Geodesy

Under the general heading of Geodesy and the subtitle Localização das Cartas Hidrográficas do Arquipélago da Madeira no Elipsóide Internacional, Mr Pires de Matos, a distinguished officer of the Portuguese navy, published a long and valuable article in the Anais do Club Militar Naval (Jan.–Feb., 1940). Its length does not allow us to reproduce it in full in these pages, so we shall transcribe only a few brief passages, in order to acquaint our readers with this subject, to whom it may perhaps be of some interest.

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In 1914 the Directorate of Geodetic and Topographical Works (Direcção dos Trabalhos Geodésicos e Topográficos) established a triangulation on the island of Madeira. Its main use and immediate purpose was the survey of the island’s chorographic map, carried out in 1915, revised and updated by the Geographical and Cadastral Institute (Instituto Geográfico e Cadastral) in 1934 and published in 1938 at a scale of 1/50 000.

The base of the triangulation was measured on the Paul da Serra (‘upland marsh’), and the survey marker of the Fort of São Tiago (St James the Greater) was taken as the origin of coordinates. This marker was built near the spot where an American Mission made observations of astronomical latitude and longitude in 1892, and the Directorate of Geodetic and Topographical Works observed the astronomical azimuth of one of the sides of the triangulation at the Fort of São Tiago.

With the exception of the azimuth, therefore, it was the astronomical observations of the American Mission that were used to calculate the geographical coordinates of the island of Madeira.

In 1936 the need was recognised to revise the 1914 triangulation, rebuild some destroyed survey markers, triangulate Porto Santo (‘holy harbour’) and the Desertas (‘deserted islands’) and link the islands of the archipelago, with the immediate aim of a chorographic survey of Porto Santo and the Desertas and a hydrographic survey of the Madeira archipelago. In this new work, carried out in 1936 by the Geographical and Cadastral Institute, some vertices of the old 1914 triangulation were abandoned. At the same time, working in parallel with the Institute, the Hydrographic Mission extended the triangulation to the cliff line and to the landing places where hydrographic plans were surveyed.

The Geographical and Cadastral Institute abandoned the old origin at the Fort of São Tiago and made astronomical observations – latitude, longitude and azimuth – at the astronomical marker on the island of Porto Santo.

Although this marker is better placed than that of São Tiago, at least as regards latitude, everything suggested that the observations would still be affected by a considerable deflection of the vertical, and that consequently, if the astronomical observations of this new origin were adopted, appreciable errors would be made in placing the whole on the ellipsoid.

If we had as the origin of coordinates a marker at which the effect of the deflection of the vertical was not felt, if the triangulations were rigorously measured and calculated, if the surfaces of the geoid and the international ellipsoid fitted perfectly, and if there were no disturbing causes or observational errors, the geographical coordinates of any vertex carried over geodetically would always equal the astronomical coordinates observed directly at it.

A single astronomical station would give us particular geographical latitudes, longitudes and azimuths for the geodetic vertices and sides. Each vertex will then have as many systems of coordinates as there are origins chosen.

The Hydrographic Mission had two systems of geographical coordinates at its disposal: the first corresponding to the 1914 triangulation, with its origin at São Tiago, and the second to the 1936 triangulation, with its origin on Porto Santo. Although the first of these two systems was used provisionally, neither was adopted in the definitive publication of the charts.

To shed more light on this problem and solve it, the Hydrographic Mission decided to make astronomical observations at the following survey marks:

The astronomical observations at these marks were made with a universal theodolite acquired in 1937 from the German firm Askania Werk A. G. Bambergwerk, listed in its catalogues as Universal Instrument A. U. Z. 21 C.

The Talcott method was adopted for observing latitude; with some 15 pairs of fundamental stars from the Eichelberger catalogue, probable errors of less than 0".10 were achieved. Longitude was obtained by comparing the chronometer states referred to Greenwich, received by radiotelegraphy and calculated by timing the chronometer at the coincidences of the rhythmic signals and the exact seconds of the chronometer, with the states referred to local time. These were obtained from about 6 series of 12 stars, 6 in each series being observed with the eyepiece to the east and 6 with the eyepiece to the west. The azimuth was obtained by observing circumpolar stars at elongation.

Once these astronomical data had been gathered, the problem was to adopt methods of calculation by which all the astronomical stations listed above could help us assign the most probable geographical coordinates to the vertices of the geodetic network under study.

In the present state of geodesy the problem, in its absolute form, cannot be solved, but that circumstance is no argument against adopting one of the various existing methods to reach the desired end. While recognising the force of the criticisms made of the hypothesis of isostasy formulated by Pratt and adopted by Faye, Airy and other scientists, we nevertheless decided to follow Hayford’s method, which rests on the hypothesis of isostatic compensation and which served to define the international ellipsoid.

Although the results obtained cannot legitimately be presented as definitive, they are undeniably of interest. Since some of the charts used for Madeira give heights and depths in metres, we have:

D = 0"000328 times the height in metres; D = 0.000202 times the depth in metres. The charts used to calculate the topographic deflections were as follows:

These charts required the construction of compartment-distribution diagrams at the respective scales, taking into account the projection used.

From the established ratio r'/r1 = 1.426, it can be seen that the radius bounding a compartment on the outside is obtained by multiplying the inner radius by 1.426.

Each pair of consecutive circles forms a ring comprising 16 compartments.

Given the scale of the charts, the innermost ring we used was number 30, bounded on the outside by a radius of 134.2 m. The outermost ring reached by the compensation was number 6, with an outer radius of 670.8 km.

From the results obtained in the Madeira archipelago we can conclude that, where astronomical data cannot be obtained at a large number of stations, it is highly advantageous to choose stations on ground that is topographically not very rugged. It should not be forgotten that rugged topography very close to a station calls for sufficiently detailed charts, which cannot always be obtained easily. The more extensive the level area around the station, the smaller the scales of the charts needed to achieve a given degree of accuracy.

On a very rugged island, the astronomical survey marker should be placed in a high area, preferably on a plateau, around which the distribution of masses is as symmetrical as possible.

In an archipelago, the island with the least rugged relief should be chosen for astronomical observations.

Under the conditions defined above, the correction applied to the observations at the origin marker seems to us sufficient.

Because the reference axes are unstable and difficult to relate to one another – which is possible only approximately, and after a great deal of work – astronomical traverses can serve us well only in mere reconnaissance work.

Variations in the deflection of the vertical are greatest in coastal regions. This shows us how dangerous it is to use an astronomical traverse in such regions to produce charts that are meant to represent the topography of the ground with a certain care and accuracy. Over distances of two hundred kilometres we may make errors sometimes exceeding 500 metres on coasts such as those of Angola and Mozambique.

The astronomical vertices could be corrected for the effects of the deflection of the vertical, but even in regions where sufficient charts are available, the method is so laborious that it is not to be recommended.

If adjustment makes us lose part of the accuracy achieved thanks to the modern instruments employed and to lengthier observations and calculations, then, according to some opinions, substantial spending on good instruments and the greater loss of time on better methods of observation and calculation are not justified.

I disagree entirely, because I believe we must not lose sight of the fact that in work of this kind there are two aims to be achieved:

First – to place the charts as accurately as possible on the international ellipsoid, in the light of the methods of investigation known at present. Second – to gather an ever larger body of data for the present and, above all, future study of the shape of the Earth.

While any modest theodolite is sufficient for the first aim, the second can be fully achieved only if we use in our work the best modern instruments possible and good methods of observation and calculation.

In my humble view, those to whom the State entrusts posts of scientific research must never forget that their services contribute to the growth of human knowledge not only through the immediate, practical purpose of their work, but also through its more distant purpose, sometimes less apparent but of great future importance.

And, besides, the truth is that the small increase in expenditure involved is no obstacle, even when the services have modest budgets.