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Independent estimates of a, J2, kME (of the solid earth), Ro (scale factor for lengths), γe (equatorial gravity) and a fixed ω are used in a weighted least squares adjustment to determine adjusted values for the variable parameters assuming a rotational equipotential ellipsoid as the figure of reference. The adjusted values are: kME = (3.9860034±0.0000023)×1014 m³/sec²; γe = 978031.69±0.77 mgals; f = 1/(298.25636±0.00147); Ro = 6363674.98±2.50 m, a = 6378139.0±2.51m and Wo (the geoid potential) = 6263681.62kgal m.

Summary Results of a preliminary gravimetric geoid computed on the Geodetic Reference System of 1967 (GRS67) are given in the form of undulation and deflection maps. For the computations, 1°× 1° mean free-air gravity anomalies formed the inner zone and 5° equal area values the outer zone around each computation point. Some of the required 1°× 1° means, together with their accuracies, were obtained by statistical prediction methods using an auto-covariance function derived from the known 1°× 1° values. The results confirm the general south-easterly down slope of the geoid undulations. The average computed standard errors are about 3·6 m in the undulations and 0·“6 in the deflections.

The paper summarizes the results of the Ohio State University geometric adjustment for the coordinates at 158 satellite tracking stations. In the solution the origin of the coordinate system is defined through the ‘inner’ adjustment procedure, whereas the orientation is through the Conventional International Origin and the Greenwich Mean Astronomical Meridian, both as defined by the Bureau International de l'Heure. The scale is implemented through Secor observations and weighted height constraints. Chord distances derived from C band radar observations and from electronic distance measurements are also included, but they seem to have very little effect. The scale selected corresponds to a best fitting ellipsoid of a = 6,378,142 m and 1/ƒ = 298.25. The average standard deviation of a single coordinate is 3.9 m. Comparisons with external standards generally show good agreements: scale agreement with dynamic satellite solutions is 1:1,600,000 and with terrestrial surveys is 1:575,000; the root mean square residual of comparisons with geoid undulations from satellite surface gravimetric solutions is 6.1 m, with an average discrepancy of only −0.3 m. The paper also presents satisfactory transformation parameters for the NAD 1927, the Australian and the South American 1969 geodetic datums.

The method of least squares collocation is briefly outlined. Its application to the determination of the astrogravimetric geoid (in fact, satellite data can be used as well) and the difficulties encountered are discussed. By the use of this method, it is possible to combine all types of geodetic data available in order to obtain an optimum solution for the geoid. It is also possible to determine consistent geodetic parameters necessary for the definition of a geodetic reference system. These parameters include the position of the existing astrogeodetic reference ellipsoid with respect to the geocenter to which the gravimetric and satellite data are referred.

By minimizing the global distance between the (quasi-) geoid and an ellipsoid of revolution, the best parameters of an ellipsoid and its location parameters are estimated. Input data are the absolute value of the geopotential at (quasi-) geoid level, a set of harmonic coefficients from satellite, terrestrial or combined observations, the mean rotational speed of the earth, and approximate values of the seminajor axis and the eccentricity of the ellipsoid. The output ranges from 6378137,63 m and 6378141,62 m for the best semimajor axis and from 298.259758 and 298.259651 for the reciprocal value of the best ellipsoidal flattening within WD 1 and WD 2. The best translational parameters are 6.4 cm (Greenwich-direction), 0.8 cm (orthogonal to Greenwich-direction), and 1.9 cm (parallel to rotational axis of the earth); the best rotational parameters are −0.2” (around Greenwich-direction) and 1.1” (around orthogonal to Greenwich-direction). The dependence of the datum of the (quasi-) geoid geopotential is studied in detail.

Absolute orientation of astrogeodetic systems, obtained by (1) computing absolute gravimetric deflections of plumb line according to Molodensky’s linear solution, (2) fitting astrogeodetic geoid sections to global geoids obtained from satellite or other data, (3) matching small astrogeodetic geoid sections with detailed gravimetric geoids, are described, and corresponding errors and problems are discussed. Experiments made in determining the absolute orientation of the European Datum are pointed out. The use of Molodensky’s formula for the transition from one ellipsoid to another (where deviations from Laplace’s condition are supposed to exist) is tested. The results are related to the geoids by Bomford, Heitz, Groten and Rummel and Vincent and Marsh. General conclusions and inferences with respect to determining geodetic datums are given.

This paper extends Clairaut's theory of rotational equilibrium to third order terms in a small parameter and is meant to be a sequel to a 1962 publication by the author bearing on the same topic. It has been feasible to obtain the Clairaut equation, which governs the deformation of the equipotential surfaces within a rapidly rotating mass in hydrostatic equilibrium, as an ordinary differential equation. This has been achieved by eliminating the two integral terms which appeared in the original formulation. It is expected that the numerical integration of this newly obtained equation will contribute toward a more precise solution of certain geophysical problems — e.g., the determination of the geoid to an accuracy of ±1 m, and the correction to the travel-time of seismic waves; it should also assist in some planetary questions like the determination of the exterior shape for the rapidly rotating planets Jupiter and Saturn.

An astrogeodetic determination of the geoid in certain parts of Canada was made with data available at the end of 1970. The astrogeodetic leveling method was used. The western and eastern parts of Canada were adjusted independently owing to the lack of data in the Province of Ontario. Five satellite stations located across Canada were chosen as initial points of the astrogeodetic geoid. Comparisons with two other astrogeodetic geoids — U.S. Army Map Service 1967 and University of New Brunswick 1973 — and one geoid obtained from a combination of gravimetric and satellite data— Goddard Space Flight Center 1972 — were made. A relative agreement of about five metres was found in every case for most parts covered by the present astrogeodetic geoid except in Newfoundland and Yukon.

With the increasing use of satellite positioning techniques, the determination of the position of local geodetic datums in a geocentric coordinate system is becoming important. This position is def...

Using the data in[1], the scale factor for lengths is derived of the geopotential model R 0 =GM/W 0 (W 0 is the potential on a generalized geoid). The resulting value, R 0 ==6 363 672.9 m, which is2 m less than the original value[5], is practically the same as that in[6].

The value of the geopotential on the geoid, W0, is computed, as well as the scale actor for lengths, R 0 =GM/W 0 , from the GEM 3 and GEM 4 systems; the systems SE I, II, III and GEM 3, 4 are compared, and the parameters of the best-fitting tri-axial Earth ellipsoid are computed.

The odd zonal harmonics in the geopotential are the terms independent of longitude and antisymmetric about the Equator: they define the ‘pear-shape’ effect. The coeffecients J 3 , J 5 , J 7 ,…of these harmonics have been evaluated by analysing the variations in eccentricity of 27 orbits covering wide range of inclinations. We use again most of the orbits from our previous (1969) evaluations, but we now have the advantage of 3 accurate orbits at inclinations between 60° and 66°, where the variations in eccentricity become very large, and 3 near-equatorial orbits, at inclinations between 3° and 15°, whereas previously there were none at inclinations lower than 28°. The new data lead to much more accurate and reliable values for the coeffecients. Our recommended set, which terminates at J 17 , is 10 9 J 3 = −2531 ± 7 10 9 J 11 = 159 ± 16 J 5 = −246 ± 9 J 13 = −131 ± 22 J 7 = −326 ± 11 J 15 = −26 ±24 J 9 = −94 ± 12 J 17 = −258 ± 19 . With this new set of values the pear-shape tendency of the Earth amounts to 44.7 m at the poles, instead of the previous 40 m, though the new geoid is within 1 m of the old at latitudes away from the poles.

Comparisons and analyses were carried out through the use of detailed gravimetric geoids which we have computed by combining models with a set of 26,000 1 deg x 1 deg mean free air gravity anomalies. The accuracy of the detailed gravimetric geoid computed using the most recent Goddard earth model (GEM-6) in conjunction with the set of 1 deg x 1 deg mean free air gravity anomalies is assessed at + or - 2 meters on the continents of North America, Europe, and Australia, 2 to 5 meters in the Northeast Pacific and North Atlantic areas, and 5 to 10 meters in other areas where surface gravity data are sparse. The R.M.S. differences between this detailed geoid and the detailed geoids computed using the other satellite gravity fields in conjuction with same set of surface data range from 3 to 7 meters.

The set of plumb lines and the set of lines perpendicular to the geoid are shown to be geodesics of the local geometry. From the properties of geodesics, it is shown that the two sets are identical. Therefore, reduction of gravity along a perpendicular is equivalent to reduction along a plumb line.

In accordance with the requirements for solving the Molodenskii boundary-value problem of the theory of the potential, a theory of topographic gravity reductions and gravity anomalies has been elaborated. Their definitions are founded on the method of removing and restoring the effect of a topographic massif, all terrestrial topographical masses above the surface of reference being considered as such. The result is Eq. (5), the term on the r.h.s. being close in absolute value to the terrain correction, the second close to the usual Bouguer reduction, and the third proportional to the potential of all topographic masses. To compute the third term one would have to know the geoid heights, however, it would be completely eliminated in formulating the boundary condition in the form of Eq. (7) and neither is it necessary for the boundary condition in classical form (1) to compute the deflections of the vertical and the disturbing potential, or the quasi-geoid heights. This can be seen from the formulae for these quantities in the “zero-approximation” (22) and (24).

(Received 1974 February 8) Summary In many theoretical rcsearches and operations in geodesy the geonietrical and physical quantities under consideration must be framed in an appropriate system of co-ordinates with simple metric. An attempt is here made to establish a one-to-one correspondence between points of the actual gravity field of the Earth and the normal ellipsoidal one both referred to geographic co-ordinates and to potential, which permits the univocal definition of anomalies and deflections, and the use of the simple metric tensor of the normal field for their analytical manipulation. The fundamental equation of physical geodesy, as well as the integrability conditions for deflections and anomalies are derived. 1. One of the main problems in both theoretical and operational geodesy is that of framing the geometrical or physical quantities observable at or near the surface of the Earth, in a system of co-ordinates of known and possibly simple metric. Traditionally, the problem is solved in two dimensions by a series of reductions of the observed values to the normal ellipsoid referred to geographic co-ordinates, the simpte metric of which is thereafter used for all subsequent planimetric computations; whereas the third co-ordinate, height, is treated separately and is referred in various ways to the geoid as the reference level surface. An extension of the foregoing concept to three dimensions has been indicated by Hirvonen (1960, 1961) who introduced the notion of the ‘ telluroid ’ as the image of the Earth’s surface in the ‘normal’ ellipsoidal gravity field; extensive use of this concept has been made in Molodenski’s theory for the determination of the actual shape of the Earth and its gravity field from observations of gravity at its physical surface. In the following an attempt is made to establish with a systematic procedure a one-to-one correspondence between points P of the ‘actual’ gravity field of the Earth, and points Q of the ‘ normal ’ ellipsoidal field, in such a way that thereafter the simple metric of this field can be used for analytical computations and practical uses. Such a procedure permits the simultaneous treatment of planimetric and altimetric co-ordinates, thus avoiding ill-defined reductions and approximations, and permits the unequivocal definition of ‘ deflections ’ and ‘ anomalies ’, for which the integrability conditions which they must satisfy are easily found.

A geodetic datum is a mathematical means to adjust and record a network of control stations. The reference surface chosen for the adjustment of the horizontal control is essentially arbitrary, but expected to fit the geoid of the area reasonably well. The geoidal separation computed from the adjusted control is an indicator of the fit. When choosing a new datum, this indicator may be used to clarify the alternatives, so that the choice will serve the purpose.

Summary Plans to place three satellites fitted with laser retro-reflectors and altimeters in near-Earth orbit in the next decade, offer a stimulus to activity in physical geodesy. Of prime interest is the definition of the oceanic geoid with sufficient precision to permit the mapping of deviations of the oceans from a level surface, as derived from considerations of the Earth's gravity field. Any such definitions in the next decade may have to be achieved in the absence of a complete representation of the Earth's gravity field by surface measurements of gravity. Another factor to be contended with in this time frame is the absence of a global satellite tracking network with both an adequate station density as well as the resolution for studies of this sea surface topography in four dimensions. The present development formulates a solution of the geodetic boundary value problem where gravity models used for orbit determination can be related to a finite entity in this solution with a view to minimizing any source of differential error in the final result. The geometry of the solution in Earth space is established. Expressions are developed for obtaining a solution from satellite altimetry. The possibility of obtaining an interim determination from a combination of satellite altimetry in oceanic regions and the available surface gravity data in continental shelf areas is discussed. The principal problems are those arising from the influence of long wavelength errors in the satellite altimetry and, to a lesser extent, in the surface gravity anomalies. It would appear that meaningful global solutions will only be possible under such circumstances if the altimeter-equipped spacecraft were tracked on a regular basis with a well-distributed network of at least 25 all-weather 10-cm tracking systems in the absence of a reliable gravity field definition. The question of resolution in regional studies is also discussed.

Kenya-Topography. Kenya lies astride the Equator between latitudes 4°N and 4°S and extends from Lake Victoria to the Indian Ocean (34°E to 42°E). Between long. 36° and 36½°E it is traversed from north to south by the great Rift Valley, the surface of which is generally 2000-3000ft. lower than the level of the plateau on each side. The plateau culminates in the Aberdare Range (13,000 ft.) east of the Rift, and the Mau Forest (10,000 ft.) and Cherangani Hills (11,000ft.) west of the Rift. Together with the isolated mountain masses of Elgon (14,000 ft.), Kenya (17,000ft.) and Kilimanjaro (19,500ft.), this topography may be expected to cause considerable local distortion in the shape of the geoid. Gravity surveys in 1933-4 indicated a negative anomaly below the floor of the Rift Valley, suggesting that it had been forced down by over-riding pressure from both sides.

In modern first-order geodetic work it is customary to measure horizontal and vertical angles, spatial distances and astronomic azimuths, latitudes and longitudes. These are projected onto an ellipsoid of reference and are then adjusted to yield positional co-ordinates on that ellipsoid. The results of spirit-levelling, which yield heights above the geoid, are commonly adjusted as a separate entity. Thus the classical method produces two distinct sets of unrelated quantities. A method for rigorously incorporating the results of spirit-levelling into a three-dimensional Cartesian frame of reference is presented. The use of the geoid and of an ellipsoid is avoided.

The principal formulae of the geophysical geodesy are based on the famous explicit expression of Stokes (1849). Up to now, there has been no method for a computation of the corresponding explicit expression of a (non-spherical) surface with masses outside the geoid. In this paper there is a solution of this problem. Another paper on this subject was presented to “Nordiska geodetmotet” in Copenhagen May 1959 (in Swedish).

Stokes 's method for d e t e r m i n i n g the form of the geoid by means of g rav i ly anomal ies rests ou two unreal ized a s s u m p tions. Assumpt ion No. ~. The geoid envelops all attracting matter. The land areas, cons t i tu t ing about one-four th of the ear th ' s surface, violate this a s sumpt ion . Assumpt ion No. 2. All observations oJ" gravity represent values of gravity as they would be on a geoid falfilling Assumplion No. t. Observat ions made at sea on a s u b m a r i n e are made so near ly on the geoid that t he i r reduct ion to the surface of the sea, that is, to the geoid, presents no ser ious diff iculty. Observat ions on land are c o m m o n l y made near the phys ica l surface of the ear th and the reduct ion of them to sea level in such a way as to get a round Assumpt ions Nos. ~ and ~ presents a p rob lem of some difficulty. The pro jec t ing land surface, which inval ida tes Assumpt ion No. t , may be d isposed of in var ious ways. For example : (a) I t may be condensed in imag ina t ion into an inf ini te ly thin surface layer co inc ident with the geoid, as He i m e r t r ecommended . The process of condensa t ion changes the form

This paper ties up with [5, 6] the fundamental notations of which have been adopted. The sets of Stokes' constants (harmonic coefficients) J n (k) , S n (k) were adopted from [4], and the scale factor for lengths, R0=GM/W0, from [5]. The equations for global meridional and parallel sections of the geoid surface W=W0 are formulated. The geoid sections are represented by best fitting ellipses, as regards the meridians always for the arcs between the equator and the pole.

Many theoretical discussions published by various geodesists during the last decades seem to have a common trend which means no less than the reformation of the very foundations of geodesy. There are three new ideas which have, in fact, no connection with each other except that they all can be expressed in a rather paradoxical and “revolutionary” form: (1) In gravimetric geodesy, the principal problem has been “the determination of the geoid”. According to the new theory, the geoid is quite unnecessary and the determination of it is considered to be an unsolvable problem. (2) In geometric geodesy, most problems have been solved by the aid of geodetic lines. From the practical point of view, the geodetic lines should be replaced by the vertical planes. (3) The theory of errors and adjustments by the method of least squares are often considered to be a special domain of geodesy. Now the theory of adjustments should not be based on the vague concept of errors, but on modern mathematical statistics.

Two models describing potential coefficient behavior are used to estimate the root mean square geoid undulation by wavelength. Four wavelength types were defined: long wavelengths: l=2 to 10; intermediate wavelengths: l=11 to 100; short wavelengths: l=101 to 1000; and very short wavelengths: l=1001 to ∞. By one representation of potential coefficient behavior the intermediate wavelength geoid information was ±5.64 m, short wavelength, ±0.66 m, and the very short wavelength, ±0.05 m. The procedures of this paper were applied to an actual residual undulation computation using detailed gravity material.

Summary The assumptions made in solving the boundary value problem are critically examined and revised expressions derived for the complete definition of the non-regularized geoid to the order of the flattening. The free air geoid is shown to be a good approximation to the non-regularized geoid. The relation between regularized geoids and the non-regularized geoid is established and the latter is shown to be no more difficult to compute than any regularized geoid. Expressions are developed for the computation of the height anomaly from free air anomalies and elevations to the order of the flattening. The significance of the zero order term is examined and its implications in assigning revised values for the parameters defining the reference system are examined.

Gravimetric methods of determining the figure of the Earth can be divided into two groups. The first employs an auxiliary surface—the geoid. The second employs the so-called quasigeoid as an auxiliary surface. Its main advantage is in that it only uses surface measurements and that it does not require knowledge of the structure of the Earth's crust. This method was treated in [1]. The results display an accuracy of the order of the Earth's flattening. The purpose of this paper is to show that this method can be applied theoretically to reach an arbitrary accuracy.

A satellite-to-satellite tracking experiment is planned between ATS-F and GEOS-C with a range accuracy of 2-meters and a range rate accuracy of 0.035 centimeters per second for a 10-second integration time. This experiment is planned for 1974. It is anticipated that it will improve the spatial resolution of the satellite geoid by half an order of magnitude to about 6 degrees. Longer integration times should also permit a modest increase in the acceleration resolution. Satellite altimeter data will also be obtained by means of GEOS-C. An overall accuracy of 5-meters in altitude is the goal. The altimeter, per se, is expected to have an instrumental precision of about 2 meters, and an additional capability to observe with a precision of about 0.2 meters for limited periods.

In the past the Eotvos torsion balance has been used only in commercial geophysical explorations and not for geodetic purposes. Even in geophysics, after approximately 30 yrs of use, it was superseded by the faster, and handier gravimeters. The lack of interest in the geodetic applications can be explained partly by the fact that geodesists of the past did not need very detailed information about the shape of the earth. The torsion balance offers several possibilities in geodetic applications, of which the most important are: The interpolation of deflections of the vertical and of geoid undulations between reference stations, where gravity material is not sufficient; the determination of the geopotential values in a given area for the purpose of leveling reductions; the determination of the radius of curvature of the plumb line for reducing astronomical coordinates to the geoid; and the investigation of curvature properties of the equipotential surface for more accurate baseline reductions to the ellipsoid of reference.

In the author's article (Wunderlich, 1966b) the anomalies of the geoid and of the heat flow were attributed to subcrustal flow. This interpretation is not dependent on whether the geoid surface, as computed by Guier (1965), is the expression of actual elevations and depressions or whether it is caused by local differences in density in the interior of the earth. If density fluctuations are responsible for the form of the geoid then these are probably the result of phase changes caused by compression and decompression in certain zones of converging and diverging flow lines in the mantle. No direct connection is recognizable between the anomalies of the geoid and the orientation of the earth's young mountain belts (see Fig.1). Therefore these mountain belts can hardly be explained by “gravitational spreading” of megaundations coinciding with the anomalies of the geoid. Fig.2 suggests instead an influence of Coriolis forces, generated by the rotation of the earth, on the subcrustal flow causing a deflection of the flow lines at every crossing of the equator. Active flow at depth is indicated by the fact that the deflection is directed towards the right in the Northern Hemisphere whereas it turns towards the left in the Southern Hemisphere.

That the accuracy of computing the undulations in the geoid is not yet as high as is desired is not due to the method itself but to the lack of gravity data. The masses of the great formations are 85 to 90 per cent isostatically compensated. The flattening ratio value, 1/298.3, obtained by the satellite method has also been obtained several times by gravity and arc-measuring methods. At the Institute of Geodesy, Photogrammetry and Cartography we have been conservative; we have used only the gravimetric method and have not extrapolated gravity anomalies more than 15° from the computation point. Uotila obtained by spherical harmonic analysis third-order harmonics similar to those obtained by O'Keefe from satellite data. The undulations associated with the pear-shaped form are of the order of ±15 meters. O'Keefe's geoid heights at the north and south poles are brought about solely by the fact that the flattening value he used (1/299.8) does not correspond to the real facts.

By postulating that the non-hydrostatic features of the Earth's gravitational potential are due to harmonic undulations of a nearly spherical internal surface with a density contrast, and that the stresses arising at this surface are of comparable magnitudes for all harmonic terms, it is found that the density anomalies responsible for the geoid must be in the upper mantle, at depths less than 1000 km.

The absolute deviations of the vertical detectable at Potsdam from astronomical-gravity comparisons differ significantly from earlier values obtained from the astronomic-geodetic observations which form the starting elements of the European triangulation network. It is possible to show, by the use of values of the deviation of the vertical extending over the whole of Europe as far as longitude 30°, and referred to the undulations of the geoid ofTanni, that the starting elements adopted at Potsdam are too large by 3″.0 in latitude and 1″.4 in longitude.

A total of 7234 Baker-Nunn camera observations of 5 satellites were analyzed to determine simultaneously 44 tesseral harmonic coefficients of the gravitational field, 36 station coordinates, and 511 orbital elements. Supplementary observational data incorporated in the solution included accelerations of 24-hour satellites and directions between tracking stations from simultaneous observations; observation equations were also written for the differences between geometrical and gravitational geoid heights at tracking stations. Several variations in relative weighting of different observational data and a priori variances of parameters were tested. The previous independent solution most closely approached was that by Anderle based on Doppler data, from which the rms discrepancy was ±0.18×10−6 for 38 normalized harmonic coefficients, or ±7 m in total geoid height. An equatorial radius of 6,378,160±5 m was obtained.

Since the presentation of “The Hough Ellipsoid” at the Toronto meeting in 1957, the astro-geodetic geoid was extended into the Caribbean, India, and Japan. The two hemispheres were tentatively connected across the North Atlantic by a reasonable assumption about the unknown geoid profile there. A series of solutions for a world ellipsoid and world datum was made, with and without enforcing the flattening of 1/298.3, with and without gravimetric orientation. The resulting ellipsoids are very small, with an equatorial radius of around 6378 160 m. The agreement between astro-geodetic and gravimetric geoid profiles is greatly improved by the small ellipsoid.

The paper discusses the errors likely to occur in calculations of the elevation of the geoid by Stokes’s formula when gravity data outside a given cap are neglected, and it is shown that they are comparable with the root-mean-square elevation. It is shown that these estimates and those of similar errors in deflexions of the vertical made earlier (Cook 1950) are not seriously affected by the assumptions made about the distribution of gravity anomalies between the harmonics up to degree 18.

Over 31,000 precision reduced optical observations of GEOS-I and II in 70 two-day orbital arcs have been used at Goddard Space Flight Center (GSFC) in a dynamical solution to determine center-of-mass coordinates for 15 tracking stations on the European Datum. Comparisons with the results obtained at Centre National d’Etudes Spatiales (CNES) give agreement of about 1.5 ppm for chord lengths. After considering a scale correction to the European Datum (ED) of 1950 to account for the absence of geoid heights at the time of its reduction, agreement to a few ppm between the CNES/GSFC and the ED chords is obtained. However, a small systematic difference between survey and satellite results remains for stations in south-eastern France and Switzerland.

The vertical gradients of gravity have great significance both in geodesy and in geophysics. In geodesy, in the free air reduction of gravity, the regional gradients should be used, rather than the customary normal gradients. The gravity anomalies, which include the free air reduction, are used in practice to determine geoid undulations, deflections of the vertical gradients, and the various coefficients of the gravitational potential field of the earth. The vertical gradients, together with the horizontal gradients, define analytically the shape of the equipotential surfaces. In geophysics, the vertical gradient, in addition to its use in computing the gravity anomalies, has great significance in exploration problems, such as the determination of rock densities. The paper describes the various methods of determining the vertical gradient and presents a map of the vertical gradients in Maine and southeastern Canada. A summary is given on the applications.

The Department of National Development of the Commonwealth of Australia kindly made available to us a list of astrogeodetic deflections of the vertical at about 600 stations in Australia, referred to the Australian Geodetic Datum. The latter is defined by the Australian National Spheroid (a = 6378160m, f = 1/298·25) and the co-ordinates of Johnston Geodetic Station (25° 56′ 54″·5515 S., 133° 12′ 30″·0771 E., height 571·2m).(1)

A set of2261 5°×5° mean anomalies were used alone and with satellite determined harmonic coefficients of the Smithsonian' Institution to determine the geopotential expansion to various degrees. The basic adjustment was carried out by comparing a terrestrial anomaly to an anomaly determined from an assumed set of coefficients. The (14, 14) solution was found to agree within ±3 m of a detailed geoid in the United States computed using1°×1° anomalies for an inner area and satellite determined anomalies in an outer area. Additional comparisons were made to the input anomaly field to consider the accuracy of various harmonic coefficient solutions.

Recent geodetical and geophysical results are discussed with regard to their possible geotectonical consequences. The geoid heights which have been observed by means of satellites are apparently not isostatically compensated; flow dynamical causes in the deeper part of the earth's mantle, which at the same time determine super-regional differences in heat flow from the interior of the earth, might be held responsible for their origin. Fluctuations of the deeper part of the mantle are to be concluded from the large-scale spatial distribution of strengthened or weakened heat flow, in the same way as secular variations of the geomagnetic field lead to movements in the outer core. Various such methods are apparently suitable for the kinematic depth sounding of the shell structure of the earth, according to the depth of the causes of the disturbance that influence the picture obtained at the surface of the earth.

The increasing use of gravity anomalies in geoid studies makes it mandatory that the tremendous amount of labor involved be streamlined and carried out by machine. A practical method is presented to cope with a mass production of deflections by carrying a uniform rectangular grid up to the immediate vicinity of all stations involved. The same method may be used to compute curvatures. Since curvature values depend on a smaller survey area than do deflection values, the curvature method may often be advantageous.

This paper deals with the geodetic applications of the gravity method. Gravity anomalies permit the absolute undulations of the geoid and the absolute deflections of the vertical to be determined. In turn, these, together with astronomical observations, form the basis for a World Geodetic System for the control of small scale maps beginning with the scale 1:100,000. They serve also, together with existing triangulations, for the correction of the dimensions of the reference ellipsoid. Additional gravity surveys are needed in order to obtain these and other minor objectives. Cooperation among scientists and agencies of different countries concerned with these problems is a prerequisite for the success of this world‐wide program.

The intervisible technique uses simultaneous optical observations of rockets or satellites for relative positioning of reference and unknown observers. The technique is developed using the method of maximum likelihood. Ten Air Force PC-1000 cameras in the southeastern United States participated in a program of simultaneous photographic observations of flashes from the ANNA 1-B geodetic satellite. Geodetic position determinations using these ANNA observations indicate that the PC-1000 camera system used with the intervisible technique is capable of extending geodetic control with an accuracy of about 10 m.