Intrinsic Coordinates in Practical Geodesy

Intrinsic Coordinates in Practical Geodesy Antonio Marussi “We have advanced no vague reasoning about the well-known properties of spheroidical triangles and geodetical lines, in a case where; in fact, there is neither any such triangle nor any such line. . . .” J . Ivory, 1828.

Summary T h e basic ideas that support intrinsic geodesy, i.e. the discipline which aims at the local description of the gravity field of the Earth by using only coordinates and quantities that have a physical reality and that are therefore accessible to actual observation, are recalled. It is shown how the integrability conditions necessary for the existence of the coordinate surfaces, and the fundamental operators, e.g. the Christoffel symbols of the second kind connecting the principal trihedra of the intrinsic coordinate system, may be expressed in terms of the curvature parameters of the field, and of gravity. An application of the theory is made to a classical geodetic problem, i.e. the generalized expansion of Legendre for the displacement of the potential (the dynamic height) along an optical path. I. Man’s natural environment forms part of a three-dimensional space. One of these three dimensions-loosely considered to be the vertical at a given placehas been instinctively differentiated from the other two; and hence it is not surprising that throughout geodesy-and particularly throughout practical geodesythis third dimension has always been treated in a different manner. As a consequence of this approach, which has profound roots in the human mind and in the historical development of geodesy, it has been necessary to use some reference surface, such as the geoid, which, although it possesses physical reality, is nevertheless inaccessible to direct observation ; or like the ellipsoid and the spheroid, which are completely hypothetical. This approach thus also carries with it the burden of making complex and badly-defined reductions involving supplementary hypotheses-reductions which vitiate the accuracy with which our present-day observations can be made. Moreover, the arbitrary distinction of two dimensions from the third destroys that unity which is the true objective of scientific synthesis. Thus it seemed desirable to develop a theory in which only those physical and geometrical quantities that are actually observed are used. Such a theory could restore to geodesy the unity which had been lost for the historical and psychological reasons explained. At the Oslo Assembly of the I.U.G.G. in 1948 I presented the outline of a general theory which would permit the description of the Earth’s gravity potential 83

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84 Antonio Marussi field in terms of a system of coordinates intrinsic to the field itself. These coordinates are the astronomic latitude and longitude and the gravity potential (or dynamic height); and in the theory they are treated in a unified manner. This theory, which I have subsequently elaborated in various papers, was originally intended to satisfy an aesthetic desire for unity, by returning the geodetic problem to its natural setting-three-dimensional space-and thereby avoiding the dichotomy between geometric and dynamic procedures, and unifying the disciplines of planimetry and altimetry. When it was first developed, the theory was not of immediate practical use, but the extremely rapid evolution both in scope and precision of geodetic measuring techniques now enables it to be applied to many problems of practical geodesy. The new techniques include the direct measurement of distance along geodesics in space, the use of targets at great altitudes such as rockets and satellites, and the facility with which gravity may now be measured both on land and at sea, and perhaps in the near future above the surface of the Earth. There have already been successful applications of a unified theory. Brigadier Hotine, in Great Britain, has developed an elegant theory of spatial geodesy, in which the three dimensions are treated on an equal footing; and the theory involves all the quantities, and only those quantities, which are or could be the subject of geodetic observation. In the U.S.A., C. A. Whitten has applied this theory to practical problems. The ingenious method of stellar triangulation already developed by V. Vaisali in Finland also falls naturally into this scheme of ideas. This is an appropriate place to observe that this geodetic revisionist movement -which has arisen in what we might call the diflerential-geometric approach to geodesy, and whose object is to gain knowledge of the actual external gravity field without debasing the measured quantities by the use of chimerical reductionshas a parallel which grew up almost simultaneously but quite independently in what we might call the integral-dynamic approach to geodesy-that dominated by Stokes’s formula. Here, too, there has been an attempt to do away with the concept of the geoid-a surface whose theoretical definition is certainly unexceptionable, but which at least on the continents is in fact inaccessible-and to use instead the physical surface of the Earth as the surface needed for solving Stokes’s problem, again avoiding those reduction terms which make the high precision of the measurements unusable. This being the case, it seems appropriate to present here the fundamental equations of my theory in a form more suited to practical use. 2. We may consider two different systems of intrinsic coordinates in the external gravity field : first, a system of general coordinates comprising astronomic latitude 4, astronomic longitude A, and gravity potential W ;secondly, a system of local Cartesian coordinates, based on the three directions North, East, and Zenith, at any point. It is well known that the latter system cannot form the basis of any general system of coordinates. One system may be more convenient than the other according to the type of problem. When, for example, it is required to study scalar or vector quantities at geometrically interconnected points (as in triangulation or trilateration), the second system would be the more appropriate; whereas the first would be preferred when the quantities are to be considered at points that are not interconnected. The fundamental differential operations are, for the first system, those of partial differentiation a/acj, a/aA, a/aW and for the second system, those of directional differentiation-a/& in the North direction, a/as, in the East direction, and

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Intrinsic coordinates in practical geodesy

85

a/ah in the Zenith direction. These operators are related by the following equations, which are fundamental to the succeeding discussion :

a as, a+ cos+ax a -a- - a as2 a+ cos+ax a -- ql-+TZ---g-. a a a _ ah a+ ah aw a

---

T

a

Kl-+--

K2

T-+--

Here ~1 and ~2 are the normal curvatures of the North and East lines, and T is the geodetic torsion of the North line (equal and opposite to that of the East line). 71 and 712, the geometric components of the curvature-vector of the line of force along the North and East lines respectively, are given by 711 =

a asl

a

-1ng;

712 =

-1ng. as2

+

The conditions necessary for the existence of the coordinate surfaces = const, A = const, W = const, lead to five conditions for integrability. These may be most concisely written :

a Tcos2+ In +-- a In-K2 = o a+ K cosgax K a K l C O S 4 +-- a l n - + + ~ t a n + K1-1na+ K cos+ah K K2

T-

~

7

a

7

awK

-

aH2

a+

7

I aH1 - --+Hztan+

cos+

-_ a K2 BWK

ah

aH1

a+

=o

a2 --(cos+ a+ax I

=

I

g - -($-I);.

To these may be added a sixth which is a form of Poisson’s equation, and which may be written :

In the foregoing expressions K = K ~ K Z T~ is the Gaussian curvature of the equipotential surface W = const, g is the gravity and w the angular velocity of rotation of the Earth. We may then put

HI

-1

a a+

= --lng,

g

Hz

a 1% gcos+ ax -I

= ___-

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86 Antonio Marussi where HI and HZare related to 71 and 72, as defined above, by the following expressions : 71 = -g(KzHl+THz) 72 = -g(THl + KlHZ). T h e first two conditions of integrability are concerned only with the parameters of curvature of the equipotential surface and their derivatives ;whereas the remaining four express the rate of change of these parameters and of gravity on passing from one equipotential surface to another infinitely close. The sets of three local Cartesian coordinates may be interconnected by means of Ricci’s rotation coefficients, which are all expressed by the five parameters of curvature of the field KI, K Z , T , 71 and 72. Similarly, the bases vectors of the principal sets at the point P, aP/a#, aP/aA and aP/aW, relative to the general coordinates #, A, W , are related by the Christoffel symbols of the second kind. Allotting the numbers I , 2, 3, to these coordinates, the Christoffel coefficientsmay be most clearly written

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If the field is symmetrical about the axis of rotation, all the preceding expressions become somewhat simplified. The derivatives with respect to the potential of the principal radii of curvature I / K ~and I/KZ, and of the logarithm of gravity, may then be written : a

1

-- =

aWK1

-($+I):;

a

a

1

--

aWKz

=

jtan+%-I)-

1

g

3. As a practical example, we shall apply the preceding to the generalized expansions of Legendre. The expansions allow the coordinates $4, A, W (which we shall now denote by yl, y 2 ,y3 respectively, so that we can use the conventional notation of tensor calculus) to be displaced along a regular arc of any curve s in space, y being the first curvature and 0 the torsion. Hence

where A* are the controvariant components of the unit vector tangent to s. Hence by Frenet’s formula

and so on, where v’ and are the controvariant components of the unit vector of the principal normal and of the binormal to s respectively. The above formulae allow these expansions to be given explicitly; in particular A‘ has the form A1 = ~1sin x cos a

cos 4 x 2 = r sin xcos a+ A3 =

+

7

~2

sin x sin a + 71 cos z sin xsin a + 72 cos x

-gcosx

where and f are the azimuth and the zenith distance of the tangent at s. By analogy v1 =

Klsin Zcos & + rsinzsin

+r / l cos 2.

In this case B and 2 are the azimuth and zenith distance of the principal normal to s, related to u and z by the orthogonality condition tanxtanZcos(u-&)+I =

0.

If the curve s of displacement is an optical path, we can assume without appreciable error that its osculating plane is everywhere vertical, in which case a = E,

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88

Antonio Marussi

and f = z + 90'. Here we shall confine ourselves to giving the explicit expression for the displacement of potential (or dynamic height) along the curve s under consideration (which we may regard as an optical path). By a somewhat lengthy but simple calculation

W

=

Wo- sgo cos zo +

sin xo - K, sin2zo- sin zzo-

a

lng +

as,

2

+(

Kl+K2+-

2"2) g

I +--

cos2x

: ( :s

-+... + ....

)

Here z is the observed zenith distance at the near end (without correction for refraction); K , is the normal curvature of the equipotential surface in the vertical plane containing s, and obviously

a

cos ct-

-=

a + sin a-. a

ds2 It is obvious that the value of each coefficient should be calculated for the near dscz

dSl

end. The preceding formula does not contain products of the curvatures and their derivatives, but retains the terms in dylds. For displacement of zenith distance we have the expression cos x

=

a go -cos zo - s ysin zo - Kasin%o- sin 2x0 -lng+ as, g

[

This formula requires the values of gravity at both ends. For the displacement of the gravity value in the system of general coordinates, we may use the ordinary Taylor's expansion lng

=

(

1

a ...) Ingo. ay

+A+-+

I n the local Cartesian system the expansion

d lng = I+s-+ ds

(

...)Ingo

may be used. In both these expansions the equations given above allow all the derivatives concerned to be expressed in terms of the surface derivatives alone. This example shows how the theory may be very easily applied to give the most rigorous solution to the problem of evaluating potential differences-a problem occurring in the most elementary surveying. I n a like manner these procedures may be applied to more complex geodetic calculations, such as the displacement of geographic coordinates along any curve in space, with little more difficulty than by classical methods, but with the advantage of providing results notably more concordant with physical reality.

The University, Trieste: 1960 October.

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References Hotine, M., 1957. Metrical properties of the Earth’s gravitational field and geodetic coordinate systems. A.I.G., Toronto, 1957. Hotine, M., 1957. A primer of non-classical geodesy. A.I.G., Toronto, 1957. Hotine, M., 1960. The third dimension in geodesy. A.I.G., Helsinki, 1960. Ivory, J., 1828. On the method in the trigonometrical survey for finding the difference of longitude of two stations very little different in latitude. The Philosophical Magazine and Annals of Philosophy (London, Dec. 1828), 432. Marussi, A., 1950. Sviluppi di Legendre generalizzati per una curva qualunque dello spazio. Rend. Acc. Nux. Lincei, Serie VIII, Vol. IX, Fasc. 1-2. (Roma, 1950.)

Marussi, A., 195I . Fondamenti di Geodesia Intrinseca. Pubblicuxioni dellu Cmnm. Geod. Ituliunu, Serie 111, Memoria no. 7. (Milano, 1951.) Vaisala, Y., I 947. An astronomical method of triangulation. Finn. Acud. Sc., 1946, 99-107. (Helsinki, 1947.)

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