The Bruns Transformation and a Dual Setup of Geodetic

NOAA Technical Report NOS 85 NGS 16

The Bruns Transformation and a Dual Setup of Geodetic Observational Equations Rockville, Md. April 1980

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(Continued at end of publication)

NOAA Technical Report NOS 85 NGS 16

The Bruns Transformation and a Dual Setup of Geodetic Observational Equations Erik W. Grafarend

National Geodetic Survey

Rockville, Md.

April 1980

u.s. DEPARTMENT OF COMMERCE Philip M. Klutznick, Secretary

National Oceanic and Atmospheric Administration Richard A. Frank, Administrator

National Ocean Survey Herbert R. Lippold, Jr., Director

CONTENTS

Abstract 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. The Bruns transformation .. .......................................... 13

3. The dual setup of geodetic observational equations .................. 32

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Appendix A. Invariant representation of Il x - x ' II-m . . . . . . . . . . . . . . . . . . . 47

Appendix B. Cartesian representation of the gravitational potential . . . . 50

Appendix C. Cartesian representation of first-order gradients of the gravitational potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Appendix D. Cartesian representation of second-order gradients of the gravitational potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendix E. Cartesian representation of the Euclidean norm of first-order gradients of the gravity potential . . . . . . . . . . . . . . . . . 61

Bibliography 65

ii

THE BRUNS TRANSFORMATION AND A DUAL SETUP OF GEODETIC

OBSERVATIONAL EQUATIONS

Erik W. Grafarend1

National Geodetic Survey National Ocean Survey, NOAA

Rockville, Md. 20852

ABSTRACT. The Bruns formula, which equates the disturbing gravity potential modulo the length of the normal gravity vector to the height anomaly, is generalized into three dimensions and into horizontal, equatorial, and inertial reference frames. It is applied to formulate the space.-1ike geodetic boundary value problem in geometry and gravity space. The Bruns transform allows a dual setup of geodetic observational equations in a network of mass points, the finite element approximation of the space-like geodetic boundary value problem, in the following sense: The observational equations can be expressed rigorously either as a function of geometric coordinate corrections alone without any gravity dependent quantity, or alone as a function of the gravity disturbing potential and its gradients alone without any geometric coordinate correction. For operational purposes, estimable quantities from reference-free observab1es are studied in geometry, gravity, and vorticity spaces. They correspond to invariants with respect to a linear similarity transformation typified by positional angles and length ratios in various vector spaces. A Cartesian series representation of the gravity potential and its gradients is given--the Cartesian coordinate system is known to be singu1arity-free--and is used for a unified Cartesian setup of observational equations.

Iprepared during a 3-month period in 1978 when the author served as a Senior Scientist in Geodesy, National Research Council, National Academy of of Sciences, Washington, D.C., while on leave from the University FAF at Munich, Federal Republic of Germany.

2

Science should be the friend of practice, but not its slave.

C.F. Gauss

INTRODUCTION

Geodesy is conventionally divided into two branches: geometric and physical. This separation has resulted in various geodetic schools, or research groups, concentrating on one or the other aspect with little intercommunication. We would like to show that geodesy is actually a unity. The two branches are only the sides of a single coin. In detail, we will prove that geodetic observational equations can be uniquely set up in either the geometric or the physical mode. For instance, a distance observation can be expressed in terms of either the coordinates of the end points of a line or the gravity disturbing potential and its gradient at these points. The proof is based on the classic Bruns formula which expresses the height anomaly in terms of the gravity disturbing potential modulo the magnitude of the normal gravity vector. The Bruns formula will be generalized into three dimensions and into various reference frames: horizontal, equatorial, and inertial.

To make the Bruns formula operational, we have to inject observable quantities. Therefore, the first section is devoted to geodetic observables. There are two perspectives from which to look upon geodetic observables. If we do not introduce an a priori reference system into the vector space of geodesy, only positional angles and length ratios are observable. They are invariant with respect to a linear similarity transformation, characterized by degrees of freedom of type translation, rotation, and scale. Referring to adjustment procedures, positional angles and length ratios are estimable quantities. This concept is applied to both geometric and physical space. For instance, we construct positiona'I angles and length ratios in gravity space from a network based on gravity vectors. The geometric quantities are a function of the length of the gravity vector and astronomical longitude and latitude at three points.

In the second section we will derive the generalized three-dimensional Bruns equation from observables that are one-point functions. These can be computed from observations once we have established a reference system for origin, orientation, and scale in any geodetic vector space. The first step will be a transformation of one-point observables into Cartesian coordinates of points on the approximate surface of the Earth, the telluroid. We will use isoparametric mappings for astronomical longitude and latitude, gravity potential, and firstand second-order gradients. The mappings are one-to-one if we use the isotropicor zero-order approximation of the gravity field. Uniqueness is lost if we use another order of approximation. The second step is formulation of the transform of disturbances of gravity into Cartesian coordinate corrections.

The third section deals with a dual setup of geodetic observational equations of one- and three-point type, either in the geometric or in the gravitational space. They refer to different formulations of the Bruns transformation based on astronomical longitudes and latitudes and gravity potential (or gravity or gravity gradient).

The appendices are a Cartesian form of series representing the gravity potential and its first- and second-order gradients.

3

The report reflects current research in space-time geodesy, especially with respect to the geodetic initial-boundary value problem and its finite element approximation, the setup of geodetic observational equations in networks of mass points.

Section 1 is influenced by the concept of geodetic invariants introduced by Baarda (1973) and estimable quantities introduced by Bossler (1973) and Grafarend and Schaffrin (1974, 1976). The isoparametric mappings of Section 2 which led to the formulation of the three dimensional Bruns transform have been partly studied by Bocchio (1976 a,b,c), Bruns (1876), Hirvonen (1960, 1961), Krarup (1969, 1973 a,b), Livieratos (1976, 1978), Marussi (1973, 1974 a,b), Moritz (1965, 1977), Niemeier (1972) and Grafarend (1972, 1975, 1978 a,b,c). The first setup of geodetic observational equations to be expressed rigorously in the gravimetric mode was by Sanso (1978 a,b) by making use of his adjoint potential. Here, we will reverse his argument exactly by employing the inverse Bruns transformation and expressing the geodetic observational potential rigorously in the geometric mode.

Geodesists have hesitated to accept the new three-dimensional mapping. Therefore we would like to make the following comments. For two-dimensional cartographic mappings the isoparametric mapping is well known, e.g. Chovitz (1952, 1954, 1956), O'Keefe (1953), Lane (1939, p. 189), Levi-Civita (1926, p. 220). Let us quote from O'Keefe (1953): "It is evident that the deformations produced by the isoparametric method are of the same order as those produced by other methods."

Another comment is on the definition of a geodetic network. Much research in geodesy has been performed in two-dimensional network analysis. Such networks are better termed mathematical networks because they do not take the gravity field into account. Here, a geodetic network consists of mass points; thus there is gravitational interaction which we cannot switch off.

1. OBSERVABLES

Having decided foundational questions, we next introduce related observations which make geodesy operational. A majority of geodesists believe that geodesy is Euclidean geometry referred to linear space with finite dimensions and Hilbertian geometry referred to linear space with infinite dimensions. What then are the basic observables?

In Euclidean geodesy position is given by vectors, for instance,

the position vector

the gravity vector

the vorticity vector

moving in space-time. Let us give an illustration of these vectors, as shown in figure 1 (p.70). Choose an origin of reference, e.g., the geocenter. The position vector extends from the reference origin to a mass point in space, e.g., the topocenter. At this point we draw the gravity vector, the rotation vector, or any other vector of reference. The corresponding vector spaces are called

4

the geometry space,

the gravity space, and

the vorticity space .

The set of all position vectors drawn from the reference origin is called the geometry space. Gravity space is constructed by a translation of the gravity vector along the position vector to the reference origin under Euclidean parallelism . In the same way the vorticity space or any other space of reference vectors is defined . Coordinates vn of a vector yare provided after we select a frame of reference �n' e . g . , the inertial frame, such that

N 1 v .. L. vn

e == VO e + v !l + • • • n"o -n -0

(Notation: vectors in Euclidean space are denoted by capital letters, or underlined small letters . )

If the base vectors are orthonormal,

1 (1)

1(2)

where 0 ij is the Kronecker symbol for an element of the unit matrix and ( . , . ) is the s1gn for the scalar product . Coordinates are recovered by

(v,e ) == - -n

n v •

(where I I· I I is the norm sign) is the relation of completeness .

1 (3)

1 (4)

It is assumed that in space-time geodesy the number of independent base vectors, which is identical to the dimension of the vector space,is (3,1).

5

In Hilbertian geodesy, position is given by vectors, for instance,

the position potential,

the gravity potential, and

the vorticity potential.

(In Hilbert space, potential is a vector.)

Figure 2 (p. 70) illustrates these vectors. Coordinates vnm of a vector v (vectors in Hilbert space carry an overbar) are provided once we have select a frame of reference, e. g.,

e nm

V2n+l -n-l P (sin cp) r

V2(2n+l)

V2(2n+l)

n

(n-m) ! (n+m) !

-n-l' r P

(n- / m / ) ! -n-l (n+ / m / ) ! r

for m=O

(sin cp) cos mA for m>O 1 (5) nm

P (sin nm cp) sin mA for m<O

where Pn are Legendre and Pnm associated Legendre functions of the first kind, and A, cp , r spherical coordinates, such that

v= 00 +n

n=o m==-n

nrn -v e • nm 1 (6)

Here, v is a "harmonic" function satisfying 1Iv=O, where 11 is the three-dimensional Laplace operator.

If the base vectors are orthonormal, as in our example,

(e . . , e,.n) 1J ,IU. 1(7)

6

e. g., the integral over the unit sphere divided by 4n. Coordinates are reproduced by

rv. e ) nm

'" !!v!! L

run v

n=o m=-n

1 (8)

1(9)

(where I I'll 12, the square of the norm, is the integral over the unit sphere of (v,v)/4n) is the relation of completeness.

The earlier question about basic observables can now be answered. Assume a network, e.g., a triangle, being constructed in a vector space. For the depiction of any vector by an arrow, as in figure 3 (p. 7l),we require an origin, direction, and a length. To remove these artificial references for translation, rotation, and scale, we need quantities which are invariant with respect to changes of these parameters. In other words, we are looking for invariants under a similarity transformation

v -+ v' T + ARv 1(10)

where T is a translation vector, R an orthogonal matrix, and A a scale factor. It is well known from analytical geometry that length ratios and angles are dual elements of the basis of invariants under the linear similarity transformation given above, e. g.,

v, /I - .. 1(11)

or

1 (12)

7

Finally, we will present three examples for basic observables in the geometry, gravity, and vorticity spaces.

EXAMPLE 1. 1 (geometry space)

Let us introduce a triangle in the geometry space constructed from position vectors Xl' X2, X3 directed from the geocenter to three mass points in space.

At first the network is observed by a theodolite tions and horizon distances (or zenith distances). are E I, E*, and F*, defined as follows;

through horizontal direcRelated reference frames

The orthonormal observational triad E' is based on the vector E3' directed from the station point at the topocenter to the target point. The base vector E2' is the normalized vector of the exterior product of the local, instantaneous gravity vector -f and E3,. El' completes the orthonormal base. The orthonormal horizontal triad E* is based on the normalized local, instantaneous gravity vector E3* at the topocenter. The base vector E2* is the normalized vector of the exterior product of the local instantaneous rotation (vorticity) vector � and the local instantaneous gravity vector -f. El* completes the orthonormal base. The "carrousel" triad F* is related to the horizontal triad E* by F* = R3(L)E*, where L is the horizontal orientation unknown such that Fl* is in the zero direction of the horizontal circle of the theodolite. To summarize, the frames are related as shown in the diagram

E* F*

� (A,B)

E'

where RE(A,B) = R2(7 - B)R3(A), and A the south azimuth, B the horizon distance, T the horizontal direction.

Now we can compute the positional angle

cos If x

( 12 13) E3, , E3, • 1(13)

8

From the diagram we read

so that

E' �<A,B)E* �<A,B)R3T<l:)F* �(T,B)F*

E3, = cos T cos B F1* + s in T cos B F2* + sin B F3*

{ <X2* - X1*) (X3* -.X1*) + (Y2* - Y1*) (Y3* - Y1*)

+ (Z2* - Zl*) (Z3* - Zl*) }

1 (14)

1 (15)

1 (16)

Here, the positional angle is represented by horizontal directions and horizon distances at the three network points and is independent of the origin, orientation, and scale of the reference systems.

Next, the network is observed by a camera through right ascension and declination. Related reference frames are E', F', and E· which are defined as follows:

The orthonormal equatorial triad E· is based on the normalized local instantaneous rotation (vorticity) vector E3. at the topocenter. The base vector E2 . is the normalized vector of the exterior product of the instantaneous ecliptic normal vector and the local instantaneous rotation (vorticity) vector. El. completes the orthonormal base. The "carrousel" triad F' is related to the observational triad E' by F' = R3(X) E' , where X is the observational orientation unknown. To summarize, the frames are related as shown in the diagram

9

E'

E' F'

1T where �(a,o) = R2(Z - 0) R3(a), and a the right ascension, 0 the declination.

From the diagram we see

so that

F' = �(a,c)E'

E3• =cos a cos 15 E1• + sin Q cos 0 E2, + sin 15 E3,

{<X2' - �,)(X3' - �.) + (Y2' - Y1·)(Y3, - y1,)

+ (Z2' - Z1,)(Z3' - Zl')}

1 (17)

1(18)

1(19)

10

(where 0ij ' Uij represent differences in 0, U between points i and j)

holds because E3 , = F3 , by definition.

The positional angle above is represented by differences in right ascension and declination at the three network points and is independent of the origin, orientation, and scale of the reference systems.

EXAMPLE 1.2 (gravity space)

Let us introduce a triangle in the gravity space constructed from gravity vectors rl, r2' r3·

The network is observed by an astronomic instrument and a gravimeter. Related reference frames are E*, E', and F' , defined as follows

The frames E* and E' have been introduced in the first example. The "carrou�el" triad F' is related to the equatorial triad E' by F' = R3 (8gr) E', where 8gr is the equatorial orientation angle, (also called Greenwich sidereal time), such that Fl' is in the Greenwich direction, the projection of the local instantaneous gravity vector at Greenwich onto the equatorial plane. The frames are related in the following manner:

E·

E*

R3 (8 ) gr F· .-------------� ..

where RE(8,¢) = R2 (� - ¢)R3(8), and A astronomic longitude, ¢ astronomic latitude, 8 sidereal time angle.

We can again compute the positional angle

cos 1 (20)

From the diagram,

E * RT3(6 ) F' gr

so that

r = - II r II E3*

cos 'l'r

{[llr211 cos AZ cos �z - IIr111 cos1A cos �11 [llr311 cos ft.3 cos �3

- IIr111 cos Al cos �1] + rllrzi! sin AZ cos �2 - IIr111 sin '\ cos �11

[lIr)1 sinA3cos�3- l I r1 1 1 sinA1 cos�1] + rllrzll sin�2

- II r 111 sin � 1] rI I r 311 sin � 3 - II rIll sin � 1] } .

11

1 (21)

1(22)

1 (23)

1(24)

Thus, the positional angle is represented by the magnitudes of the gravity vectors and astronomic longitude and latitude at three network points, and is independent of the origin, orientation, and scale of the reference systems.

EXAMPLE 1. 3 (vorticity space)

Let us introduce a triangle in the rotation space constructed from rotation (vorticity) vectors nl, n2, n3•

The network is observed with respect to a frame f' uniformly rotating with rotational speed w. F', defined in the second example, and f' are related by F' = Rc(y,S,a) f', where RC(y,S,a) = Rl(a) R2(S) R3(Y) are rotation matrices, and a,S,y Cardan angles (Grafarend et al., 1979: p, 208).

12

The positional angle

1(25)

can now be computed, taking into account

F3• = (sin a sin y + cos a sin 8 cos y) fl' 1 (26) - (sin a cos y + cos a sin 8 cos y) f2•

+ cos a cos 8 f3•

1(27)

cos 'l' = I In2-nll 1-1 I In3-nll 1-1 1(28) w

1[lln211 (sin a2 sin y 2 + cos el2 sin 82 cos Y 2)

- IInlll (sin ell sin Yl + cos ell sin 131 cos Yl)J [lIn311 (sin el3 sin Y3 + cos el3 sin 133 cos Y 3)

- IInlll (sin ell sin Yl + cos ell sin 131 cos Yl)J +[lln211 (sin el2 cos Y 2 - cos el2 sin 132 cos Y2) - IInlll (sin ell cos Yl - cos ell sin 131 cos yl)J

[!!n3!! (sin el3 cos Y 3 - cos el3 sin 133 cos Y 3) - IInl!! (sin ell cos Yl - cos ell sin 131 cos Yl)l +[lln211 cos el2 cos 132 - II nlll cos ell cos 1311

[II n311 cos el3 cos 13 -3 IInl!! cos ell cos 131] I

13

is represented as a function of the magnitude of the rotation (vorticity) vectors and the three Cardan angles at three network points and is independent of the origin, orientatio� and scale of the reference systems. 2. THE BRUNS TRANSFORMATION

Operational geodesy uses observables as input data and coordinates of the position vector as output data. This input-output relation is called the Bruns transformation, originally presented in its linear form by Bruns (1878) when referring to the horizontal frame. The Bruns transformation classically yields the height anomaly from the disturbing potential divided by normal gravity. Thus it transforms the "observable" disturbing potential into the vertical coordinate called "height." Now, we will present a three-dimensional generalization of the Bruns transformation which can be used in both terrestrial and satellite geodesy.

The idea of the Bruns transformation is the following: Let a vector field V be observed, for instance,at a point P on the Earth's surface. Decompose the vector field into a normal part, whose structure is known ( which approximates the real vector field),and into a disturbing part:

2 (1)

The normal part vp at P can be linearized by a Taylor series with origin at a point p of the telluroid:

2 (2)

where 02 indicates second-and higher-order terms. If we know the approximate position vector 15, we can determine the "displacement vector" P-E from

v p 2 (3)

Figure 4 (p.7l) illustrates the vector field in geometric space. Let us call the two-point functions, Vp - vp = �v and P-E = 6x, anomalies of the vector field and the position vector, respectively, so that

2 (4)

14

We can choose �v

where

a which we refer to as the isoparametric mapping of oVp � ��

B

BOv, 0'1

-1 C

2 (5)

2 (6)

the Brun's matrix, which is the inverse of the gradient of the vector field at point p. Here we have assumed that grad vP is a regular matrix excluding rank deficiencies with respect to inj ectivity. In practice, singularities appear and have to be treated separately.

Thus far the Bruns transformation 2 (5) of vector field disturbances into position vector anomalies is coordinate-free. Its form with respect to geodetic reference frames is the following:

�Yi. " B"o'1 2 (7)

�"'6.* B*o'1 2 (8)

�£S.0 BOov 2 (9)

B* �(A,¢) B" 2 (10)

2 (11)

��" = [�x", �y", �z·]T displays the coordinates of the "displacement vector" in the Earth-fixed equatorial triad f·, ��* = [�x*, �y*, �z*]T, and the corresponding coordinates in the horizon triad e*, ��o = [�xO, �yO, �zo]T in the "fixed" or inertial (ecliptic) triad eO, where �, v are Eulerian angles,as shown in the following diagram:

o e

e'

e*

15

f'

f*

Instead we could represent the "displacement vector" in the "network" frames (written by capital letters). In the theory of mappings the two frames are called Eulerian and Lagrangian; thus we have here the Eulerian description. Before we show examples of the thus far abstract Bruns transformation, we first note another remarkable property. In many applications, the vector field of observables is "conservative;' i. e. ,

div OV 0, rot ov o 2(12)

at least insofar as we are outside of the masses. A consequence of the Bruns transformation 2(5) is then

div 6.� 0, rot 6.e 0; 2(13)

thus ov and 6.e can be expressed as the gradient of a scalar potential. If we introduce

16

ov grad ow, 2(14)

grad ox B grad ow 2(15)

follows. ox will be called the adjoint potential. might be surprising that the "displacement vector" vector is a "harmonic" function, div 1I2!; = div grad quite "natural. "

In the first instance, it which leads to the position ox = 0, but the result is

Another basic assumption we have made is that we know the approximate position vector p, but from where? If we have settled a convention about origin, orientation,-and scale of a geodetic network to, for example, the geocenter, directions to extragalactic objects and a unit length in geometry space, how can we find the a.pproximate position vector p in a geodetic reference system? The factor of uncertainty is introduced by the fact that nearly all geodetic observables depend on the gravity field whose coordinates in Hilbert space (e. g., coefficients in a spherical harmonic representation of the gravitational field) are unknown. Fortunately, there are geodetic observables that are gravity free, like positional angles and distance ratios. Only because of this can geodesy be made operational: coordinates in Euclidean and Hilbert spaces can be determined. This general statement will be verified in example 2.1.

EXAMPLE 2.1 (longitude, latitude, potential) Let us introduce the isoparametric mapping

w p

where A and � are astronomic longitude and latitude, W the scalar gravity potential, A and ¢ geodetic longitude and latitude, and w the scalar normal gravity potential (better known in geodesy by the letter U). Longitude and latitude are spherical coordinates in gravity space defined by

and

A arc tan r Ir y x

arc tan y /y y x

arc tan r I Ir 2 + r 2 z VJ X Y

2(16)

2 (17)

2 (18)

2(19)

17

2 (20)

where (rx' ry' rz)' (Yx' yy' yz) are Cartesian coordinates of the gravity vector r, y, respectively, in the "Eulerian" frame f·. With respect to a chosen reference system, (A, <P, W) are "observable," whe�eas (A, cp, w) are "computable," as in the representation of the potential given in the first section.

A zero-order approximation of the actual gravity potential is

w 2 (21)

where gm is the product of the gravitational constant and the mass of the model terrestrial body, II�II = " 2 2 2 ' the length of the vector x. Vx + Y + z

aw/ax

(and similarly for y and z).

A arc tan y/x

and W

are the corresponding zero-order mapping equations.

x = � cos A cos W 2(28)

(excluding, of course, ¢ = ± n/2). If we know (A, 1>, W) and (gm) from observed quantities, Cartesian coordinates of the approximate position vector can be computed. But how can we call the quantities (A, 1>, W, gm) observable?

Astronomical longitude and latitude are quantities derived from observations related to geodetic astronomy. Fundamental catalogs and a variety of reductions (precession, nutation, polar motion, aberration, parallax, etc.) are involved. The setup of an observational equation in geodetic astronomy is not routine and assumes, strictly speaking, approximate a priori information about the position of an observer in geometry space. From gravimetric leveling we obtain only potential differences. In order to be able to derive absolute potential, the reference system should contain sufficient information in its definition. In addition, if we introduce length observations and we extend the isoparametric mapping to the isometric case in geometry space by

2 (29)

for example, by imposing equal length of an observational line between two points on the Earth's surface and the corresponding points on the telluroid, we will be able to determine a value for gm. More details will be given in section 3.

The next step is a computation of the Bruns matrix.

y

C·

m· "

[aA/3X - 3¢/3x

3w/ax

3/../3y 3¢/3y 3w/3y

aA/3Z] 3¢/3z 3w/3z

2(30)

2(31)

- � 2 2 x +y xz

W sin It - gm cos ¢

W - -- cos It sin ¢ gm

W2 - --2 cos It cos ¢ (gm)

x +22 x +y yz

gmy ( 2 2+ 2)3/2 x +y z

+ � cos It gm cos ¢

W • • • '" - -- S1n it S1n '" gm w2

- --2 sin It cos cp (gm)

B"

A*

o

gmz ( 2 2 2)3/2 x +y +z

o

W -- cos ¢ gm w2 --2 sin ¢ (gm)

19

20

-1 B* C*

where the dot, asterisk, and circle denote equatorial, horizontal, and ecliptic coordinates, respectively.

A zero-Order approximation of the vector 6�* is

l:!.x* o 2(32)

l:!.y* o 2 (33)

2 (34)

which corresponds to the original Bruns formula, because within the zero-order approximation

y

') W'-gm

Finally , figure 5 (p . 71) illustrates the isoparametric mapping in the curvi-linear gravity and Cartesian geometry space.

EXAMPLE 2.2 (longitude, latitude, gravity)

Let us introduce the isoparametric mapping

A , P <l>p

2 (35)

2(36)

21

where A and � are astronomic longitude and latitude , r the length of the gravity vector, A and ¢ geodetic longitude and latitude , and y the length of the normal gravity vector. We refer also to (A, �, r) as the spherical coordinates cf the gravity vector. Alternatively , a Cartesian representation of 2(36) is the vector identity

or

r x

r y

r z

in an " Eulerian" frame f·. Because

r r cos A cos � x

r r sin A cos � Y

r r sin � z

the spherical and the Cartesian mappings are equivalent (excluding again ¢ = ± Tf/2).

Corresponding to the first example , astronomical longitude and latitude

2(37)

2 (38)

2 (39)

2(40)

2(41)

2 (42)

2(43)

are observed by astronomical instruments , and the length of the gravity vector by gravimeters. In addition , the scale of gravity space has to be included in the reference system.

22

A zero-order approximation of the actual gravity potential is

'OW/'ox y

and similarly for y and z.

A arc tan y/x

are the global mapping equations. They can be inverted into

x = 6 ffiffi Vr- cos A cos cjl

y � sin A cos cjl

z = vI!f- sin cjl

2(44)

2 (45)

2 (46)

2 (47)

2(48)

2 (49)

2 (50)

2(51)

(excluding ¢ = + TI/2). Compare 2(26) , 2(27) , 2(28) to 2(49) , 2(50) , 2(51) to see that only the "radial" component has changed from gm W-l to Igm r -1

A "Cartesian" proof of 2(49) , 2(50) , 2(51) follows. Starting with global mapping equations

r = Y = _ gm II� 11-3 x x x

r = Yy = _ gm II� 11-3 y y

and

r = Y _ gm II� 11-3 z . z z

We write these in the general form:

or

A = a

(x2+/+z2)3/2

B = a

(x2+/+z2) 3/2

Insert 2(57) into 2(55) and 2(56) to derive

C A=-x z

C B = - Y z

-1 x = AC z

-1 y = BC z

x

y

23

2(52)

2(53)

2(54)

2 (55)

2 (56)

2 (57)

2(58)

2 (59)

2(60)

2(61)

24

Writing 2(57) as

leads to

x =

Y =

z =

r r -l x z z

r y -1 r z z

-1 C az

(±) V;; r (r 2 + r 2 + r 2)-3/4 gm z x y z

-1 C az

-1 C a

t'7 cos It cos cp

= f T sin It cos cp

l7 sin CP

A first-order approximation of the actual gravity potential is

where w is the length of the rotation (vorticity ) vector.

dW/'dx = y x

2 (62)

2(63)

2(64)

2(65)

2(66)

2(67)

2(68)

2(69)

2(70)

Equations 2(70) , 2(71) , 2(72) can be written in the general form:

A = I- 2 2 a

2 3/2 + � L<x +y +z ) J x

B = r= 2 2 a

2 3/2 + J y L<x +y +z ) J

or

a C = z

(2 2 2

)3/2 x +y +z

Insert 2(75) into 2(73) and 2(74) to obtain

C A = (- + b ) x z

B (� + b) Y z

x = Az (B + bz) -l

y = Bz (C + bz) -l

which , together with 2(57), is written as

25

2(71)

2 (72)

2 (73)

2 (74)

2 (75)

2 (76)

2 (77)

2(78)

2 (79)

26

leads to

2 2 2 3 (x + Y + z ) -2 2 2 C a z

This is an equation of the tenth order, i.e., of the form

10 9 z + az + . . . + [3 o

2(80)

2(81)

2(82)

Equation 2(82) gives a set of solutions for z, then 2(78) for x, and 2(79) for y. Thus we have inverted 2(38), 2(39), 2(40) in gravity space into equations in geometry space. Of course, the inversion is not single-valued, but the solution space can be easily obtained.

The next setup will be a computation of the Bruns matrix.

C '

oy

x - tll2!;112

+ 2w2

xy

xy i - tll2!;112

+ 2w2

xz yz

to a first-order approximation.

2(83)

2(84)

xz

yz 2 (85)

27

Because the gravity vector field to the first order is conservative, div oy = 0, rot oy = 0 leads via the Bruns transformation to div 6� = 0, rot 6� = O. Thus if oy is taken from the space of spherical harmonics, so is 6�.

Figure 6 (p. 72) illustrates the isoparametric mapping in the Cartesian gravity and geometric space.

EXAMPLE 2.3 (longitude, latitude, gravity gradient)


P ¢p' W . . lJ w .. p lJ 2 (86)

where A and � are astronomic longitude and latitude, Wij second-order gradients of the actual gravity potential (i,j ranging over x,y,Z), A and ¢ geodetic longitude and latitude, and Wij second-order gradients of the normal gravity potential in the horizontal trlad. Specific gravity gradients Wxz' Wyz' Wx and W6 = Wyy - Wxx are assumed to be measured by a torsion balance, or any �ij by a Gradiometer.

The first problem is to find a representation of model gravity gradients in the equatorial triad. Because of the transformation f· � e* = RE (A,¢) f·, the first-order gradient tensor (grad y) can be determined by

(grad '£)* 2 (87)

For instance ,

wx* z* 3 gm II �11-5 2 (x - z2) cos A sin ¢ cos ¢ 2(88)

+ 3 gm II �11-5 xz cos A cos 2¢

+ 3 gm II �11-5 xy sin A cos ¢

+ 3 gm 11�11-5 yz sin A sin ¢

28

w y*z* = + 3 gm 1 1� 1 1-5

3 gm II � 1 1-5

-I- 3 gm 11�1 1-5

+ 3 gm 1I �1I-5

w x*y* = + 3 gm I I xll

-5

- 3 gm 1I� 11-5

+ 3 gm II � 1I-5

+ 3 gm I I �1 1-5

3 gm 1I� 1 1-5

- 3 gm I I �1I-5

2 2 (x - z )

xz sin cp

xy cos A-

yz c os A-

2 sin A-x

2 sin A-y

2 sin A-z

xy cos 2A-

xz sin 2A-

yz cos 2A-

sin A- sin cp cos cp

c os 2CP

cos cp

sin cp

cos A-• 2

cp Sl.n

cos A-

A- 2 cos cos cp

sin cp

sin cp cos cp

cos cp ,

If we choose the zero-order approximation of the normal gravity potential

w = + gm II �II -l

the gr�vity gradients are given by

Examples of 2(92) are

w • • x x

w • • x y

- 3 gm I I �11-5 xy

2(89)

2 (90)

2(91)

2 (92)

2(93)

2(94)

If we are interested in the zero-order approximation 2(91), the isoparametric mapping equations 2(86) can be summed up to be

A = arc tan y/x

w . . = w.. (x, y, z) 1.J 1.J

The general solution can be represented by

Scale is taken from

II�II

x = II� II cos A cos 4>

y

�V • • 1.J

II � II sin A cos 4>

z = I I � II sin 4> •

gm 11�11-3

f .. ( A,4» 1.J

VJ�· 1.J

29

2 (95)

2 (96)

2 (97)

2 (98)

2 (99)

2 (100)

2 (101)

(no summation) 2 (102)

where fij ( A,�) is a specific expression, an example of which, for i=x and j=z, is shown below. Substituting equations 2 (98) to 2 (100) into 2 (88) yields

2 (103)

2 . 2 3 2 . 2 } + cos A sin 4> cos 4> cos 24> + S1.n A cos A cos 4> + sin A S1.n 4> cos 4>

= gm II x 11-3

f (A, 4» • - xz

30

Compare 2(26) - 2(28) , 2(49) - 2(51) and 2(98) - 2(103) in order to see that only the "radial" component has changed in the following way:

3

� l' Wij

f . . (A, <jl) 1J

2 (104)

2 (105)

(no summation ) . 2 (106)

Thus any coordinate of the Cartesian tensor of gravity gradients is as easily chosen as another.

In addition to the isoparametric mapping of 2(86 ) , I tried one with only the gravity gradients mapped isoparametrically, but the mapping equations turned out extremely nonlinear and I have been unable to invert them .

The next step is the computation of the Bruns matrix.

v

[aA. lax a<p/ax aw . .lax 1J

Ov

aA. lay a<p/ay aw . . lay 1J

a"Aldz ] a¢ laz aw .. /az • 1J

2 (107)

2(108)

The first two rows of the matrix C' were computed within 2(31) ; in addition

aw . . /ax, aw . . /ay, aw . . /az 1J 1J 1J

have to be computed. Using 2(92) we arrive at

2 (109)

31

aw . . /ax 2 (110) 1J -

Finally, figure 7 (p . 72 ) illustrates the isoparametric mapping in the generalized gravity and geometric space.

Let us summarize what the examples tell us. If we refer to the isotropic (zero -order) approximation of the normal gravity field

w

we can represent zero-order coordinates of telluroid points by

x = I I�I I cos A cos <ll

y I I � I I sin A cos <ll

z = II�II sin <ll

where scale is taken from a quantity referring to the gravity field, like I I �I I, as given by 2(104) , 2(105) , 2(10 6 ) . For a higher order normal gravity field the (x, y, z) representation is more complicated as can be inferred from 2(78) , 2(79) and 2(82) . In addition, the examples emphasize the nearly arbitrary choice of the isoparametric mapping p � P. As we will see in the next chapter, the isoparametric mapping

w p

leads to the Stokes approximation of the geodetic boundary value problem and its finite element form.

32

3. THE DUAL SETUP OF GEODETIC OBSERVATIONAL EQUATIONS

Once we have decided upon the reference system in either the geometry , gravity , or vorticity space , we are able to set up geodetic observational equations . In general , these depend on coordinates in these spaces . Let us assume for a moment we know the approximate coordinates such that we can linearize observational equations . The quantities "observed minus computed" �P - yp can be represented by the gradients with respect to these coordinates , such as

y - y -P -p (grad yp) 6� + oy x - p -p 3(1)

where oy is the disturbance vector . There are geodetic observational equations which depend on coordinates only o f the geometry space , but , in general , they are a f unction of gravity space coordinates of Hilbert type , e . g . , spherical harmonic coef ficients .

To present the idea of dual setup of geodetic observational equations in a simple way , we will start with a priori parameters which describe the normal gravity field, e . g . , (gm) .

If we use the dual Bruns transformation

6x B oy

ov C 6x

3 (2)

3 (3)

which expresses coordinate corrections in the geometry space in terms of disturbances in the gravity space (with BC = I, det B # 0, det C # 0) to replace geometric coordinate corrections by gravimetric coordinate disturbances and vice versa , we arrive at the observational equations

y - y -p -p

y - y -P -p

3(4)

3 (5)

These depend either on geometric or on gravimetric unknowns . Thus we have two alternatives in adjusting a geodetic network , a geometric mode or a gravimetric mode , as shown by the following examples .

EXA}�LE 3.1 (geometry space)

As shown in example 1.1, geometric positional angles or length ratios are independent of a reference system with degrees of freedom for translation, rotation, and scale. The linearized observational equations read

I'lxl I'lYl I'lzl

I'lx2 I'lY2 I'lz2

I'lx3 I'lY3 I'lz3

33

3 (6)

where the matrices Ali are functions of the coordinates (xl' Yl' zl' x2' Y2'

z2, x3' Y3' z3) at the points (Pl' P2' P3), and X�12 represents either positional angles or length ratios. If horizontal directions and horizon distances have been observed, they are correlated observations, in general. ( See formula 1 (16).)

The vector I'l� of geometric coordinate corrections can now be transformed into gravimetric coordinate disturbances by the three-dimensional Bruns formula , e.g., 2 (31), 2 (34) or 2 (108).

(a) Isoparametric mapping of type longitude, latitude, geopotential:

3 (7 )

OA 3 (8)

34

(Grafarend (1978a) : formulas 1(21), 1(22) ).

x x X312 - l:':312

x "'l:':312

OAl o� 1 O\Jl oA2 0¢2 oW2 oA3 0<1> 3 oW3

3 (10)

Thus, we have found that the observed geometric positional angles or distance ratios depend now only on the disturbing potential ow and the coordinates dioW of its gradients. Equations 3(6) and 3(10) are dual.

If we know scale, distance observations can be approached in the same way .

(b) Isoparametric mapping of type longitude, latitude, gravity:

oy

OA 3(8) o¢ 3(9)

(Grafarend (1978b ): formulas (1. 38) ),

3 (ll)

3 (12)

or

(B' is, of course, a general notation, and not equal to B' in 3(11). )

oy . 1 d. OW 1

35

3 (13)

3(14)

3(15)

Thus, we have found that the observed geometric positional angles or distance ratios depend now only on the coordinates dioW of the gradient of the disturbing potential. If we know scale, distance observations can be approached in the same way .

36

(c) Isoparametric mapping of type longitude, latitude, gravity gradient:

0>.. 3(8) ocjJ 3(9)

ow, , 1.J

d ,d, ow 1. J

' y x = [A B I A12 B2 I A13 B3J= °-312 11 1

0>.. 1 oh ow� , 1.J 0>...., L..

3 (16)

3 (17)

3 (18)

Thus, we have found that the observed geometric positional angles or distance ratios depend now only on the coordinates dioW and didjOW of the first- and second-order gradients of the disturbing potential.

If we know scale, distance observations can be approached in the same way.

EXAMPLE 3.2 (gravity space) :

As we have seen in example 1.2 gravimetric positional angles or length ratios are independent of a reference system with degrees of freedom for translation, rotation, and scale. The linearized observational equations read

37

3 (19)

where the matrices A2i are functions of the coordinates (Yx1' Yy1' Yz1, Yx2'

Yy2' Yz2' Yx ' Yy , Yz ) at the points (Y1' Y2' Y3) and yr represents 3 3 3 312

either positional angles or length ratios in gravity space. If astronomical

longitude, latitude, and gravity have been observed, they are correlated

observations, in general. (See formula 1(24).)

The vector oy of gravimetric coordinate corrections can now be transformed

into geometric coordinate corrections by the inverse of the three-dimensional

Bruns formula, e.g., 2(84).

I1x 1 I1Y1 I1z 1 I1x2 I1Y2 I1z2 I1x3 I1Y3 I1z3

3(20)

3 (21)

38

Thus , we have found that the observed gravimetric positional angles or length ratios depend now only on geometric coordinate corrections. Equations 3(15) and 3(Zl) are dual .

The next step is to assume that we know orientation and scale in gravity space. The linear observational equations for astronomical longitude and latitude, the length of the gravity vector, and potential differences read

fl - >- (a>-/ax) f1x + (a>-/ay) f1y + (a>-/az) p P P P

 - ¢ P p

r - y P P

(a¢/ax)

(ay /az )

Wz - WI - (wZ

- wI)

+

+

P

f1x + (a¢/ay) f1y + (a¢/az) P P P

p f1x + (ay/ay) f1y + P

(ay /az) P

(aw/ax) f1xl (aw/ay) f1l Pl PI

(aw/a z) f1xZ + (avl/ay) f1/ Pz PZ

OWZ - oW

l'

f1z + 0>-

f1z + o¢

f1z + oy

(aw/az) PI

+ (aw/az) Pz

(a) Isoparametric mapping of type longitude, latitude, geopotential:

B" [EJ 0>- 3(8) o¢ 3(9 ) oy 3 (lZ )

f1z1

f1zZ

3 (ZZ)

3 (Z3)

3 (Z4)

3 (Z5)

3(Z6)

6w

2 ow + P Cl ow --= Clp

where

p Y -1/2

(Grafarend (1978a ) : formula 1(23)).

o o o

6y -p Yp

39

Clw 3 (27) Clp

The observational equation for the length of the gravity vector is weI] known; it is the boundary condition for the "harmonic" potential ow. Even better known is its zero-order approximation based on 2(21)

2 Cl - ow + - (o w) r Clr

(Grafarend (1978a) : formula 1(26)).

- 6y 3(28)

Thus, we have found that the observed length of the gravity vector depends only on the disturbing potential ow and the coordinate Clow/Clp or Clow/Clr of its gradient vector , where r = I I�I I .

The dual formulation is obtained if we make use of the inverse three-dimensional Bruns transformation

ow 3(29)

Employing the summation convention over i x , y , z ,

40

f - y -p -p

3 (30)

3 (31)

where the vector of geometric coordinate corrections ll� is a "harmonic" function as long as we measure in empty space. Again we have found that the observed length of the gravity vector depends only on the geometry vector ll� and its gradient. Equations 3(27) (or 3(28)) and 3(31) are dual.

(b) Isoparametric mapping of type longitude , latitude , gravity

where

A p

1>u � fp

Wp - w p

0), 3 (8)

o<p 3(9)

oy 3(12)

- ), ll)' P

- <p P

ll<P

- y p lly

0

0

0

llw = ow + £. Cl ow 2 Clp

3(32)

3 (33)

p

(Grafarend (1978b): formula (1.39)).

-1/2 Y

4 1

The observational equation for the potential is well known; it serves alternatively as the boundary condition for the "harmonic" potential ow. For the zero-order approximation 2(21) we find

r a ow + - - (ow) 2 ar 3(34)

Thus, we have found that the absolute potential depends only on the disturbing potential ow and the coordinate aow/ap or aow/<lr of its gradient vector where

r = I I � II

For the dual formulation we mention only that we have to integrate oy in order to arrive at ow.

(c) Isoparametric mapping of type longitude, latitude, and gravity gradient

0\ 3 (8)

o¢ 3(9)

ow .. 3(17) 1J

3(35)

42

w -p

r -p

1I. p

<l> p w . . p

1J

w P

Y p

>.. b.>" 0 P

<P p M 0

w . . P b.w . . 0 1J 1J

(grad w) B ' [EJ + ow p 3 (36)

(grad Y )p

B ' [EJ + oy 3 (37)

Thus, we have found that the observed potential and length of the gravity vector depend only on the disturbing potential and coordinates of its firs� and second-order gradient.

Another set of geodetic observations depends on the gravity field , e . g . , astronomical azimuth A and horizon distance B . Their observational equations are structured according to

A12 - a12

+

B12 - S 12

+

1 1 (aa12/d x )b.x +

2 2 (aa 12/ax )b.x

1 1 (as l/ax )b.x

2 2 (a s u/ax ) b.x

+

+

+

1 1 (aa 12 lay )b.y

2 2 (aa 12/a y )b. y

1 1 (a S 12 lay )b.y

2 2 (a s 12/ay )b. y

+

+

+

+

1 1 (aa 12/az )b. z

2 2 (aa l/a z )b. z

1 1 (a S 12/a z )b. z

2 2 (as l/a z )b.z

3(38) + oa12

3 (39 )

+' o S 12

where a , S are zero-order approximations of A , B and oa . o S are subj ect to the Laplace condition.

L [��J

Here, we assumed that the base vectors F ' and f' are parallel in the Euclidean sense. Hence, the observational equations will read

3 (40)

43

ll Xl llYl [" lJ X12 - Y12 ll Y12 A ll zl + L 3 (41 ) Y ll x2

o¢l llY2 ll z2

The equations can be formulated in the geometric mode if we transform o A , o¢ according to the inverse three-dimensional Bruns formula of types (a) , (b) or (c) . In all cases we will obtain

3 (42 )

The dual representation is found to be

3 ( 4 3 )

where

O A I O A I O A I o¢l 09 1 c¢l 1 oWl oY l oW o 0 3 (44)

lJ (i) o y o A2 (ii) O y o A2 (iii) o y 0 A2

0¢2 0¢2 0¢2 oW2

2 oY2 oW o 0 lJ

if we choose different isoparametric mappings 0 At this point we introduce the unknowns describing the gravity field of the

Earth, for instance , the mass density virials

I , 1 . , 1 . 0 ' Ilo Jo k , o l lJ

4 4

given in the appendix. They appear in the form

11• Xl. I I x l 1 -3, I x x ij i j 3(45)

and must fulfill the Laplace differential equation. Because of this condition , the degree of freedom is 2n+1, where n is the order of approximation in a series for the gravitational potential, e.g. , for n = 2 only 5 coefficients of the six

are independent as a result of tr Iij = O .

The quantities "observed minus computed" YP - lp are represented by

Ip - lp = (dl/ dX)p�X + (d l/d Y)p�y + (dl/d Z)p�Z

+ (dy/d I ) 0 1 - 0 P 0 + (dl/d l1)p 0 11 + (dl/d I2)p 0 12 + (dl/d I3)p 0 13 + (dl/d I11)p 0 111 + (dl/d I22)p 0 122 + (dl/d I12)p 0 112 + (dl/d I13)p 0 113 + (dl/d I23)p 0 123 + • • • + ol

3 (46)

3 (47)

By the symbol "p" we understand the or�g�n of the Taylor series , a set of approximate coordinates of points in the geometry space and of approximate gravity field parameters. How can these be determined?

To answer this question we present an example.


[6.>" M 6.w 6.y �]

applied to a zero-order approximation

of the gravity field.

w

II arc tan Y x

arc tan i;;:;;;;z==;;.

fx2

+ i

w gm

45

3(48)

46

are the global mapping equations . The general solution can be represented by

x =

y

z =

gm

I I � I I cos A cos 

I I � I I sin A cos 

I I �I I s in 

�

-... fiFr

m -W - l �

Thus we have derived approximate values for (x , y , z ) and gm :

x =

y

z =

gm

W cos A cos r

W sin A cos r

W s in r

ACKNOWLEDGMENT

3 (L.9)

3(50)

3(51)

3(52)

3 (53)

3 (54)

I wish to express my apprec iation. to the Nat ional Geodetic Survey (NGS) staff for their warm hospitality . Inspiring discussions were held with John D . Bossler , Bernard Chovitz , and John G . Gergen of NGS , Hyman O . Orlin of the Nat ional Academy of Sciences , and last , but not least , Petr Van{tek , Vis iting Senior Scientist in Geodesy of the Nat ional Research Counc il . Their cooperation is gratefully acknowledged .

APPENDIX A.--INVARIANT REPRESENTATION OF I I x - x ' I I -m

In terms of Hilbert invariants

i = I I � I I

i ' I I � ' I I

i" (x, X' )

we will set up series for

Set

I I � _ � ' I I -m = . -m 1 [ . , 2 . , J 1 + -1- - 2 � i" E1. . 2 1 2 1

m

i , 2 . , E: = - - 2 � i" . 2 1 1

( HE:) 2 = 1 - l :m

2 E: + m�rZ) E:2

m (m+2 ) (m+4) 3 + m (m+2) (m+4) (m+6 ) 4 3 : 8 E: 4 : 16 E:

5 - 0 (E: )

= L; n=o

(_l )n m (m+2 ) (m+4) • • • (m+2n-2) E:n n : 2n

4 7

(Al )

(A2 )

(A3)

(A4)

(AS)

(A6)

4 8

i" < 1 ; thus the series are convergent within the sphere determined by Und�r this assumption we can represent (A6) by

00 I I �-� ' I I -m = L

n=o

I I �I I < 1 I I � I I .

(A7)

where P� (i") are Gegenbauer (ultraspherical) polynomials (Abramowitz and Stegun, 1964 : p. 7 74 )

1

m PI (x) = mx

m m ? Pz (x) = 2 [ (m+Z)x� - 1]

pm ( ) _ m (m+Z) x [ (m+4)xZ - 3] 3 x - 6

P:(x) = n\ {m (m+Z) ... (m+Zn-Z)xn

n (n-l) m (m+Z) ... (m+Zn-4)xn-Z Z

(A8)

(A9)

(AlO)

(All)

(AlZ)

4 9

(Al3)

+ n (n-1) (n-2) (n-3) m(rn+2) • • • (rn+2n-6)xn-4

2 · 4

+ (_1)£ n (n-1) · · · (n-2£+1) m (rn+2) • • • (rn+2 (n-£ -1)] xn-2£ } ,

2 · 4 · · · 2£

o 1 . . . < .!!. " - 2

Despite the identical notation, these should not be confused with associated Legendre polynomials .

5 0

APPENDIX B . --CARTESIAN REPRESENTATION OF THE GRAVITATIONAL POTENTIAL

In the appendices B-E , we char.ge to index notation (xl , x2 ' x3 ) in place of (x , y , z) and utilize the summat ion convention .

Let

(Bl)

be the Newtonian representation of the gravitational potential . In terms of Hilbert invariants

holds if I I � ' I I I I � I I

represented by

i I I ?El l

i ' I I � ' I I

i" (x'A' )

I I � I I I I � ' I I

i , n i-n-l P (i") n

(B2)

(B3)

( B4)

(B5)

< 1. Under this assumption the gravitational potential can be

-f g i-n-l fdx' i , n Pn ( ill ) p(?E' ) (B6) n=o

where Pn are the standard Legendre polynomials .

In cart es ian coord inates the Hilbert invariants read

i I I � I I � 1 1

rX�X� i ' = I I � ' I I = 1 1

x . x� i" 1 1 =

f x . x . fx� X� 1 1 1 1

and the related Legendre polynomials pn ( il l )

etc .

P ( i" ) = I o

, , I , , 3 x . x . x . x . - - x x x x P ( i" )

1 1 J J 3 p p q q 2 = "2 -----'� ........ �'---x-:-�-x-:-� '--'''--........... .-..

(x . x ! ) (x x ) (x' x' ) 1 1 P P q q

(B6) can be wr it ten

3 2 2 + 35 (x x ) (x' x' ) r r s s

5 1

(B7 )

(B8)

(B9)

(BID)

(Bll)

(BI2 )

(B13)

(BI4)

In cartes ian coord inates the Hilbert invariants read

i = I I � I I = � 1. 1.

i ' = I I � ' I I = rx � x � 1. 1.

x . x � i" 1. 1.

t' x . x . fx � X � 1. 1. 1. 1.

and the related Legendre polynomials P� ( i" )

etc .

P ( i" ) = 1 o

P ( i" ) = x . x . ' .- (� �) 1 1. 1.

, x . x � 1 x ' x ' 3 x . x . - - x x P ( i" )

1. 1. J J 3 p p q q = -2 2 �� x' x ' Q, Q,

x . x � P ( i" )

5 1. 1. 3

= "2 .r::-::-, Xj Xj t' xkxk

(x . x ! ) (x x ) (x ' x ' ) 1. 1. p P q q

(B6) can be writ ten

3 2 2 + --35 (x x ) (x ' x' ) r r s s

5 1

(B7 )

(B8 )

(B9)

(BIO)

(Bll)

(B12)

(B13)

(B14)

5 2

where

00 U (x) = L un

n=o

U _ _ --lg=ffi::...-. = _ _----<g:2.--_ o 1/2 10 .rx:x-: (x . x . ) , --i --i � �

gx . ul = - --��-..,./- f dffi' X � :;

3 2 � (x . x . )

U3 - -

+

] ]

5g

2 (xQ, xQ, )

3g 2 (xQ, xQ, )

7 /2

3 /2

x . x . � fdm' x � x � xk � ] � ]

x . cS · k � ]

dm' x � x � � �

(BI5)

(BI6)

(BI 7 )

(BIB)

(BI9)

and

etc .

I o f dm m

I . f dm X . � �

I . . �J

I . . 1. 0 �.1 l\.N

5 3

(B20)

(B2l)

(B22 )

(B23)

(B24 )

(B25)

In summar iz ing , we can represent the gravitat ional potential in Cartes ian coordinates by

u (X) = - gm _ � __ 1_' 3_._.-.:.. • ....o.(...,.2n_-_l....:.)..,......,_

�� , (X . X . ) ( 2n+l) / 2 gXil

,Xj Xj n=l n . J J

X . • • • x . �2 �n

(B26)

5 4

where

+ n (n-1) (n-2 ) (n-3 ) ( ) 2 2 . 4 . ( 2n-1 ) ( 2n-3)

X . X . • • • X . 0 i i 0 i i � Xk 11 12 1n_4 n-3 n-2 n-1 n K

£ n (n-1 ) • • • (n-2£+1) - • • • + (-1 ) -2-. 4-:-.-.-.�2:':"£ ..;.:(2::.....n..;::::.-..!:-1�) 7'(2:-:n:":':_'-::-3�) -=--• • ";::::'.�(�2n---2-£-+--'--1 )

£ = O , 1 , • • • ,� � .

(B27 )

5 5

APPENDIX C . --CARTESIAN REPRESENTATION OF FIRST- ORDER GRADIENTS OF THE GRAVITATIONAL POTENTIAL

Let

( Cl )

b e the Newtonian represent ation of the f ir st_order gradients o f the grav itat ional potent ial . In terms of Hilber t invar iants

I I x ' I I hold s if ---

I I xi i

i I I � I I

i '

i" = _--=(x=.,..'-"x ..... '-<.) __ I I � I I I I � ' I I

00 I I x - � ' 1 1 -3 = L

n=o

< 1 . Under this assump tion the f ir st-order gradients of the

gravitat ional po tent ial can be represented by

00 d U/ d xi = + L

n=o i-n-3 f dx ' i , n p3 ( i" ) (x . -x . ' ) p (x ' ) g n 1 l

where P� are Gegenbauer polynomials ( see (A8 ) - (AI 3 ) )

(C2 )

(C3 )

(C4)

( CS)

(C6)

5 6

p3 0 (x)

p3 1

(x)

p3 2 (x)

p3 3

(x)

p3 4

(x)

1 = 3x

3 2 = -( Sx -1) 2

S 2 = -:::x ( 7x - 3 ) 2

IS 4 2 = --(2lx -14x +1) 8

P� (X) , :: � . 5 . . . ( 3+2n-2 ) Xn

n (n-l ) 3 . S • • • ( 3+2n-4 ) xn-2

2

+ n (n-l ) (n-2 ) (n- 3 ) 3 . S • • . ( 3+2n-6 ) xn-4 2 · 4

+ (-1 ) £ n (n-l ) • . • (n-2£+1 ) 3 . s . • • ( 2n-2£+1 ) X

n-2� 2 · 4 • • • 2£ J

n 0 , 1 , • . . < 2" •

(cl)

( C8 )

I f we refer to Equations (BlS ) to (B27 ) as the Cartesian representat ion o f the gravitational potential , w e will arrive at

or , in general ,

00

-� n=1

00

where

1 · 3 • • • ( 2n-l )n n ! g I I x l l - ( 2n+l ) I . . .

111 . . . In_1

5 7

(C9)

(CI0)

(Cll)

(CI2)

(Cl4)

5 8

APPENDIX D . --CARTESIAN REPRESENTATION OF SECOND-ORDER GRADIENTS OF THE GRAVITATIONAL POTENTIAL

We ref er to equat ions (C9 ) to (C14 ) as the Car tesian representat ion of the f irst-order gradients of the gravitational potent ial . Here we will c ompute second-order gradient s .

2 a u / a x . a x . 0 1 J

2 a ul /a x . a x . 1 J

2 a u,) /a x . a x . � 1 J

2 a u/ a x . a x . 1 J

= - 3g 1 1 � 1 1 -5 I 0 1 I I � 1 1 2 0

ij] [x . x . 3 1 J

= + 3g I I � 1 1 -5 (xi I . +

x . 1 . )

=

=

J J 1

_ 15g I I � 1 1 -7 � Ik [xiXj 1 I I � 1 1 2 0

ij] 5

3g 1 1 � 1 1 -5 1 . . 1J

- l5g I I � 1 1 -7 xk (x . J

+ 105 -2- g 1 1 � 1 1 -9 xkxQ,

_ 15 g I I � 1 1 -7 � Iikj

+ 105 -2- g 1 1 � 1 1 -9 �xQ,

1 . k + x . 1 . k) 1 1 J

IkQ, 1 [x . x . - 7 1 J

(x . Ij kQ, + x . 1 J

I I � 1 1 2 0 ij]

IikQ, )

(Dl)

(D2 )

(D3 )

(D4 )

or, in general,

2 a u/a x . a x . 1 J

(J() - L n=l

1 · 3 . . • (2n-1 ) (n-1 ) n n ! g

+! 1 · 3 • . • (��-l) (2n+1) n g I I � I I - (2n+3) n=l

(J() +L 1 . 3 • • . (2n-1;�2n+1) (2n+3) g 1 1 � 1 1 - (2n+S )

n=l

. (x . . . . x . I . . ) (x . x . - 2 1+3 I I � 1 1

2 0 . . ) 11 1n 11 · · · 1n_1 1 J n 1J

5 9

(DS)

(D6)

6 0

where I I � I I = �xk � . It is a "ni ce" exerc ise to prove

tr (a 2u l a x . a x . ) o 1 J

o .

APPENDIX E . --CARTESIAN REPRESENTATION OF THE EUCLIDEAN NORM OF FIRST-ORDER GRADIENTS OF THE GRAVITY POTENTIAL

6 1

Let w = u - �2 (x12 + x 2 ) be the scalar part o f the gravity potential such

that y2 = ( dW/ dxi) (dW/ dxiJ is the square of the Euclidean norm of its f irst-order gradients . We ref er to (C14 ) as the Cartesian representation of the gravitational potential . The formula

dW -- = dX . 1 + g I I � 1 1 -3 x . I 1 0

00 _ � 1 · 3 . . ��2n-l )n g I I � I I - (2n+l) n=l

00 +� 1 . 3 . • . (2:"!1 ) (2n+l) g 1 1 � 1 1 - ( 2n+3) n=l

2 - w x 0 • a a l

(where Greek indices range over 1 , 2 only)

contains four terms and we designate it as a + b + c + d . The four-term scheme leads to

+ 2ab + 2ac + 2ad + 2bc + 2bd + 2cd .

Explicitly , it has the form

(El )

6 2

00 + � n=l

00 � 1 · 3 . . . (2n-l )n 1 · 3 • . • ( 2m-l)m g2 � n ! m! m=1

13

, . ) J 1 · . . Jm-1

x . . . . x . x . . . . x . 11 In_l J 1 Jm-1

+ g2 � � 1 · 3 . . . (2n-l ) (2n+l) 1 · 3 • • . (2m-l) (2m+l) � � n ! m! n=1 m=1

+ w4 ( x x ) a a

2 00 - 2g � 1 . 3 • • ��2n-l )n 1 1 � 1 1 - ( 2n+4 )

x . . • • x . I n=1 11 In_l 0

00

(E3 )

I I . . + 2g2 � 1 . 3 • • • (2:"!1 ) (2n+l) 1 1 � 1 1 - (2n+4 ) n=1

x . . . . x . 0 11 . • • 1n 11 In

or

- 2g w2 I I � I 1 -3 (x x ) I a a 0

()() ()() 2 - 2g � � 1 . 3 • . • (2n-l ) n 1 · 3 • • . (2m-I) (2m+l)

n=l m=l n !

. I I � 1 1 - (2n+l) I I � 1 1 - (2m+3)

+ 2w2g ± 1 . 3 • • ��2n-l )n 1 1 � 1 1 - ( 2n+1 ) n= l •

m!

6 3

_ 2w2g � 1 . 3 • • • (2n�1 ) (2n+1) 1 I � 1 I - (2n+3) (xa xa) x . . . . x . I . . �

1 n . 11 1n 11 · · · 1n

+ g2 I I x l 1 -6 I . I . - 1 1

+ 3g2 I I x l 1 -8 x . x . I . I . - 1 J 1 J

x x i j

I ki

I kj

(1::4 )

6 4

+ IS 1 1 !S 1 1 -1O

+ 22S I l x l l -12

4 -

? I I !S 1 1 -

6 + 4g�

- 6g 2 I I !S I I-8

+ 9i I I !S 1 1 -8

+ w 4 (x x ) a ().

- 2gw 2 I I �" -3

x . x . xk Ii Ij k 1 J

x . x . Xk XQ, I . . 1 J 1J

x . I I . 1 0 1

x . x . I . I . 1 J 1 J

x . x . I I . . 1 J 0 1J

(x x ) I a a 0

+ 2gw 2 1 1 !S 1 1 -3 Xi I . 1

+ 6gw 2 " � , , -S I . x . X 1 a 1a

- 6gw 2 I I !S I I -S (x x ) I . x . 1 a a 1

Ik£

- ISgw 2 I I !S 1 1 -7 (x x ) I . . x . X .

+ 0 ( 1 . ' k) 1J

up to third-order terms .

1 J a a 1J

BIBLIOGRAPHY

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6 5

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6 6

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6 7

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6 8

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7 0

Figure 3 . --Tr iangular network in a vector space .

w

(1\. , <P , W)

I I I I

�---- I � <p I

(A , p , w)

v v

Figure 4 . --The d isplacement vector .

z

p

I'------� y

x

Figure 5 . --I soparamet ric mapp ing in the geometr ic and the grav ity space for long itud e , lat itude , and potent ial .

r y

I I

�---- I � r '" I Y "' ...J

z

�----� y

Figure 6 . --Isop arametric mapp ing in t he geometric and the gravity space for long itude , lat itude , and gravity .

w xz

(fI. , ¢ , W ) xz

I I �--- I � ¢

, I '�

(A , p , w ) xz

x

z

F igure 7 . --I soparametric mapp ing in the geometric and the gravity space f o r long itude , lat itude , and gravity gradient .

7 1

r y

I I

�---- I � r '" I Y "' ...J

z

�----. y

Figure 6 . --Isop arametric mapp ing in t he geometric and the gravity space for long itude , lat itude , and gravity .

11.

w xz

(11. , ¢ , VI ) xz

I I �--- I � ¢

.......... I .......... �

(A , p , w ) xz

z

F igure 7 . --I soparametric mapp ing in the geometric and the gravity space f o r long itude , lat itude , and gravity gradient .

7 1

(Cont inued f rom ins i de front cover)

NOS NGS-3 Adjus tment of geodetic field data using a s equential method. Marvin C. Whi t i ng and Allen J . Pope, Ma rch 1 9 7 6 , 1 1 p p ( PB2 539 6 7 ) . A sequential adjus tment is adopted for use by NGS field

pa r t i e s . N O S NGS-4 Reducing the profile of sparse symme t ri c ma t rices . Richard A. Snay, June 197 6 , 2 4 pp ( PB-

2584 7 6 ) . An algori t hm for improving the profile of a sparse symmet ric mat rix is introduced and tes ted against the widely used reverse Cuthi ll-McKee algorithm.

NOS NGS-5 National Geode tic Survey dat a : availabi l i t y , explanation, and application. Joseph F . Dracu p , Revi sed January 1 9 7 9 , 4 5 p p ( PB80 1 1 8 6 1 5 ) . The summa ry gives data and services available f rom NGS , accuracy of survey s , and uses of specific dat a .

N O S NGS-6 Determination of North Ame rican Datum 1983 coordinates of map corners. T. Vincent y , October 1 9 7 6 , 8 p p ( PB262 442 ) . Predicti ons of changes i n coordinates of map corners are detailed .

NOS NGS-7 Recent eleva t ion change in Southern Calif ornia. S . R . Holdah l , February 1 9 7 7 , 19 pp (PB265-94 0 ) . Veloci ties of elevation change were det e rmined from Southern Ca l i f . leveling data for 1 9 06-62 and 1 9 5 9- 7 6 epochs .

NOS NGS-8 Establi s hment of ca libration base lines . J oseph F. Dracup , Cha rles J. Froncz ek , and Raymond W. Tomlinson, Augu s t 1 9 7 7 , 22 pp ( PB2 7 7 1 3 0 ) . Specificati ons are given for establ ishing calibra t i on base line s .

N O S NGS-9 Na t i onal Geodetic Su rvey publications on surveying and geodesy 1 9 7 6 . September 1 9 7 7 , 17 pp ( P B2 7 5 1 8 1 ) . Compi lat i on l i s t s publications authored by NGS s taff i n 1 9 7 6 , source availab i l i t y f or out-of-print Coas t and Geodetic S u rvey publications , and subscription information on the Geodetic Cont rol Data Au tomatic Mailing L is t .

NOS NGS-IO Use o f ca l i bra t i on base lines . Charles J . Fronc z ek , December 1 9 7 7 , 3 8 p p (PB2 7 9574) . Detai led explanation a l l ows the user to evaluate elect romagnetic distance measuring inst rument s .

NOS NGS- l l Applicability o f array algebra. Richard A . Snay , February 1 9 78 , 2 2 p p (PB2 8 1 1 96 ) . Conditi ons required for the t ransforma t i on f rom mat rix equations into computationally more efficient array equations are considered.

NOS NGS- 1 2 The TRAV- I 0 horizontal network adjus tment program. Charles R. Schwarz , April 1 978 , 52 pp ( PB28308 7 ) . The design, objective s , and specificati ons of the horizontal control adjus tment program are present e d .

N O S NGS- 1 3 Ap plica t ion of three-dimens ional geodesy t o ad jus tments o f horizontal network s . T . Vincenty and B . R. Bowring , June 1 9 78 , 7 p p ( PB28667 2 ) . A method is given f or adjusting measurements in three-dimensional space without reducing them t o any computational surface.

NOS NGS- 1 4 Solvabi l i ty analysis of geodetic netwo rks using logical geome t ry . Richard A. Snay , Octobe r 1 9 7 8 , 29 p p (PB2 9 1 2 8 6 ) . N o algori thm based solely on logical geome t ry has been f ound that can unerringly dis t inguish between solvable and unsolvable hori zontal network s . For leveling networks such an algori thm i s well known.

NOS NGS- 1 5 Goldst one validation survey - phase 1 . Wi lliam E. Ca rter and James E . Pettey , November 1 9 7 8 , 44 pp (PB2 9 2 3 1 0 ) . Result s are given for a space system validation study conducted at the Goldstone, Calif . , Deep Space Communi cati on Complex.

NOS N�S- 1 6 Determination of North American D atum 1 98 3 coordinates of map corners ( s econd predi ction) . T. Vincent y , April 1 9 7 9 , 6 pp ( PB2972 4 5 ) . New predicti ons of changes in coordinates of of map corners are given.

NOS NGS- 1 7 The HAVAGO three-dimens i onal adju s tment program. T. Vincenty , May 1 9 7 9 , 18 pp (PB297069 ) . Th e HAVAG O computer prog ram a d justs nume rous kinds of geodetic observati ons for high precis i on special surveys and ordinary survey s .

NOS NGS- 1 8 Det ermination o f ast ronomic p os i ti ons for Ca l ifornia-Nevada boundary monuments near Lake Tahoe . James E . Pettey , Ma rch , 1 9 7 9 , 22 pp (PB3 0 1 2 64 ) . As tronomic observations of the 1 2 0th meridian were made at the request of the Cal if . S t a t e Lands Commission.

NOS NGS- 1 9 HOACOS : A program for adju s t ing horiz ontal networks in three dimens ions . T. Vincent y , July 1 9 7 9 , 18 pp ( PB301 35 1 ) . Horizontal networks are ad justed s imply and effici ently in the height-cont rolled spatial system without reducing observati ons t o the ellipsoi d .

NOS NGS-20 Geodetic leve l i ng and t h e sea level s lope along t h e Ca lifornia coa s t . Emery I . Balazs and Bruce C . Douglas , September 1 9 7 9 , 23 pp (PB80 1 2 0 6 1 1 ) . Heights of four local mean sea levels for the 1 9 4 1 -59 e p och in Ca lifornia are de termined and compared from five geodetic level lines obse rved ( leve led) between 1 9 68-78 .

NOS NGS-21 Haystack-We s t f ord S u rvey. W. E . Ca r t e r , C . J . Fronc z ek , and J . E . Pettey, September 1 9 7 9 , 5 7 p p . A special purpose survey was conducted f o r VLBI t e s t comparison.

NOS NGS-22 Gravime t ri c tidal loading computed f rom integrated Green ' s funct ions . C. C. Goad, October 1 9 7 9 , 1 5 pp. Tidal loading is computed using i ntegrated Green ' s functions .

NOS NGS-23 Use of auxi liary ellipsoids in height-controlled spatial adju s tment s . B. R. Bowring and T . Vincent y , November 1 9 7 9 , 6 pp. Auxi liary e l lipsoids are used in ad jus tments of networks i n t h e height-cont rol led three-dimens ional system for controlling heights and simplifying t rans formation of coordinat e s .

N O S NGS-24 Determination o f the geopot ential f rom satell i t e-to-satellite tracking data. B . C . Douglas , C. C. Goad, and F. F. Mor r i s o n , Janua ry 1 980 , 32 p p . The capability of det ermining the geopotential from satellite-to-satellite tracking is analy z ed .

(Cont inued o n ins i de back cove r )

N O S 6 5 NGS

NOS 6 6 NGS 2

NOS 6 7 NG S 3

NOS 6 8 NGS 4

NOS 70 NGS 5

NOS 7 1 NGS 6

NOS 7 2 NGS 7

NOS 7 3 NG S 8

NOS 7 4 NGS 9

( Cont i nu e d )

NOAA Techni ca l Repo r t s , NOS / NGS subs e r i e s

T h e s t a t i s t i c s of res iduals a n d the det e c t i on of out l i e r s . Allen J . P o p e , M a y 1 9 7 6 , 1 3 3 pp ( P B258 4 2 8 ) . A c r i t e ri on f o r r e j e c t i on o f bad geode t i c data i s d e r i ved on the bas i s o f re s i du a l s f rom a s i mu l taneous lea s t-squares a d ju s tment . S u brout ine TAURE i s inc lude d . E f f e c t o f G eoceiver observa t i ons upon t h e clas s i c a l t riangu la t i on ne two r k . R . E . Mo o s e and S . W . Henrik s e n , J u n e 1 9 7 6 , 6 5 p p ( P B 2 6 09 2 1 ) . T h e use o f Geoce iver observa t i ons i s i nve s t ig a t e d as a means o f imp roving t r i angu l a t ion network a d ju s tment resu l t s . Algori t hms f o r comput ing t h e geopot ent i a l us ing a s imp le-layer dens i t y mod e l . F o s t e r Mo r r i s o n , Ma rch 1 9 7 7 , 4 1 p p ( P B2 6 6 9 6 7 ) . Several a l g o r i t hms are deve loped f o r compu t i ng w i t h h i gh accu racy the gravi t a t ional at t ra c t i on of a simp le-dens i t y layer a t a r b i t ra r y a l t i t ud e s . Compu t e r p ro g ram i s i n c lude d . Tes t r e s u l t s of f i r s t-o rder c l a s s III l e ve l i ng. Cha r l e s T . Wha len a n d Emery Bala z s , November 1 9 7 6 , 30 p p ( GP OD 003-0 1 7-0 0 3 9 3- 1 ) ( P B 2 6 5 4 2 1 ) . Spe c i f i ca t i ons for re leve l i ng the Nat i onal ve r t i ca l cont r o l ne t were tested and the re s u l t s published. Se lenocent r i c geode t i c r e f e rence s y s t em . F rede r i ck J. Doy l e , At e f A . E la s s a l , and James R. Luca s , February 1 9 7 7 , 53 p p ( PB 2 6 6 0 4 6 ) . R e f e rence s y s t em was e s t a b l i sh e d by simu l taneous ad jus tment o f 1 , 2 3 3 met r i c-camera p h o t o g raphs of the lunar surface f rom wh i c h 2 , 6 6 2 t e r ra i n point s w e r e pos i t ioned. App l i ca t i on of d i g i t a l f i l t e r i ng to s a t e l l i t e geodes y . C. C. G oa d , May 1 9 7 7 , 73 pp ( P B-2 7 0 1 9 2 ) . V a r i a t i ons in the orbi t of GEOS-3 were ana ly z ed f o r M2 t i da l harmonic coe f f i c i ent va lues that p e rturb t he orbits of a r t i f i c ia l s a t e l l i t es and the Moon . Sy s t ems f or t h e d e t e rmina t ion of polar mo t i on . Soren W . Henrik s e n , May 1 9 7 7 , 5 5 p p ( P B 2 7 4 6 9 8 ) . Methods f or d e t e rmining polar mo t i on a r e des c r i be d and t h e i r advantages and d i s a dva ntage s comp a r e d . Cont r o l leve l i n g . Cha r l e s T . Wha len, M a y 1 9 7 8 , 2 3 pp ( GPOD 003-0 1 7 -004 2 2- 8 ) ( PB 2 8 6 8 38 ) . The h i s t o ry of the N a t i onal ne twork of geo d e t i c cont r o l , f rom i t s o r i g i n in 1 8 7 8 , is pres en t e d i n addi t i on t o the la t e s t observa t i onal and computat ional procedu r e s . Su rvey of the McDona l d O bs e rva t o ry ra d i a l l i ne s cheme by r e l a t i ve la t e r a t i on t e chniqu e s . W i l l iam E . Ca r t e r and T . V incent y , June 1 9 7 8 , 3 3 pp ( PB 2 8 7 4 2 7 ) . Resu l t s of exp e rime n t a l a p p l i c a t ion of the " ra t i o method " of e le c t romagne t i c d i s tance measurements a re g i ven f o r high res o lu t ion c rus t a l d e f orma t ion s tu d i e s i n the v i c i n i t y of the Mc Dona ld Luna r Las e r Ranging a n d Harva r d Ra d i o As t ronomy S t a t i on s .

N O S 7 5 NGS 1 0 An a l g o r i thm t o comp u t e t h e e i genve c t ors of a symme t r i c ma t r i x . E . Schmi d , Augu s t 1 9 7 8 , 5 pp ( P B 2 8 7 9 2 3 ) . Method desc r i be s compu t a t i ons f o r e i g enva lues and e i g enve c t o rs o f a s ymme t ri c ma t ri x .

NOS 7 6 NG S 1 1 T h e appl i ca t i on o f mu l t iquadr i c equa t i ons and p o i n t ma s s anoma ly mod e l s t o crus t a l movement s t u d i e s . R o l land L . Hardy , November 1 9 7 8 , 6 3 pp ( PB 2 9 3 5 4 4 ) . Mu l t iqua d r i c equat i ons , bo t h ha rmo n i c and nonharmoni c , are su i t able as geome t r i c p re d i c t i on func t i ons f o r s u r f a ce d e f o rma t i on and have p o t ent i a l i ty f o r us age i n a na ly s i s of subsu r f a ce mass r e d i s t r i bu t i on a s s o c i a t e d w i t h cru s t a l moveme nt s .

NO S 7 9 NGS 1 2 O p t i mi z a t i on o f h o r i zontal control ne tworks by non l i near p rog rami ng. Denn i s G . M i l be r t , Augu s t 1 9 7 9 , 4 4 p p ( P B80 1 1 7 9 48 ) . Seve r a l h o r i z ont a l geode t i c control networks a r e o p t im i z e d a t mi nimum c o s t whi le ma i nt a ining des i r e d accu racy s t andar d s .

N O S 8 2 NGS 1 3 Fea s i b i l i ty s tudy of t h e c o n juga t e gradi ent me thod f o r s o l v ing l a r ge sparse equa t i on s e t s . Lothar G rund i g , February 1 98 0 , 2 2 p p . M e t h od i s su i t a bl e f o r cons t ra i ne d a d j u s t ment s o f t r iangu la t i on networks but n o t f o r f ree a d j u s tmen t s .

N O S 8 3 NGS 1 4 T i d a l c o r r e c t i ons t o geode t i c quant i t i e s . P e t r Vanlcek , February 1 9 8 0 , 3 0 p p . Corr ect i ons for t i dal f o r ce a r e f o rmu lated and t i dal aspects r e l a t ing t o geodesy a re d i s c u s s e d .

NOS 8 4 NG S 1 5 App l i ca t i on o f s p e c i a l var iance es t ima t o r s t o geode s y . J ohn D . Bo s s le r a n d R o b e r t H .

NOS NG S

Hans o n , F e brua r y 1 9 80 . S p e c i a l va riance e s t i ma t o r s , one invo lving the use of noninteger degrees o f f re e dom, are ana l y z e d and app l i e d t o leas t-square adjus tmen t s of geode t ic cont rol networks to de t e rmine t h e i r e f f e c t ivenes s .

NOAA Manua l s , NOS / NGS subs e r i e s

Geode t i c bench ma rk s . L t . R i chard P . F l o y d , S e p t embe r 1 9 7 8 , 5 6 p p ( GPOD 003-0 1 7- 0 0 4 4 2 - 2 ) ( P B 2 9 6 4 2 7 ) . R e f e rence gui d e p rovi des spe c i f i ca t i ons f o r h i gh ly s table bench ma rk s , i nc luding chap t e rs on i ns ta l l a t i on p roce du re s , ve r t i c a l i n s t a b i l i t y , and s i t e s e l e c t ion cons i d e r a t i o n s .

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