I1. C O ~ L P T E S R E N D U S DE L'ASSE!~IBLEE GEN]~RALE D'OSLO SECTION V . - - (]E01DE

G. BOMFORD

PROCEEDINGS OF THE MEETINGS OF SECTION V - G E O I D

P r e s i d e n t : D r . J. DE GRAAFF HUNTER; Secretary: Brig. G. BOMFORD.

A.-

AGENDA

The agenda of the Section and the times of meeting were as below :

Tues~. y. 24 August. l) Presentation of Brig. BOMFORD'S Report on the Deviation of the Vertical. 2) Presentation of Dr HEISK~NE~'S Report on the work of the Helsinki Isostatic Institute. 3) Presentation of Dr W. D. LAMBERT'SReport on Earth Tides.

Wednesday. 25 August. 4) The treatment of gravity results for use with Stokes' theorem. 5) Consideration of the distribution of gravity observations necessary for the successful application of Stokes' theorem. 6) Consideration of. the determination of the geoid from a combination of observations of gravity and of deviation of the Vertical.

Tuesday. 26 August. 7) Resolution regarding the observation of geoidal profiles, 8) Presentation of the following papers : a) Prof. A. MAaUSSI. Fondements de g~om~trie diff~rentielle absolue du champ potentiel terrestre.

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b) Prof. Dr. K. MADER. The determination of a geoid elevation from measurements with the torsion balance. c) Prof. Dr. A. PREY. La d4termination des d4viations de ta verticale sans compensation de r~seau.

B. - -

TUESDAY

24 A U G U S T

1. Brig. BOMFORD presented his report (1) on the Deviation of the Vertical. A s u m m a r y had been circulated which was taken as read, and the verbal report was mostly with reference to a proposed resolution urgeing the observation of trans-continentaI lines of closely spaced astronomical stations in order to determine continuous geoidal sections across the earth's main land areas. He said : During the past nine years few observations of the deviation of the vertical have been made, but the necessary triangulation has made very great progress, notably the extension of the Russian triangulation to Vladivostock, the connection of Egypt and Iraq to the European system by new triangulation in Turkey, and the extension of the triangulations of the U.S.A. and Canada to the Behring Straits and the Aleutian Islands. The completion of the African arc of the 30th meridian, and the extension of triangulation on the North American datum through Central and South America are also planned for completion during the next three or four years. This progress in triangulation makes it possible to envisage the early possibility of making the astronomical observations necessary for the determination of continuous geoidal profiles throughout the length and breadth of the five continents. (See Figure A, p. 382). Another direction in which important progress has been made is in the measurement of distances (tri-lateration) by RADAR. This may not yet have reached geodetic accuracy, but there is reason to hope that it may presently be possible to measure lines 500 miles in length, with an accuracy of 1:100.000 or better. With the help of RADAR it m a y then be possible to cover the gaps between the Netherland East Indies and Australia, and between Canada and Scotland via Greenland, Iceland and the Faroe Islands. There are two objects in observing these long lines of geoidal sections : (a) To enable base-lines to be correctly reduced to spheroid level on a single world spheroid. A case oecured in India and Burma where a base-line in Southern Burma, which is 10 feet (t) Cf. Bulletin Gdoddsique, n ~ i3, pp. 249-258.

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above sea level, was found by geoidat section to be 347 feet above the adopted spheroid, with a consequent error of reduction of I :60.000. Such discrepancies would not occur in small countries, nor would it have been so great in India if a more modern spheroid than EVEREST'S had been in use, but without n u m e r o u s astronomical observations w h o can say h o w m a n y hundreds of feet t h e geoid at Vladivostock m a y be above or below a spheroid w h i c h has been oriented b y astronomical observations at an origin in Europe ? This is not a question w h i c h can be answered b y calculations based on isostatic or other theory. Observations are necessary, either astronomical or of the intensity of gravity. (b) For scientific purposes the determination of the true figure of the earth, not a bi-axial or tri-axial spheroid, but an odd-shaped object of only approximately spheroidal shape. It is true that gravity observations and STOKES' formula constitute an alternative method w h i c h should be vigorously prosecuted, but w h e n it is possible to measure the figure of the earth by direct means, there can be no justification for w o r k i n g only on the indirect. In an incomplete state the two methods m a y be able to supplement each other, and w h e n each gives a complete solution their agreement m a y be a source of confidence or their disagreement a fruitful source of study. The system of w o r k recommended requires the spacing of astronomical stations (both components) at intervals of t0-20 miles in long continuous lines as has been done in India and Burma, where a n e t w o r k of over 9000 miles of such lines has been observed. W h e n the triangulation has been completed these geoidal sections can be observed quickly and easily, and in f a v o u table circumstances 25 stations a m o n t h have been completed b y a single party. The spacing of t0 miles is advised in m o u n tainous c o u n t r y and a m a x i m u m of 20 miles in plains. It has been found that with a spacing of 50 or 100 miles very serious errors m a y occur. W h i l e ' t h e w o r k most urgently called foc is the observation of the section lines s h o w n in Fig. A, it w o u l d be of value if every c o u n t r y in w h i c h geodetic triangulation exists, w o u l d observe two or more section lines across its length and breadth, connecting with those of its neighbours. Delegates present are invited to vote in favour of a resolution w h i c h will be proposed on 26th. They are also urged to press their Survey Departments and Governments to take action on the resolution, and it is hoped that m a n y present will be able to procure instruments, take the field, and make the necessary observations themselves.

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a. BOMFORD

2. ~r. HEISK~NEN, in presenting his report on the work of the Helsinki Isostatic Institute said : a) Situation des travaux de l'Institut Isostatique d'Helsinki concernant le calcul des dIdments du gdo~de. As in Bulletin gdoddsique (n ~ 7, ,t948, p. 50-54) already has been mentioned, Dr. L. TANNI has by the aid of the mean gravity anomalies of the squares 5 ~ X 5 o on base of the STOKES' formula computed the universal and continental parts of the undulations N of the geoid at more than 200 points along a broad strip around the Earth. Later o n also the regional part of N in Europe has been computed by the aid of the mean gravity anomalies of the squares ~o X {~ The N-values of the compensated as well as of the actual geoid are given in tables and in maps, which all will be printed before the 0slo-Assembly. The accuracy of the obtained N-values is sufficient for reducing the triangulation base lines from the geoid to the spheroid. As to the gravimetric computing of the components ~.and of the deflection of vertical at the triangulation points in Europe, the study is under work, but the gravity and map material of the Isostatic Institute is hardly enough for the accurate gravimetric computing of ~ and ~ in the mountainous regions, where and ~ were most important for the correction of the closing errors of the measured triangles. In the level lands and in the regions with not too rough topography the effect of ~ and ~ to the measured directions is, of course, practically zero. b) Ddtermination du gdo~de par les ddviations relatives de la verticale. The fine structure of the actual geoid can be very well computed by means of the deflections of vertical. The universal, continental and perhaps also the regional parts of N are best to compute gravimetrically and the small undulatiofls (the fine struc'fure) trigonometrically, if a suitabie triangulation net with enough m a n y astronomical observations is at one's disposal. The gravity anomalies can, of course, give the fine structure of geoid, too. The ~ and ~ method alone has in estimation of the geoid undulations only a small significance, then unsuitable deflection components to and ~o at the initial point as well as wrong reference ellipsoid can induce the undulations of geoid, which do not need to be real. An exemple of this is the geoid of the Alps and of the southeastern part of Europe, computed by A. MARUSSI (Saggio del geoide nel Mediterraneo centrale e nella Zona Alpina, Nota

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preliminare, ~944). This geoid rises in Greece about 30 m above the spheroid at Rome, w h a t hardly can be possible. It seems to me that the reason of this rise of the geoid is : t) the fact that the used ellipsoid of BESSEL is tOO small, i.e. the equator radius to small, and 2) the fact that the topographic-isostatically computed ~o and no values at the initial point Rome, namely ~o -- ~- 2"55 and no ~---- § ,2"69 are, because of the large masspositives of the western Mediterranean basin, too large. Perhaps slight negative values of ~o and no were nearer the truth. In computing of the fine structure of geoid such local geoids as the geoid of MARuss~ are however very important. 3. Dr. W. D. LAMBERT'S Report on Earth Tides had been circulated before the meeting, and was taken as read. 4. The meeting then proceeded to the discussion of these reports. 5. Dr. VENING-MEINESZ invited consideration of the system of KRAS~0WSK~-MOLODE~SKI, w h e r e b y the deviation of the vertical is observed at (say) every 200 kin, while intermediate values are deduced from them in combination with observed values of gravity covering a radius of perhaps t50 km. Dr. VENI~a-MEINESZ gave a short exposition of the method. 6. Dr HEISKANEN remarked that it w a s difficult to k n o w the fundamental deviations at the origin of a survey, b u t several delegates pointed o u t that this was a matter for arbitrary definition, so long as no insistance was placed on the centre of the reference spheroid being at the earth's centre of gravity. 7. Dr de GRAAFF HUNTEa, refering to the KR•SNOWSKI-MOLODENSKI system, remarked that the observation of perhaps ~0 intermediate values of the deviation is probably easier than 70,000 square k m of gravity survey. Brig. BOMFORD agreed, but welcomed the method as a means of getting geoidal sections across sea gaps, w h i c h triangulation might be able to bridge b y RADAR, but in w h i c h intermediate deviation stations might be impossible. 8. Prof. BOA~A asked w h a t probable errors were acceptable in the proposed geoidal sections. Brig. BOMFORD replied that in India they had been content with _ 0".3 of r a n d o m error, and -- 0s.02 (of time) of systematic error for one,observer for (say) 2 months. But that this error w a s expected to be r a n d o m as between different observers, and even for the same observer after a visit to his base station for redetermination of his personal equation. 9. Brig. HOTINE remarked that it was a w k w a r d that ignorance of the form of the geoid produced error in the triangulation, w h i c h in

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turn would produce error in the deviation. He thought an initial determination of the form of the geoid by STOKES' method desirable. t0. Dr de (}RAAFF HUNTER asked Dr HEISKANEN how much of the earth's surface was still devoid of gravity survey, and obtained the reply 91,5 %. tt. Dr. V~T. D. LAMBERT said : MIKHAIL0V visited Washington last Spring. He is as much geodesist as astronomer, and seems to be the contact man for those subjects with the outside world. He told me that the Russians, using the old Bessel spheroid (known to be much too small), and starting their computations at their original origin at Pulkova near Leningrad, had found that by the time their triangulation reached the Pacific, the depression of the spheroid below the geoid, and the consequent error in reduction of ba~e lengths to the spheroid, were extremely large. This was of course to be expected. In mountainous regions, where lines of sight m a y be elevated or depressed by considerable amounts, the deviation of the vertical enforces a correction to horizontal angles. This correction m a y be fairly large, especially as in such regions the deviation may also be large. i2. Prof. Dr. A. PREY presented his paper La determination de dgviations d~ la verticale sans compensation de rgseau. This paper having been circulated among the delegates will not be reproduced in the Bulletin Ggodgsique.

C. - -

WEDNESDAY

26 A U G U S T

13. Dr de GRAA~ HUNTERopening the discussion on the treatment of gravity results for use with STOKES' theorem, said : Geoids are sea-level surfaces of the attraction and rotation forces of the matter, or of a portion of the matter, in the earth. The natural geoid, or simply, the geoid is the sea-level surface of the whole of the matter in the earth. (Some of this matter is external to the geoid.) Topography is expressed in terms of heights found by spirit-levelling or some approximation thereto, such as triangulation; the heights purport to be geoidal. By calculation we can remove the effects of compensated topography on deflections and g-anomalies. In so doing we affect the. potential. The residual matter is bounded by the geoid; but this is not a level surface of the residual matter. Consider the matter from the other end, starting with a

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Cogeoid or compensated geoid which bounds all the matter of which it is the seaqevel surface. Apply to this compensated terrestrial topography; and compute the elevation, ~h, of the geoid at any point above the co-geoid (vide HUNTER and BOMFORD, Bull. Geod., n ~ 29, pp. 20-26). Observe that in this case compensation is reckoned from the co-geoid and not from the geoid. The separation of geoid and co-geoid (which is easy to compute) is unlikely to exceed 20 or 30 metres. As a physical realis the difference between the two geoids as a basis for compensation is small. If however heights from the co-geoid are used in the (reversible) artifice of removing effects of compensated topography from observed deflections and g-anomalies, advantages follow; for example l) the residual matter is precisely adapted for use of Stokes' theorem and for the calculation of deflections of the vertical from gravity anomalies; 2) the complications dealt with by VENI~G MEINESZ in Bull. Geod., n. s6r., n ~ l, and the Bowie indirect effect are eliminated.

Definition. The co-geoid bounds and is a level surface of the attraction and rotation forces of all residual earthly matter after removal of the effects of topography compensated from this surface. This definition is more precise than that in earlier papers

(HUNTERand BOMFORD,Survey of India Geodetic. Reports III and F, t929 and Bull. Geod., n ~ 29, i93'1) written at a time w h e n little attention had been paid to the indirect effect. It it now thought important to particularise the upper level from which compensation exists - - not vaguely sea-leveh The term Co-geoid may suggest compensated or corrected according to taste but it is thought to be more convenient than the full compensated geoid. Father LEJAY has preferred the term isostatic geoid to the original compensated geoid. The abbreviation Co-geoid might also suggest computation geoid, which indeed is its very object. Summarising, the procedure proposed is l) calculate the rise of the geoid, ~h, due to compensated topography at selected points. N.B. Ordinary .spirit-levelled heights, h, m a y be used as approximation to compensated heights H - - h ~-~h. 2) Convert all map heights from h to H by a suitable system of charts, based on (l): map depths d to : D :d

~ h . t~

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G. BOMFORD 3) Eliminate compensated topography, using compensated geoidal heights H. 4) Apply free air height correction for H.

14. Dr VE~NO ME~NESZ remarked that he went into this carefully when making his observations at sea. At first he thought that the mass ~h should be compensated, but later he became doubtful and concluded that it was not compensated. If the objeeti.ve was to find the geoid bv STOKES' theorem, he thought it was quite satisfactory to compensate ~h or to move it below the co-geoid. But if the co-geoid was itself considered to be an object of interest, e.g. as a guide to the figure of the earth, that would then not be good, because matter would have been added which did-not exist in nature. Also the centre of gravity would have been moved oft the earth's axis, the rotation would become uneven, and normal gravity formulae would become inapplicable. Such gravity anomalies could not then be used for discussing the question of the triaxial ellipsoid, or the earth's mechanical equilibrium. Dr VENING MEINESZ mentioned his own formula, namely ~h ~ 0,11~ (Th --[-2,95 h 2) where T is the thickness of the crust, T and h both being expressed as fractions of the earth's radius. He claimed that its use avoided all the foregoing inconveniences. ([5. R. P. LEJAY found himself able to agree in great measure with both Drs de GRAAFF HUNTERand V E ~ G MEINESZ,(laughter), although he did not quite follow all that the latter had just said. He thought that it was not moving the l a y e r ~h that moved the earth's centre of gravity, so much as the removal of the topography and compensation. t6. Dr VENINO MEINESZ, interposing, disagreed. Removing the topography and compensation admittedly moved the centre of gravity, but the layer ~h brought it back to its original position. 17. R.P. LE,~AY disagreed and further discussion produced n o reconciliation of the two different points of view. Dr VENIN~ MEINESZ then had to leave to keep a previous engagement, and further prosecution of the subject was postponed. 18. Dr de GRAAF~ HUNTER, introducing the next item of the agenda, said that in 1935 he had given much consideration to what would be a minimum useful distribution of gravity stations for the application of STOCKES' theorem. The answer largely depended on how far a single station could he .~aken to typify the whole of (say) a 5 ~ X 5 ~ square. Three separate areas required consideration : a) that within (say) 30 km of the station. This could now be dealt with easily by gravimeter;

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b) between 30 km and 20 ~ In a country w h i c h happened to be gravimetrically surveyed, such as U.S.A., India, or the Netherlands East Indies, adequate data might exist; c) beyond 20 ~ Only 8 % of the earth was as yet surveyed, and adequate data clearly did not exist. Before STOKES' theorem could usefully be applied it w a s essential to k n o w what error was liable to result from any given scarcity of data. It was also necessary to consider w h a t accuracy was needed. Should a computed deviation be correct to l " or 5" or w h a t ? W o u l d the present discrepant European origins be satisfactorily reconciled if deflections at them could be computed with an error of l " ? Clearly not. He had attempted to a n s w e r these questions in the Philosophical Transactions o[ the Boyal Society in 1935, but was not entirely satisfied with the result. 19. Mr C. A. WHITTEN reported some w o r k recently done in the U.S.A. Three existing astronomical and geodetic stations lay on a line about 100 miles long, in the centre of an area of general gravity survey about 300 b y 300 miles. The deviations at each computed b y STOKES' theorem, and the observed A-G values, w h i c h varied considerably + and - - , were correctly reproduced b y STOKES within U'. The effect of successive zones on the computation were noted, and as the radii reached about 100 miles, the effects at the three stations were seen to be getting smaller and more similar. It w a s concluded that little change would have occured, if it had been possible to take the computations out further. The result gave some confirmation of the accuracy of Ihe North American fundamental datum. 20. R . P . LEJ'AY stated that the outer zones were liable to have serious effects, and that it could only have been b y chance that correct results were obtained. Relative values at the three stations should be determinate, b u t a considerable liability to c o m m o n error must remain. 2t. Dr NhRGAARD asked w h a t was the effect of the 100 metres nearest the station.

22. Mr WHITTEN. Practically nil. 23. Prof. BAESCHLIN. W h e n the zones get out as far as the Pacific, very large areas will be found with appreciable anomalies of constant sign. 26. Mr WHITTEN agreed that the outer zones ought not to be neglected, but it w a s no good including zones w h e r e the data w a s not reasonably adequate. 25. Prof. •AESCHLIN. DO delegates generally agree that w h e n gravity observations are used for STOKES' theorem, they are best reduced

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G. BOMFORD

isostatically, in order that a single station may most accurately typify a large area ? 26. R.P. LEJAY, Dr de GRAAFF HUNTER and expressed general agreement.

Brig. BOMFORD

27. Mr B. C. BROWNE said he had recently been studying a large number of gravity results in the United Kingdom and in the sea to the south-west of it. Over the Atlantic topographical anomalies averaged + 300 mgals while isostatic anomalies were generally between + 20 and - - 30. Viewed broadly, the area was 90 % compensated, but the outstanding 20 or 30 mgals were not negligible, and they extended over large areas. In the U.K. itself rapid changes of 5 or i0 mgals frequently occured in distances of i0-20 miles, and no isostatic correction could smoothe them out. It was noticeable that the scatter of anomaly values was rather greater over that part of the Atlantic than over the U.K. The mean deviation over the sea was -----i6 regals. He would not like to be quoted as agreeing with Prof. BAESGHLII~'S suggestion. D. - - THURSDAY 26 AUGUST 28. The following resolution which had been discussed on August 2~ was proposed by Brig. BOMFORD,and passed for recommendation to the plenary session without further discussion. Resolution

The International Association of Geodesy l~esolves : a) that when the deviation of the vertical is observed for the purpose of determining the form of the geoid, such observations are best spaced at intervals of generally between ~5 and 30 km along continuous lines, and : b) that as the progress of geodetic triangulation makes it possible, (as is already the case to a great extent), such geoidal sections should be observed at least along the following general lines : 1) From the Arctic Ocean through Turkey and Egypt to South Africa, 2) From the British Isles eastward through Europe and the U.S.S.R., across Asia to the Pacific, and eventually to the Behring Straits, 3) From the Behring Stairs through Alaska and southwards through North and South America, with a branch line through the U.S.A. to eastern Canada,

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4) From Turkey through India to Singapore, with eventual extension to and through Australia. 29. Dr K. MADER distribued copies in English and French of his paper La d~termination d'une dldvation du gdo~de d'apr~s des ~mesures [aites a la balance de torsion d'EStvSs, and gave a verbal s u m m a r y . The paper described w o r k done in Austria to determine the form of the geoid underlying an isolated rectangular mountain mass. 30. Dr VENIN~ MEINESZ asked whether when evaluating the double integral, the first integration did not introduce an u n k n o w n value ~U of , so that the geoidal heights were determined with respect to a ~y reference surface w h i c h w o u l d be inclined to the spheroid at an u n k n o w n angle. 3t. After discussion, it w a s agreed that such an u n k n o w n constant would arise as the result of u n k n o w n distant masses and anomalies, but that Dr MADER had obtained correct relative values of the deviation within the area he w a s studying. 32. Prof. A. MARUSSI presented his paper Fondements de gdomdtrie di[fdrentieUe absoIue du champ potentiel terrestre, and gave a full s u m m a r y of it. This paper is published in this n u m b e r of the Bulletin Geoddsique (p. 4tt-440). 33. Dr de GnA,~F~ HUNTER invited Dr VENING-~[EINESZ to give a s u m m a r y of the w o r k of Baron de Vos van STEENWIJK on the determination of the geoid from STOKES' theorem as applied to gravity observations in the Netherlands East Indies. 34. Dr VENING-MEINESZ explained that Baron de Vos had obtained one-metre contours of the co-geoid, and that in places they had the very steep gradient of 30 metres in 200 kin. The contoured geoidal chart w h i c h the Baron had drawn, did not go up to the margin of the gravity survey, but would of course be influenced b y the u n k n o w n gravity anomalies outside it. His map. might properly be raised or lowered, tilted, or even twisted, but the sharp irregularities s h o w n w o u l d be correct. Computationally, a s s u m i n g the gravity data used to be correct and fully representative, his values of the deviation should be correct to 0".5. It w a s almost impossible to assess the possible error arising from the u n k n o w n part of the gravity field, but 1" might perhaps be the w o r s t error to be feared. 35. Dr de GRAAFF HUNTER thanked the speakers and declared the meetings of the section closed.