ON ISOSTASY AND GRAVITY

M. Polikarpov.

The present article, being a supplement to Prey’s review, “The Theory of Isostasy, Its Development and Results,” published in Advances in the Physical Sciences (Vol. VI, issue I), presupposes the reader’s acquaintance with the fundamental propositions of the theory of isostasy.

The material for the article offered to the reader was the extensive “Investigation of Gravity and Isostasy,” carried out by Heiskanen1 (W. Heiskanen). As a result of his calculations, Heiskanen arrived at a number of interesting conclusions both concerning the state of equilibrium of the most remarkable regions of the earth’s crust in the gravitational respect (the Caucasus, the Harz, the Alps, etc.) and concerning the form of the earth as a whole. A more detailed exposition of Heiskanen’s work (than was given in the above-mentioned article by Prey) is justified by the fact that the author dwells chiefly on the picture of the distribution of the force of gravity in Europe, which is of extraordinarily great interest from the geophysical point of view.

Investigations of the force of gravity in the United States have undoubtedly demonstrated the necessity of introducing, for each station, corrections both for the influence of the relief of the entire terrestrial globe and for the isostatic compensation of this relief. For American stations this work, carried out by Hayford and Bowie, made it possible to draw a whole series of extremely interesting and important conclusions.

For Europe and the Caucasus the same work was recently carried out by Heiskanen, but, unlike the American investigators, who used for their calculations the generally known hypothesis of Pratt, Heiskanen introduced the isostatic reduction for the European and Caucasian stations not only according to Pratt, but also made use of the hypothesis of the English astronomer Airy.

1. Introduction. Concerning the isostatic compensation of the earth’s crust, as is known, there are two hypotheses. According to the first hypothesis, the density of the rocks lying beneath mountains is less than the density of the rocks beneath oceans, so that the excess of surface masses in the first case and the deficiency of masses in the second are correspondingly compensated by subsurface masses extending down to a certain definite depth, where the so-called “isostatic layer” is located. Below the isostatic layer there is already hydrostatic equilibrium, so that each unit of the surface of this layer experiences the same pressure. The second hypothesis says that beneath the earth’s crust there is a layer of dense lava in which mountains and continents float, like icebergs in the sea. According to the latter supposition, isostatic compensation should occur beneath oceans sooner than beneath mountains. Thus, the first hypothesis is characterized by the depth of occurrence of the “isostatic layer,” the second by the thickness of the floating earth’s crust and by the difference of densities between the earth’s crust and the layer of lava.

The first hypothesis was put forward by Pratt, who, having calculated the deflection of the plumb line caused by the Himalayan range, found that the value obtained was greater than that actually observed. To explain this fact he assumed that beneath the Himalayas there is a deficiency of mass, which causes a reduction in the deflection of the plumb line. The first to put forward the second hypothesis was the English astronomer Airy.

All geodetic investigations of isostasy show that assumptions concerning isostatic compensation of terrestrial masses reduce both the magnitude of deflections of the plumb line and the magnitude of anomalies of gravity.

At the present time the doctrine of isostasy is no longer a mere hypothesis, but may be regarded as an established theory. The only unresolved question is: which of the hypotheses (Pratt’s or Airy’s) better corresponds to reality. For this reason, all attempts to compute the depth of the isostatic layer according to Pratt, and the thickness of the earth’s crust and the density of lava according to Airy, are of great interest.

Geodesists make exclusive use of Pratt’s hypothesis, since, while leading in a geodetic sense to the same results as Airy’s hypothesis, it is considerably simpler for mathematical treatment. Moreover, Hayford and Bowie assume that at the earth’s surface there is complete compensation, i.e. that every elevation and every depression, however small they may be, corresponds underground to a deficiency or an excess of mass.

Under these assumptions Hayford, in 1909, obtained from deflections of the plumb line in the United States a depth of the isostatic surface of 113 km, and, secondarily, in 1910, likewise from deflections of the plumb line, 122 km. Bowie, on the basis of deflections of the plumb line and measurements of gravity in the United States, gives the value 96 km. Helmert obtained his value, 118 km, by a somewhat different method. He investigated changes in gravity at steep coasts and computed the depth of the isostatic surface that could best explain the gravity anomalies present there. All the computed depths of the isostatic surface agree with one another fairly well.

Hayford computed tables by means of which it was possible to introduce a topographic-isostatic reduction for the deflection of the plumb line, if only one topographic reduction is known. As for the topographic-isostatic reduction for gravity, in the works of Hayford and Bowie there are tables by means of which one can obtain the topographic and isostatic reduction for an isostatic depth of 113.7 km, if the mean heights of the separate zones surrounding the station are known. Bowie, in addition, computed formulas and gave tables that make it possible to determine compensations for any depth, if these are known for 113.7 km. Thus Bowie gives tables for depths: 42.6, 56.9, 85.3, 127.9, 156.25, and 184.8 km.

Pratt’s hypothesis meets with objections chiefly from geologists, since the depth of the isostatic layer calculated according to this hypothesis is considerably greater than that which can be admitted from the standpoint of geology and geophysics. But for geodesists this assumption of Pratt’s, as has already been said above, is very useful, since it gives a fairly simple method for computing isostatic compensation.

Geologists take a different point of view. They begin their reasoning with geological facts and theories and investigate to what extent their conclusions are confirmed by geodetic data. Geologists consider Airy’s hypothesis the more correct one. This follows from the works of Kober, Kosmat, and Born. Geophysicists also find Airy’s hypothesis more probable, and the well-known German geophysicist Wegener based his theory of floating continents on Airy’s hypothesis.

Up to the present time no attempts have been made to compute gravity anomalies and deflections of the plumb line on the basis of Airy’s hypothesis, nor has any comparison of it with Pratt’s hypothesis been carried out.

The hypothesis of Pratt, refined by Hayford and Bowie, explains very well the anomalies of gravity and the deflections of the plumb line in the United States. But if we consider the map of gravity anomalies remaining after the introduction of the orographic reduction, it may be observed that in some places in Europe, and especially in the Caucasus, the remaining anomalies cannot be explained by means of isostasy. Thus, it turns out that almost the entire Mediterranean Sea has a positive anomaly reaching up to \(0.1\ \mathrm{cm/sec^2}\), while the Caspian Sea has a negative gravity anomaly of the same magnitude.

In any case, there is no doubt that Europe and the Caucasus belong to regions that are highly interesting from a geophysical point of view and deserve detailed isostatic investigations.

Heiskanen sets himself the aim: 1) to study the gravity anomaly in the Caucasus and in certain interesting regions of Europe under the assumption of isostasy, 2) to clarify the question of which of the hypotheses, that of Pratt or Airy, better explains these anomalies, and finally, 3) to compute, with the aid of all stations for which isostatic reductions are available, a formula for the normal distribution of gravity on the earth and for the flattening of the terrestrial spheroid.

2. Method of reduction.

The numerical values of the accelerations of gravity for stations subjected to isostatic reduction were taken by Heiskanen from the well-known work of Borrass. In this work Borrass gave all observations of gravity known to him in the so-called “Potsdam system,” so that the numbers given by him are quite comparable with one another.

In order that the observed values of gravity \(g\) may be compared with the theoretical values \(\gamma_0\), the quantities \(g\) must be reduced. Depending on the degree of accuracy with which one wishes to take into account the causes affecting the measured gravity, four reductions are used: 1) the “free-air” reduction (Freiluftreduktion), 2) the Bouguer reduction, 3) the purely topographic reduction, or 4) the topographic-isostatic reduction.

The “free-air” reduction, often called the “height reduction,” transfers the observed value \(g\) from the plane of observation to sea level, taking into account only the height of the station. This correction will be:

\[ \frac{2H}{R}g = 0.0003086 \times H, \]

where \(H\) is the height of the place of observation in meters, and \(R\) is the radius of the earth in the same units. Quantities reduced by this formula are usually denoted by \(g_0\).

To obtain gravity at sea level, one must take into account the attraction of the rock masses lying between the point under investigation and sea level.

In addition, the irregularities of the terrain around the station must be taken into account, i.e. a further “terrain” correction (orographic) must be introduced. This correction arises because at the summit of an isolated mountain gravity is less than on a plain of the same height, since in the first case there are absent the rock masses which in the second case surround the observing station, rising up to the height of that station. This correction, therefore, must be positive; but for plain stations it must have the same sign, since the masses lying above the station diminish the magnitude of gravity. The positive correction \((g' - g)\) may be neglected in localities with weakly expressed relief, but in mountainous regions it must be taken into account, since it sometimes exceeds \(0.020\ \mathrm{cm/sec^2}\).

If we combine all three corrections, we obtain the Bouguer reduction:

\[ g''_0 = g + \frac{2H}{R}g\left(1 - \frac{3}{4}\frac{D}{D_m}\right) + (g' - g), \]

where \(D\) denotes the density of the intervening layer of rocks and \(D_m\) the mean density of the earth.

The Bouguer reduction should be preferred to the simple “free-air” reduction, since in the former case the influence of the intermediate layers lying between the observation station and sea level is taken into account. This reduction can be computed just as easily as the “free-air” reduction (if, of course, one disregards the “local” correction, which in most cases is very small): for this it is sufficient to multiply the “free-air” reduction by the factor \(\left(1-\frac{3}{4}\frac{D}{D_m}\right)\).

The Bouguer reduction will be the more correct the larger the region surrounding the station for which the orographic correction is taken into account. But since usually the terrain correction is computed for a region of 100–200 km, and since the Bouguer reduction does not take the curvature of the earth into account, it is not always sufficiently accurate. In order to obtain a reduction that would take account of the irregularities (both continents and oceans) of the entire terrestrial globe, it is necessary, after determining the mean heights of all zones surrounding the station, to compute their influence. This will be a purely topographic (orographic) reduction. Its formula is written as follows:

\[ g_t = g + \frac{2H}{R}g + \delta g_t, \]

where \(\delta g_t\) denotes the influence of the topography of the entire terrestrial globe.

If, in addition to the attraction of all external masses, we also took into account the action of underground masses compensating the deficits or excesses of surface masses, we would obtain the “topographic-isostatic” reduction:

\[ g_i = g + \frac{2H}{R}g + \delta g_t + \delta g_i, \]

where \(\delta g_i\) corresponds to the attraction of the underground compensating masses. Depending on which hypothesis of isostasy we adopt, we obtain either the Pratt–Hayford reduction or the Airy reduction.

To obtain normal values \(\gamma_0\), Heiskanen used Helmert’s formula (1901):

\[ \gamma_0 = 978.030(1 + 0.005302 \sin^2 \varphi - 0.000007 \sin^2 2\varphi), \]

where \(\varphi\) is the geographical latitude of the observation station.

Depending on whether we compare \(\gamma_0\) with \(g_0\), \(g_0''\), \(g_t\), or \(g_i\), we obtain anomalous values of gravity corresponding to the “free-air” reduction (Bouguer), the purely topographic reduction, or the topographic-isostatic reduction.

In this work, for the computation of the topographic-isostatic reduction, Heiskanen applied Hayford’s method, which consisted in dividing the whole earth into separate regions by concentric circles (the station being taken as the center) and by radii issuing from the center; for each of these regions the mean height or mean depth was determined. The effect of these separate regions was computed with the aid of Hayford’s tables. It should be mentioned here that the depth of the isostatic layer of 113.7 km is reckoned in Hayford’s tables not from the surface of the sea, but from the physical surface of the earth. The first zones, nearest to the station, bear the letter designations \(A—O\), and the remaining ones the numerical designations \(18—1\).

The heights and depths were established in the usual manner: transparent paper with the zones drawn on it, at the scale of the map, was laid over the map, and the mean height of each zone was read off.

Usually the effects of the zones \(A—O\) (the outer radius of zone \(O = 166{,}700\) meters), and often also of zones \(18—9\), were established separately for each station. Only in the reduction of stations lying close to one another was interpolation resorted to. Since the effect

ON ISOSTASY AND GRAVITY

Since zones 8—1 generally vary little from point to point, the influence of these zones was calculated for only one station; the remaining stations were determined by interpolation. If near stations adjusted by Heiskanen there were stations previously computed by Bowie, as for Harrah, the Giant Mountains, etc., then the influence of the more distant zones, sometimes beginning with the 13th zone, was determined by interpolation from Bowie’s data.

Hayford and Bowie carried out their calculations under two assumptions concerning the character of compensation. The first hypothesis, called “local compensation” (lokale Kompensation), assumes that the compensating underground masses are uniformly distributed down to a certain depth and lie strictly beneath the compensated irregularities of the earth’s surface. According to the second hypothesis—the hypothesis of “regional compensation” (regionale Kompensation)—the compensating masses extend not only in the vertical but also in the horizontal direction, and large areas of the earth’s surface are compensated at once, not individual insignificant parts. Bowie uses three assumptions, according to which “regional” compensation extends to 18.8, 58.8, and 166.7 km from the station, while beyond these limits ordinary “local” compensation takes place. As a result of his calculations he comes to the conclusion that the first two assumptions lead to almost the same results as “local” compensation, and that the third assumption gives worse results than the assumption of simple “local” compensation.

Let us turn to Airy’s hypothesis.

According to Airy’s hypothesis, the lighter earth’s crust floats in a heavier lava-like layer. The first is called the “Sal” layer; the second layer—the “Sima” layer, after the chemical elements that enter into them. The earth’s crust is submerged in the “Sima” layer to a lesser depth beneath the oceans than beneath mountain ranges. If we assume that there is no compensation at sea level, then it is easy to understand that beneath mountains there is negative compensation, while beneath deep oceans there is positive compensation.

To facilitate the calculations, Heiskanen assumes that the densities of the “Sal” and “Sima” layers are constant and that one layer passes into the other abruptly. For the mean density of the earth’s crust the value 2.67 is taken, and for the difference in densities between the above-mentioned layers the values 0.2, 0.3, and 0.6 are adopted. It may be objected that this hypothesis of homogeneous layer densities does not correspond to reality, and that the density of the layers should be a linear function of depth, but this latter assumption changes the general result very little.

Heiskanen carries out his calculations under the following four assumptions, collected in the table given below. In the first two columns are given: the assumed thicknesses of the earth’s crust, corresponding to sea level, and the differences in densities between the “Sal” and “Sima” layers. If the mean ocean depth is taken as 3680 m, and the mean height of the continental masses as 800 m, and if it is considered that \( \frac{2}{3} \) of the earth’s surface is covered by water, then for the mean thickness of the earth’s crust we obtain the values given in the 3rd column, while the thickness of the earth’s crust beneath the oceans and the continents will be different; their values are collected in the last two columns:

We see that the average thickness of the earth’s crust, under these assumptions, fluctuates between 35 and 60 km, which is quite probable.

To obtain the Eris and Heiskanen reductions, auxiliary tables were computed for the influence of the zones \(A - O\) as a function of the mean height of these zones. In zones 12–1, the reduction according to Hayford’s hypothesis with the depth of the isostatic layer \(2T\) has almost the same value as the reduction according to Eris’s hypothesis with the mean depth of compensation \(T\); thus in these cases one may use the tables of Hayford and Bowie. Heiskanen did so.

The results of the calculation of the various reductions and of the corresponding anomalies are collected in a whole series of tables. The following anomalies are usually given: 1) “in free air,” 2) Bouguer + topographic, 3) Hayford (113.7 and 156.3 km), 4) “regional” (regionale) and 5) Eris—for the 1st and 2nd assumptions (sometimes also the 4th).

3. Anomalies of the force of gravity.

a) Let us first consider the Caucasian group V, containing 8 stations on the Caspian Sea, where the forces of gravity are considerably smaller than the theoretical value.

Here we find, on Zhiloi Island and in Baku, the largest negative anomaly, reaching \(0.120\ \mathrm{cm/sec^2}\) (these anomalies correspond to Hayford’s 1st hypothesis—113.7 km). From these stations the region of negative anomaly extends in all directions. To the south the negative anomaly ends unexpectedly only between the station Alat (\(-0.077\)) and the station Lenkoran, lying only 130 km south of Alat, where there is already a positive anomaly \(+0.085\). To the east (the region of negative anomaly does not extend) on the eastern shore of the Caspian Sea, opposite Zhiloi Island, we find an anomaly \(+0.055\). To the north the negative anomaly extends as far as Petrovsk (\(-0.031\)), but already for Astrakhan we have a positive anomaly \(+0.048\). A small negative anomaly is also present to the west of Zhiloi Island and Baku, inland, as in Shemakha \(-0.007\) and in Grozny, in the Northern Caucasus, \(-0.017\ \mathrm{cm/sec^2}\).

Consequently, the negative anomaly of gravity and the mass defect at the Caspian Sea are a purely local phenomenon, occupying a very limited area.

The other groups include the following stations (for the Caucasus): group I—13 stations of the Caucasus mountain range, II—17 stations of Armenia, III—12 stations between the main range and Armenia, IV—12 stations of the Northern Caucasus, and VI—9 stations on the Black Sea.

Since mountain stations, almost everywhere and for every method of reduction, give mean values of anomalies different from the corresponding values for stations situated in lower places, it is of interest to express in the form of a linear function the dependence of these anomalies on height. Such formulas, corresponding to 9 different methods of reduction, were derived by Heiskanen for the Caucasian stations.

For the “free-air” reduction, the Bouguer reduction, and the purely topographic reduction, the mean deviations of the individual anomalies from their mean (group) value are considerably greater than for the other methods of reduction, in which underground compensation is taken into account. The linear dependence of these anomalies on height is also much more significant than for the isostatic reductions.

We may summarize the foregoing as follows: reductions without the assumption of isostatic compensation do not correspond to the course of the variation of gravity in the Caucasus.

Both reductions according to Eris (according to the 1st and 2nd assumptions) give somewhat better agreement with the observations than any other reduction; thus, the mean deviations of the individual anomalies from their mean value will be considerably smaller than with other methods of reduction, and the term depending on the height of observation, under the 1st assumption, disappears completely and is very small under the 2nd assumption.

From both assumptions of Airy, the first gives less good agreement with the observations than the second. Although the heights of the Caucasus and Armenia vary between 0 and 2,000 m, the thickness of the earth’s crust, according to Airy’s first assumption, is 77–104 km.

From all that has been set forth above we may conclude: in order to explain the variations of gravity in the Caucasus it is necessary to take into account the existence of isostatic compensation. The topographic-isostatic reduction according to Airy, with a thickness of the earth’s crust of 77–104 km, best agrees with the observations. If we wished to obtain equally good agreement by using the reduction according to Hayford, it would be necessary to operate with depths of isostasy of about 250 km, or to assume the existence in the Caucasus of “regional” compensation. The Apennine peninsula and its environs are regions with local mass defects.

b) The first group of European stations, studied by Heiskanen, comprises 11 stations of the Harz. Heiskanen set himself the goal of verifying the generally accepted opinion that these mountains are isostatically uncompensated. All 11 stations were chosen so as to cover the entire Harz region with a uniform network. To resolve this question, it was necessary, using the reduction of Hayford or Airy, to regard only the Harz region as uncompensated, and to compare the anomalies obtained under this condition with the above-mentioned anomalies of Hayford or Airy.

The mean values of the anomalies in the Harz and its environs show equally good agreement with each other both under the assumption that the Harz is compensated and under the assumption that it is not compensated. The mean deviation of the anomalies from the mean value and the dependence of the anomaly on the height of observation, under the assumption of non-compensation of the Harz, are somewhat smaller than under the usual reduction according to Hayford, but the difference here is so small that on the basis of this material nothing can be decided definitively; all the more so because the reduction according to Airy agrees just as well with the observations as does the reduction under the assumption of non-compensation of the Harz.

On the basis of all his computations Heiskanen arrives at the following conclusion: gravity in the Harz, after the introduction of the isostatic reduction, is normal, and the Harz does not constitute any special exception to the general doctrine of isostasy.

Although geological data indicate that the Harz is not compensated, investigations of gravity do not confirm this.

The treatment of 8 stations in the Giant Mountains, which entered into the second group of European stations, shows quite undoubtedly that these mountains are fully compensated, so that Helmert’s assertion of incomplete compensation of the Giant Mountains was not confirmed. It is even possible to consider that in the Giant Mountains there is rather an excess of compensation than a deficiency of it.

Passing to the stations in the region of the marginal subsidence of the Alps. Geologists assert that not only mountain chains, but also the regions of their marginal subsidences are compensated (Kossmat). Thus, in the region of marginal subsidences the anomaly is negative, while in the region of mountains it is positive (relative to the surroundings). To examine this question, the distribution of gravity was studied, besides the marginal subsidence of the Alps, also in the same regions of the Carpathians and the Caucasus.

Ten reduced stations were selected along the marginal subsidence of the Alps from Vienna (16°21′.5) to Freiburg (7°50′.9). The mean values of the anomalies for these stations, under the reduction according to Hayford, are: +0.025 and +0.026, and under the reduction according to Airy: +0.030, −0.031, and +0.032 cm/sec².

Investigations of gravity in the region of the marginal subsidence of the Alps and in the Alps themselves compel the conclusion that the anomalies computed according to Hayford and Airy for the marginal subsidence will be positive with respect to the ridge itself, so that the above-mentioned assertion of geologists is not confirmed by measurements of gravity in the environs of the Alps; here, too, there is rather a deficiency of compensation than an excess of it.

The study of gravity in the Carpathians and in the Caucasus, as well as in their surroundings, has shown that, in general, in the region of marginal subsidences of mountain chains, an excess of compensation is not observed; on the contrary, the variation of gravity here is normal and fully follows from isostatic compensation.

Of the other stations reduced by Heiskanen, the first group consists of the stations of Italy. These stations show that gravity is increased here. All stations have a positive anomaly greater than \(+0.060\ \text{cm/sec}^2\), and at the station Brindisi the anomaly reaches even \(+0.110\ \text{cm/sec}^2\).

All these stations are concentrated in a very small area. Other stations on the Mediterranean Sea and its islands—in Algeria, Sardinia, Corsica, and Dalmatia—show that the entire region from Spain to Dalmatia is a region of excess gravity. Thus, beneath the Mediterranean Sea there is positive overcompensation. It is possible that the deficiency of gravity in France, Spain, and inland Algeria is connected with the excess of masses beneath the Mediterranean Sea.

The station Sörvaagen in Norway, reduced by Bowie, with its anomaly of \(+0.134\), gives a not entirely correct picture of the course of gravity in Scandinavia, since already at \(50\ \text{km}\) from this station there is an anomaly of \(-0.050\). Throughout Scandinavia the anomaly is generally negative, which is connected with the fact of uplift of the continent1.

The processing of 11 stations on Spitsbergen showed that the gravity observed there is normal and corresponds to the doctrine of isostasy. These stations are of great interest in view of their considerable latitude.

4. Comparison of Hayford’s Hypothesis with Airy’s Hypothesis

Above we have already had occasion to be convinced that, for the Caucasus, the observed values of gravity correspond somewhat better to Airy’s hypothesis than to Hayford’s hypothesis, which assumes that the depth of the surface of isostasy is \(185\ \text{km}\). Heiskanen set himself the aim of testing, on more extensive material, which of these hypotheses better corresponds to reality.

The Pratt–Hayford hypothesis explains the variations of gravity in the United States quite well. Heiskanen computed the same stations also according to Airy’s hypothesis. The 56 mountain stations in the United States reduced by Bowie were processed by Heiskanen according to Airy’s 1st, 2nd, and 4th assumptions.

The mean anomaly values of these stations show that the reduction according to Hayford (\(113.7\ \text{km}\)) and the first reduction according to Airy give equally good results, while the two other reductions according to Airy give somewhat worse results.

The linear dependence of the anomalies on the height of the observation point will be the smallest for the first two reductions according to Airy, so that in this sense the mentioned reductions will be better. But the linear term in all these methods of reduction is so small that this criterion cannot have very great significance.

The mean anomaly values of mountain stations, stations on the ocean shore, and lowland stations lead Heiskanen to conclude that Airy’s hypothesis, with an approximate thickness of the Earth’s crust of \(50\ \text{km}\) (corresponding to sea level), explains the course of the variation of gravity in the United States somewhat better than any assumption of Hayford–Pratt.

Further, Heiskanen tested Airy’s hypothesis at the following stations: 16 stations in the Alps and 10 in the region of marginal subsidences of these mountains. Since

6 stations in the Austrian Alps had a somewhat different character than the remaining 10; therefore they were set apart as a special group.

The depth of the isostatic surface of 107 km, according to Hayford’s hypothesis, and the thickness of the Earth’s crust of 41 km, according to Airy, are quite sufficient to explain the gravity anomalies in the Alps and in the region of their marginal depressions.

If we wished to explain the negative anomaly of the other 6 stations in the Austrian Alps by means of isostatic compensation, we would have to operate with smaller depths of isostasy and smaller thicknesses of the Earth’s crust; thus, to an isostatic depth of 85.3 km there correspond the anomalies: \(+0.003\), \(+0.007\), \(-0.002\), \(+0.013\), \(+0.37\), while to Airy’s 4th assumption: \(+0.017\), \(-0.013\), \(+0.07\), \(-0.002\), \(+0.022\), and \(+0.041\ \mathrm{cm/sec^2}\).

But it is possible that beneath the Austrian Alps there is a small region of mass deficiency (between \(10^\circ\) and \(13^\circ\) longitude and \(2^\circ\) in latitude). The stations Spittal and Hohenmantel (longitudes: \(13^\circ.5\) and \(15^\circ.2\)), with their large positive anomaly, show that these mass defects gradually pass into Hungary—into a region of excessive compensation.

The mean deviations of the anomalies (from their mean value) in the Giant Mountains and the Harz, computed according to Airy’s hypothesis, prove to be somewhat smaller than the corresponding quantities computed according to Hayford’s hypothesis. But these regions are too small and cannot give an exhaustive solution to the question of the advantages of one hypothesis or the other.

Heiskanen summarizes his computations as follows: in all the places investigated—on the Caucasus, in America, and in the Alps—the Airy hypothesis explains the course of the variation of gravity just as well as, or even somewhat better than, Hayford’s refined Pratt hypothesis; moreover, the depth of compensation—according to Pratt’s hypothesis—and also the thickness of the Earth’s crust—according to Airy’s hypothesis—do not have the same value over the whole globe.

These conclusions contradict the opinion expressed by Helmert, who asserted that Airy’s hypothesis cannot explain the course of the variation of gravity that takes place on high mountains.

On oceanic islands, as is known, large positive gravity anomalies are obtained if Hayford’s hypothesis is adopted with an isostatic depth of 113.7 km. Thus Bowie found at the station St. Georges (in the Bermuda Islands) an anomaly of \(+0.080\), at Jamestown (St. Helena Island) \(+0.120\), and at Honolulu (Hawaii Island) \(+0.075\). If, however, one abandons the depth of 113.7 km and operates with a depth of 156.3 km, then for these stations one obtains the anomalies: \(-0.003\), \(+0.060\), and \(+0.022\ \mathrm{cm/sec^2}\). Approximately the same results can be obtained using Airy’s hypothesis, if only suitable thicknesses of the Earth’s crust and differences of densities between the Sial and Sima layers are chosen.

Heiskanen concludes from this that oceanic islands are not absolutely overcompensated, but that the anomalies on them are explained by the choice of one hypothesis or another.

From the table given by Heiskanen it is evident that Airy’s 2nd and 3rd hypotheses, which differ from one another only in that according to the 2nd hypothesis the difference of densities between the Earth’s crust and the lava layer is \(0.3\), while according to the third it is \(0.6\), lead to almost identical results, so that the thickness of the Earth’s crust corresponding to sea level has far greater significance than the difference of densities.

5. Derivation of the formula for the normal distribution of gravity and the flattening of the Earth.

With the aid of the isostatically reduced stations, Heiskanen derived a new formula for the distribution of gravity on the terrestrial globe, and from this formula determined the flattening of the Earth.

Since isostatically reduced stations, especially in Europe, are in most cases situated on mountain plateaus and in mountains, a certain number of coastal and lowland stations were reduced by the author in an abridged manner, and these stations were added to the above-mentioned stations that had undergone exact reduction.

Stations on plains, as, for example, in Northern Germany, Russia, Finland, etc., where the terrain is fairly level, can be reduced easily. The local reduction here is usually so small that it may be entirely neglected, while the purely topographic reduction, which is added to Bouguer’s reduction in order to obtain the topographic-isostatic reduction, is easily determined in flat countries; it changes so little from point to point that it needs to be determined only for some stations, whereas the majority of stations can be determined by interpolation.

If, on the contrary, the stations lie near mountains or the deep sea, as do the stations of Norway, France, England, the Iberian Peninsula, and on the Mediterranean and Red seas, then the isostatic reduction must be determined exactly for each station.

All reductions were computed by Heiskanen according to Hayford’s hypothesis, assuming the depth of isostasy to be 113.7 km.

Since errors of measurement and errors of reduction at different stations are often not the same, the anomalies obtained could contain different errors; therefore not all stations were taken into account in deriving the formula for gravity, and not all observations were taken with equal weight. Thus, for example, Germany, Denmark, and Hungary, with their numerous, well-determined stations, could have completely drowned out the influence of other, less well studied regions.

In order to enter small regions with a large number of stations into the computation with not very great weight, all stations lying in a region bounded by \(1^\circ\) in latitude and \(1^\circ\) in longitude were combined into one station, with weight \(=1\), and the coordinates of the midpoint of the region were taken as the coordinates of the area.

In all, Heiskanen derived six formulas for the distribution of gravity: 1) the 1st formula—for European and Caucasian stations and one Algerian station (from a total of 283 separate areas), 2) the 2nd—from all the listed stations, as well as stations on the coasts of Africa and the shores of the Red Sea (from a total of 335 areas), 3) the 3rd—from American stations (234 areas), 4) the 4th—from Asian stations (87 areas), and the 5th and 6th—from all stations of Europe, Africa, America, and Asia (from a total of 656 areas). In deriving the 6th formula the Earth was taken to be a triaxial ellipsoid, and in the 5th—a biaxial ellipsoid.

From these computations it is evident that the coefficients of formulas (1) and (2), obtained from European and African stations, differ considerably from the coefficients of formulas (3) and (4)—for America and Asia. This means that in Europe and Africa, at the same latitudes, a different gravity prevails than in America and Asia, so that the distribution of gravity depends not only on geographical latitude but also on longitude.

To determine the dependence of gravity on geographical longitude, the last formula was derived from all stations, with the Earth considered a triaxial ellipsoid. This formula has the following form:

\[ \gamma_0 = 978.052 \left[ 1 + 0.005285 \sin^2 \varphi - 0.000007 \sin^2 2\varphi + 0.000027 \cos^2 \varphi \cdot \cos(\lambda - 18^\circ) \right], \]
\[ \pm 3 \qquad\quad \pm 6 \qquad\qquad\qquad\qquad\quad \pm 5 \]

where \(\varphi\) denotes geographical latitude, and \(\lambda\)—geographical longitude.

Thus, if the Earth is considered a triaxial ellipsoid, then the major axis of the equator must pass \(18^\circ\) east of Greenwich, and the minor axis of the equator—\(72^\circ\) west. The difference between these axes will be: \(690 \pm 75\) m.

Formula (5) was derived from all stations under the assumption that gravity does not depend on longitude; therefore the sum of the squares of all 656 anomalies according to this formula

= 635,300, and by formula (6) = 554,300, so that the latter formula gives smaller anomalies.

The quantities \(\left(\dfrac{1}{f}\right)\), reciprocal to the compression of the earth, corresponding to the first five formulas, are expressed as follows:

\[ \frac{1}{f} = \begin{array}{ccccc} 293.8, & 291.5, & 299.8, & 299.2, & 297.4 \\ \pm 1.1 & \pm 0.7 & \pm 1.1 & \pm 1.4 & \pm 0.5 \end{array} \]

and by formula (6) the mean value is \(\dfrac{1}{f}=296.7\), while the smallest and largest values will be: 294.3 and 299.0 (corresponding to both axes of the equator).

\[ \begin{array}{cc} \pm 0.6 & \pm 0.6 \end{array} \]

For comparison, below are given previously derived formulas for the normal distribution of gravity on the earth’s surface.

\[ \text{1. }\gamma_0 = 978.030\left(1+0.005\,302\sin^2\varphi - 0.000\,007\sin^2 2\varphi\right) \]

\[ \text{2. }\gamma_0 = 978.052 \left[ 1+0.005\,285\sin^2\varphi -0.000\,007\sin^2 2\varphi +0.000\,018\cos^2\varphi\cdot\cos^2(\lambda+17) \right] \]

\[ \begin{array}{ccc} \pm 3 & \pm 5 & \pm 4 \end{array} \]

\[ \text{3. }\gamma_0 = 978.039 \left( 1+0.005\,294\sin^2\varphi -0.000\,007\sin^2 2\varphi \right). \]

\[ \begin{array}{cc} \pm 4 & \pm 12 \end{array} \]

The first of these formulas is the well-known formula of Helmert, derived by him in 1901 (this formula was used by Borras in his work).

The second formula was also derived by Helmert—in 1915. He obtained this formula on the assumption that the earth is a triaxial ellipsoid; in doing so the earth’s surface was divided into 410 parts, but Helmert did not take into account stations lying near mountains, nor coastal stations situated near great depths. The stations were reduced only for height.

The third formula—Bowie’s—was derived from 252 topographic-isostatically reduced stations (or groups of stations), lying for the most part in America, whose surface was divided into 11 zones, each \(3^\circ\) in latitude.

Although Helmert derived his second formula from plain stations, reduced only for height, this formula coincides almost perfectly with the 6th formula of Heiskanen, obtained both from plain stations and from mountain and coastal stations subjected to topographic-isostatic reduction. The first two terms in both formulas are completely identical, and only the third term differs. The numerical coefficient of this term, according to Helmert, is \(18\cdot10^{-6}\), and according to Heiskanen’s formula is \(27\cdot10^{-6}\). In addition, according to Helmert the major axis of the equator lies \(17^\circ\) to the west, while according to Heiskanen it lies \(18^\circ\) to the east of Greenwich. That these terms differ so strongly from one another is explained mainly by the fact that Helmert did not take into account the stations in the Caucasus, the Mediterranean Sea, and the Red Sea, which were included in Heiskanen’s computations.

Bowie’s formula is less precise, since he used only stations lying within a 33-degree zone (in latitude), so that the determination of the term \(\sin^2\varphi\) is not reliable. This formula does not correspond to the variation of gravity in Europe.

The reciprocal quantities of the earth’s compression, \(\dfrac{1}{f}\), according to Helmert’s first and second formulas, will be equal to: \(293.2\pm9.5\) and \(296.7\pm0.4\), and according to Bowie’s formula: \(297.4\pm1.0\).

How well Helmert’s latter formula expresses the course of the variation of gravity over the entire earth’s surface is difficult to judge, since the regions studied

cover only an insignificant part of the entire globe. If a sufficient number of stations, reduced isostatically, were available in South America, Africa, Asia, Australia, and on the oceans, it would be possible to derive a quite reliable formula; but that is a matter for the future.

For Europe and its immediate environs, Heiskanen has drawn a map of anomalies corresponding to the Hayford reduction (113.7 km). The anomalies are expressed in units of 0.001 cm/sec².

On this map, one is struck by the remarkably regular distribution of the anomalies. If we compare this map with Kosmat’s map, on which the Bouguer anomalies are given, we notice that, first, the absolute magnitudes of the anomalies are considerably smaller and, second, they are distributed more regularly under the assumption of isostasy than without this hypothesis. For example, the strong negative anomaly in the Alps and the Carpathians has almost completely disappeared, while the positive anomaly over the Mediterranean Sea has become considerably smaller.

Large mass deficiencies are observed in Scandinavia, France, Spain, Algeria, Galicia, and in the region of the Caspian Sea, while significant excesses of mass occur in Germany, Bohemia, Hungary, and over the Mediterranean Sea.

A very remarkable strong positive anomaly \((+125\) and \(+70)\) occurs on the island of Lofoten, near the region of negative anomaly located on the Scandinavian Peninsula.

The negative anomaly—in the Alps—and the positive anomaly—in the Caucasus—will be considerably smaller if, in the Alps, we operate with smaller depths of compensation, and in the Caucasus—with greater depths of compensation, as has already been indicated above.

1. While the present article was in press, a new work by Heiskanen was received, devoted to a more detailed study of gravity in Norway (W. Heiskanen. Schwerkraft und isostatische Kompensation in Norwegen. Veröf. des Finn. Geodet. Inst. № 5, 1926). The author subjected 29 stations in southern Norway and 17 stations in northern Norway to topographic-isostatic processing according to the same plan as in the work reviewed here, and the results presented here were fully confirmed. ↩↩