An exact solution for the gravity curvature (Bullard B) correction

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1179-1184 ◽  
Author(s):  
T. R. LaFehr

The complete Bouguer reduction includes, in addition to the simple Bouguer slab correction (Bullard A), both curvature (Bullard B) and terrain (Bullard C) corrections. A new closed‐form formula for the curvature correction is derived for which the calculated values differ from those published by Swick by more than 0.5 mGal for high elevations. These corrections reduce those of an infinite slab (Bullard A) to that of a spherical cap having a surface radius of 166.7 km. The spherical cap produces a lesser effect than the infinite slab because of the “truncation” of that part of the slab above the earth and extending to infinity, but it produces a greater effect than the slab because of subslab earth resulting from curvature. The physical significance of the correction lies in the combination of these two differences, which are each a function of elevation. The Bullard B surface radius (166.7 km: outer radius of the Hayford‐Bowie system) is reaffirmed by the exact formula to be appropriate for exploration surveys. Three series approximations are presented and compared, but the exact Bullard B formula is very efficient and easy to program for routine data processing.

Geophysics ◽  
1979 ◽  
Vol 44 (8) ◽  
pp. 1464-1464
Author(s):  
J. R. Hearst ◽  
R. C. Carlson

Our equations (3) and (4) are correct. They represent the difference between the attraction of the shell viewed from [Formula: see text], the outer radius of the shell, and [Formula: see text], its inner radius. (The attraction of the shell viewed from [Formula: see text] is zero.) On the other hand, equations (5) and (6) of Fahlquist and Carlson represent the difference in attraction of the entire earth from the same viewpoints and thus, as they say, include a free‐air gradient term. However, their equation (5) would be correct only if the mean density of the earth were equal to that of the shell, and the free‐air gradient obtained by their equation (10) is correct only under these circumstances.


2019 ◽  
Vol 49 (4) ◽  
pp. 511-523
Author(s):  
Róbert Kysel ◽  
Andrej Cipciar ◽  
Kristián Csicsay ◽  
Lucia Fojtíková ◽  
Martin šugár ◽  
...  

Abstract The National Network of Seismic Stations of Slovakia (NNSS) consists of eight short period and five broadband permanent seismic stations and a data centre located at the Earth Science Institute of the Slovak Academy of Sciences (ESI SAS). The NNSS recorded and detected 11704 seismic events from all epicentral distances in 2018. Totally 86 earthquakes originated in the territory of Slovakia in 2018. This paper provides basic information on the configuration of the NNSS, routine data processing, seismic activity on the territory of Slovakia in 2018 as well as macroseismic observations collected in 2018.


2021 ◽  
Vol 51 (4) ◽  
pp. 373-389
Author(s):  
Róbert KYSEL ◽  
Andrej CIPCIAR ◽  
Martin ŠUGÁR ◽  
Kristián CSICSAY ◽  
Lucia FOJTÍKOVÁ ◽  
...  

The National Network of Seismic Stations of Slovakia (NNSS) consists of eight short period and six broadband permanent seismic stations and a data centre located at the Earth Science Institute of the Slovak Academy of Sciences (ESI SAS). The NNSS recorded and detected 11229 seismic events from all epicentral distances in 2020. Totally 96 earthquakes originated in the territory of Slovakia in 2020. This paper provides basic information on the configuration of the NNSS, routine data processing, seismic activity on the territory of Slovakia in 2020 as well as macroseismic observations collected in 2020.


2019 ◽  
Vol 49 (1) ◽  
pp. 1-10
Author(s):  
Róbert Kysel ◽  
Andrej Cipciar ◽  
Zuzana Chovanová ◽  
Kristián Csicsay ◽  
Lucia Fojtíková ◽  
...  

Abstract The National Network of Seismic Stations of Slovakia (NNSS) consists of eight short period and five broadband permanent seismic stations and a data centre located at the Earth Science Institute of the Slovak Academy of Sciences (ESI SAS). The NNSS recorded and detected 10888 seismic events from all epicentral distances in 2016. Totally 87 earthquakes originated in the territory of Slovakia in 2016. This paper provides basic information on the configuration of the NNSS, routine data processing, seismic activity on the territory of Slovakia in 2016 as well as macroseismic observations collected in 2016.


2019 ◽  
Vol 49 (2) ◽  
pp. 195-206
Author(s):  
Róbert Kysel ◽  
Andrej Cipciar ◽  
Zuzana Chovanová ◽  
Kristián Csicsay ◽  
Lucia Fojtíková ◽  
...  

Abstract The National Network of Seismic Stations of Slovakia (NNSS) consists of eight short period and five broadband permanent seismic stations and a data centre located at the Earth Science Institute of the Slovak Academy of Sciences (ESI SAS). The NNSS recorded and detected 10 719 seismic events from all epicentral distances in 2017. Totally 73 earthquakes originated in the territory of Slovakia in 2017. This paper provides basic information on the configuration of the NNSS, routine data processing, seismic activity on the territory of Slovakia in 2017 as well as macroseismic observations collected in 2017.


2020 ◽  
Vol 12 (17) ◽  
pp. 2848
Author(s):  
Jing Huang ◽  
Huadong Guo ◽  
Guang Liu ◽  
Guozhuang Shen ◽  
Hanlin Ye ◽  
...  

Spatio-temporal characteristics are the crucial conditions for Moon-based Earth observations. In this study, we established a Moon-based Earth observation geometric model by considering the intervisibility condition between a Moon-based platform and observed points on the Earth, which can analyze the spatio-temporal characteristics of the observations of Earth’s hemisphere. Furthermore, a formula for the spherical cap of the Earth visibility region on the Moon is analytically derived. The results show that: (1) the observed Earth spherical cap has a diurnal period and varies with the nadir point. (2) All the annual global observation durations in different years show two lines that almost coincide with the Arctic circle and the Antarctic circle. Regions between the two lines remain stable, but the observation duration of the South pole and North pole changes every 18.6 years. (3) With the increase of the line-of-sight minimum observation elevation angle, the area of an intervisible spherical cap on the lunar surface is obviously decreased, and this cap also varies with the distance between the barycenter of the Earth and the barycenter of the Moon. In general, this study reveals the effects of the elevation angle on the spatio-temporal characteristics and additionally determines the change of area where the Earth’s hemisphere can be observed on the lunar surface; this information can provide support for the accurate calculation of Moon-based Earth hemisphere observation times.


The hypothesis that the liquid core of the Earth represents a phase-change at high pressure (and suitable temperature) of the mantle material is further investigated. A more accurate series of two-zone models have been computed, and also a new series of three-zone models. The change of overall radius as between an original all-solid Earth and the present size is shown to be at least 370 km. In the outer regions, greater pressure may be needed with rising temperature to effect the transition to denser crystal form (associated with the 20°-discontinuity), and from this cause acting alone slight expansion of the Earth would result but to an extent less than one-tenth the overall contraction. Epochs of rapid contraction (mountain-building eras) could thus be separated by longer intervals of very slow expansion. The initial liquefaction of the central regions brings about pressure increase at the boundary of the core that renders the Earth unstable in that about 6 per cent of the entire mass liquefies extremely rapidly to cause a sudden collapse of the planet as a whole. The accompanying decrease of outer radius is about 70 km. Thereafter the planet remains thoroughly stable and contracts only slowly. The total contraction to date would have reduced the moment of inertia by a factor about 4/5, and the corresponding reduction in rotation period (through conservation of angular momentum) would be an effect comparable with tidal friction. The contraction also leads to release of gravitational energy at an average rate comparable with that from radioactive sources. An important consequence of the phase-change hypothesis is that the melting-point gradient changes sign after sufficient depth, thereby permitting melting of the central regions to occur at moderate temperatures explicable by a reasonable content of radioactive elements.


2020 ◽  
Vol 50 (4) ◽  
Author(s):  
Róbert KYSEL ◽  
Andrej CIPCIAR ◽  
Kristián CSICSAY ◽  
Lucia FOJTÍKOVÁ ◽  
Martin ŠUGÁR ◽  
...  

The National Network of Seismic Stations of Slovakia (NNSS) consists of eight short period and five broadband permanent seismic stations and a data centre located at the Earth Science Institute of the Slovak Academy of Sciences (ESI SAS). The NNSS recorded and detected 11,487 seismic events from all epicentral distances in 2019. Totally 91 earthquakes originated in the territory of Slovakia in 2019. This paper provides basic information on the configuration of the NNSS, routine data processing, seismic activity on the territory of Slovakia in 2019 as well as macroseismic observations collected in 2019.


1876 ◽  
Vol 166 ◽  
pp. 315-324 ◽  

In a paper read before the Royal Society, May 1874, I pointed out that the upward diminution of temperature in the atmosphere (known to exist under certain circumstances) must refract and give an upward direction to the rays of sound which would otherwise proceed horizontally; and it was suggested that this might he the cause of the observed difference in the distinctness with which similar sounds are heard on different occasions, particularly the very marked advantage which night has over day in this respect. At the time at which that paper was written no direct experiments or observations had been made to verify the truth of this suggestion, and therefore its probability rested on its reasonableness. Since that time, however, I have carried out a series of observations and experiments which, although far from complete, throw some light on the subject, besides revealing some remarkable facts. I hope to be able to continue the investigation; but since its nature is such as to render the chance of bringing it to any thing like a final conclusion very uncertain, it seems to me that it may be well to publish an account of what has been already done; and this is the object of the present communication. In order to render the object of the various experiments clear, it may be well to recapitulate here some of the theoretical considerations previously explained. It will be remembered that the idea that the variations of temperature would cause refraction of sound occurred to me while making experiments on the effect of wind upon sound, from which it was shown that when sound proceeds in a direction contrary to that of the wind, it is not, as had been thought, destroyed or stopped by the wind, but that it is lifted, and that at sufficiently high elevations it can be heard to as great distances as in other directions, or as when there is no wind—thus confirming the hypothesis first propounded by Professor Stokes and afterwards by myself, that the effect is owing to the retardation of the velocity of the wind near the earth, which allows the sound moving against the wind to move faster below than above, and thus causes the fronts of the waves to incline upwards, and consequently to move in that direction. Having clearly shown that this was the case, it became apparent that any thing which would cause an upward diminution in the velocity at which sound proceeds would cause a similar effect to that of the wind and lift the sound, and that since the speed of the sound depends on the temperature of the air in which it is moving, an upward diminution in the temperature must cause such an effect. That such a diminution of temperature does very often exist was proved by Mr. Glaisher’s balloon ascents in 1862, in which he found that when cloudy the mean rate of diminution for the first 300 feet was 0°·5 for each 100 feet, and that when clear it was 1°, and that on some occasions it was greater and on others less than this. A variation of 1° in the temperature of the air alters the velocity of sound nearly 1 foot per second, so that with a clear sky the sound instead of moving horizontally would move upwards on a circle of 110,000 feet radius, and with a cloudy sky on a scale of 220,000 feet radius. This rate of refraction is very small compared with that caused even by a very moderate wind; and consequently in order to verify it by experiment it is necessary to observe sounds at much greater distances. This renders the experiment very difficult to carry out; and to make it worse we have no means of determining what the upward variation of temperature is, which therefore can only be surmised by the behaviour of the sound.


1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.


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