to the Efttvtts-formulas for the terrain effect [e OTVOS, 1906]: + [0.0379€ (^+*5 )+ 0.006le (z3+z7 ) +
+ 0.0221c (z2+z^+z6+z8) + 0.0l60ic (z2+z6 ' V z8^r,5m +
+ [0.13046 (zx-z^ + 0.09225 (V z8“V z6 )]r -5m +
+ [0.01173 (V z4- + O*00^ 1 (z2+z8"z4“z6>3r-2Qm + + [0.00108 (2^ 25) + 0.00077 (z2+z8"z4 'z6^r»50m + + [ 0.00028 ( 21- 25 ) + 0.00020 (W V^^r-lOOm +
+ [- 0.000015 (2x- 25) + 0.000010 (z2+z8"z4‘ z6) 3r - 100Qm + + [0.001747 (2^ - z^) + 0.001235 ( 2^ + zg - 2^ - z |) +
+ 0.001300 (z^&5 + z5z6 - zQz1 - z ^ g ) +
+ 0.000538 (ZgZ^ + Z6Z7 - ZgZ3 - 27Z8)]r>i5m
5.77 K +
+ [0 .0379k (z3+z7) + 0 .0061k (z-j+z^) +
+ 0 . 0 2 2 1 k (z 2 + z ^ + z 6+ z q ) + O.0l 60e (z 2 + z 6“ VZ8^ r a5m + '
+ [0.13046 ( 23-z?) + 0.09225 (V W V W m +
+ [0.01173 (z3-z7) + 0-00831 (Z2+V z6”z8 ) 3r=20m + + [0.00108 (23- 2? ) + 0.00077 (z2+V z6' z8^ r - 50m ‘+
+ [0.00028 (23-z7) + 0.00020 (V V z6"Z8)]r=100m + + [- 0,000015 (z3-z?) - 0.000010 (22+2r z6-z8)]ra|1000m+
+ [0.0001747 (z7 - z3) + 0.001235 (Zg + Zg - z^ - z2) +
+ 0.001300 (z6z7 + Z7Zq - ZgZ3 - ZgZ^) +
+ 0.000538 (z5z6 '+ z8z1 - z^z5 - *1*2 )]Vm0 BL z3 + z5 - - [O.O819IV ( z ;l - Z7 ^ r = 20m - [C.03181 (z - Z3 + Z5 ' Z7 ^ r = 5 Q m " - [0.03077 (z1 - z3 + Z5 ' z7 ^ r a l 00m - [0.001*27 ^ - z3 + z5 ' z7 r=1000m ...(60) [0*2^131 (Zg “ Z|^ + Zg - Zg) xy r=5m + + [0.01*097 (Zg - Z^ + Zg - Zg ^r=2 0 m +
+ [0.01591 (Zg - Z^ + Zg - Zg) r = 5Q ® + zk + z6 - z + [0.01539 (z, ralOQm 4
+ [0 .00211*- (z, Z|. + Zs - z 8 ralOOOm
_3 These formulas were computed with h a 100 cm and a » 1.8 gr cm . _ o If the density differs from a a 1.8 gr cm , the results should he multiplied hy . The indices of the values z designate the dif ferent azimuths as seen in Figure 30. The equations (60) give the terrain effect in E units if z is in centimeters. We should mention t t that in the expressions giving V and V the constants 5*77 are cor- . xz yz rected values. In the original publication these constants are er roneous (7 *69). If we investigate these formulas we find that the terrain effect is much larger for the gradients WA and W than for Lt Ajr the gradients Wxz and W ^. This is so because in equations (58) t t "t t in the numerators of VA and V r is cubed while in V„_ and V „ A xy xz yz r is squared.
The second important set of formulas was derived by Schweydar.
Starting with equations (58) he assumed the z a f (a,r) function to 79
be equal to the Fourier series in expression (51)* Using the
same symbols as in equation (60) the Schweydar-formulas are [SCHWEYDAR,
192k -1925]:
vl8ZoC+£c'CXZ
V* sEb-B+Eb'B' ^ ...(61) = S e E
2V^y = L d D
where
c 5 V Z1 1 p = z6 " z2
b = z? - Z3 , q = zQ -
c*=p-q, f=z1 +z5
b'=p+q, g=z3 +z^
e = f - g , h = z ^+zQ
d = h - i , i=Zg+Zg
C = B = f1 (h, r, a)
C ' = B' = C = B
E = D = f2 (h, r, a) .
Table k contains the constants B, C, D and E taken from the computa
tions of Dr, I. Haaz. The values were determined for h = 100 cm and
a = 1.6 gr cm . In deriving the formulas (6l) Schweydar kept only the first order terms of the Fourier series. The secondorder terms are very complicated expressions and they may be neglected if the 80
T 1*5 3 5 10 20
B * C 0,424 0,134 0,052 0,016 0,00412
% E = D 0,554 0,370 0,263. 0,168 0,0712
t r 30 40 50 70 100
B = C 0,00105 0,00041 0,00027 0,00018 0,00009
E S D 0,02939 0,0l6l 0,0137 0,0123 0,00944 •
Table 4 81
terrain is not very rough. In case of very rough topography
the second method of Schweydar should be used. This method can be
found in the literature [SCHWEYDAR, 1927]•
Tables 5 and 6 are an example of the computation based on
formulas (6l) and the constants of Table k. Table 7 was designed to
speed up similar computations and gives the terms: c C, c r C', b B,
b 1 B 1, e E and d D.
Starting with assumptions similar to Schweydar's, but keeping
the second power terms of the Fourier series, Numerov of the Insti
tute of Astronomy in Leningrad has derived formulas which are wide
ly used in the Soviet Union today. Using the same symbols the
Numerov formulas are [NUMEROV, 1931] •
For smooth topography use eight azimuths (Aa =
4 - a-707 (B2 - B 4 - B6 + B e ) + (Bx - b5)
= 0*707 (Ba + B 4 - B6 - Bg) + (B3 - B7 ) . ...(6a) VA = Ax - A3 + t^-A7
Vxy = 0.5 (Ag - + Ag - Ag)
For rough topography use sixteen azimuths (Aa = 22.5°):
+ 0*353 (B3 + B? - Bu - B15) + 0.191 (B2 + Bg - B 10 - B lg) 82
LEVELING RECORD Station N°: 271 Area s Gttncz
r | Instrument N o . : +93 Instrument Height: 157 r i--- -- me “I ' "■""2. ..3“ " ... 4’. .. 5"" .6~ me ters 0° *+5° 90° 135° 180° 225° 2?CP 315° ters 1.5 157 157 156 157 157 158 157 157 1.5
3 157 156 156 157 158 159 157 158 3
5 157 153 155 158 160 157 163 158 5
10 156 t 15*+ 15^ 1 0 * 156 160 170 159 10 1 c
20 175 157 153 159 163 160 171 158 20
30 . 163 1^9 165 163 163 162 170 163 30
ko
50 1 70
100
Date; August 2, 195^ Observer s Ivan Mueller
Maps k^66l/k Readings In centimeters
Table 5 83
THE TERRAIN EFFECT
c b c ’ b' m BaC rC • bB P q B'aC1 c'C1 b'B' 5-1 7-3 6—2 8-4 p-q p+q
1.5 ,424 0 +1 +0.4 +1 0 .300 +1 +0.3 +1 +0.3 3 ,134 +1 +1 +1 +0.1 +3 +1 .095 +2 +0.2 +4 +0.4
5 .052 +3» +2 +8 +0.4 +4 0 .037 +4 +0.1 +4 +0.1 10 ,.018 0 0 +16 +Q.3 +6 -5 .013 +11 +0.1 +1 0 20 00412 -12 0 +18 +0.1 +3 -1 .00291 +4 0 +2 0 30 ,.00105 0 0 +5 0 +13 0 .00074 +13 0 +13 0 40 ,0004l .00029 50 ,.00027 .00019 70 .00018 .00013 100 .00009 ,00006
ZcC ZbB G=0 1.6 Zc'C* Zb'B +0.8 1 +0.3 +1.3 +0.7 f g e , h [ 1 d m eE e=aD dD 1+5 3+7 f-g 4+8 2+6 h-i
1.5 314 313 +1 +0.6 .554 313 315 -1 -0.6 3 315 313 +2 +0.7 .3702 315 315 0 0 5 317 318 -1 -0.3 .2634 316 310 +6 -1.6 10 312 324 -12 -2.0 .1678 323 314 +9 +1.5 20 338 324 +14 +1.0 .0712 317 317 0 0 30 326 335 -9 -0.3 .0293 326 311 +15 +0.4 40 .0161 50 ,0137 70 .01232 100 .00944
ZcC +0.3 ZbB +1.3 ZeE +2.3 ZdD +3.5 Computed by; Zc'C1 +0.7 Zb'B' +0.8 -2.6 -0.6 Ivan Mueller * t t t Date: Vxy +1.0 Vyz +2*1 VA -0.3 2Vxy +2.9 Oct. 3, 1954
Table 6 I 3[qei 1 9 10 19 16 .8 b 17 1b 20 19 06.0 6 20 54.5 4 15 13 11 18 .2 4 14 100 5.1 17 16 .4 b 13 13 45.9 5 14 11 16 10 .0 0 0 8 4 9 .5 0 7 1.5 5 90.3 0 .9 0 3 .6 0 2 1 6 07 140 70 40 0122 70 50 30 20 0 9 .7 1 4 .0 0 0 7 6 5 .a o 3 2 1 .0 8 .3 0 32. -39 .7 2 .3 8 .4 2 2.1 .4 0 .2 1 .2 7 .7 5 .0 3 .8 4 .6 7 .5 1 .5 5 e ✓ o • t .3 3 .7 2 .4 5 .9 3 .7 4 80.6 0 .8 1 .2 4 .8 3 41.1 .4 3 .4 0 .0 3 .5 2 2.1 30.4 0 .3 1 .5 1 175 1° 88 52 35 .0 0 . 2-4 0.1 .1 2 5-37 .5 2 .2 0 .7 0 3-31. -97 .7 1 -35 -33 .4 2 .3 2 .8 0 9- 5 -2 .9 0 .0 2 8-21. -149 .9 1 -52 .8 1 .7 1 .4 1 0 -8 .0 1 1 1 -28 .0 1 .5 1 .7 1 9-40. -41 .7 0 -14 .9 0 o.a .9 1 0 .0 1 3-39 .3 1 .2 1 .6 1 .9 1 33-4 .3 0 .5 0 .3 1 .5 1 .2 1 . 1-2 0.1 ,0 0 .7 0 244 4 524 349 175 157 105 5 3 70 2° 5 3 0-1 22 -2 0 -2 -14 -12 -31. -29 l l - -17 -55 5 .8 1 -50 -44 .5 1 -41 6 -3 2 .4 1 -27 -25 -47 -20 -16 -22 -12 -IB -8 7 -10 5-6 -9 60.2 0 •6 367 1 280 210 262 105 °4° 3° .0 0 0 • 4-4-5 .4 0 -154 -34 -26 .4 0 .3 0 .1 0 0.0. .8 0 5-42 .5 0 .9 0 0 -5 .6 0 7-57 .7 0 .0 2 .0 2 6-2 561.6 -566 -126 .6 1 .8 1 1.9 o.a .9 0 .6 0 9 1 1.1 .4 1 7-134 .7 1 b . l .5 1 .2 0 0.1 4-5-109 -25 .4 0 .3 1 -58 .0 1 5, .5 0 1.3 1 *c .3 0 1.1 490 210 5 3 526 438 350 140 699 -3 0 13-497 -113 -108 4-11 12-635 -142 -532 -119 -111 -103 -102 17-704 -157 9-13 0-2 36 -3 , 8 - 3 -65 -19 66 -6 _ 3 -7 0 7 “ -88 -91 875 5 420 350 262 613 175 -80 020 10 -74 6 281.1 -278 -63 -47 36 -3 30 -3 -52 -84 -19 10 q 5° <: 1051 210.8 0 -291 0-17 301.0 1 -360 -120 -472 270.7 0 -257 11.0 26 .8 1 1.7 -448 .6 1 -424 -400 -394 3 860 736 -497 -463 -108 -669 -326 -223 -600 -376 -352 371.3 1 -327 -254 -230 -206 -181 -157 -133 315 210 -20.0 0 0-12 - -85 -51 6° J 9ft 600.2 0 0 -6 030 20 1228 491 245 6 422 368 614 .0 0 .3 0 .2 0 . 28-2 . 700-1170 0 0 -7 0.1 -520 -208 0.1 9-3939 .9 0 3291 -1319 .9 0 .6 0 .5 0 028 .0 2 .5 0 8-1 -2080 -816 .8 0 ‘.60 .0 2 7° 9 1 .9 1 9-912 .9 0 .7 0 .3 0 4-432 .4 0 1.4 .5 1 .4 1 .0 1 1.1 1.1 .8 1 23 1.5 0.1 1.9 1.3 1.7 I •£ 451504 1405 984 281 562 702 -1458 8° -1180 -1041 -1010 « C» '« B -69 0 -485 -624 -902 -763 37-867 -347 22 -720 B - 0 27 25 711.4 27 .8 9 24 23 -624 24 22 21 11-367 .0 0 -141 0-122 0-48 28 25 67.8 7 26 21 4 -2 -336 -527 040 30 191234 1109 1 5 389 353 317 7 2 503 529 475 792 .6 2 . 8 3 0 7 851 770 705 633 . a 1. 7. .9 6 6. .3 6 7. 1 9° 4 5 3 5 2 6 5 10.6 11.9 10.2 1836 -1 32.9 2 .3 9 0-173 .9 8 -2326 -2571 -1102 -1591 -1346 1.5 3648 2600 2947 2254 .3 0 1215 1562 -857 1908 620. 918-1420 8 1 -9 .2 0 -612 1763 2.1 .4 2 .0 2 7-77 .7 2 .5 2 802 .3 2 .2 2 40 0 1 12° 11° 10° 3.8 .3 551 .5 3 -49 .4 3 .6 3 .2 3 1 8-41 .8 2 3 1361 1944 .1 .0 0 8-4420 .8 0 .4 0 .2 0 60 -6 .6 0 .7 0 .5 0 -68 6 -6 2.1 8 -5 9-3487 .9 0 .6 0 .4 0 .0 1 -71 .8 0 .7 0 .5 0 7 031154 1063 972 -74 .3 2 -63 0.1 3-1286 .3 0 .0 1. 432.2 2 3 -4 542. -158 .8 2 4 -5 2 -5 4 4-136 .4 2 -47 -45 5 1408 162309 2126 0-260 .5 2 -3854 -3120 -5460 -4940 -2730 -1820 -3900 425 -2340 0-183 .3 2 -3380 .7 2 -4290 -2860 .6 2 -1300 -2754 15 25 .4 0 -2387 -2554 -1653 638 .2 2 .4 2 -2020 50 . -550 .6 2 .5 2 .7 2 .3 2 2.1 50 1616 — — 924 - _ * 462 - 693 3 14° 13° -147 -619 -141 -125 -152 19521 -130 -119 219 173 211 9 -876 196 -806 180 165 203 108 3510 1 5 -3 300.0 0 -390 0 1 20 .10 -1950 -8 O'.O 0-283 18 3-3640 .3 0 -1987 -3122 10 -852 2493 100 70 70 1745 1246 748 997 9 453 499 -738 -772 -910 -979 942.7 2 -944 -842 692 9 -6 632.6 2 -643 -667 570 7 -5 546 595 .4 0 . 0 20 .5 0 .3 0 .5 0 . 0 .2 0 0.1 2679 0240 1072 1875 1340 804 15° -3900 2 1 i5 2 .8 2 .1 2 .6 2 2 2 .3 2 .1 2 .5 2 .4 2 .8 2 .7 2 .2 2 .3 2 .4 .2 -8580 -5460 -7020 -2340 0-780 -1560 0-520 -4680 -2600 -5720 100 70 100 50 20 30 100 98 96 94 92 84 82 80 832. .2 3 2 .6 2 3 90 88 86 72 62. .0 0 2 28.1 76 20 46 427.4 7 2 78 74 70 68 66 62 60 56 58 38 36 34 39 .1 8 28 26 12.2 24 22 424.4 44 42 .2 2 2 40 32 64 48 52 50 54 16 14 19 18 17 .3 8 *5 13 12 16.1 11 10 8 4 5 1.1 3 2 7 9 1 6 .4 5 5 222. .8 5 1 .2 2 2 .2 3 3 2121.5 1 2 32.1 .8 8 2 18.5 .7 7 2 9.9 29 21.0 3is;5 ; s i .3 3 2 25.5 .0 1 3 812.6 2 1 .8 8 1 16.6 15.5 26.6 9913.3 3 1 19.9 .9 8 14.4 13.3 17.7 11.1 10.5 10.0 .9 8 46.3 6 .4 9 .8 7 .6 0 .2 7 .5 1 .6 6 .2 2 .3 3 .8 2 .5 5 .4 4 .9 3 .0 5 .7 1 .0 7 3 .3 3 .6 3 2 2 .8 1 3 31.1 .6 9 2 .9 8 2 .6 6 18.4 2 .9 5 2 .2 5 2 .4 24 .7 3 2 .9 2 2 .0 0 2 .4 0 3 .7 0 2 14.1 .4 7 11.1 10.4 .8 1 1 17.0 9213.7 19.2 16.3 14.8 17.8 .4 7 .9 5 .7 0 ;4 0 .0 7 .7 6 .6 9 .2 5 .4 4 .0 3 64.0 4 .6 5 .8 4 .1 1 .2 2 72.6 2 4.1 .7 3 .8 1 .6 2 32.4 2 .3 3 .5 1 .8 1 3 .8 5 2 .8 4 2 .2 4 2 22.1 .6 1 2 21.1 .3 5 2 .7 3 2 117.0 7 .6 1 1 11.1 20.5 11.4 .9 7 1 .3 6 2 7411.1 17.4 16.3 12.1 19.5 .7 9 15.3 13.2 14.2 19.0 16.8 8 -6 .0 8 12.6 14.8 10.0 10.5 95.7 5 .9 8 45.4 5 .4 8 .5 9 .9 7 .8 5 .8 0 .5 0 .3 0 33.4 3 .3 5 .7 4 .5 4 .2 4 .8 6 .7 3 .2 3 .0 5 34.0 4 .3 6 .4 3 .9 2 2.1 .0 1 1.3 . .0 1 1.6 5 .13.1 6.4 16 10.1 16.1 .8 5 1 15.4 15.1 11.7 .7 0 1 12.4 8-316 .8 6 1 14.8 14.4 14.1 30-189 13.0 13.4 12.1 10.4 27-4 0-358 .0 0 1 -147 12.7 .7 8 .4 8 .7 7 .4 7 . -77 .4 9 9.1 .4 6 .0 6 7 -3 .7 4 .4 4 .5 0 .7 6 .3 0 .2 0 .3 2 020 10 .2 3 .0 3 .8 2 .7 2 .5 2 .0 5 .7 3 .2 2 .7 0 .0 2 .8 0 .2 1 1.7 .3 1 . -16 1.8 . -13 1.5 -301 0 6 -2 -288 -273 -245 -203 -175 -133 358 5 -3 -344 -329 2715.0 5 1 -231 -217 -161 -119 -104 940. -228 .6 0 4 -9 57 -5 88 -8 .4 4 6 -6 3 -6 -91 82 -8 -74 -71 80 -8 493. -419 .4 3 9 -4 9 -2 6 -2 20 -2 543.8 3 .6 3 4 -5 2 -5 6 -4 3 -4 2 -3 60 -6 96 -9 -85 27 -2 4 -2 3 -2 -21 -40 -35 8 -1 -9 8 10 -1 -6 5 -14 .8 0 2 -1 -17 1-2 -4 -3 -7 6.2 6 . .0 9 4 .0 1 2 0 .0 0 2 0-494 .0 4 1 .0 5 2 .0 4 2 .0 3 2 .0 2 2 .0 8 1 .0 6 1 7.0 17 12.0 11.0 30-460 13.0 6- 0 6 -1 .6 4 .0 8 .0 7 .0 9 .3 0 .0 3 .0 4 6-8 .2 0 .8 6 .4 6 8-201 .8 5 .0 5 .4 5 .2 3 .4 2 .4 0 .2 4 0.1 .5 0 0-208 .0 6 .8 4 .7 0 .2 5 .9 0 .6 5 .6 1 0- .0 2 .8 2 .6 2 2- 8 -7 .2 2 .2 1 35-40 2 -2 .6 0 .9 1 .8 1 .7 1 .4 1 .0 1 5- 2 -5 .5 1 .3 1 1.1 -732 -664 .0 5 1 -962 -527 -562 180 8 -1 -187 -835 -801 -767 -698 -630 -595 426 2 -4 3 3 -1 -391 -214 25-465 5 -255 a - -221 -167 13-1 .0 5 -316 -173 33- .0 9 8 8 -5 -323 -194 -870 -117*21 9-11 -126 -192 -112 -105 146 4 -1 -139 13-279 -153 -289 -71 85-5 4- -202 2 8 -1 0 8 -1 .4 2 -155 8 -9 5 -8 9 -4 63-114 6 -6 3 -6 9 -5 6 -5 92 -9 6 -4 22-27 -15 42 -4 2-5 18 -1 3 7 1.1 -71 -39 -25 -35 9 -2 32 -3 30 Z m -837 . -1459 -1398 -1334 -1274 -1025 -1583 -1522 -1210 -1149 -1085 01 .2 0 10-16 28-34 - .0 2 0 3 --1 -391 -292 -341 7612.0 -776 .2 3 -204 0 -713 4286.8 6 .6 6 8 2 -4 -416 -403 -304 -366 395.2 5 -329 -354 -217 *121 8 0 -1 -102 -900 -229 -167 -142 -254 -242 -527 -378 -652 274.2 4 -267 -9 4 90 -9 65 -6 2 -5 58 -5 -96 -83 -77 -46 050 40 .0 1 2 0 .0 0 2 .0 3 2 .0 2 2 0 .0 9 1 .0 0 1 .0 6 1 .0 4 1 7.0 17 0.0 10 13.0 11.0 .1 0 .3 0 42-229-36 26-32 .4 0 .5 0 .0 7 .7 0 .0 4 .6 0 .2 6 .0 8 .8 4 8- 2 -6 .8 0 4-401 .4 5 .9 0 .4 3 .6 3 -211 .0 3 .8 2 .6 2 .8 3 .6 4 .4 4 4-474 .4 6 .8 5 .6 5 .2 2 .2 1 .0 6 .0 1 .8 1 .6 1 .5 1 .4 1 .3 1 .9 1 .7 1 -1568 -1496 -1204 -1350 -1276 -1130 1058 -1 41-47 -1860 -17U8 -1715 -1642 -1422 19-25 11-18 33-40 -478 0 3 4 - 4-10 912-1014 2 1 -9 0 4 -2 -984 9 9 -2 837- 3 3 -9 7 3 -8 8 2 -3 546 4 -5 3 0 -5 372- 3 1 -4 2 7 -3 692 9 -6 -255 -142 -127 -153 -135 -113 -270 -105 -284 313 1 -3 49-511 -459 3 4 -3 -620 8 8 -4 -495 -445 -357 -766 -430 -386 416-462 6 1 -4 12-158 -142 -120 226 2 -2 -186 -197 -167 -54 76 -7 69 -6 -83 98 -9 -91 21-28 45-52 32 216*26 .2 0 13-20 -2070 -1907 -1257 -1338 -1095 37-44 -1177 -1990 -1826 5-12 171.4 1 -117 -1QB 607 0 -6 770 7 -7 -284 -125 303.6 3 .2 -300 3 -267 332 3 -3 -101 -349 689 8 -6 .6 6 .4 6 -543 7 2 -5 7 9 -3 852 5 -8 6 4 -4 -r133 -170 -150 -235 -219 -316 -381 -365 -559 1 1 1 1 1 -251 70 60 -6 -85 -69 -77 iiiii -93 .0 5 2 .0 4 2 .0 3 2 .0 2 2 .0 1 2 . .0 0 2 .0 2 1 .0 3 1 .0 5 1 .0 6 1 .0 4 1 .0 1 1 0 .0 9 1 .0 8 1 .0 7 1 112 -1 .0 0 1 .1 0 327-37 .3 0 Oi5 .4 0 659-68 .6 0 .7 0 8- 0 -9 .8 0 .9 0 .0 1 .0 3 .2 1 .6 4 .4 4 .0 7 .8 4 -121 1.1 .8 5 0 4 -5 .0 5 .0 9 .4 5 .2 5 6- 3 0 -6 .6 5 0- 6 2 -2 .0 2 .5 1 4- 4 6 -2 .4 2 .8 2 .2 2 8-412 .8 3 -143 .3 1 .0 4 .2 6 .0 6 .0 8 7-185 .7 1 .4 3 .6 1 .8 1 6'- 6 8 -2 ' .6 2 2-455 .2 4 .8 6 .9 1 38-47 48-58 1748 4 7 -1 1323 -1 -1642 1430 -1 -1217 -2277 170 -2 960 -1 852 -1 -1006 2490 -2 -1535 -2065 -2700 -2596 -2383 6-15 100 153 5 -1 -152 -111 -100 -174 -164 206 0 -2 -196 -391 9 4 -3 7 0 -3 -243 793 9 -7 7 8 -6 -624 582 8 -5 370 7 -3 328 2 -3 -434 666 6 -6 -497 476 7 -4 646 4 -6 -731 -709 8 1 -5 -561 900 0 -9 79 -7 85 VA = °*5(Ai“A5+A9“A13)+°»373(A2-A^-A^+Ag+A10-A12-A1^+Aig) 00.(63)
VXy= 0.25 (A3-Ay+Ai i " A ^ )+0.177( Ag+AJ+-Ag-Ag+Al0+A12- A^li” Aj.6 ^ where the constants A^, Ag .... A^g, and Bg .... B ^ are func tions of the following type for the azimuths 1, 2 .... 16,
n 2 2 2 Aa = 2 EMz1 + Nz1+1 + Pz± + Qz1+1 + R (zi+1 “ zi) ^azimuth a
Ba = S [M'Zi ♦ r s 1+1 + P-.8 + «-z2+1 + B- (z1+1 - „
The constants M, N, P, Q, R, M', N', P 1, Q 1 and R 1 are functions of 2 k , a, 7T, r^, ri+i an<* k* Table 8 gives the values of the summa tions in the brackets above, as functions of z for each radius.
The table was computed with h a 100 cm and cr » 2 gr cm . The quan tities in the brackets are symbolized in the Table for the radii
1.2, 2 .... 50 m by A^-g, Ag .... A^q, and so forth. For example,
A1.2 = ^0+1.2+ Nzl.2+2+ PZ0+1.2+ Qz1.2+2+ R ^Z0+1.2 ” zl,2+c ^ '2 2 2 2 2 2 and
A10 = Mz6+10+ N10+l8+ Pz6+10+ Qz10+18+ R ^Z6+10 " 210+18^ 2 2 2 2 2 2
Using these symbols the values A and B for the azimuths 1, 2 OC u , boo* £L2T61
A a = Ea1.2 + A2 + A3 + A 1j.5 + a6 + A10 + Al8 + ^O^azirauth a
Ba = tBx.2 + B 2 + B 3 + + ®6 + B 10 + Bl8 + B 503azHnuth a 86
z Ag z Al ,2 B l a2 B2 A3 B 3 A4,5 B4*5 jn m -0.20 -- — -7,0 -4,0 -0,50 — -- -10,0 -3.0 18 — -6,4 -3,6 0.45 — — - 9.1 -2.6 16 -- — -5,7 -3,2 40 -- — - 8.2 -2.3 14 — -- -5,1 -2,8 35 - - 7.2 -2.0 12 -- -k;.k -2 ok 30 - 8,9 -3.5 - 6.2 -1.7 ! 10 -4,3 -4,3 -3,7 -2,0 25 - 7.5 -2.9 - 5.2 -1.4 08 -3.0 5 -3,5 -3,0 -1,6 20 - 6,1 -2.3 - 4,2 -1.1 06 -2,7 2,7 -2,3 -1.2 15 - 4.6 -1.7 - 3.2 -0.8 04 -1,8 1,8 -1,5 0,8 10 - 3.1 1.1 - 2.1 -0.5 i 02 -1.0 0.9 -0.8 -0.4 - 05 - 1.6 -0.5 - 1.1 -0.3 ; 0.00 0. 0 0 0 0.00 0 0 0 0 + 02 +1.0 +0,9 +0.8 +0,4 + 05 + 1.6 +0.5 + 1.1 +0.2 04 +2,0 +1.8 +1.6 0,8 10 + 3.3 +1.1 + 2.2 +0.5 06 +3,0 +2,7 +2 *k 1.2 15 + 5.0 +1.6 + 3.3 +0.7 08 +4,1 +3,8 +3,3 1,6 20 + 6,8 +2,1 + 4,4 +0,9 10 +5»2 +4.8 +4,2 2.0 25 + 8.6 +2.6 + 5.6 +1.2 12 —— +5,0 2.4 30 +10.4 +3.0 + 6.7 +1.4 Ik — — +5.9 2.9 35 —— + 7*9 +1.5 16 —— +6,8 3,3 40 —— + 9.1 +1.7 + 18 — -- +7,8 3,7 +0,45 — +10.2 +1.9 0.20 — -- +8.7 +4.1 +0.50 — +11.5 +2.0
Table 8
(Continued on the next page) 8?
— --- ! z z z in a6 % A10 B 10 m A l8 Bl8 m A50 B 50
-1,0 _ _ -16,1 -2,5 -2.0 -25,5 -2,7 -5,0 -28,0 “2.5 0,9 -- -14,6 -2,2 1,8 -23,0 -2,? 4,5 -25,3 -2.1 0,8 — -13,0 -1,9 1,6 -20,5 -2,0 4,0 -22,5 -1,7 =0,7 -- -11,4 -1,6 1,4 -17,9 -1,6 3*5 -19*7 -1.4 0,.6 -1Q«9 -2,o 3 - 9,8 -1,3 1,2 -15,4 -1,3 3*0 -16,9 -l.l 0,o 5 - 9.o 2 -1.8 - 8.2 -1,0 1,0 -14,1 -0,8 2,5 -14,1 -0,8 0,4 - 7,4 -1,4 - 6.6 -0,8 0.8 -10,3 -0,8 2.0 -11.3 -0.6 0,o 3 - >.6 -1.0 - 5,0 -0,6 0,6 - 7,7 -0,5 1.5 - 8.3 -0.4 0,2 - 3,7 -0.6 - 3,3 -0,4 0.4 - 5,2 -0,3 1.0 - 5*7 -0.2 0..1 - 1.9 -0.3 - 1.7 i-0.2 -0,2 - 2.6 -0.11 -0,5 - 2.8 -0.1 0.00 0. 0 0. 1 0 0,0 0. 0. 0.0 0 0 +0.1 + 1,9 +0,. 3 + 1.7 +0,2 +0,2 + 2.6 +0,1 +0.5 + 2.8 0. 0..2 + 3,8 +0.6 + 3,3 +0,3 0,4 5,2 +0,2 1.0 + 5,7 +0.1 0,3 + 5,8 +0,8 + 5,0 :+0,4 0,6 7,8 +0,3 1,5 + 8,5 +0.1 0,4 + 7,8 +1.0 0 + 6,7 +0,5 0,8 10,4 +0,3 2.0 +11, 4 0 0,5 + 9,8 +1,2 + 8,4 +0.6 1,0 13.0 +0,4 2.5 +14,2 -0.1 O.06 +11.8 +1.3 +10.1 +0,7 1,2 +15,7 +0, 4 3*0 +17,1 -0,2 0.7 -- — +11,8 +0,8 1.4 +18,3 +0.3 3.5 +19,9 -0.4 0,8 — — +13,6 +0,9 1.6 +21, +0,2 4,0 +22.8 -0,5 0,9 -- — +15,3 +0.9 1.8 +23,6 +1.0 4.5 +25*7 -0,3 1.0 -- -- +17.0 +0.9 +2.0 +26.3 +0.0 +P e 0 +28.5 -1.1
Table 8
(Concluded) 88
For azimuth 8 (cUg = 315° in equations (62) and ag = 157-5° in equations (63) ) Bg is the sum of the values g, Bg «... B^q from
Table 8,along the azimuth 8, for each intersection. Similarly, is the sura of the values A^ g, Ag .... Afrom the table for all the radii along the azimuth 1 (0^ = 0° in both equations (62) and
(63) )-
Having the values A^., Ag .... A ^ , and B^, Bg .... B ^ for the different azimuths, the terrain effect can be computed by means of equations (62) or (63).
An example, based upon equations (62) is given in Table 9»
2.332. Mechanical Methods
Three kinds of mechanical devices are used to determine the terrain effect: Slide rules, special levelling rods, and the gravi- tacheometer. All of them are based on equations (62) and (63 ). The goal is to determine the constants A^, Ag .... A^g and B^, Bg ....
B^g for the azimuths l-l6 . Having these constants the terrain effect is corrputed by means of equations (62) or (63).
The slide rules are designed to give the constants A and B as functions of z. There are two slide rules for each radius, one for the constant A and one for B. In case of eight azimuths, for in stance, there are 8 X 2 = l6 slide rules. This method does not speed up the computation significantly and so is not used in practice.
In order to speed up the computations, special levelling rods were designed. There is one levelling rod for each radius, giving
v {. 89
RELATIVE ELEVATIONS (z - h) IN CENTIMETERS
”\" q 1 / 2 Z ' 3 / 4 ^ 5 / 7 / 8 ^ / 9 P ° neters? /W > 1 8 0 ° ^ 2 5 ° / 2 7 P 1.2 + 2.0 2.2 - 2.1 + 1.5 + 3-0 + 6.0 + 9.1 + 3-5 2 2.5 4.3 - 3.7 + r.i + V.7 + 8.5 + 7-8 + 5.0 3' 2.9 7'-4 - 6.0 - 0.5 + 7-2 +10.1 + 10.8 + 5.8 4-5 f 1..1 10.5 - 11.5 - 1.6 + 8.2 +14.5 + 19-7 + 9-0 6 - 1 14 - 17 - 3 + 9 +19 + 17 +12 10 0 23 - 30 - 10 +16 +30 + 28 +20 18 f 2 50 - 62 - 28 +24 +4l + 48 +38 f i 50 26 148 -201______V -102 - 5 +17 +131 +99
THE TERRAIN EEEECT r Ag \ a let ere A i A 3; *5 6. . ^7 A 8, 1.2 +1.0 - 1.1 - 1.0 +0.8 + 1.5 + 3-0 + 2-5 + 1.8 2 +1.0 - 1.6 - 1.4 +0.4 + 1.9 + 3-5 + 3-2 + 2.0 3' +1.0 - 2.3 - 1.8 -O', a + 2*3 + 3-3 + 3-6 + 1.9 4.5 +0.2 - 2.2 - 2.4 -0-.3 + 1.8 + 3-2 + 3-0 + 2-0 6 -0.1 - 2.5 - 3'-2 -0.6 + 1.8 + 3-6 + 3,2 + 2.3 10 0.0 - 3'. 8 - 5-0 -1.7 + 2.7 + 5-0 + 4.7 + 3-2 18 +0.3 - 6.5 - 8'.0 -3.6 + 3'-l + 5-3 + 6.2 + 5-0 50 +1-.4 - 8.3 -11.3 -5-7 + 0.3 + 0.9 + 7.4 + 5-7 2 +4.7 -28.3 -34.1 -0.9 +14.8 " +27.0 +33-0 +23-9
r leters I B2 B3- B4 B5 b 6 B 7 b 8 1.2 I[ +0.9 -l';0 -1.0 +0.7 +1.4 +2.7 +2-3 +1.6 2 ! +0.5 -0.9 -0.7 +0.2 +0.9 +1.7 +1.6 +1.0 3' +0.3 -O'. 8 -0.6 0.0 +0.8 +1.1 +1.2 +0.6 4.5 ■.0 -0.5 -0.6 +0.1 +0.6 +0.7 +0.6 +0.4 6 .0 -0.4 -0.5 +0.1 +0.3 +0.6 +0.5 +0.4 10 .0 -0.5 -0.6 +0.2 +0.3 +0.6 +0.6 +0.4 18 .0 -0.4 -0.5 +0.2 +0.1 +0.2 +0.2 +0.2 50 ,0 -0*4 -0.6 +0.2 +0.0 +0o 0 +0.1 +0.1 2 +1.7 -4.9 -5-1 +0.1 +4.4 +7-6 +7-1 + 4 4
V* = 0.707 (-4.. 9 - O.l - 7..6 + 4..7) + (1-7 - 4.4) = - 7.ol E
Vyz = Oo7°7 + Oo1 " 7“6 ' + (“5o1 " 7,1)= "2^ 3 E V ^ = 4.7 + 34.1 +■ 14.8 - 33.8 =» 19.8 E Vt = 0.5 (-28.3 + 8.9 + 27-8 - 23-8) = - 7.7 E
Table 9 90 eight rods and restricting the used radii to these particular values. On one side of the rod the constants A are painted; the other side shows the constants B. Each constant A and B is narked on the rod at the corresponding elevation z, had from Table 8.
In the telescope of the levelling instrument we read the constants A, or by turning around the rods the constants B, instead of the ele vations .
The gravi-tacheoroeter is a theodolite with a special vertical circle. The theory of the instrument is complicated [SOROKIN, 1953]*
The vertical circle is graduated in terms of A and B found in Table 8.
This is possible because A and B are functions of radius r and rela tive elevation z. The influence of r is eliminated by choice of radii such that the ratio of two consecutive radii is constant. The z is related to the observation angle through the tangent functions; thus the values A and B are also similarly related to the observa tion angle. In practice, the intersections of the circles and a radial line are marked with stakes and the gravi-tacheometer is pointed at each stake in turn and the values of A and B had at such pointing are summed to get A^ and B^ related to the azimuth of that ct 05 particular radial line. If this is done in each azimuth formulas
(62) or (63) can be used to find the whole terrain effect. 2*3^. THE CARTOGRAPHIC EFFECT
2 <.3^1° Analytical Methods
In order to determine the cartographic effect we divide the
topographic masses— beyond the limit of the terrain effect--into
mass-plates using vertical and horizontal planes. The vertical
planes radiate from
the observation Worth point. They divide
the area in a man
ner similar to the
radial lines used
when computing the
terrain effect.
The horizontal Figure 31 planes are spaced
equal distances,
Az, to one another, starting with the horizontal plane through the
observation point. The topography is represented on the used maps
by means of contour lines. Each of the horizontal planes contains
the contour line corresponding to the elevation of the plane. We
assume that the contour lines can be replaced by circular arcs run
ning between two adjacent radial planes. We also assume that the
topography steps up or down along these arcs, i.e., the transition
of the topography from one contour line to the next is not con
tinuous (Figure 3l)« This way, the topographic masses are divided 92 i , into plates. Each plate is generally wedge-shaped and is bounded by the two radial-vertical planes on its sides and a segment of a cylin der wall placed along the contour line on the nearer border. Two horizontal planes on the top and on the bottom define its vertical extent. The fourth side of the plate, i.e., the outer border is imagined to be at infinity (Figure 31)• The sum of the effects of the horizontal plates gives the cartographic effect.
The effect of one plate can be computed from equations (57)•
The integration limits are: - 0^, r - o°, and z^ - z^. Here r is the distance of the contour line having relative elevation,z^, from the observation point. The angles and are the azi muths of the radial planes. The relative elevations of the horizon tal planes are z^ and Zg. The value h in formulas (57) can be neg lected with respect to z and r. The new formulas for one plate are:
k 2 .°2 -C7° ; - n r ,cos n g ^rr2a \ = -3* I f f -°a -3*5/2 ar dz (r2 + z2 )5/2 '
3.2 rQ^r<9°f2t- va rr^ sin s m 2adu. , , , xyVV = J J J o o R/O a, r z. (r2 + z2 )5/2 ' ...(61^) y „ up o^: ■tr r r r " or0 r2 z cos au vxz = f f f ~~~o o.f^/o ^ Ql r zi (r2 + z2 )5/2
„k 2 ra2 f°®fz2 ao rr 2 z sin a ^ ^ = t o d r d z 93
After integration and simplification we get:
„k ... 2 A • A cos 2a VI = - 3k c Az sin Aa A r
2Vk „,_22 A_ sin 2a = 3k a Az sin a xy .(65) ,1c 3 , 2 A/ 2\ , A a cos a ■ " 2 k aA(z } 6iny 7 “
,1c 3 , 2 A /_2^ , A a sin a V* = - ^ k a A(z ) si n ----- *— yz 2 2 r where r is the distance of the z^ contour line from the observation
point,
Ace = a2 - 0^
Az = Zg - 2l
A(z2) = z| - zf
rv - ° 1 + z
Compute these values for each plate in each sector and the sum will he the cartographic effect.
The number of sectors to be used is eight for the gradients
and and sixteen for the values and . The distance to xz yz A xy which we must compute depends upon the topography, but generally does not exceed a few hundred kilometers.
This computation is very laborious since a single point re quires about seventy pages of computation and, therefore, one does not use equations (65) in this form. However, equations (65) are the base of a widely used mechanical method to be introduced in the next chapter. 2.3^2. Mechanical Methods
The mechanical method of computation is based on equations (65).
These expressions can be written in the following form:
= -A Az cos 2a
2V* =s +A Az sin 2a xy ...(66) xz a -B A(z2 ) cos 0!
= -B A(z2 ) sin a yz
where ^ 2 A = — k o sin Aa *■
B = — „ k2 a sin — 2r 2 2 These values depend, if we disregard the constants k , a and Aa,
only upon the distance r which is the real variable. Dr. I. Haaz has constructed a special scale to facilitate mechanical solutions
[HAAZ, 19^7]• At one edge regular linear distances of the scale were replaced by the corresponding values A; at the other edge by the values B. This way, by placing the scale on the map and in the mean azimuth a of a wedge and putting the zero index on the observa tion point, we can read the A and B values at each contour line which crosses that mean azimuth. Then compute the influence of each plate by means of formulas (66) and the sum will give the car tographic effect.
It is also possible to replace the linear divisions of the scale with the values A sin 2a, A cos 2a, B sin a and B cos a.
The computation will he speedier, but we will need several scales for the different azimuths.
Table 10 shows an example for the computation of V^z and from eight azimuths. The computation of the and is similar but it has to be executed in sixteen azimuths.
2.3^3• Graphical Methods
The most significant graphical methods for computation of the cartographic effect were developed at about the same time by Haalck
[HAAI0K, 1928; HEILAND, 1928], Jung [JUNG, 1927], and Samsonov
[SAMSONOV, 1931] working independently. All methods use the same principle. They divide the surface of the Earth into compartments by means of radial lines and circles. The radii and the azimuths were so chosen to insure equal influence by each compartment having a certain mean elevation. The cartographic effect of a compart ment (a^ - a^, Tg - r^) can be determined by means of equations
(58) by changing the limits of the integrals. Neglecting again the value h, with respect to z and r, the influence of one sector and after integration becomes:
V* O | k2 m (i- - |-) (Bin 20^ - sin St*,)
2V£y = § ^ 02 (°os ^ " eos ^ 1 2 ...(68)
■ ? k 2 0 2 2 < h - h (si n “ i - 3 i n “ a 5 k ri ra
V = - - k2 az2 ( ~ - ^j) (cos cl - cos a ) y 4 rg
We will take the value as an example. If we want the influence of each compartment to be constant, we have to select the quantities rl* ^2, ai and a2 50 they the following relation:
1 1 (------) (sin 2a. - sin 2a.) = constant 1 r2 To obtain this we assume that
sin 20^ - sin 2a^ = ^ =» constant where n is .an even number. We also assume that
-— — = cn = constant rl r2 3 2 where c is a number to be selected to insure that k cn is a very small number, for instance 0.00001. In this particular case 97 Table 10
Contour Line BA(z2 ' sin a z z2 A(z2 ) Elevations b ^(z 2; cos a ■ H H-H (H-H0 )2 A(H-H0 ) B sin a + m 10m B cos a (10m)2 (10m)2
Hq - 170 - -- ... - r - l60 1 1 1 0.25 - Q .25 140 3 9 8 0.12 - 0.96 a a oc 130 4 16 7 0.05 0.35 140 3 9 -7 0.02 0.14 - 160 1 1 -8 0.01 0.08 - 140 3 9 8 0.004 - 0..03 1.37
H = 170 -- - #* - o 160 1 1 1 o,.o4 - 0..04 140 3 9 8 0.01 — 0..08 a = 45° 130 4 16 7 0.003 0.02 140 3 9 -7 0.002 0..01 - 160 1 1 -8 0.001 0.01 1 0.12
Hq = 170 - - - m - - 160 1 1 1 0.02 0..02 a = 90° 140 3 9 8 0..006 0.05 120 5 25 16 0.003 - 0.05 0.12
Eq = 170 - - - . , - - . r ■ 160 1 1 1 0..40 “ 0..40 0=135° l40 3 9 8 0.10 0.80 120 5 25 16 0.004 - 0.06 1.26 1
(Continued on next page) 98
Table 10 (Concluded)
Joritour Line 2 z z e ^(Z2 : sin a Elevations a(z2 ) b a ( z 2; cos a H H-H A(H-H0 ) B sin a (H_Ho)g + m 10m B cos a (10m) (10m)2
H0 - 170 - - - -r -- 160 1 1 1 0..70 . - 0.70 o=l80° 1^0 3 9 8 0..15 1.20 170 0 0 -9 0.012 0.11 r 120 5 25 25 0.003 «* 0.08 1.87
ii o - - w . o
150 2 k k 0..05 - 0.20 2=225° 170 0 0 -k 0..005 0.02 .200 3 9 9 0.001 - 0.01 0.19
**
ii -
o - - -
-tq - 160 1 1 1 0,.02 - Q.02 2=270° 180 1 1 0 0,. 005 - 0..00 200 3 9 8 0.001 - 0.01 0.03
H o = 170 -- - ■ • r • - .. 160 1 1 1 0.10 - 0..10 35=315° 1^0 3 9 8 0..07 0.56
160 1 1 -8 0.02 0.l6 - 180 1 1 0 0.001 - o , . o o 0.50 i N 99
z s 1 m will cause each compartment to have a cartographic
effect of 0.00001 a E. The total effect will be
= 0.00001 a £ z ...(69)
In the case of gradient the radii and azimuths have to xz be selected to insure that the following conditions are fulfilled:
sin - sin ^
12 - 1 2 = cn rl r2
In this case the total influence will be 2 = 0.00001 a Z ...(70) xz 2 ' '
When the radii and azimuths are decided upon, templates
can be drawn on transparent paper, one for and one for V^z.
We place the templates on the map with the template center over the
observation point and a = 0° to the North and estimate by eye the mean elevations of each compartment. With these estimates and
V^z can be computed using equations (69) and (70) respectively.
The V s can also be computed from equation (69) but before xy estimating the mean elevations of the sectors, the template designed for should be rotated ^5° in either direction. With this rota tion we switch from sin 2a to cos 2a as required in equations (68).
Similarly, V^_ can be conputed from equation (70) if the mean ele- vations are estimated using the template designed for V^z but rotated 90° in either direction. Rotation switches the values from sin a to cos a as required by equations (68).
2.35* THE TRANSFORMATION OF THE TORSION BALANCE MEASUREMENTS
The gradients W„, W , W and W measured by the torsion A xy xz yz * balance refer to a coordinate system whose originis at the center of the swinging beam, the + x axis points to North, the + y axispoints to East and the z axis coincides with the torsionfiber, i.e., with the vertical. 'Let us transform the gradients into a new (n,s,t) coordinate system of common origin, whose t axis coincides with the z axis of the former system. The n axis forms an angle a with the x axis clockwise, and the s axis is perpendicular to n and t. The relation between these two coordinate systems is given by the following transformation equations:
x s n cos a - s sin a
y = n sin a + s cos a
z = t
The partial derivatives of the x, y and z coordinates computed from these transformation equations are 101
The first partial derivatives of the potential function W in the new coordinate system are
dW dW cbc dW dy dW dz dW dW , 35 = K 3 5 + 5 ? 5 5 + 3 J 3 5 " S CO80! + 5^ slno aw aw ax aw ay , aw az aw . „ ^ aw „ Si = S S i + ^5 ? + Ste5i= ' S Bina + ^ cos“ aw aw ax aw ay aw az aw St ■ 53? a t + 5$F I t + 5? St 3 5£
The second partial derivatives are
afw _ (~) ~ + — (~) ^ + — (—) — 2 an ax an an ay an an az an an a2u p a^w p a2w = cos a + sin a + 2 -2-2- sin a cos a ax ay axay
a^w _ a_ (aw} ax + a_ ^aw^ ay + a_ ^aw^ az _ as2 ax as as ay as as az as as
a ^ . 2 a ^ 2 0 A , „ = — 5 sin a + — 5 cos a - 2 --- sin a cos a ax ay axay
a ^ _ a_ (aw} ax + a_ ^aw^ ay + a_ ^aw^ az _ a ^ at2 ax at at ay at at az at at az2
a ^ a (aw^ ax + a ^aw^ ay + a ^aw^ az asSn ax an as ay an as az an as
= - ^ 5 ) sin Ct cos a + --W (cos2 a - sin2 a) ay ax axay a S a /dw> &c , a ,aw, ay ^ a ,aw» az S 5 t ■ S (55' St + 55? W 5t + 5J (35} 5E "
■ s s . coa ° + H ? sln “
a2# a ,aw, ax a ,aw> ay a ,aws az 5S3t ■ S W 5t + 57 (5i} It + 5F (SS' 5t " a 2 !} a^w = " 5c5z sln “ + 5J5? 008 “
From the first and second expressions above (see previous page) we get
A . afw , A (sln2 „ _ c o b 2 -a) + A (cos2 a - sin2 a) - ds2 dn &t dy
- 4 cos a sin a dxdy
Considering the expressions above, the transformation equations with the usual abbreviations are 103
2.4. The Determination of the Instrument Constants
2.41. THE INSTRUMENT CONSTANTS
In order to compute the gradients from the readings obtained
during the observations we have to know the instrument constants.
According to expressions (30) these constants are:
DK 2D m h i a is ------Da — --- T T
where D is the distance between the swinging mirror and the scale
(in case of photographic registration the distance between
the mirror and the photographic plate) expressed in units
of the scale divisions.
K is the moment of inertia of the swinging system with
reference to the rotation axis,
r is the torque constant of the torsion fiber,
m is the mass of the lower weight.
h is the vertical distance between the lower weight and the
beam, i.e., the length of the wire suspending the lower
weight.
i is half the length of the beam.
The values i, h, D and m can be determined by means of direct IT measurements. The quantities — and t are determined indirectly.
In the following chapters we shall deal with the determination of these values [e OTVCS, 1906], 10i|-
2.1+2. THE DETERMINATION OF THE QUANTITY, -jp
First we restrict our investigations to the horizontal variometers. The quantity - will he computed from the period of the swinging system.
The differential equation of the mathematical pendulum in a homogeneous gravity field is
& + g sin cp = 0 (71) dt where i is the length of the mathematical pendulum, g is the gra vity and (p is the amplitude. The solution of this equation in cases of small amplitudes gives the period of the pendulum as
• • • (72)
The period of a physical pendulum is described by the same equa tion but here Z is the so-called "reduced length" of the physical pendulum,
(73) where K is the moment of inertia of the pendulum with reference
to its swinging axis,
ra is the mass of the pendulum,
s is the distance between the center of gravity and the
swinging axis.
Substituting equation (73) into expression (72) the swinging 105 period of a physical pendulum is
K T m s g
Let us imagine a special kind of a physical pendulum which swings about a torsion fiber instead of a friction-less axis. The moment caused by gravity keeps the pendulum swinging about its axis
(Figure 32). Since the amplitude, cp, is small this moment is
F = m g s sin cp a m g s cp
The torsion resistance of the fiber,
T(p, is equal to this moment at any instant; so
rep a m g s cp and, therefore,
t a m g s .
Substituting this into expression Figure 32 (7*0 the swinging period of the torsion pendulum in a homogeneous gravity field is
Comparing this equation with expression (72) of the mathematical pendulum we can see that both give the period T. Thus, we may replace quantities g and & in equation (71) by K and t respectively and get the differential equation of a torsion pendulum in a
homogeneous gravity field,
(76) dt The period of the swinging system is not constant; it increases
due to friction and the resistance of the surrounding air. The
period on agreement with equation (j6) can he computed from
[MIHAJLOV, 1939]:
T (77) (1 + 0.5372 x2)1/2
where T' is the observed period and \ is the so-called "loga
rithmic decrement," which is the logarithm of the ratio of two con-
secutive amplitudes (\ = log — ). This reduction usually is done 1 2 1/2 by means of tables where the value (1 + 0.5372 X ) ' is given as
the function of the consecutive amplitudes [KOHLRAUSCH, 1927, Table
31]. Substituting the reduced period T from equation (77) into ex- pression (75) the value — can be computed,
The differential equation (76) of the pendulum can not be used for the torsion balance because it was derived from the assump tion that the gravity field is homogeneous. We know that the gra vity field is not homogeneous in the vicinity of the swinging sys tem of the variometer. This non-homogeneity causes the rotation of the swinging system, i.e., produces the torque moment F in 107
equation (10) or in a more advanced form, in expression (28).
This torque moment has an opposite sign to that of the torsion
resistance moment and, therefore, it has to be subtracted from
equation (76). The differential equation of the torsion balance in
a nonhomogeneous gravity field is ,2 K ^ - 2 + rcp-F-O ...(79) dt*
where F is identical to the F in expression (28), i.e.,
F » J K WA sin 2a + K W cos 2 a - ihm VJ sin a + ihm cos a 2 A xy xz w,_ yz _(20)
In using this expression we should remember that it was derived with the assumption that the change of gravity is linear in the
field of the torsion balance. In equation (28), a is the azimuth
of the swinging system when in equilibrium. If aQ designates the
azimuth in a torsion-free position then
a » aQ +
nometric functions in equation (28) can be written as follows:
sin 2 a a sin 2(aQ + cp) a sin 2otQ cos 2q> +
+ cos 2 a sin 2cp 2 sin 2 a + 2cp cos 2a o o o
cos 2a a cos 2(aQ + cp) » cos 2aQ cos 2
- sln 2a sin 2cp 2 cos 2a - 2cp sin 2a^ o 0 0 108
sin a ■ sin (aQ + cp) =* sln a0 cos cp +
+ cos a sin cp s sin a + cp cos a o o o
cos oc = cos (aQ + cp) s cos &o cos cp -
- sin a sin cp s cos a - cp sin a o o o
Using these values in equation (28), and (28) in expression (79),
then (79) becomes ,2 K — ■-? - F + r'cp ■ 0 at2 ° where
Fq = i K WA sin 2C + ihm W _ cos a yz o t ' « t - K cos 2aQ + 2K Wxy sin 2ocq + ihm Wxz cos aQ + + ihm W sin oc ...(8l) yz o The torque moment, Fq, acting at the swinging system when it is in the torsion-free position (a = aQ) is obviously zero. There fore, the differential equation (79) finally becomes 3 K 2 -12 | + T'(pn 0 ...(82) .2 dt Comparing this equation with expression (76 ) it follows that the period of the balance is ...(83) Comparing this with equation (75)* it can be seen that 109 equation (78) should be replaced by t-~ 4 — w K T The left side of this equation contains the gradients through equa tion (8l). In order to eliminate these gradients from the equation (81|-) we observe the swinging periods in four azimuths. First we observe in an arbitrary azimuth a and then in azimuths a + 90°, o o a Q + l80°, and ctQ + 270°. If we designate the corresponding periods T, and the values t', with subscripts 1 to If, then the mean value of the quantities in equation (81f) from four observations Is: 2 i-(t- + T* + T' + T{) - 2- (ig + K + ^2 + h) •••(85) k K 1 a 3 . U T®. t| T® T® The quantity (t ^ + + Tjp can be computed by substituting the azimuths a , a + 90°» a + 180°, and a + 2 7 0 ° consecutively 0 0 0 o into equation (8l) and summing the four equations obtained. The result is, t | + - IfT With this, equation (85) becomes T 7f /I 1 1 1 \ fQ£l\ — (-p + “ p+ -p + -p) ...(86) K If T* T* T* t £ From this formula the quantity - can be computed. The periods, K T, are reduced periods, which should be computed from the observed periods, T', by means of equation (77)• lid a a 0° a a 180° a 0 270c o % - 90° 0 0 l l ml4s llm9s i i V i A i 8 Observed 10s 8s 128 8s periods 12s 9S 9S 8s Mean observed T* n V , i i “98 = n V » n V - periods 672.5s 669. 0s 670. 0s 669. 0s ' Amplitude «p 2.57 2.45 2.51 2.46 Reduced T 646s 643 s 644s 644s period (From Kohlrausch1 s Tables) T ^ ( 1 ■ 1 • i * JL ^ i * SJr 6k62 6k3* 644 2 644 2 " ^2022 - » 42022 CGS. T Table 11 Ill In the case of the curvature variometer, since h » 0, equation (8l) becomes r 1 a r - K WA cos aot0 + 2K Wx y sln 2aQ ...(87) The mean values of the quantities in equation (84), from observa tions in two azimuths,are — (Ti + rl) » —— (— g + 2K 2 TJ Tg Observing in azimuths aQ and a Q + 90°, from equation (87) we get rj + Tj o 2r and, therefore, i = + ...(88) K 2 T* Tg where T.^ and Tg are the reduced periods. A computational example is shown in Table 11. The computation is based on equation (86). 2.43. THE DETERMINATION OF THE TORQUE CONSTANT,yt 2.431. The Swinging-period Method This method is similar to that for the determination of the If quantity -. The difference is that the regular swinging system sus pended on the torsion fiber is replaced by a brass cylinder (Figure 33)* The weight of the cylinder should be approximately equal to that of the swinging system in order to apply the same Figure 33 Figure 3^ 112 113 tension to the wire. The cylinder is twisted around its verti cal axis and then relaxed and allowed to swing. In order to com pute the reduced periods, T, the amplitudes, are observed by means of a mirror mounted on the cylinder in con junction with a telescope-scale device similar to the one in Figure 5. Since the gravity field associated with the cylinder is much smaller than that of the swinging system we can assume this gravity field to be homogeneous. Therefore, equation (78) gives the proportion ~ in the form K ^2 ...(78) K T o where KQ is the moment of inertia of the cylinder and all the masses attached to it (mirror, screws, and so forth). In order to determine the torque constant t from equation (78) we should know the value Kq. If the cylinder were truly homo geneous and without any attachments, its moment of inertia would be K = ~ m r2 o 2 o where m is its mass and r is its radius, o Since the mirror and,;the other attachments change the shape and the mass of the cylinder considerably, thus making KQ an unknown, this formula can not be used as it stands. In order to circumvent the uncertainty in the value KQ we use the following procedure. We take a homogeneous cylindrical ring or collar having no attachments, the inner radius of which is equal to the radius of the cylinder it surrounds. The moment of inertia of this ring is determined hy K' = | «• (R2 - r2 ) (89) where m f is the mass of the ring, R and r are the outer and inner radii respectively. We fit the ring on the cylinder (Figure 3*0 and let them swing together. The reduced period, T^, of the cylinder . w H • and the reduced period, Tg, of the combined cylinder and ring, are expressed by Kq + K' and T, 2 T These expressions are used to eliminate KQ so that the torque con stant may be computed, thus, • • • (90) The reduced periods, T^ and Tg, should be determined as the mean of many reduced periods had by reducing observed periods by use of (77)* The moment of inertia, K 1, is computed from equation (89). 2 .U32. The Cavendish Method In the Cavendish method the cylinder suspended by the fiber is replaced by the beam of a curvature variometer. The swinging system of the torsion balance can also be used by lifting up its suspended weight and placing it on'the end of the beam. A heavy mass, usually a lead sphere, is brought close to the beam which swings the beam from its equilibrium position, 5q. After the . Figure 35 2r Figure 36 115 relaxation period has passed the beam will have stopped in a new equilibrium position, 8. The angle of rotation (& - &Q) can be observed either by visual methods or by photographic registration. In both cases we know (Figure 5) that 5 - 5 = n " no ° 2D If the horizontal attraction component of the sphere on one weight of the beam is T , then the torque moment caused by this force is T & (Figure 35)* In the equilibrium position this moment is equal to the torsion resistance of the fiber, t(5 - 5q), i.e., TX£ . T ( 8 - eo) - (n - n0 ) ...(91) Let us determine the attraction component, T . The attrac- tion of the sphere can be substituted for by the attraction of a mass-point situated at the center of the sphere and having the same mass as the sphere. The horizontal attraction component of the sphere upon one of the weights is T = cos 0 p2 where M is the mass of the sphere and the other symbols correspond to those in Figure 35- It can be seen from .the same figure that cos P = ^ and, therefore, T =k2M/ \ d m ...(92) P3 Let us assume that the center of the Bphere and the center 117 of the weight on the beam are in the same horizontal plane (Figure 36). In this case 2 a2 ,2 P = A + t; and 2 dm = a 7T r d £ where the symbols correspond to those in Figure 36. Substituting these values into equation (92) we get z +2 T = k2 M ir a r2 A / a % x -f (A2 + E2)3/2 z 2 + 2 t -3/2 T a k M ttu^s f (l+^p) x ”2 A After integration we get T=2ka * § ------1 l/s ...(93) ( l + ^ } o where m is the mass of the weight on the beam (m = a r ir z). Substituting equation (93) into expression (91)» after simplifi cations, we get [EOTVOS, 1906] : T = Uk2 2 M ------w 4 V *.>p • & > Since the quantities on the right side of this equation are all known the torsion constant, t , can be computed. Below there is an example of a computation utilising the Cavendish method: 118 The mass of the sphere M = 13142 gr. The mass of the weight m = 28.4-5 gr. The height dimension of the weight z = 6 cm. The distance "between the mirror and the scale D = 16 cm. The horizontal distance "between the sphere and the weight A = 12 cm. The half-length of the beam I = 20 cm. The reading, with and without the sphere n-nQ = 2 cm. Equation (94) with these data gives 2 -2 -1 t =0.5085 gr cm sec radian With this value of t and with the result of the computation in Table 11, - = 4-2022 CGS and assuming that the length of the wire sus pending the lower weight is 59*3 centimeters, the Instrument con stants of that particular instrument are a = ^ - 7 6 X 42022 = 3,194,000 CGS b^ = 2 h M ) 2 X 59.3 X 20 X 28.45 X 76 c ™ CGS v The determination of the values - and t should be done in T the laboratory. The manufacturers give the instrument constants in the certificate of the instrument. 119 3. THE APPLICATIONS OF THE TORSION BALANCE MEASUREMENTS IN GEODESY 3*1 Introduction and the Problem of the Analytical Determination of the Equipotential Surfaces of Gravity The Bonnet law in differential geometry states that if any six quantities at a point and upon a surface fulfill three given re lations, then these quantities are fundamentals and they define the surface at this point. The six quantities are the so-called "Gaussian Fundamental Quantities." The three relations to he ful filled are the two Godazzi-Mainardi relations and the Gaussian "teorema egregium." Here we will only refer to these relation^ as the details can be found in any differential geometry book [BAESCHLIN, 19^7} BIEBEHBA2H, 1932]. The six Gaussian fundamental quantities are 2 E = 1 + p F s pq (95) 2 G = 1 + q The quantities E, F and G are the so-called First Order Gaussian Fundamental Quantities, while the quantities L, M and N are the Second Order Gaussian Fundamental Quantities. In equations (95) 120 the symbols are dz 3z „ a2z « = g| 8 = | ^ — <56) m = vf 1 + . -ap J-+ qny +t = S2Z _ ay2 where z = f (x,y), I.e., the equation of the surface is given in explicit form. The quantities r, s and t are the so-called "Monge Quantities." In our case the equipotential surfaces are given in the implicit form, W (x,y,z) = constant. Let us compute the first and second derivatives of this W (x,y,z) function, assuming that z is a function of x and y. The first derivatives are aw aw az . 5 ? + 5? S = 0 ...(97) aw aw az _ 5 3 ? + 3 J 5 £ = 0 The second derivatives are a ^ L „ aS/ az ^ a ^ /az%2 L aw a2z n — 2 + c — + — 2 '— ' + § = u ax axaz ax az ax az ax a ^ + a ^ az + . a ^ az + . a ^ az + . aw a2z = o ,..(98) axay a«az ax ayaz ax az ax az axay a ^ 0 a ^ az ^ a ^ ,azv2 1 aw a2z n — o + 2 +— 2 *— ' ------2 " ay ayaz ay az ay az ay In our coordinate system where + x is towards the North, + y is 121 towards the East and z is positive downwards i = i h ° 85,4 i = g With these, with equations (97) and (98)> and after reductions we get bz _ Q d2z _-l dSf dx dx2 g dx2 - n ^ 2 “ i d2^ (qqV Sy " &xdy g dxSy S2z _-l dy2 g By2 With the quantities in (99) expression (96 ) gives -1 b2^ -1 „ p = o r = = - Wxx . g 8x g 4 = ° 8 =*5 sl? “ ’ 1 V ...(100) m = i t =-i ^ =-i W g 3y g w With (100) the Gaussian Fundamental Quantities in (95) become -1 E = 1 L =“- Wxx. g F = 0 M ’I Wxy "*y ...(101) G a l N - r W, g vyy In other words, if we know the quantities W , W„ , W and g at xx xy yy 122 any point on the equipotential surface then It Is possible to describe the surface by analytical means.Gravity, g, Is con sidered to be known. The torsion balance gives the quantities W xy and WA = W - W„ • From the Laplace equation we can compute the A yy xx value (w + W ) if W and the rotation speed of the Earth, o>, is xx yy zz given W + W * 2a>2 - W xx yy zz From this equation and from the measured quantity, WA ■ W - W A yy xx the values W and W can be computed. With W and W and with xx yy xx yy the measured quantities, W and g the Gaussian Fundamental Quan- xy titles, i.e., the surface can be determined theoretically. In prac tical application, two difficulties arise: (l) We know from the theory of the potential that the equi potential surfaces of the Earth's gravity field (geops) can be divided into two groups, (a) Geops which are outside of the Earth's physical sur face, i.e., which enclose all the masses of the Earth. On these sur faces the potential W and its derivatives are finite and continuous. Therefore, this type of geop is an analytical surface, i.e., it can be described by the methods explained above. (b) Geops which are, in whole or in part, inside of the physical surface of the Earth. On these surfaces only the potential W and its first derivatives are finite and continuous. The second partial derivatives have discontinuities where the density of the 123 mass cut by the equipotential surface changes. On those parts of such a geop where the density of the surrounding mass is constant, the potential and all its derivatives are finite and con- , I tinuous. These surface-pleces can he described by analytical means. Therefore, the geops inside of the Earth actually are mosaics of the analytically describable surface-pieces. The transitions between these surface pieces are without breaks, splits or edges. Our measurements are always carried out on the equipotential surfaces which do not enclose the whole Earth. Therefore, the measured quantities refer only to that part of the equipotential surface which is outside of the masses, i.e., whose border line Is the intersection of the equipotential surface and the topography. (2) The accuracy of the quantity (W and W ) determined xx yy from the Laplace equation, i.e., from the vertical gradient W__, ZZ is lower than that of the measured quantity, WA = - W„ . be- A xx yy cause presently we do not have a method to give the vertical gra dient, W „ with the accuracy of torsion balance measurements. We ZZ have drawn similar conclusions in Chapter 1 when we described the variation of gravity by analytical means. It can be seen that there is a theoretical possibility of describing the shape of the equipotential surfaces or at least the shape of the surface-pieces, and the variations of gravity by analysis. Whether this is possible in practice depends upon the accuracy of the determination of the vertical gradient. If we are able to determine this gradient with the accuracy of the torsion balance results, then our problem is solved. As we shall see in Chapter ^ this accuracy has not been completely achieved yet. However, we are able to investigate the detailed shape of the geops in the vicinity of the observation points in a less com plex way, and to solve many other important geodetic problems as will be described in the following chapters. When we introduce the different applications of the tor sion balance measurements we omitted the determination of the gravity differences between two stations. This application caused the torsion balance to become widely used in geophysics, since before the development of gravimeters it was, together with the pendulum, the only field instrument with which to observe gra vity differences. However, in the nineteen thirties the cheap er and faster gravimeters were developed and superceded torsion balances. Therefore, the determination of gravity differences by means of the torsion balance has historical significance only. 3.2. The Total Horizontal Gradient The torsion balance measurements give the derivatives W xz and W^. These derivatives are the rate of change of gravity in the horizontal plane xy. The derivative Wxz = ^ is the rate of change of gravity in the + x direction(North); tie derivative ^ ^ = fj (g) - |£ is in the direction of 125 the + y.axis (East). These derivatives are called "the hori zontal gradients of gravity." The horizontal gradients are vector quantities and their re sultant can be constructed as shown in Figure 37. The resultant, dg Gr (g) * » is the so- x called "Total Horizontal ♦ Gradient," which gives the I / I ✓ direction and the quan ✓ tity of the maximum rate of change of gravity in xz the horizontal plane [ECTVCS, 1906]. The quan y tity of that rate of Figure 37 change, i.e., the size of the total horizontal gradient from Figure 37 is r - s — Gr (g) w 102 V w xz* yz ...( ) The azimuth of that gradient can be computed from the same figure; so W. tan a - ..•(103) w*z Expressions (102)and (103) give the relations between results of the measurement and the size and direction of the total horizon tal gradient. The total horizontal gradients are usually drawn on the maps as vectors to the observation points. Those maps which contain these vectors at the observation points are the so-called Figure 38 "Gradient Maps." Figure 38 shows such a map of an area above an underground salt-dome. It can be seen that the total gradients point toward the surrounding masses of larger density. The horizontal gradient of gravity in an arbitrary direction can also be computed from the results of the measurement. If P is the azimuth of the arbitrary direction, n,and a is the azimuth of the total horizontal gradient then the horizontal gradient in the direction n is (Figure 39), §g - COB (p - a) = |s cos a cos fJ + ^ sin a sin p ...... (lOi*-) but since cos a = ir r and ✓ / / / n expression (10^) becomes (105) which is naturally the ■>- y same result as in the Figure 39 transformation equation 3»3» The Horizontal Directivity or the Sphericity Coefficient In addition to the horizontal gradients the torsion balance measurements also give the derivatives, W and W. = W - W . . xy A yy xx These values are generally called the ’’curvature gradients of gravity" because with them we are able to determine some curvature properties of the equipotential surfaces. According to differential geometry the radii of curvature in the principal directions at any point on a given surface are r + t - [(r - t)g + Us2]1/2 1 2 (rt - s2) ...(106) r + t + [(r - t)2 + lis2]1/2 2 2 (rt - s2 ) where the values r, t and s are the Monge Quantities given in section 3«1* If the surface is given in the implicit form W(x,y,zj = 0, then the Monge quantities can be computed from ex pression (100). Prom the equations in (106) we get i- - i = 2 (rt - s2)( -— i , - P1 p2 (r + t - [(r - t)2 + lte ] ' 1______r+"t + [ (r - t)2 + 4s2]1/2 After reductions thlB equation becomes Substituting the Monge Quantities r, t and s from equation (100) we get ft f—-* - —) = [fw - w pi p2 « ** V or R - 6 - twf + (awxy)2]1/2 ...(107) The quantity, R, called the "Horizontal Directivity" or"Sphericity Coefficient," can be computed from the measured values and [EOTVGS, 1906]. If the equipotential surface-plece at the investi gated point is a part of a sphere then p 1 = and, therefore, R » 0. If the geop is not a sphere then R > 0. The coefficient R increases with the difference between the principal curvatures, i.e., the R represents the discrepancy between the equipotential surface and a sphere. For this reason R is called the "Sphericity Coefficient." R is always positive since in expressions (106) p.^ is always smaller than pg. It is easy to show that if in addition to the quantity ( W - W ) the quantity (W + W ) is also given, then fromequa- yy xx yy xx tions (100), (106) and (107) we can also compute the principal radii of curvature in addition to the curvature differences. After deri vation we get 130 Since the quantity (VT + W ) can be computed from the Laplace xx yy equation, which includes the vertical gradient, W „ , this attempt ZZ , depends again on the accuracy of the vertical gradient. The azimuths of the principal directions are also important curvature properties of the equipotential surfaces. These azimuths can also be computed from the curvature gradients. We know from dif ferential geometry that the curvature of a normal section in an ar bitrary azimuth X is 1 2 2 - = r cos X + 2s cos X sin X + t sin X ...(108) We are looking for the azimuth of the normal section in the princi pal direction in which the curvature — is maximum (p ■ p^). This azimuth cam be found from equation (108) after differentiating and setting the result equal to zero 2 2 -2r sin X cos X - 2 sin X + 2s cos X + 2t sin X cos X » 0 Substituting the values r, t and s from (100) into here, after re ductions, we get r tan 2X = - - — * ...(IO9 ) % where X is the azimuth of the principal direction with the ->>2W,xy smallest radius of curvature. The sphericity coefficient \ can be imagined as a fictitious vector with length R and with Figure kO azimuth 2K in a (-W^, 2Wxy) coordinate 131 system as shown in Figure 40. Both equations N (107) and (109) can also be derived from * the geometrical properties of that figure. In practice the fictitious vector, R, is drawn at the observation points on the map of the surveyed area as shown in Figure 4l. Those maps which contain these sphericity co efficients at the observation points are the so-called “Curvature Maps." Usually the to tal gradients and the sphericity coefficients Figure 41 are drawn in one figure as shown in Figure 42. Those maps which contain both the total gradients and the sphericity co efficients are the so-called “Gradient- Curvature Maps.” Such a map is shown in Figure 43. The curvature maps can be de !r(g) veloped for another type of graphical representation which shows the princi pal curvature lines of the equipotential surfaces. This can be done by drawing a Figure 42 set of curves tangent to the fictitious vectors R and then a second set perpendicular to them. Along these two sets of lines the principal curvature radii are minimum and maximum, respectively, and they give a very detailed picture of the geops. This type of map is shown in Figure 1+4 [EBTV&S, 1906]. ^ ■ / \ x , V ^ *rf y l { V t *t*M *»e V < * Figure 43 1000m 0 1 0 JO ZO 30 *0 SO 60 70 80 90 100 _n --1 1-- 1-- 1-- 1-- 1-- 1-- 1— ! 1-- 1 30 ' CGS Scale of the Quantities R '''''■'■yC. o Values in E Units 30 o 20 20 U> ZO SO iP Oo 30 Figure 45 135 Figure 1*5 shows the same area as Figure but the cur vature properties of the equipotential surface are represented in yet a different way. Lines of equal sphericity coefficients were drawn by means of interpolation. Along one particular contour line the curvature difference in the principal directions, i.e., the discrepancy between the equipotential surface and a sphere is constant [e OTVOS, 1906]. 3 .^. The Radius of Curvature of the Plumb Line Let points A and B be on the equipotential surface W = con stant in the plane of the meridian, i.e., in the xz plane, at an infinitely small dis tance, dx, from one C another d z '-dz (Figure k6)i The normal, W+dW i.e., the tangent to Figure k6 the plumb line at point A is on the plane of the figure* If we go downwards on the plumb line with the infinitely small distance, dz, we find point A' which is on the equipotential surface W + dW ■ constant. 136 This equipotential surface Intersects another plumb line through point B at B*. The Infinitely small distance between B and B f is dz’. The distances between the two equipotential surfaces at A and at B, respectively, are dz =» — and dz* = g 8 where g and g' are the gravity values at A and at B, respectively. From these equations we get the well known relation dz' g dz = g' From this and after subtracting one from each side we get ...(no) i Since both points, A and B, are in the xz plane the gravity at B is g' = g + dx With this, expression (110) becomes dz1 - dz 1 dg - " 3 5 ------or dz' - dz 1 /-,-1 -1 \ dx " g 7 S ...(111) From Figure k6 it can be seen that ^ ■ tan € S e ...(112) where e is the angle between the tangents at A and at A'• The tangents intersect at point C. This point is considered to be the 137 center of a circular arc passing through A and A'* This arc Is substituted for the projection of the plumb line in the xz plane. In this case the distance CA = r is the radius of curvature-com- ponent of theplumb line in the xz plane, i.e., in the meridian plane. Since the angle € is small it can be seen from the figure that dz a er Replacing the left side of equation (ill) with (112), and putting the above expression for dz into the right side, we get 6 = - ?■ i 6 rx Assuming that the gravity at A and at B is the same, we get after reduction ® g r x a — a ” — ix Wxz By a similar derivation it can be proved that the radius of curva ture component in the yz plane, i.e., in the prime vertical plane, r , is y r a y yz The radius of curvature of the plumb line at point A is the resul tant of the perpendicular components r and r [EOTVOS, 1906], x y r • (r| + ,± ...( u 3 ) xz yz The radius of curvature lies in the tangent plane of the plumb line at point A. The azimuth, a, of that plane can be computed 138 from the following expression tan a = —^ = ...(ll^) rx WXZ Comparing this equation with expression (103) we can see that they are the same; therefore, the radius of curvature of the plumb line falls into a vertical plane, which also contains the total hori zontal gradient* This fact is to be expected because the equi- potential surfaces bend towards one another in the direction where the change of gravity is maximum. The tangent plane of the plumb line at A intersects the equipotential surface trough A in a curve. The radius of curvature of that curve, at A, is the same as the radius of curvature of the plumb line because they are perpendicu lar to each other. Therefore, it is possible to determine the cur vature of the equipotential surface in the direction of the total horizontal gradient, i.e., in that direction where the change of gravity is the maximum. 3.5. The Reduction of Astronomic Coordinates to the Geold, to the Telluroid a n d to the Quasi-Geoid The astronomic latitude is the vertical angle between the normal to the geop, i.e., the tangent to the plumb line at the ob servation point, and the plane of the equator. The astronomic, longitude is the horizontal angle between this same normal and the plane of the Greenwich meridian. The astronomic coordinates refer to the geop of the observation point. The normal of the geoid has a different direction than the normal of the geop because the tangents of different points on the plumb line are not parallel. Therefore, if we want the astronomic coordinates to refer to the geoid, we have to reduce the observed astro-coordinates to account for the angle between the normal to the geoid and the normal to the equipotential surface of the observation point. Let us investigate this problem first in the plane of the meridian. In Figure 47 the plumb line of the observation point A intersects the geoid at point P. The projection of the angle between the tangents of the plumb line at A and P in the tangent plane of the plumb line at A is AS. If H is the orthometric height of point A and r is the radius of curvature of the plumb line, then AS can be computed from the following expression (Figure 47), tan A S S AS ...(115) or with expression (113)* AS = 2 (w2 + w2 )1/2 ...(115) g v xz yz' The azimuth of the tangent plane of the plumb line at A, contain ing this angle,can be computed from expression (ll4), W tan a = xz From Figure 48 it is obvious that W XZ COS Of a <4 + 4)1/2 .(ll6) w 1 sin a o — a— — o— p*rr Figure 47 fjr Figure 48 11(0 I k l The latitude correction Acp is equal to the projection of the angle A6 on to the xz plane, i.e., on to the plane of the meridian; so Acp = A5 cos a With the aid of expressions (115) and (ll6) this equation becomes AV = | Wxz ...(117) Similarly, the projection of the angle A5 on to the yz plane, i.e., on to the plane of the prime vertical, is equal to the pro jection of the longitude angle on to the same plane, A\ cos cp. A\ cos With the aid of expressions (115) and (ll6) this equation yields A\ cos cp = - W „ ...(118) g yz v ' Since the quantities H, g, Wxz and W ^ are all known from measure ments, expressions (117) and (118) give the reductions of the astro- coordinates from the observation point to the geoid. The gravity, g, in the formulas can be replaced by the normal gravity, y, from the used gravity formula without any loss of accuracy. In using these formulas we should remember that in their derivation we have assumed that the curvature of the plumb line is constant between points A and P. Since the curvature depends upon the second partial derivatives of the potential this assumption holds only if there are no significant changes in density along the plumb line between the observation point and the geoid. If this is not the case, then 142 according to the Poisson equation, the second partial deriva tives have discontinuities where the density changes and, there fore, the curvature can not he considered as constant. This diffi culty can he avoided hy applying the "new theory of gravimetric geodesy" described separately he Hirvonen [HIRVONEN, 1959]> Molo- densky [MOLODENSKY, 1945], de Graaff-Hunter [DE GRAAFF-HUNTER, 1958] and hy others. In this case the secondary surface is not the geoid hut the telluroid (Hirvonen), or the quasi-geoid (Molodensky). Formulas (117) and (118) can he used for the reduction of the astro- coordinate s from the observation point to the telluroid, or to the quasi-geoid, hut the orthonietric height, H, has to be replaced hy the so-called "height anomaly," or by the "normal height" instead. 3.6. The Densification of a Net of Deflections of the Vertical For use in astronomic levelling for the Bruns-Villarceau method [BAESCHLIN, 1948] of synthetic determination of the geoid, and for solution of other geodetic and geophysical problems, it is necessary to have a fairly dense net of deflections of the vertical. The required density depends upon the mass distribution close to the physical surface of the Earth. The requirement is that the change of the deflection of the vertical components between two neighboring stations should be linear. To obtain such a dense de flection net by astronomic or by gravimetric means is very laborious. Ikz The following method solves this problem by giving the de flection components with any density in a surveyed area, where both deflection components | and tj are known at one station, and at least / one of the deflection components is also known at a second station,, This means that in the whole area we need, in addition to the torsion balance measurements, two astro-latitude determinations and one astro-longitude determination* We have mentioned earlier that the origin of our (x,y,z) co ordinate system is at the observation point with its x axis oriented towards the North, i.e., the x axis is tangent to the local meri dian, and its z axis coincides with the local vertical. Therefore, the relative orientation of the coordinate system is different at each station. For the sake of simplicity let us.use only one co ordinate system for a small area of about 0.5° X 0.5°. We choose an arbitrary station 0, with astro-coordinates $ and L as our re ference station, and we use the coordinate system based on this point over the whole area. Hence, at any point in our area the z axis is parallel to the vertical at point 0 and the x axis is parallel to the tangent of the meridian at point 0. Figure shows the xz plane of a coordinate system as de fined above and through an arbitrary point A-^. The origin of the system is at A^. The z axis is parallel to the vertical at the reference station 0 and, therefore, it does not coincide with the direction of the gravity vector g at point The vector A^G is the projection of the gravity vector on to the xz plane. The ll*l* 'difference between the vec +x tor A^G and the gravity vector is negligible. In this case the A^H vector is the gx component of g. The line E - E is the intersec tion of the equatorial t plane and the xz plane. If the astro-latitude of the reference station is and Figure 1*9 the angle between the ver ticals at 0 and at is then the latitude at A1 is ^ a $ + A From Figure 1*9 it is obvious that - gx = g Bin A ^ or since A is a small angle, g'X 1 g After a similar approach the angle difference in the yz plane, i.e., in the prime vertical plane is g AT i- COS =s - — ...(119) 1 1 g where AL^ is the astro-longitude difference between the reference station 0 and A^. Expressions (119) give the North and the East components of the angle’ between the geop normals at 0 and at A^, respectively. We can 1^5 similarly determine the North and the East components of the angle between the geop normals at 0 and another arbitrary point Ag, £rt>g and ALg. The quantities (A4>g - and (ALg - AL^) are the North and the East components of the angle between the geop normals at Ag and at A^* Atg - A*1 = “ — (gx2 ” 6xl) = - “ (wx2 " Wxi) ^ ^ ...(120) 008 °m - - Syl) = - ^ (V V where g^ and-0 are the mean gravity and the mean astro-latitude between and Ag. The quantities (A A^. The gxl, gxg, g and gyg are the gx and gy gravity components at A^ and Ag. Similar expressions can be derived for the North and the East components of the angle between the spheroidal normals at A^ and at Ag : A < * 2 - V 008 ■ - rm < v ■ V where the quantities (Acpg - Aq^) and (A\g - A^1) are the geodetic latitude and longitude differences between points A.^ and Ag. The differences between the astronomic and geodetic latitude and longitude differences in expressions (120) and (121) are the differences of the deflection components between the points A^ and Ag. Ik6 Without any loss in accuracy we can assume that in the surveyed area 7m * 6^ and. q>m - Subtracting expressions (121) from (120), [fc»8 - m 2) - - ACP^] gm - - (Wx2 - Wxl) + (Uxa - uxl) E (ALg - A\£ ) c o s (pm - (AL^ - A ^ ) cos + (wx2 - wxl) + (ux2 - uxl) ^ 2 ' A n ...(122) (H2 ‘ \) Ai (W^ - Wyl) + (Uy2 - Uyl) W - U is the potential anomaly, AW. Using the abbreviations, Ati2 - ^ - 62 and At112 =* \ “ n2 * (122) becomes em A *21 = " a Wx 2 + A Wxl ...(123) gm *,2i - - A W ^ + A Wyl Let us assume that the points and A2 are close enough together so the variation of the gradients between them is linear. We introduce a new (n,s5t) coordinate system whose t axis coincides with the z axis, the n axis goes through points A^ and A g, and the s axis is per pendlcular to n and to t -(Figure Figure 50 50). 147 Let us find the integral of the gradient Wng between points A1 and Ag* The coordinates of points A^ and Ag in the new coordinate system are (s = 0, n = n ^ and(s = 0, n => ng), re spectively. With these coordinates the integral we seek is I I s b ^ ■ (H 52 ‘^ l = ws2 ' wsl ...(12^ nl Since the change in Wng is assumed to be linear the trapezoidal rule gives = I [ + 1 (ns • “l 5 = ni = 2 ^Wnsl + Wns2^ ^n2 “ nl^ = T12 •••(125) We know from the transformation equations in 2.35 that W„ a* - W sin a + W cos a s x y Let us use this transformation equation in (124) + ( V - Wyl) cos ci^g ...(126) where QL^g is the azimuth of the direction A^ Ag. A similar expression can be derived for the ellipsoid, i.e., for the normal gravity field. " - tracting( 127) from (126) we get T1S - = (- A Wxg + A «xl) sina 12 - (- A Wyg + A Wyl) oos 0^ ...(128) Substituting eaqaressions (123) into (128) and using the ab breviation A T^g = T12 - T^g we get A T12 - 8m A *21 sln “ L2 - 8m A ’'ai 008 “ ls " * (129) In this equation the left side can be computed from expression (125) where the anomalies A W rather than the observed values ns W are used, ns A I = | (A Wnal + A Wns2) (ng - nx) The gradient anomaly A Wng in the new coordinate system can be computed from the observed gradient anomalies by means of the transformation equation in section 2.35 A Wng = ^ A sin 2a + A cos 2a Special caseB of this transformation are: a a 0° or 180° A W = A W ns xy a = 90° or 270° A "ns " - A V a = or *5° 225° A H r.s " I awA a = 135° or 315° A W ne = -l AWA If we take a new third point A^, thus forming a triangle with points and Ag, then equations similar to (129) can be evolved 1^9 for the sides and AgA^, A t13 “ «m A 63l Bln°i 3 ’ A ^31 cos “ 13 --(130) A T23 - A S32 8ln a23 ‘ Sm A I32 cos °23 ‘ *'(131) The sum of the deflection component differences along the sides of the closed triangle, A.^ Ag A^* should he zero % therefore, two more equations can he writtens A + A + A £32 = ® ■•..(132) A T)g1 + A t)31 + A n32 » 0 ...(133 ) This way in a single triangle we have equations (129)> (130), (131), (132) and (133) giving five equations and six unknowns, (A iQ1, A A £31* A A ^32 and A ^31^* The only P°ssible solution requires us to express five unknowns as functions of the sixth. Let us consider the difference between the North components at the first triangle side, A 6-,, as an unknown quantity, - — . dl A 6gl - - ^ ...(a*) In this case after solving equations (129) “ (133)* the unknowns are [E&TVflS, 1906] s AT.. + u sin 0Lo A ”2 1 ------^ ~c o s " ^ g * 150 flPa3c°8 8ln Og!- g,AT|al cos o ^ c o s A! 32" *m sin ( o ^ - a23) ..*(136) An3a= - M 238ln ajl+^AT31+ 8 ^ gl Bln O ^ - g ^ °°B O^ B i n ttg3 8m sln (“ 31 - Formulas (136) are applied using the logically appropriate indexes for the quantities A g ^ and A tj^, also for the deflection differences, A g ^ > A g^g, A A T|^g in the triangle Ag A^ Aj^ (Figure 51) ° Applying this computational method to a A1 Figure 51 progressive triangle chain we can get the deflection component differences for the sides of the whole chain. Since formulas (136) contain the presently unknown quantity A gg^ and this quan tity appears in every expression for the deflection component differences within the chain, we can not use these formulas directly. In order to determine the unknown quantity, A gQ1 = _ H_ f we need the g deflection components at a minimum of two points within the concerned area. These g values can be obtained 151 either by astro-latitude determinations or by computations from gravity anomalies. If these two points are by any chance on one triangle side, then according to expression (13*0 the difference between the observed values, A 621 a 6g - ^ , is the desired quantity, - — . If the two astro-points are not neighbors then the Sm known £ values will appear in the equations and the quantity A has to be computed by solving them. After finding A equa tions (135) and (136) give all the deflection component differences, A £ik » 6^ - !k and A T|ik = - tj^ , along the sides of the tri angles. Starting with the known | value of one of the points where the astro-latitude has been determined the deflection component, |, can be computed at each corner of the triangles. In order to com pute the deflection component, tj, since the equations give only the A T^k = tj^ - Tik differences, we need an tj value at least at one corner to start with. This can be obtained either by astro-longi tude determination or by computation from gravity anomalies. Starting from the station with the initial T|, using the computed deflection component differences, the T| quantities can be determined at each corner of the triangles. With this method we are able to determine the deflection components to any density in an area where the deflection component i is known at least at two points and tj at one point. If these three initial t and tj components are gravi- metrically computed absolute deflections, then the deflection components computed at the comers of the triangles by this method are also absolute values. If the initial deflections are astro- geodetic deflections then naturally the results of this method are also relative deflections. The gravity, gm , in the formulas can be replaced by the corresponding normal gravity values without any loss of accuracy. The relative accuracy of this method is very high and depends mostly upon the distances between the neighboring stations. The absolute accuracy depends upon the accuracy of the initial $ and tj values at the reference stations. The situation is similar to that of relative gravity measurements where the relative accuracy is also very high due to the high sensitivity of the gravimeters. The abso lute accuracy, however, depends upon the accuracy at the base sta tion where the absolute value of gravity is determined by means of pendulums whose accuracy is smaller than that of the gravimeters. Tables 12 euid 13 give comparisons of these accuracies. Table 12 shows some results of an experiment by Professor K. Oltay [EOTVOS, 1909]. The deflection components were computed from torsion balance measurements at forty stations. At seven of these points astro-latitudes were also determined and the astro-geodetic de flection components, computed. Table 12 gives both the computed and the measured deflection components at these seven stations. The reference stations were Pankota and Zabrany. At Mikalaka comparison was also made between the rj components. The mean error of the astro- geodetic ( components was - O.V. The mean error of the deflection components computed from the torsion balance measurement was also -0.4 153 TORSION BALANCE ASTRO-GEODETIC DIPFE FENCES -11 No. STATION ! t" I" p" -p" 5t b ^TB AG \ g TB AG ‘TB'^AG +1 o' 0 • + 1 1 Pankota 0 a 0 0.0 2 Vllagos + 0,.8 + 0.3 + 0,5 3 Kuvin - 2,. 6 - 2,-9 + 0..3 k Paulis- - 8,. 6 - 8.7 + 0.1 5 Zabranv - 8.1 ■ , - 8..1 , 0.0 . , 6 Mikalaka - 1..9 - 5-6 - 2>5 - 5.3 + 0.6 - 0.3 7 Nagyhalom - 2.3 - 1.5 - 0.8 ■ Table 12 TORSION BALANCE ASTRO-GEODETIC DIFFERENCES No. STATION P* . p1* 1 TB ^TB 5AG ^AG 5TB 5AG TB ‘AG 1 Rentovo - 6,5 + 1,A - 6,5 + l.k 0,0 0.0 2 DolgoprucLnaya - 2,2 - 3 > - 1,7 - 0,5 3 Lobneya - 0,8 - 3.-9 - 0,8 0,0 k Ozereskoye + 0,1 - 3.-5 + 0,2 - 0,1 5 D m ltrlyevka + 1,0 - 3.. 8 + 1,0 » 0,0 . 6 Rakovo | + 3.5 - h.i + 3.5 - 3.6 0.0 - 0.5 Table 13 15^ Table 13 shows some results of an experiment hy V. A. Rasansky near Moscow [KASANSKY, 1936]. He computed the deflection components at fifty-two stations.- Six of these were also astro-con- trol points. Stations Rentovo and Rakovo were the reference sta tions. The mean error of the astro-geodetic deflections and of the computed deflections were - 0.5" and - 0.3" respectively. 3.7. The Determination of the Geoid With the method described in the previous chapter, deflec tion components can be obtained to any specified density. After reducing the deflection components to the geoid as described in 3.5» relative geoid undulations can be obtained by means of the well-known method of astronomic levelling. If the three initial deflection components required by this method are absolute quan tities, i.e., the resulting deflection net is absolute, and we know the undulation of the geoid, at least, at one point. Then the absolute undulations at any point can be also computed. In this way a very detailed picture of the geoid can be obtained. The de tails depend upon the density of the deflection net, I.e., upon the torsion balance measurements. In practical computation the results of the torsion balance measurements are interpolated to the corner points of a rectangular grid system oriented to the North and the East (Figure 52 ). The 155 n+1 M,N n-1 m-1 m m+1 Figure 52 y=East deflection components | and n are also computed at these corner points. In this way the triangles used in the computation of the deflection components are all right isosceles triangles. Therefore, equations (13.6) will reduce to much simpler forms. In one square of the grid system the triangles can have four different positions. In each position there are three possible designation systems, 156 depending upon which corner of the triangle is considered to be the initial one. Finally in each system there are two possi bilities for direction in going around the triangle, i.e., the desig nations of the corner points can be made clockwise or counter clock wise. This gives ^ X 3 X 2 = 2h different possibilities. The forms taken by equations (136) in the twenty-four cases are given in Table ik. After computing the deflection components 6 and tj at Xj xy each intersection point, the undulation at an arbitrary point (x = n, y = m) can be computed from the following formulas: ( x=n-1 y=m-l ) N =N-i) s i + l n r ...(137) n 'm ° x=0 y=0 y J The relative undulation is rx=«n-l y=m-l A N = N - N = - £ n,m n,m o E S „ + 2 Hxv J -..(138) x=0 xy y»0 7 where Nq is the undulation at the given reference point, x = y = 0. The I is the spacing of the grid system. Nn m is the computed undula tion at the intersection of the North-South gridline y = n, and of the East-West gridline x = m. The relative accuracy of this method depends, upon the accuracy of the deflection components and upon the spacing of the grid system. The absolute accuracy depends upon the undulation of reference point, Nq . 157 Table 1^ Cast 'gm S32 "®niT*32 Figure °31 a23 1 180 & 315 AT23 ^ V ^ a A AT31+Tl21gm r 2 90 -AT 315 23 AT23"A T31 21gxn‘i‘ir^21gra ‘ 3 90 180 AT31+^21gm 'AT23 k 270 135 & "AT31+^218m -AT23 '^2-*AT 31"62A 5 135 0 - ^ 2 3 ^ 3 1 V2+62igm+ri21gm • AT23 IN. ;6 0 270 at23 -AT31+T)21gm 3 2s 7 315 180 at23-at3i ■!-.2+tal8m-«ialen -at23 3 8 180, 90 "AT23 • AT31+Tl21gm 90 , 9 ^ 2 315 A T 31+S21gm AT23 ^ ’^31" 1 0 10 135 -AT23 ^ + A T 31-t,2lSm 'AT31+Tl21gm 11 135 270 -AT23-fAT31 'fe+l 2lgm+7*21gm ! ^ 2 3 ( 12 270 0 -AT31+521gm Ar23 0 13 t? 225' AT23 ^ “AT31+T|21em "AT3l'fr,21gni 225 90 Ifc ■ ^ 2 3 -AT23+AT31 ^ " 6 2lsm+TJ21sm 90 0 AT . 15 V AT31+S2lem 23 2 1 3 180 -AT - ^5 T,21gm 23 j V - ...... - ... ' {Continued on Next Page) ■ ro o 00 uo CU VJ1 VO IV) o VJI VJ1 C O ■ u> al 1 (Concluded) 1^Table LU x>. CO JO to CO LO ro to CO co ro iv> co co ro im 159 3*8. The Determination of the Geopotential Value and the Shape of the Geops Applying equations (12*0 and (125) for potential anomalies, we get vsh ‘ l + (A (n2 ' V ” ^ 139) From the transformation equations in 2.35 we have A W = - A W sin a + A W cos a b x y - ...(1*40) A W = ^ A WA sin aa + A W ,, cos 2a ns 2 A xy The computation is performed in a grid system as in Figure 52. Along the North-South lines where a = 180°, equation (139) 1560011168 Along the East-West lines of the grid system where a = 270°, equation (139) becomes (A Wx^x,y " ^A Wx^x,y-1 “ ' 2 ^ A Wxy^x,y-1+ ^A Wxy^x,y^ Points x, y and x-l,y or-x,y and x,y-l are two consecutive inter section points of the grid system along the North-South and along the East-West lines, respectively, where the gradient anomalies A W have been interpolated to. xy Equations (l*H) and (1*42) give the differences, i\— "y'x,y "yyx-l,yJ "x^y " ^ x’x^y-l-** Starting from a reference point x = y = 0 with A W = A W„ a 0, x y and always proceeding from North to South, and from East to West, the quantities A Wx . and A W^. can be computed for an arbitrary intersection point (x a n, y a m) by simple summation. yon-1 ...(143) X a n - 1 (AW) -' E [(AW) - (A W ) . ] y n,m x=Q y'x,y v y'x-l,yJ Having the quantities, A W and A W at each intersection x y point of the grid system, the relative potential anomaly at the intersection point (x a n, y = m) can be computed by integration Xan yam (8AW)“. m - J (A W*}x.y + J g ( Xa n <8 A W >n.» - f KXaO t(A Wx>x,y + (A "x^-l.yl + yam Z + yaO'S [(A Wy}xsy + (A V x . y - l 1} The relative potential anomaly divided by the local gravity gives the separation between the geop and the spherop through the reference point. 8 nn>m - (8 A w)n,m ...(11*5) ®n,m l6l If during the whole procedure we operate with the de rivatives of the potential, W , W ..•., and so forth, rather xy x than the derivatives of the potential anomalies A Wxy , A W x ...., and so forth, then equation (lMf) gives the relative potential 6 W with reference to the initial point x = y = 0. If the geopotential at this point is known, then the geopotential at the point (x ~ n, y =» m) is u = u + (5 w) n,m o ' ^ m where (5 W)n m is computed from (1^*0, and u q is the geopotential at the reference point. The geopotential plays a very important role in modern geodesy. i— n--- rj— Figure 53 shows the quantities g ^ | + vp , i.e., the products of the total deflections and the local gravity values, as vectors at the intersection points of a grid system. The contour lines shown are relative equipotential-anomaly lines rather than geop separations, i.e., they connect the equal S A W quantities of expression (l^). The contour lines give the relative potential anomalies with reference to station 2103. According to equation (1^5) the magnitude of each contour line is the product of the rela tive geop separation 8 N and the local gravity g. When computing the deflection components, the deflection component difference A 5,^, between points B and C was chosen to be 14-. 7" arbitrarily. At the right side of the figure the heavy line represents the border of a mountainous area. It can be seen that the defleetion-vectors point towards the large masses of the mountains. '+* C O S.LI6 1-OS K \ Kilometers ’•*7 10~ ^ CGS ^ 10~ f XX 6 9 3i° Hf3i° *\ Figure 53 ° ° 1 t)^ ^ 1 1 1 1 ^ 1 1 1 1 ^ Seconds ^ 1 1 1 1 ^ 1 1 1 t)^ 1 ^ +. ^ h— * ^ \ T^ I t O O O m O i Relative Potential Anomalies Potential Relative 162 Scale of the quantities the quantities Scale of Scale of the deflections deflections of the Scale V 163 3»9« Correction of Observed Directions for Deviations (Azimuth Correction) In a first order triangulation net the observed directions should be corrected for deflections. The well-known Helmert formula for the azimuth reduction is [HEIMERT, 188^]: ^ 1 2 ^ 1 2 * " ^ 1 sin a12 ■ n l COS °b-2^ where Ah^g is the elevation difference between observation station 1 and observed mark 2. s^g is the horizontal length of the direction 1-2. and are the deflection components at observation point 1. a12 is the azimuth of the direction 1-2. The investigation of Vening Meinesz proved that the following for mula gives better results [VENING MEINESZ, 1953]• h2 (A|ai sin °12 ' A,|2i 006 “a a 5 ...(i W) where hg is the elevation of observed mark 2. Atg^ and Ar)g^ are the deflection component differences between 1 and 2. A£gi = |g - &1 . Neither the first Helmert nor the Vening Meinesz corrections can be applied until the deflection components at the station have been determined either by astro-latitude and longitude measurements or by 16k computation from gravity anomalies* This laborious procedure can be shortened by using the following method. We divide the length s12 into sections of such lengths that within each section the change of the gradients shall be linear. If the deflection component differences between the terminal points of a section, i, i+1, are ^ and ^ > then ex pression (lh-6) can be written in the following form: hp i=n-l A£*12 = ^ ^i+l,i 8ln ai2 " AT)i+l,i 005 ^ 2 ^ * •*( — ?) Since i+1, according to expression (129) AT A5 i+i,i sin “ia - 008 aia ’ where according to expression (125)* ATi,i+l ■ I I AWns = | AWA sin 2a12 + AWxy cos 2a12 ...(lpO) Substituting equation (1^+9) into (1^8) and then the result thereof into (lVf) we get, after reductions, *“ 12 * i + • • -h 5 D After measuring the curvature gradients W and WA at the termination points of the sections, computing the gradient anomalies AW = W - U and AW a = Wa - U A , and transforming them xy xy xy A A A into AW by means of equation (150), expression (15l) gives the 6X1 desired azimuth correction. Gravity gj^ can be replaced by the normal gravity value without any loss in accuracy. Note that no astro-latitude and longitude measurements nor gravity anomalies are necessary. 3.10. The Effect of the Inner Circle upon the Gravi metric Determination of the Absolute Deflections The absolute deflection components can be computed from gravity anomalies by means of the Vening Meinesz formulas [VENING MEINESZ, 1928]: -.1 tt 27r |" = -i— f J Ag f'(V) cos A d\|f dA kuy 0 0 „ tt 2tt T)" ---- - r j J Ag f ' (Y) Sin A d\|r dA 1+uy 0 where Ag is the gravity anomaly at a point defined by the local spherical coordinates i and A. The y is the mean valueof gravity on the Earth's surface.The f '^)is the so-called Vening Meinesz function. 166 The Vening Meinesz formulas in this form can not be used close to the computation point because there the function f ' (>|r) is infinite. In the practical computations the surface of the Earth is usually divided into four zones: 1. 0 < if < 5 km 2. 5 km < ♦ < 3° 3. 3° < i|r < 20° k. 20° < * < 160° The computation method in each zone Is different. The pro cedure in zones 2 to 4 is based on the original form of the Vening Meinesz function. The effect of zone 1, the so-called "inner circle" is significant. The effect of this zone can be computed from the following formulas [HEISKANEN-VENING MEINESZ, 1958] : I" » - 0.105 rQ a ...(152) y . - 0.105 r0 a ^ where rQ is the radius of the inner circle. The quantities are the horizontal gradient components of the gra vity anomalies at the computation point. They are usually com puted from gravity anomalies assuming that the rate of change of the anomalies within the inner circle is constant, (Rice and Kasansky methods). This assumption is not always permissible and it can be avoided by using the following procedure. We know that 3 sf ■ ^ - lb ■ wxz ■ uxz * a w m S H " 3y (g ' 7) ■ I? (H ' = V ‘ U ya = A V Substituting these into equations (152) we get = - 0.105 ro A W XZ .«.(1^3) r>"1 = - 0.105 r o AW yz Measuring the horizontal gradients W and W „a t the station by xz yz ° means of a torsion balance, and computing the gradient anomalies AWxz = Wyz - Uxz and AWyz = Wyz - Uyz, expressions (153) give the effect of the inner circle upon the deflection components. The effect of the inner circle upon the deflection com ponents at higher elevations is not too significant. However, deflections can be computed from the formulas of Hirvonen [HIF.VONEIv, 1952]. After alterations similar to those above, the formulas of Hirvonen can be written in the following form: ...(15^) T|' a - 0.105 A W ^ rQ ro where t « . The h is the elevation of the computation point. From these formulas the effect of the inner circle upon the deflection components at a point having elevation h can be computed. The gradient anomalies, and should be determined as in equations (153). 169 4. THE DETERMINATION OF THE VERTICAL GRADIENT OF GRAVITY 4.1. The Determination of the Vertical Gradient by Measurements 4.11. MEASUREMENTS BY THE CLASSICAL METHOD The classical method of measuring the vertical gradient is to measure the weight difference of a mass at two different levels. If the weights of a mass, m, at the upper and at the lower levels are p1 and pg , respectively, and the magnitudes of gravity at these levels are g^ and gg, then according to Newton's second law P! = glm P2 = g2m and, assuming that the change of gravity with elevation is linear, we get Wzz ■ ^ - S-Lli •••(*»} dz dz Ah Ahm where Ah is the elevation difference between the levels p.^ and . Pg. The measurement is carried out by means of a two-armed balance whose dishes are separated by the elevation difference Ah. The mass is placed into the upper dish and the balance is balanced by 170 putting scaling weights into the lower dish. Then the mass and the scaling weights are switched, placing the mass into the lower dish and the scaling weights into the upper one. Since the weight of the mass at the lower position is larger than at the upper posi tion, the balance will not be in balance after the switch. Ad ditional scaling weights are necessary in the upper dish to re gain the balanced state. These additional scaling weights give the magnitude of the weight difference pg - p1 sought. Having this difference and measuring the height difference Ah between the dishes and knowing the mass of the weight m, equation (155) gives the vertical gradient. Four experiments have been carried out using this method [HEIM3HT, 1910]. First, in l88l Jolly in Munich, Germany, measured weight difference of a 5000 gram mass of mercury. The elevation difference between the dishes was 21.005 meters. The gradient obtained was W__ = 2950 E. zz The second experiment using this method was performed by M. Thiesen in 1890 at Breteuille, France. He measured the weight difference of a 1000 gram mass of platinum. The elevation dif ference was 11.^79 meters. The gradient obtained was W s 3030 E. ZZ The third experiment was carried out by Scheel and Diessel- horst in Berlin, Germany in 1895* They measured the weight dif ferences of a 1000 gram mass of copper at three height differences.: 29.7311 1*1-.055* and 7-599 meters. The average gradient 171 obtained was VJ « 2890 E. ZZ The last determination of a vertical gradient by this method was carried out by Richarz and Krieger-Menzel in Spandau, Germany, in 1898, They measured the weight difference of a mass at a relatively small elevation difference, 2.2628 meters. The obtained gradient was Wzz a 2890-E . It follows from the description of the classical method that we can not expect a high accuracy in the vertical gradient so obtained. The mean errors of the four experiments cited above vary from - 60 E to - 2^0 E. Therefore, this method has historical value only for geodesy. The results of the four experi ments are tabulated in Table 15 in order to compare them with the results obtained by other methods. k.12 MEASUREMENTS BY MEANS OF GRAVIMETERS The determination of the vertical gradient by means of gravimeters is also based on equation (155)• However, instead of measuring the weight difference,pg - p^, we measure the gravity difference, g^ - between the two levels by means of a gravimeter. w „ . - ...(155) zz Ah Ah The gravity differences between the levels can be measured in buildings or on observation towers. The vertical gradient is very sensitive to the neighboring masses; therefore, the observed 172 value should be corrected by the terrain effect. The cor rection to be applied is *®t2 " ^ t l Ah where Ag^2 and are the regular terrain corrections computed at the upper and at the lower level, respectively. If the measurements are carried out in buildings then in addition to the terrain correction, the attraction of the building should be considered. Depending upon the used gravimeter the mag netic Influence of the building may also call for a correction. Therefore, the measurement of horizontal and vertical components of the magnetic field is necessary. Out-of-doors the Earth's magnetic field is sufficiently constant so that their differential effects are negligible. The determination of the vertical gradient by means of gravi meters is associated with the names of three geophysicists: Siegraund Hammer, Stephen Thyssen-Bornemissza and B. K. Balavadze. Siegmund Hammer [HAMMER, 1938] of the Gulf Research and De velopment Company, Pittsburgh, Pennsylvania, was the first scientist to determine the vertical gradient by this modern method. In 1938 he determined the vertical gradients at five stations in the United States using a Gulf (Wyckoff-Hoyt) gravimeter, measuring the gravity differences between two levels in high buildings. These experiments were carried out in the Cathedral of Learning of the University of Pittsburgh, in the Gulf Building, Pittsburgh, Pennsylvania, in the 173 Washington Monument, WaBhington,D.C., and in the Empire State and Chrysler Buildings in New York City. The vertical components of the magnetic field were determined simultaneously with the gravimeter measurements. The effect of the horizontal component of the magnetic field was eliminated by always placing the swinging system of the gravimeter in the same magnetic azimuth by means of a compass. The accuracy obtained by Hammer varied from - 3 E to - 17 E. The results are given in Table 15. The attraction of the building and of the neighboring buildings could t not be computed with the required accuracy. To compute the exact influence of a building is almost impossible. This is the main drawback of determining the vertical gradient in buildings. Hammer also made a very detailed gravity survey around the Washington Monument and computed the vertical gradient anomaly, AW,, from the ZZ gravity anomalies obtained in the survey (see section h-.3)- He found that the computed vertical gradient anomaly was about 66 per cent smaller than its measured value. The difference was ac corded to inaccurate measurements, but since the mean error of the measurements was - l*v E and the anomaly found was + 39 E it is more likely that the difference should be blamed on inaccurate corrections. The purpose of Hammer's measurements was to find suitable gravimeter calibration lines. Stephen Thyssen-Bornemisza, "independent geophysicist" (as he calls himself) may be the most persevering investigator since 19^3 in the field of determination of the vertical gradient. His 17H work has heen mostly in Cuba and Canada. In the beginning he tried to measure the gravity difference at small elevation differences in order to work out a suitable field measuring procedure [THYSSEN- BORNEMISZA, 1950]. He used a small Askania gravimeter with a reading accuracy of0.05 milligals and measured the gravity dif ferences between elevations of 25, 50, 100, 112 and 150 centimeters. The average accuracies obtained at the different elevation dif ferences were as follows: Ah cm. - E 25 2U0 50 96 . 100 55 112 55 150 HO The conclusion was that a 150 centimeter elevation difference using the Askania gravimeter was not large enough to obtain the needed accuracy in the horizontal gradients. A similar conclusion was drawn for the 3H0 centimeter elevation difference by Rosenbach of the Houston Technical Laboratory who obtained an accuracy of - 30 E [ROSENBACH, 195 H]. In 1955 Thyssen-Bornemisza carried out another field experiment in Alberta, Canada [THYSSEN-BORNEMISZA, 1 1956, 1957 (2), 1958]« Using a Worden gravimeter (reading ac curacy 0.01 milligals) he measured the gravity differences at eleven stations between the ground, and the top of a portable steel tower 380 centimeters tall, making three to eight repeti tions. The observer stood on a ladder completely separated from the tower. Measurements were carried out only in the relatively calm periods between gusts of wind and under constant temperature conditions. A terrain correction was computed for each point. The results are given in Table 15. The average accuracy obtained was - 10 E, with a variation from - 0 E to - 28.4 E. The average ac curacy was equal to or smaller, than - 10 E at eight stations, and larger than - 20 E only at two stations. In the Soviet Union B. K. Balavadze [BALAVADZE, 1955 (2), 1957] of the Georgian Academy of Sciences carried out experimental field work in West Gruzia in 1951 and 1952. Using a Norgaard quartz gravimeter (accuracy ^ 0.05 milligals) he measured the gra vity differences between the ground and the top of permanent steel observation towers of 14 to 44 meters elevation. The measurements were taken during windless nights to insure constant temperature and no vibration. Terrain corrections were computed. The average accuracy of the measurements at eighteen stations was - 29 E. The average accuracy at ten stations with elevation differences of X 30 centimeters and higher was - 18 E. At the other eight stations with elevations smaller than 30 meters, the average accuracy was - 43 E. The results at the stations with elevations of 30 centi meters and higher are given in Table 15. As did Hammer, Balavadze 176 Table 15 Elevation Name Place Year Range WZZ: in E Method in Meters Jolly Munich 1881 21.005 -2950 Classical Germany 1 M. Thiesen Breteuille 1890 11.1*79 -3030 Classical t France' Scheel and Berlin 1895 29.. 731 Classical Diessel-1 Germany 14.055 -2890 Classical horst 7.-599 Classical Rlcharz Spandau 1898 2.2628 -2850 Classical and Germany Krlegar Jtenzel S. Hammer Pittsburgh 1938 L37.8 -3060 t'4 Gulf Gravi Pa* j.U.S.A. . meter tr II II 1938 i58..3 ' • -3.048 - 6 tr Washington 1938 152.. 2 -3124 - 14 II D.C . .U.S.A. ii II New1 York, ‘ 1938 290.6 -3128 t 3 N.Y.,U.S.A. Balavadze West Grusia 1951 30.53 -2940 - 18 Norgaard Quar: Soviet Union Gravimeter No. 5 n ft " No,. 17 1951 20,. kk -3234 - 39 If II •15 1951 40.34 -3113 - 5 ii II 14 1951 4l,.04 -3258 - 18 ii II ” 13 1951 43.-96 -3278 - 11 ii II k 1951 40.45 -2925 - 5 ti II 6 1951 43,. 26 -3054 - 7 it II 16 1951 41,. 00 -3259 - 20 ii 18 1951 19.-71 It 24.67 -2957 - 34 VI » 19 1951 23..90 41.58 -2982 - 19 II v.- 1 (Continued on Next Page) 177 Table 15 (Concluded) - Name Place Year Elevation Range . in E Method zz in Meters Rosenbach Houston 1 9 5 3 3 . ^ 0 - 3 0 6 3 - 3 7 Borden Texas', Sravimeter U.S.A. 11 Thyssen- Alberta 1 9 5 5 3.80 3 1 ^ 9 .6 - 9 . 7 Borne- Canada No.l mis^a n ti 2 1 9 5 5 3.. 80 2 9 9 2 .f l t 8.1 tr ir 3 1 9 5 5 3.80 3 1 5 0 , 9 - I 3 .i* ir tt if 1 9 5 5 3.. 80 3182,8 ± 6,6 + CO «—I tr 1 • IV 5 1 9 5 5 • 3.. 80 3 0 3 6 , 6 tt tt 6 1 9 5 5 3.80 3112, 2. - 23.2 tt 7 1 9 5 5 3,. 8 0 3 2 2 1 , 0 t i f , 6 it it 8 1 9 5 5 3.. 8 0 3 2 1 8 , if * 5 . 9 11 11 9 1 9 5 5 3,. 8 0 2 9 6 4 ,7 i 2 8 . if ff ti " 1 0 1 9 5 5 3 .-8 0 3160,1 - 0, it 11 " 1 1 1 9 5 5 3 .8 0 2807.2 - 3 .7 178 also computed the vertical gradient anomalies from gravity anomalies by the Hofmann and Malovitsko methods (see section 4.3) > and found that they are much smaller than the observed anomalies. His explanation was that the observed vertical gradient anomalies * depend upon masses very close to the surface, which do not reflect properly in the gravity anomaly picture. Therefore, the computed gradient anomalies are only the regional part of the total anomaly. This explanation seems very reasonable. >.13. MEASUREMENTS BY MEANS OF SPECIAL INSTRUMENTS 4.131. The Haalck Horizontal Pendulum [HAAICK, 1942] The determination of the vertical gradient by means of the Haalck instrument is based on the Laplace equation. The deriva tives W and W are measured and then the vertical gradient is xx yy computed from W = 2co2 - (W + W ) ...(156) zz ' xx yy' ' The instrument,itself consists of a horizontal pendulum which is suspended by the in'var or quartz tapes, OqA and OgB, as shown in Figure 54. This pendulum swings about the axis (^Og. The pen dulum is in an indifferent position when the axis O^Og coincides with the tangent to the plumb line at Og. If the O^Og axis and the vertical do not coincide then the equilibrium position of the horizontal pendulum is in the plane determined by the O^Og axis and the vertical. If we swing the instrument about point Og and perpendicularly to the plane previously mentioned with an angle of dtp, then the amplitude of the pendulum 6 , measured in B the horizontal plane through 0^, can be determined from Figure 55 (see next page). tan 5 « t a n d £ c o s _ i sin i where i is the angle between the Figure 5k O-jOg axis and the vertical. Since the angles 5, i, and dtp are small the equation above becomes dtp 6 = sin i (157) Let us use an (x‘, y', z') rectangular coordinate system con nected to the caBe of the instrument. The origin is at point Og. The z' axis coincides with the vertical at Og. The x 1 and y' axes are perpendicular to each other and to z 1. The horizon tal angle between the +y' axis and the pendulum is 7 , the bearing. Let us place two horizontal pendulums in the same case. The coordinates of the point Og of the first pendulum should be 180 (0,0,0) , and of the second pendulums (*i . y; » ) Vertical at 0, (Figure 56). The pro jections of the angle sin i tan cos between the verticals, at the 02 points of the two pendulums, :os 1 upon the vertical planes x'z' and y'z' are q>x, and CP , » - [x' W' + X ' g o XX + y' W* + z' W' ] jO Xy 0 X 2 ...(158) qp , = - [x' W* + y' g o yx + y' W' + z' W ’ ] o yy 0 yz Let us assume that the axes (0,0,0) O^Og of both pendulums form the same angle i with the z 1 axis. We also assume that Figure 56 181 due to Inaccurate levelling the z' axis connected to the case forms an angle e with the true vertical. The projections of this angle upon the planes x'z' and y'z* are e* and e.' : respectively. x y If the hearings of the horizontal pendulums are 7^ = 0 and 7g » 7 (Figure 56), then the amplitudes of the pendulums hased on equa tion (157) are *■, - - r h - 1 s*n l , ...(159) B cos 7 + f y sin 7 + < & ’. cos 7 + 9 y sin 7 2 = sin i Using the same kind of telescope-mirror device for reading the po sitions of the pendulums as used with the torsion balances, then n - n 5 = -----2 1 2D ...(160) n'- n' &. = 0 2 2D where D is the distance between the mirror and the scale at the telescope. The nQ and n^ are the readings at the equilibrium posi tions of the pendulums. The n and n 1 are the corresponding readings at the 6. and positions. j. d If 7 is zero, i.e., the pendulums are parallel to the y* axis, then from equations (159) and (l60) after reductions we get ' 2S _i t(n' ‘ "i5 ' (n ’ no )] ...(161) Since the left sides of equations (153) and (160) are the same, we 182 can write *i + * 0 Hiy. + 2C WAz ■ - »i> * <“ - -0» ...(162) The relations between the gradients in equation (162) and our regular gradients in the + x = North and + y - East coordinate system, from section 2,35 are given by the coordinate transfer, o o W' a - W sin a cos a + W (cos a - sin a) + xy yy xy + W sin a cos a WV, =* Wv_ cos a - W,_ sin a . •. (163) !xz xz yz 2 2 VP. b w,_, sin a - 2W„„ sin a cos a + V/ cos a xx yy xy xx where a is the azimuth from North of the y' axis of the instrumental coordinate system. Equations (162) and (163) are the principal equations of the horizontal pendulum. The parallel pendulums can be placed in three different rela tive locations within the case (see Figure 57 on the next page). Pendulum No. 1 Pendulum No. 2 x ’ y£ ■i x2 4 Z2 Position 1 0 0 0 0 0 xo Position 2 0 0 0 0 y0 0 Position 3 0 0 0 0 0 zo In the first position from equations (162) and (163) 183 we get W» 3 - [(n * - n ) - (n' - n )] = ww sin2 a - xx x 2D ' 1 ' o o' yy o , 2 - 2W sin a cos a + W cos a xy xx Measuring in azimuth a = 0° , or a = 180°, from equation (l6^) we get ...(165 wxx - ¥ 15^ [<»’ - “) - - ”0 )] o Measuring in azimuth a = 90° or a = 270°, from equation (16U-) we get W = 3. [ (n‘ - n) ~ n' - n)] ..,.(166) yy xq2D lv ' o /J Position 1 In equations (165) and (166) i, xq , y1 i > D and (n1 - n ) are instrument ' o o constants; the normal gravity can he substituted for gravity g. After reading the n 1 and n values, equations (165) and (166) give y' Position 2 (1 derivatives and Wyy. Wgz can be computed from equation (156). Measuring both in azimuths y0 x' a = ^5° and also in a - 135°* from equation (l6h-) we get, respec Position 3 y' tively, 1 > 5 ' V + V • V - ■ SX p g b [(n'' n) ■ K’ n0 )J 1,2 ...(167) t I (Wyy + W**> + Wxy - Figure 57 - V a g 1 [(»'- ») - K - “o)] o 18k From these (W + W ) and W can be computed. Again the yy xx xy vertical gradient is given by equation (156). Position No.ldiscussed earlier is the only one which interests us when the problem is the determination of VT_. However, for the z z sake of completeness we also discuss the other two positions. In the second position from equation (162) and (163) we get Wxy " g2y^D 1 C(n' ' “ (ni " no )] = ' Wyy sin a cos 0 + p p + W (cos a - sin a) + W sin a cos a ...(168) xy xx Measuring in a » 0°, or a =* 90°, or a » 180°, or a ■ 270° from equation (168) we get w „ g-g f e i [(n * - n) - (n* - n )] ...(169) xy 2yQD ' 7 ' 0 o /J \ Measuring in a s 135°, or a = 315° from equation (168) we get 12 Having the instrument constants and the measurement results, W and WA cam be computed from equation (169) and (170) respec- xy (4 tively. In the third position from equations (162) and (163) we get [(n1 - n) - (n^ - nQ )] = cos a - W sin a ...(171) o Measuring in azimuth a ■ 0°, from equation (171) ve get Wx* - ¥ 5 ^ K”' - n> - K - V ] -<172) o 185 Measuring in azimuth a ■ 270°, from equation (171) we get wyz - &T W 1 [ Having the instrument constants and the measurement results, Wxz and can be computed from equations (172) and (173)# re spectively. It can be seen that the Haalck Instrument, with horizontal pendulums in the second and in the third positions, corresponds to the curvature variometer (Chapter 2.1l) and to the gradiometer (Chapter 2.13), respectively, as far as measurement results are concerned. Experimental models using similar principles were con structed by Schweydar [SCHWEYDAR, 1921] and Hecker. These instru ments had very large dimensions. The distances x , y^ and z^ o o o are about one hundred centimeters long; therefore, the whole instrument is larger them a cube with sides of 100 centimeters length. The obtained accuracy was - 5 E. The instrument is very interesting in its theoretical as pects but it is not practical for field measurements. Its big advantage is that it is not sensitive to levelling errors, (it can be seen that the levelling error, e , fell out when deriving equation (l6l) ). 186 4.132. Other Instruments In addition to the Haalck horizontal pendulum, various other instruments were constructed to determine the vertical gra dient of gravity. All of the instruments introduced in this chap ter have a lower accuracy than the Haalck horizontal pendulum and are not used in the field. We deal with them briefly for the sake of completeness and because of their historical significance. Berroth in 1920 introduced a balance which swings about a horizontal axis [BERROTH, 1920]. This instrument gives very low accuracy mostly because the friction at the ends of the swinging axis could not be sufficiently eliminated. Sadovskiy in 1930 developed the Berroth instrument into a more precise model, but his conclusion was that* the instrument is not applicable to field work [SADOVSKIY, 1930]. Schmerwitz in 1931 [SCHMERWITZ, 1931] and Lettau in 1937[LETTAU, 1937] designed horizontal pendulums similar to the one used by Haalck. Schmerwitz used a single pen dulum in his instrument; however, it can not be used in the field because of its high sensitivity to inexact levelling. Haalck, as we saw, avoided this difficultyby using two parallel pendu lums. During the derivation the levelling error € dropped out of the mathematics (see equations (l59)-(l6l) ). Finally, Wegener in 1939 tried to construct an instrument to measure the vertical gradient [WEGEMER, 1939]• The principle is to measure the rate of the sinking of a floating system into a fluid at different elevations. 187 k.H*. CONCLUSIONS 1. Presently the only practical method for measuring the vertical gradient of gravity is to measure the gravity difference between two levels by means of gravimeters. 2. An accuracy of - 5 - - 10 E can be obtained by using a modern gravimeter of 0.01 milligals reading accuracy, by repeating the measurements three to five times and by respecting the following conditions: a. The gravity difference should be measured on separate permanent or portable towers, such as the Bllby Tower, (not in buildings) with heights not smaller than 3*5 - ^*0 meters, far from surrounding buildings, trucks and other objects. b. The platform of the observer should be separated from the tower to avoid vibration. / c. The observations should be carried out at night or under constant temperature conditions. d. Absolute lack of wind is necessary. e. Terrain and drift corrections should be applied to the measured results. 3. The observed vertical gradient depends mostly upon the masses close to the surface of the Earth. Therefore, the observed gradient is representative only at the surface and changes very rapidly, decreasing with height. For this reason the observed . 188 vertical gradients should be applied in geodesy only after t very careful consideration. It should not be used in applications where the decrease with height plays an important role, for in stance when reducing gravity values to the geoid, and so forth. In these cases the regional values of the vertical gradients give better results. The determination of the regional values will be explained in 14-.3 and in k,5. The observed vertical gradient should be used when the application is associated with the observation station, for instance to determine the radius of curvature of the geop at the observation point, and so forth. I*. Further experiments are necessary to find a method to determine the vertical gradient with the same accuracy as the horizontal gradients can be had. 2. The Normal Vertical Gradient Equation (156), derived from the Laplace equation gives - (WXX + V — (156) From differential geometry we know that on a surface i - + A - „ (r + t) ...(17U) Hx Hy r and t are the Monge Quantities (see section 3*1)# and p and p. are the radii of curvature in the perpendicular normal sec- y tions x and y,respectively. 189 From equations (100) we get r + t 3 - (W + W ) g v xx yy' Substituting this Into (17*0 w® ge-t According to differential geometry, the sum of the curvatures in two perpendicular normal sections of any surface equals the sum of the principal curvatures, i.e., 1_ 1_ _ 1_ + i_ px py P1 p2 Substituting this into (175), and then the result into (156) we get VT Z2 = g (£- + ^-) Pg + 2co2 ...(176) ' The normal vertical gradient can be obtained by applying (176) on the surface of a rotating ellipsoid. On the surface of this ellipsoid W a U g - 7 P1 = M P2 = N Therefore, equation (176) becomes vz* - y (i+ + •" 190 From the geometry of the ellipsoid it is well known that a(l - g2) M (1 - e2 sin cp)3/2 N “ Ty 2 7"2 3 7 2 (l - e sin cp) ' Substituting these values into (177)» neglecting the e and higher terms we get after reductions J (2 + e2 - 2e2 sin2 cp) + 2u>2 ...(178) Z Z ct Expressing the eccentricity, e, by the flattening, aj the normal gravity, y , b y ... the Clairaut formula, y = 7e ^ + (f q sin2 ^ 2a 2 7 where q = — — } remembering that co = -^ q j after neglecting' 7e 2 k terms such as a,qa and sin cp, from equation (178) we get, 2y Uzz - [1 + a + q - (3a - | q) sin2 cp] ...(179) 0r 27 (1 + a + q) 3a - | q p U = — ------[1 - = sin cp] ...(180) zz a 1 + a + q v 2 1 or since sin cp = - (1 - cos 2cp) from equation (l80) we get y (l + a + q) 3a - § q 3a - I q V z z ------1 ------[1 - 2(1 + a + q) + §Cl~ a~ ~ q)' 008 ^ This equation after reductions becomes 7e (2 - a - \ q) 3a - | q U zz ------S-- (i + cos 2cp) ...(181) a 2 - a - ^ q 191 The data of the International Ellipsoid and the International Gravity Formula are: a a 6,378#388 meters a a 0.003367 - 7 a 978.049 gals © q a 0.00346783 With these data equation (l8l) becomes U = - 3085.5 (1 + 0.000712 cos 2 At cp a 1+5° this equation gives U _ » - 3085*5 E. Helmert's zz original value for U is-3086 E. [HEIMERT, 1910]. It is applied ZZ to the whole surface as an average. Cassinis [CASSINIS, 1937] and Hofmann [HOFMANN, 1949] starting with different assumptions de rived results as in (182). If we substitute the extended form of the normal gravity formula 2 7 = 7 e t 1 + (| q - ' O S - qp!) sin2 qp - (| qa - g-) sin2 2 7e(g + I <1 - go2- 10i) [ 1 + a 5 a cos 4q> ] (l8la) With the data of the International Ellipsoid and of the 192 International Gravity formula this equation becomes U = - 3085.504 (1 + 0.0007354 cos ap - 0.00000065 cos 4 Other authors derived formulas which also Include an elevation term, i,e., they give the normal gradient on the surface of the Earth rather than on the ellipsoid. Jeffreys [JEFFREYS, 1952] derived the following formula Uzz a ¥ (1 + 6 + q " 2e sin2 9 “ a^ After some alterations this formula can be written as follows U = izl1 + [1 + — cos 2cp - r l zz a 1 + q a(l + q) J Inserting the data of the International Ellipsoid and of the Inter national Gravity Formula we get U = - 3085.5 (1 + 0.000712 cos 2cp - O.OOOI467 h1™) E ...(183) ZZ This same formula is used by the United States Coast and Geodetic Survey [HAMMER, 1938]. Hammer, using Helmert’s formula [HEBERT, 188U],. computed the following expression for the International El lipsoid [HAMMER, 1938], U = - 3085.5 (1 + 0.000736 cos 2* - O.OOOVfO h1™) E ZZ where i|r is the geocentric latitude. Lambert [LAMBERT, 1930] and Rudzki also developed similar for mulas. For the constants in formula (183) Rudzki [RUDZKI, 1905] obtained different values. His analysis is correct but a numerical error has slipped into the final state of his calculations. 193 De Graaf Hunter [DE GRAAF HUNTER, 1959] recently pub lished a formula developed by him from spherical harmonics U = - 3088,0 (1 - 0.0013 sin2 cp - 0.00022 h10”) E zz This formula can be written in the following form U - - 3086.0 (1 + 0.00065 cos 2cp - 0.00022 h1™) E zz Itcan be seen in (183) that the elevation term is very small. It gives about a 1 E correction per 200 meters elevation. The normal vertical gradient at high elevations can be de rived from Hirvonen's formula [HIRVONEN, 1959] . A C 1 2 3 51 2 1 i* 7=7° I^b! 56 +S , '554e- + TIe + + cos ap (| e2 - | 1 + | q eS + i e V 2 5 2 1 Ik - cos 2cp (g q e - ^ e )- - | [3 + | q. - | e2 + cos 2q> ( | e2 - q)]^ Inserting the data of the International Ellipsoid and the Inter national Gravity Formula this equation becomes 7 = 7o - 0.3085507 h - 0 .000220? h cos 2 + 0.0000000725^ h2 + 0.00000000011 h2 cos 2cp where 7 has to be in milligals and h in meters. 19^ After differentiation we get Uz2 = ^ - 0.3085507 - 0.0002207 cos 2cp + + 0.0000005 COS2 2cp + 0.0000001^51 h + 0.00000000022 h cos 2cp or, in our usual form, U„ « - 3085.507 (1 + 0.0007153 cos 2 Formulas (182), (183) and (18^4-) give the normal vertical gra dient of gravity on the International Ellipsoid, on the physical surface of the Earth and at high elevations above the Earth, re spectively. 4.3. The Computation of the Vertical Gradient Anomaly from Gravity Anomalies M l - THE PRINCIPLE OF THE COMPUTATION The anomaly of the vertical gradient by definition is AW« - Wzz - V zz •••(185) where W and U _ are the true and the normal values of the zz zz vertical gradient. It is also true that AW = AV zz zz because the potential of the centrifugal force has no anomalies. 195 The principle of the computation of the anomaly AV „ follows, zz It is known that the potential M 1 anomaly AV at a point A, caused by a mass anomaly Am at point M, (Figure 58 ) is *Am 2 Am AV = k ...(186) 7 2 2x1/2 Figure 58 (p + z ) ' where p and z are the coordinates of Am. It can be proved that the same potential anomaly can also be expressed as A V ■ b I 5s " •••(187) F where Ag is the gravity anomaly at cm arbitrary point P caused by the Am mass. The r is the distance between A and P (Figure 59)* The symbol F on the Integral means that the Integral should be extended over the whole surface of the Earth, or at least as far as the gravity anomalies have an Worth Influence on AV. It is understood that if Ag is the gravity anomaly caused by all of the disturbing masses rather than by the mass Am alone, then AV in equation (187) A y is the potential anomaly caused also Figure 59 by all of the disturbing masses. 196 Differentiating (187) with respect to x we get VX Sc 27T fj ^ The second partial derivative is A V= ~ a v = - — / Ag [S- (— )2 + ^ & ] df ...(188) “ 3x x 2ir P dx r 5x From Figure 59 , 2 2 a / 2 r » (x + y ) ' The partial derivatives of r with respect to x are ^ = x (x2 + y2 )1/2 = p » cos a §-| -r '1 - xr"2 §£ „ i (l - £ §£)„. i [1 - ( ^ ) 2 ] = Sx Sc r r Sx r Bx .-.(189) = - (l - ^cos2 a) r Substituting equations (189) into (188) we get A V - — / (3 cos2 a - 1) df ...(190) Sir P r After similar derivation for AV we get yy 0 AV - i- / (3 sin2 a - 1) df .••(191) ^ 21T F r3 The Laplace equation ...( 192) 1 9 7 with df ■ r dr da -equation (192) becomes . r'2ih A AVzz “ / / dr da ...(193) zz 2w o o t‘2 t .where r' is the limit of the computation,(i.e., 100 - 200 kilpmeters). k.32. THE PRACTICAL COMPUTATION The integral (193) can be divided into two parts. The first part gives the effect of an inner circle with radius r ■ 1-2 kilo- o meters. The second part gives the effect of the annulus between r and r '. o r 2tt r 1 2?r Avzz ■ o O ^farcloi-— f f dr da • 2f r 2tt r o r o We divide the surface of the area under consideration into zones with circles of radii rQ ri+1*.•. r*. If the average gra vity anomaly in a zone between r . and r . . is Ag. . ,, then the 1 1-rJL 1 1 14*1 effect of that zone is i+1 2tt a _ _ - — f f dr da - Ag. . _ (— i- ) _ J J 2 i,i+l 1 2F ri 0 r ri+i ri Using t——: — --- — the effect of the zones from r to r' is ^i.i+i ri+i ri 0 i-r'-l " E ^ i i +l F ^ ...(195) -i-r 1,1+1 “ i,i+l o From (19.^) And after solving the first Integral and replacing the 198 second integral by (195) we get Ag i=r'-l 1 AV = - 2 - 2 Ag, . , ------...( 1 9 6 ) zz r U r ' Ar, , , o o i,i+l The different methods for practical computation used by several authors have the general form of equation (196) and dif fer from each other mostly in the selection of the radii. The methods used in practice are the following: In 19I49 Hofmann, starting from the Bruns' term and using spherical harmonics derived the following formula [HOFMANN, 19^9], ^zz “ I “S2 + • (0-667 ^1,3 + 0,133 ^3,5 + 0,057 *®5,7 + + 0.052 *S 7>11 + 0.039 ^ i11>17 + 0.019 0i 17j25 + + 0.012 ^ i25>35 + 0.010 ^ 35>55 + 0.009 ^655, 105) 10"8 c o s where R is the radius of the earth in kilometers, AgQ is the average gravity anomaly in milligals within the inner circle of r = 1 kilometer, and the values Ag. . , are average gravity O 1) 1+JL anomalies in milligals in the zone from r = 3-k;Qorae-ters to r - ^+1jC3_i01Qgters # Malovitsko derived a similar formula [MALOVTTSKO, 1951] in 1951* , ^ - ^ ll2 *i0 - & s Ag Ib in milligals. 199 Similar formulas were also derived by Baranov [BARANOV, 1953, 195*0, Chramov [CHRAMOV, 1935], and Evjen [EVJEN, 1936]. The main drawback of the above formulas is that the effect of those subterranean masses close to the computation point is not sufficiently considered* The gravity anomalies used in the com putation do not reflect these masses. Since the vertical gradient is very sensitive to these masses, the gradient anomalies AV ZZ computed by this method can not be considered as total anomalies, but as the regional part of the anomalies only. This difficulty can be partly overcome when a detailed gravity survey is carried out around the computation point and within a circle with a radius of about one kilometer. The effect of this inner circle can be computed from Malkinas formula [MALKIN, 1930] • r d2g d2g (AVzz\),r = " “ + ^ ' o 4 axo oyo d2g d2g The quantities — § and — § should be computed for the computation ax2 ay2 point by double numerical differentiation from the observed gravity values. Similar formulas were derived by Green [GREEN, 1871] and by Jeffreys [JEFFREYS, 1938]* If this local survey procedure is not followed the results will be only the regional parts of the anomalies. Table 16 contains the vertical gradient anomalies com puted by the Hofmann and by the Malovitsko methods at the Balavadze's points listed in Table 15. The anomalies had from the observed gradients are also.shown. It can be seen that the computed 200 A W In E zz Station Hofmann Malovitsko Observed No. 5 - lk.O - 14.4 - 171.0 IT i - 7.0 - 7.4 + 148.0 15 ! - 4.6 - 11.7 + 27.0 14 - 6.3 - 3.5 + 172.0 13 - 11.4 - 14.2 + 192.0 k - 63.5 - 62.5 - 161.0 • 6 + + 9.5 VO VO - 32.O I 16 + 17.8 + 19.6 + 173.0 18 + 53.8 + 55.4 - 129.0 19 - 15*5 - 19.8 - 104.0 Table l6 201 values are much smaller than the observed ones because the former ones represent only the regional part of the anomalies. It can also be seen from the table that the values computed by the different methods check fairly well. U.33. REMARKS REGARDING THE APPLICATIONS The gravity anomalies used in the computation formulas pur port to represent all of the masses between the physical surface and the ellipsoid. Therefore, the best gravity anomalies would be the surface anomalies as computed by subtracting normal gravity from observed gravity (may or may not be corrected for the terrain effect). The free air anomalies computed by the regular method can be considered as surface anomalies. In this case the computed gradient anomalies, AV = AW added to the normal gradients U ZZ ZZ zz from (183) give the vertical gradient W on the physical surface. ZZ If the gravity anomalies used in the computation were re duced to the geoid (isostatic or Bouguer anomalies), then adding the computed gradient anomalies to the normal gradients from (182) we get the vertical gradient on the geoid. In geodetic applications both vertical gradients can be useful, depending upon the nature of the problem to be solved. For instance, for the reduction from the physical surface to the geoid the second type of gradient should be used, but when in vestigating the shape of the geop at the observation point the first type of gradients are thought to give better results. 202 In computing the vertical gradient anomaly from gravity anomalies reduced to the geoid the selection of the type of gra vity anomalies has to be done with great care and according to the problem to be solved. 4.^1-. The Computation of the Relative Vertical Gradient from the Torsion Balance Measurements t The computation of the vertical gradient from the torsion balance measurements is carried out in an x,y grid system as shown in Figure 52. The gradients W and WA are determined from the xy lj observed values by interpolation at the intersection points. The change of these gradients between two consecutive intersection points is assumed to be linear. The gradients W , W and W at any point can be expressed xx yy xy by the following series: Wxx ■ <»„>o + S W*x + h ** + h W*x dz + V ■ V o + 5 V + h V ay + h wyy az ...... (197) Wxy ■ Vo + h V + h V dy + 5z Wxy dz + (W„v )_, (W ) and (W ) are known quantities at a point 0, in- xx o yy o xy o finitely close to the investigated point. If these two points are in a horizontal plane and in the x direction then, since dy a dz a p, equations (197) become: W» c “ ( V o + S WXx ax+ V ■ (V o + 5 wyy + - • W = (W ) + W dx + •»• • xy ' xy'o dx xy Prom here $-a (w - w ) = -J- a w = Ww—22— ’ wxx 22---- ’ c^ww-Zy_?— ) 0’ (wx ^ x U°.. ] Sx yy xx ^x A dx WAA - (WA) ' A'o dx ..(198) and also a Wxy - (WxA ox W xy = W xxy « " dxJa" ..(199) If the two points are in a horizontal plane hut in the £ direction then dx = dz = 0, and from equation (197) we get: W = (W ) + I- W dy + .... xx v xx'o dy xx ' V ■ V. + h V ay + -• V + t ? wx y dy + " " Prom here we get a (w , - w N) = 2 a- wA » -22 wyy---- “ Wxx22— “ -—[( 2QL2----22_2_V o - WA " Wxy - (Wxy>o ^ Wxy ~ Wxyy dy Mi. Mi. x _ Z l ...(201) P(X,Y) i Applying expressions (198) - (201) to the consecutive x-1 corner points P1, Pg, P^ and p3 ' P^ of our grid system, we y-1 Figure 60 y get (Figure 6o):j (w - W ) = ^ x , y ~ ^WA^x,y-l l f WA ' xyy xxx'A £ I - w . (w - w )„ - (^ )ic>y. ' dy A yyy xxy'B & ...(202) ~ 2W (2W ) - (2V ^ , y ~ (2Wxpx , y - l dx xy xxy'A i 8 / \ ^xy^x.y ” ^^xy^x-l.y —- 2w = (2w )_ = — y *y x dy xy xyy'B j where the indices x,y and so forth, outside the parentheses, are the coordinates of the point to which the particular gradient refers. The quantities (tf - W )A, (2W )A and (W - W )_, (2W )_ xyy xxx'A’ v xxy'A ' yyy xxy'B' v xyy'B refer to the center points A and B, respectively. The same quanti- tities can he computed at the center point P(X,Y) of the P^P^P^P^ square by taking the mean of the terms of (202) computed at B and D, and at A and C, respectively. Thus, ^Wxyy " Wxxx^X,Y “ 2£ ^ WA^x,y ” ^WA^x,y-l + ^WA^x-l,y (WA^x-l,y-l * ^Wyyy ” Wxxy^x>Y “ 21 ^ WA^x,y " ^WA^X"l,y + fWA^x,y-l " ^WA^x-l,y-l ^ .(203) (2Wxxy>X,Y = h t(2V x , y - (2V x , y - l + (2Wxy)x-l,y " ^xy^x-ljy-l ^ (2Wxyy>X,Y “ b [(2Wxy}x,y " (2Wxy}x-l,y + ^ x y W - l “ (2Wxy^x-l,y-l ^ The Laplace equation at point P (X,Y) is (Wzz^X,Y " ~ ^Wxx^X,Y ” ^Wyy^x,Y Differentiating this equation with reference to x and y we get ^zzx^x>Y ** " ^xxx^X,Y ’ ^xyy^X,Y * ^Wxyy ” ^xxx^X,X ^2Wxyy^x,Y ..(201^) ^Wzzy^X,Y “ ' ^Wxxy^x,Y " ^Wyyy^X,Y “ “ ^Wyyy ' Wxxy^X,y 206 Substituting the quantities from (203) into (20^) we get (Wzzx^X.»Y " ^ WA " ^ x y ^ y " ^WA + 2Wxy^x,y-l + + < W w ~ h [- (WA + « V , , y + - In other words, having the torsion balance results WA and W £mk A jr interpolated to the corner points of the square quantities W and W can be computed at the center P(X,Y) zzx zzy of the square by expressions (205). Having the quantities W„_„ ZZX and W at the center point of each square, the relative vertical zzy gradient 5W at any point P (M,N) can be computed by integration. ZZ If the coordinates of the reference station are X =* Y = 0 then the relative vertical gradient is X=N Y=M (vW x >y ^ + y{0 or in the grid system, using the trapezoidal rule, (&Wzz^ N,M " 2 ^ Wzzx^X,Y + ^Wzzx^X-l,Y^ + + Y»0 [ ^ ZZ^ X >* + ^ z y h , X ’l • (206) \ If at the reference point X m Y ■ 0 the vertical gradient is 207 known, then the vertical, gradient at point P (N,M) is h.5. The Computation of the Regional Vertical Gradient Anomaly from the Deflections of the Vertical From equation (123) we get - 8m (‘a - 51> ■ - ^ x h ...(123) - <^2 - V ■ P ( x ^ ) and P (x2,y2) and AWx and AWy are the anomalies of the first partial derivatives of the potential there. Dividing the equations in (123) by (x^ - x^) and again by (y - y^) we get *2 ~ S1 _(AWx^2 ~ Bringing the two points infinitely close to each other equations 20 8 (208) become ...(209) ' 8m ^ - |F AWy " AWyy The Laplace equation is valid not only for the second partial derivatives of the potential but also for the anomalies; therefore, ve get AW22 - - ( A W ^ + AWyy) ...(2X0) Substituting (209) into (210) we get AWzz = ®m[ ! + | ] Applying this equation in a grid system (Figure 60) where the de flection components | and t) were determined at the intersection points, we get the vertical gradient anomaly at the center of the square from the following equation (AWzz}x,Y * 21 ^*x,y “ *x-l,y) + ^x,y-l" 6x-l,y-l^ + + ^ x , y " T,x,y-1 ^ + ^x-l,y " \-l,y-l^ Another form of this equation is ^AWzz^X,y ” 2 l + ^x.y ^ ■ I’^x-l,y + ^ ' <1^x,y-l " - <*+ " W - i 1 - (S12) where (AW _)v v is the vertical gradient anomaly at the center of Z Z A j X the square PjPgP^P^, 209 ; t) are the deflection components at the corner points indicated by the indices of the parentheses, ^ isthe average gravity value within the square, £ isthe length of one side of the square. The £ length should he selected carefully because in the derivation of (212) it is assumed that the change of the deflection components is linear over that distance .The length £ can vary in the compu tation hut depends upon the variation of the deflections. The gravity value can he substituted for by the normal gravity value without any loss of accuracy. The deflection components at the corner points can he determined by interpolation from any relative deflection inthe vertical net. We do not need absolute values because in equation (212) we have deflection differences only. Thus, the deflection net can be either astro-geodetic or a net de termined as described in chapter 3 .6 . If the deflections were reduced to the geoid, then the anomaly computed from (212) and added to the normal gradient had from (182) gives the vertical gradient. If the deflections were not reduced to the geoid, then equation (183) rather than (182) should be used when computing the normal gradient. In both cases, however, - U + AW zz zz zz We mentioned earlier that the vertical gradient is very sensi tive to the masses directly below the computation point. Naturally 21 0 the astro-geodetic deflections do not show the influence of these masses. Therefore, the vertical gradient anomaly computed . by this method can he considered only as the regional part of the total anomaly. Figure 6l shows the vertical gradient anomalies in central Europe as computed by this method. The astro-geodetic deflection components used in the computation were taken from Maps No. 1 and No. 3 of the recent publication of Helmut Wolf [WOLF* 1956]. The computation, using about 2,000 squares, has been carried out by the students of the Division of Geodetic Science, The Ohio State Univer sity, as a laboratory exercise in Physical Geodesy under the supervision of the author. The anomaly values obtained from Figure 6l should be deducted from the normal gravity computed from (182). 4.6. Applications From the previous chapters it is clear that the vertical gradient of gravity has great significance in geodesy. There is no use to emphasize that it is better to use the computed regional gradient (see 4.3 and 4.5), rather than the normal vertical gradient, t in the free air reduction of gravity. This is equally true when we reduce the observed gravity to the geoid, or when we compute *013 •ml' •*» a REGIONAL ent n ie d a r g es ie l a m o n a 2 1 1 the gravity at elevations above the physical surface of the Earth. In addition, the observed vertical gradient of gravity (see 4.12, 4.3 and 4.4) together with the torsion balance measurements can determine the principle radii of curvature of the equipotential surface (see 3»3)t and also the shape of the equipotential surface j at the observation point by analytical means (see 3*1)* We do not deal here with the geophysical applications of the vertical gradient. Papers about this subject can be found in the third section of the Bibliography. 212 BIBLIOGRAPHY 1. References Baeschlin, C. P. Einfflhrung In die Kurven und Flachentheorle. Zurich, 19^7. Baeschlin, C. F., Lehrbuch der Geod&sie. Zurich, 19U8 . (Pp. 690-721) Balavadze, B. K. "K metodeke opredeleniya vertikalnovo gradienta sily tyazhesti." Izv.Ak.Nauk.S.S.S.R.Seriya geophysitseskaya No. 1. 1955. Balavadze, B. K. "K voprosu o vitsisleniy vertikalnovo gradiente sily tyazhesti." Izv.Ak.Nauk. S.S.S.R.Seriya geophysitseskaya No. 5- 1955. Balavadze, B, K. "0 vliyanii topograficheskikh mass no vertikalnyy gradient a sily tyazhesti." Akad.Nauk Grunzinskoy S.S.R. v.19. No. 1. 1957. Baranov, V. "Calcul du gradient vertical du champ de gravity on du champ magnetique m^sure a la surface du sol. "Geophysical Prospecting. V. 1. No. 3» 1953* Baranov, V. and Tassencourt, J. "Some remarks on the errors in the calculation of the vertical gradient of gravity." Geophysical Prospecting. V. 2. 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"Diagramme zur Bestlmmung der Terrainwirkung fur Fendel und Drehwaage und zur Bestlmmung der Wirkung. zveidimenslonaler Massenanordnungen." Zeitschr. fur Geophysik. (1927. Kasansky, V. A. "Prakticheskly opyt gravimetrlcheskovo vyvoda uklonenij otyesa y formygloida." Trudy CNIIGAiK.V.II. Mbskpw. 1936. Klupathy, J. "B4rA Eotyos Lir&nd fold kutat&sai." Ur&nl~a.7 k. Budapest. 1906. Kohlrausch, B,. Praktische Fhysik. Leipzig. 1927. Lambert, W. D*. "The reduction of observed values of gravity to sea level." Bui. Geod. No. 26. 1930. \ Lambert, W. D., and Darling, F. W. "Tables for determining the form of the gepid. and its indirect effect on gravity." Special Fubl,. 199. U.S. Coast and Geodetic Survey. 1936. Lambert, W. D. "The gravity field for an ellipsoid of revolution as a level surface (i-IIl)." Technical Papers No. l6l,, 716-1» and 716-2. Mapping and Charting Research Laboratory; OSU. Columbus.. 1952, 1957* and 195#. Lancaster Jones, E. "Principles and. practice of the gravlty-gradiometer Journal of Sci. Instruments. V. 9. 1932. Lettau, H.. "Das Horizontal doppelpendel." Zeitschr.f.Geoph. V.13. 1937 Malkin, N. R. "Zavisimost mezsdu gradlentamy njutonovskovo' potentlala na pleskosty V premenyy k isledovaniyu grayitationnych y magnitnich anomalyy." Izv.Ak.Nauk S.S.S.R.Phys.Math. 1930. Malovitsko, A. K. "Isledovanlya po analiticheskorau prodelzsenlyu grayitationnych anomalyy." Trudy Novosjbirsky In.ta Inzs. geodezyy. aerophotosemky 1 kartografyy. Y, IV. 1951* Michajlov, A. A. Theoriya flgury zemly y gravlmetrlya. Moskow. 1939* Mikola, S. "B&rA Eotyos L6r&nd nehlzslgi kutat&salnak jelentos&girol." Ur&nla, 19 k. Budapest. 1918. Mblodenskiy, M.. S., Basic Problems of Geodetic Gravimetry. 19^5* Trans lation. U. S. Office of TEchnlcal Services. Washington, 1959. Nettleton, L. L. Geophysical prospecting for Oil. New York. 19|W>. 216 Nikiforov, P. "Noviy typ gravitationnovo variometra s korotkim periodam" Izv. Ifist. prikladnoy geophys lkl. V. 3. 1927* Numerov, B. "Redukciya nablyugenniy gravitationnovo variometra za topografiyuo" Trudy Glavnovo geologorazvedotsnovo upravleniya. V. 36. 1931. Numerov, B. "Gravitations variometer mit drei Waagebalken." Bulletin de 1 1Institut Astronomique. No. 30. 1932. Oertling, L. The Eotvos torsion Balance. London. 1935.* Redey, I, "A potenciilelmllet alkalmazAsa a geod4zi&ban." Tlrk4p4szeti Kozlem4nyek kih.onfuzete.13. sz. Budapest. 19^9* Renner, J. "Untersuchungen uber Lotabveichungen." Acta Technlca Ac. Sci. Hungarica. Tom XV. Fasc. 1-2. 1956. Roseribach, 0. "Quantitative studies concerning the vertical gradient and second derivative studies of gravity interpretation." Geophys. Prosp. V.2.No.2. 1951*-. Rudzki, P. "Sur la determination de la figure de la terre d 'apres les measures de la gravit&." Bui. Astr. V. 22. 1905* Rybar, I. A Gr. 5 typusu gravitAciAs k4szullk. Kezirat. Budapest. 19 ^ Rybar, I. "Eotvos, torsioh balance model E-5^." Geofisica Pura e Appjicata. V. 37. 1957. Sadovskiy, M. A. "Velitsina dg/d.z y eye znatseniye v geophysice." Trudy Inst, prlkl. geoph.V.6. 1930. Samsonov, N. "Grafltseaskiy metod utsets vllyannlya topografitseskich i mass pp rpbljugeniya s gravitationnim variometrom." Trudy GGRU V.36. 1931. SChmerwitz, G. "Erhohung. der Empfidllchkeit der Hebelwaage durch ein Horizontalpendel." Zeitschr.f.Geoph. 1931. Schveydar, W. "Die photographische registrierende Eotvos*sche torsions- waage der Firma Car Bamberg in Berlin Frledenau." Zeitschr. fur Instrumentenkunde. 1921. Schveydar, W. "Lotschvankung und Deformation der Erde durch Flutkrafbe.' Zentralbureau. der Intern. ErdmesBung.Neue Folge.No. 38* Berlin. 1921. II Schveydar, W. "Uber eine neue Form der Drehvaage." Zeitschr.f.Geophys. 1926. 217 Schveydar, W. "Die topographisehe Reduktion bei Schweremessungen mittels elner Torsionswaage." Zeitschr.f.Geophys.V.l and V. 3. 1 9 2 V 2 5 ^ 1927. Sorokin, A. V. Gravimetriya y gravlinetrltseskaya razvedka. Moskow.195: Thyssen Bomemlsza, S. "Uber die mAglichkeit den vertlkalen, Schweregradienten mit Gravimeter zu messen." Beitr. angew. Geophys. v. 11. 1950. Thyssen Bomemlsza, S.,and Stackler, W. F „ "Observation of the vertical gradient of gravity in the field." Geophysics.V. 21. No. 3. 1956, and V. 23. No. 2. 1958 (Letter to the Editor). Thyssen Bomemlsza, S. "Das zvelstufige Gravimeter Mess verfahren." Erdol und Kohle. V. lO.No.l and No. 6. 1957* Vening Meinesz, F. A. :lfA formula expressing the deflection pf the plumb, line ip the gravity anomalies." Proc. Roy.Ac.Amsterdam V.XXX. 1928. II Ilf Vening Meinesz, F. A. Physical Geodesy. I and II. Konlnkl. Nederl. Ak.van Watenschappen.Amsterdam. Series B ,56. No.l. 1953. Wegener, K. "Bemerkungen zu dem vertlkalen Gradienten der Schvere." Zeitschr.f.Geophys. v. 1*5. 1939. Wolf, H. "Versuch Einer Geoidbestlmraung im mittleren Europa aus tf Astronomisch-Geodatischen Lotabveichungen." Deutsche Geod. Kommission bei der Bayerlsche Ak.D. Wiss.Reihe A. Heft. No. 18 218 2. Other Publications in Connection with the Topic of This Paper Abelskly, M. Y. "Opredelenlye momenta inertsii i tsentrobezlmogo momenta krutilnykh vesov gravitatsionnogo variometra." Akad. Nauk S.S.S.R. Izv. ser. geofiz. No. 7 1958. ii * Apsen, B. Uber die Ausrechnung der zweiten partiellen Ableitungen des Schwerepotentials aus der Drehwaagebeobachtungen nach der Methode der kleinsten Quadrate." Zeitschr. f . Geoph. V. 17. 19te. Ackermann, H. A. and Dix, C. H. "The First Derivative of Gravity." Geophysics, V. 20. No. 1. 1955° Barton, D. C. "Gravity Measurements with the EtttvfisBalance." Physics of the Earth.V. II. Bull. Nat. Research Council No. 78. 1931- Barton, D. C. "Accuracy of Determination of Relative Gravity by Torsion Balance." Trans, of the Soc. of Petr. Geophysl- cists.V.3. 1932. Belluigi, A., "Sulla Bilancia di EfttvOs." Bull, della Soc. Seis- mologica Italians. 1927 Belluigi, A. "Su l'uso del variometrio di gravita." L'Industria Mineralia. 1927. Benfield, A. E. "Note on the variation of gravity with Depth." Zeitschr. f. Geophys. V. 13* No. ^4—5- 1937* Bragard, L. "Courbure moyenne des sections principales et gra dient vertical de la pesanteur en un point d'un elllpsolde de revolution." Soc. Royale Sciences Liege Bull. V. 26. No. 3. 1957. Egyed, L. Geofizikai alapismeretek. Budapest. 1955- Etttvfis, R. "VizsgAlatok a gravitAciA jelensAgeinek kfireben". Term. Tud. Ktizl. 20 k. Budapest. 1888. Etitvtts, R. "A Szt. Gelldrt hegy vonzA erejAre vonatkozA vizsgdla- tok." Term. Tud. Kflzl. 21 k. Budapest. I8S9. 219 Efltvtis,R. "Uber die Anziehung der Erde auf verschiedejne Sub-. stamen." Math* und Naturw. aus Ungam. V. 8 . 1890. Etttvfis, R. "Etude sur les surfaces de niveau et la variation de la pesanteur et de la force magnitique." Rapports prlsentAs au Congres International de Physique a Paris. Tom 3 . 1900. Etitvbs, R. "Megfigyelesek a Balaton jegen." Math, es Phys. Lapok* V. 10. Budapest. 1901. Efitvtis, R. "A nehAzsAgriil es a ffildi mAgneses erttrttl." Term. Tud. Kflzl. 35 k. Budapest. 1903. E8tv8s, R. "Programme des recherches gravimetrique dans les regions vesuviennes." ComteB redus des stances de la premiere reunion de la comission permanente de Sels- mologie rAuni a Rome. 1906. Efitvtts, R. "Die Niveauflfichen und die gradienten der Schwerkraft am Balatonsee." Resultate der Wiss. Erforschung des Balatonsees. I. Band I. I. Teil. Geophysikalischer Anhang. Budapest-Wien. 1908. ---- Etitvtis, R . .. "Uber Arbeiten mit der Drehwaagen." Verh. d.XVII. allg. Konferenz der Intemat. Erdmessung in Hamburg. 1912. Etttvfis, R. Oeoflzikai kutatAsok KecskemAt kttmyeken. Urania. Budapest. 1913* E8tv8s, R. "Experimenteller Nachweis der Schwere&nderung, die ein auf normal geformter Erdoberflfiche in Ostlicher Oder westlicher Richtung bewegter Kftrper durch diese Bevegung erleidet." Annalen der Physik. V» 59* 1919* Etttvtis, R. "Beitrflge zum Gesetz der Proportionalitflt von Tr&gheit und Gravit&t." Annalen der Physik. 1922. Facsinay, L. GravitAcios mArdsek As az izosztAzia. Budapest. 1952. Geoloso, G. "La Bilancia di torsione di Etitvtts a registracione fotografica." La Miniera Italians. 192^. M _ Helmert, F. R. "Uber die Reduktion der auf der physikalischen Erdo- berflMche beobachteten Schwerebeschleunigungen auf ein gemein- sames Niveau." Sitzungsberichte der Ron. Preuss. Ak. der Wiss. i902. - — :------———" 2 2 0 Hoffmann, W. "Uhbersuchungen fiber den Vertikalgradienten der Schwere." Dissertation. Bonn. 19*+8*. Hondros, D. 0 streptos sygos EfttvKs-Schveydar. Athen. 1932. Hopfner, F-. "Die EBtvtisf sche Schwerewaage und ihre Ei.chgung." Zeitschr. des Qsterreich. Ing. und Architekten. 1927. Jeffreys, H. "An application of the free air reduction of gravity." Qerl. Beitr. zur Geophys. V. 31* 1931. Jordan-Eggert • "Theorle der Drehwaage von Etitvfis" Handbuch der Vermessungskunde, Stuttgart. 1923* Kaula, W. M. 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"Bemerkungen zur vorstehenden Erwiderung von H. Haalck." Beitr. z„ angew. geophys. V. 10. 19*6• Schweydar, W. "Die Bedeutung der Drehwaage von Etitvfis fxlr die geo- logische Forschung, nebst Mitteilung der Ergebnisse einiger Messungen." Zeitschr. f. prakt. Geologle. No. 11. 1918. Schweydar, W. "Die Drehwaage und ihre Bedeutung fur die 2 2 6 Anffindung von Bodensclhatzen *'M Die NaturwisBenschaftep,. (Pp. 1 6 0 -1 6 5 ). 1918. Selem, A. M. and Monnet, C. ‘'Application of vertical gradients t and comparison of different geophysical methods in. a dif ficult area." Geophys. Prosp. V. 1. No. 3« 1953* 227 AUTOBIOGRAPHY I, Ivan Istvan Mueller, was b o m In Budapest, Hungary, January 9, 1930* I received my secondary-school education in the public schools of Budapest, Hungary. In the years 19^*+-4-7 I attended the State Architectural Trade School in Budapest, which granted me a Certificate of Architecture in 19^7 • In 19**8 I started my under graduate training in the School of Architecture and Civil Engineer- i ing of the Technical University of Budapest, where I received the Diploma of Engineering in 1952. While in residence there, I was Assistant in the Department of Engineering Mechanics in the year 1950-51, and in the Department of Mapping Engineering (Geodesy) in the year 1951-52. In September, 1952, I was appointed Instructor of Geodesy in the same department. In September, 195^, I was promoted to Assistant Professor. I held this position until December, 1956, when I left Hungary. In the years 1952-5& I was also preparing to get the Hun garian equivalent of the Ph.D. degree. I arrived in the United States on January 8 , 1957* In the years of 1957 and 1958 I worked as a De sign Engineer at C. H. Sells Inc., Consulting Engineers and Surveyors in Hew York City. In January, 1959, I was appointed Instructor in the Division of Geodetic Science at The Ohio State University. I held this position till June, i960, while completing the requirements for the degree Doctor of Philosophy. My professional activities include contributions to several scientific magazines and memberships in the following socie ties: Sigma Xi; American Congress on Surveying and Mapping; American Geophysical Union; Society of Exploration Geophysicists; Society of American Military Engineers.