Far-zone gravity field contributions corrected for the effect of topography by means of molodensky’s truncation coefficients

A spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton’s integral. In this study we utilise Molodensky’s truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky’s truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data.     

An approach to refined mapping of the anomalous gravity field in the Earth’s polar caps

This work describes the different sets of instruments and methodic approaches for testing the models of gravity anomalies by repeated airborne gravimetric surveys in the polar cap regions of the Earth. The survey design including the specifications for flying the survey profiles and the arrangement of the base stations in polar areas is described, and the necessary modifications of the airborne gravity metering complexes for high-latitude measurements are suggested.     

On the gradient of curvature of the plumblines of the Earth’s normal gravity field and its isocurvature lines

This paper presents an approach to determine the gradient of curvature of the normal plumblines at a point P above the ellipsoid and introduces a new geometrical object which is the isocurvature line. The assumed facts are the coordinates of the point P and the formula for the normal gravity potential U. For the determination of the gradient of the normal plumbline curvature k at the point P we define a small circle on the meridian plane of P whose center is at the point P. The circle has the radius of one meter and interior D. In this circle we construct a curvature replacement function to approximate the curvature function k. This replacement function is a quotient of polynomials hence it is easy to find its partial derivatives at the point P. For the construction of replacement function we make the assumption that in the interior of the circle D the first order partial derivatives of U behave linearly and the second order partial derivatives have constant values which equal their value at the point P. Then we set the gradient of the curvature function to be equal with the gradient of the aforementioned replacement function at P. An isocurvature line of the normal gravity field passing through a point P is a curve such that the value of the function of the plumblines’ curvature k is constant and equals k(P). We give a formula to find the direction of the isocurvature line on the meridian plane and we prove that there are infinitely many isocurvature lines passing through the point P and they all lie on a special surface, the isocurvature surface.     

Comparison of two different solutions to Molodensky’s G1 term

The main purpose of this paper is comparison of two different approaches of solution to the Simple Molodensky’s Problem, the Molodensky’s Approach and the Analytical Continuation Approach, based on numerical computation. Although these approaches have been described theoretically by several authors, e.g. Molodensky et al. (1960), Heiskanen and Moritz (1967), Vaníček (1974), Moritz (1980) and Holota (1991, unpublished results) and theoretical proof of equivalence was given by Heiskanen and Moritz (1967), Moritz (1971), Ecker (1971) and Pellinen (1972, unpublished results), only very few practical experiences about the differences between particular solutions and computational efficiency exist. In this paper we compare the above two mentioned approaches in terms of the G₁-effect on quasigeoid. Both quasigeoid solutions are tested by the independent GPS/levelling approach and are also compared with the previous quasigeoid model of Slovakia where the G₁-term has been approximated using the classical terrain correction. The effect of the G₂-term is also numerically estimated, revealing that it might be significant for the precise quasigeoid determination. Some practical comments and recommendations are given at the end of the paper.     

Comparative study of temporal variations in the earth’s gravity field using GRACE gravity models in the regions of three recent giant earthquakes

Comparative analysis of coseismic and postseismic variations of the Earth’s gravity field is carried for the regions of three giant earthquakes (Andaman-Sumatra, December 26, 2004, magnitude M _w = 9.1; Maule-Chile, February 27, 2010, M _w = 8.8, and Tohoku-Oki, March 11, 2011, M _w = 9.0) with the use of GRACE satellite data. Within the resolution of GRACE models, the coseismic changes of gravity caused by these seismic events manifest themselves by large negative anomalies located in the rear of the subduction zone. The real data are compared with the synthetic anomalies calculated from the rupture surface models based on different kinds of ground measurements. It is shown that the difference between the gravity anomalies corresponding to different rupture surface models exceeds the uncertainties of the GRACE data. There-fore, the coseismic gravity anomalies are at least suitable for rejecting part of the models that are equivalent in the ground data. Within the first few months after the Andaman-Sumatra earthquake, a positive gravity anomaly started to grow above the deep trench. This anomaly rapidly captured the area of the back-arc basin and largely compensated the negative coseismic anomaly. The processes of viscoelastic stress relaxation do not fully allow for these rapid changes of gravity. According to the calculations, even with a sufficiently low viscosity of the upper mantle, relaxation only covers about a half of the observed change of the field. In order to explain the remaining temporal variations, we suggested the process of downdip propagation of the coseismic rupture surface. The feasibility of such a process was supported by numerical simulations. The sum of the gravity anomalies caused by this process and the anomaly generated by the processes of viscoelastic relaxation accounts well for the observed changes of the gravity field in the region of the earthquake. The similar postseismic changes of gravity were also detected for the region of the Tohoku-Oki earthquake. Just as in the case discussed above, this earthquake was also followed by a rapid growth of a positive postseismic anomaly, which partially counterbalanced the negative coseismic anomaly. The time variations of the gravity field in the region of the Maule-Chile earthquake differ from the pattern of changes observed in the island arcs described above. The postseismic gravity variations are in this case concentrated in a narrower band above the deep trench and shelf, and they do not spread over the continental territory, where the negative coseismic anomaly is located. These discrepancies reflect the difference in the geodynamical settings of the studied earthquakes.     

Determination of the Earth’s free core nutation parameters by using tidal gravity data

The Earth’s free core nutation (FCN) is a retrograde eigenmode which is attributed to the interaction between the solid mantle and the liquid core of the rotational elliptical Earth.     

Nonzonal structure of the response of the global field of the Earth’s atmospheric temperature to solar activity

The work describes the results of calculations obtained with the Atmospheric Research Model (ARM) general circulation model. The temperature response of the troposphere and middle atmosphere to variations in UV solar radiation were found to have a large-scale wave structure when planetary waves at the lower model boundary were taken into account. In the present paper, the results from the processing of global temperature fields with three databases (ERA-20C, NOAA-CIRES 20th Century Reanalysis, v2, and NCEP/NCAR Reanalysis I) are provided. Analysis of the differences of the mean monthly temperature global fields (January and July) between the maxima and minima of three solar activity cycles (21, 22, and 23 cycles) also demonstrated their nonzonal structure. It was shown that the amplitude of this difference in January in the stratosphere (10 hPa) can be 7–29 K in the Northern Hemisphere. In July, this effect is prominent in Southern Hemisphere. In the troposphere (500 hPa), a nonzonal temperature effect is present in both the Northern and Southern Hemispheres; the amplitude of the effects amounts to approximately 5–12 K. In conclusion, we discuss that the mechanism of solar energy impact on atmospheric temperature discovered by numerical modeling is supported after reanalysis data processing.     

Role of atmospheric circulation in the baric field response to the Earth’s crossing of the interplanetary magnetic field sector boundaries

The Earth’s crossings of the magnetic sector boundaries are accompanied by changes in the magnetosphere, ionosphere, and troposphere. We considered the baric field’s response to the crossing of the inter-planetary magnetic field (IMF) sector boundaries during a geomagnetically quiet period. The IMF sign is shown to affect atmospheric pressure in high-latitude regions. The efficiency and sign of the relationship vary during the year. The baric field response to the Earth’s crossing of the IMF sector boundaries is most distinct during equinoxes. It is shown that, during a geomagnetically quiet period, the circulation processes in the atmosphere drive the changes in the atmospheric pressure when the Earth passes from one IMF sector into another.     

Large-scale aseismic creep in the areas of the strong earthquakes revealed from the GRACE data on the time variations of the Earth’s gravity field

The refinement of the accuracy and resolution of the monthly global gravity field models from the GRACE satellite mission, together with the accumulation of more than a decade-long series of these models, enabled us to reveal the processes that occur in the regions of large (M_w≥8) earthquakes that have not been studied previously. The previous research into the time variations of the gravity field in the regions of the giant earthquakes, such as the seismic catastrophes in Sumatra (2004) and Chile (2010), and the Tohoku mega earthquake in Japan (2011), covered the coseismic gravity jump followed by the long postseismic changes reaching almost the same amplitude. The coseismic gravity jumps resulting from the lower-magnitude events are almost unnoticeable. However, we have established a long steady growth of gravity anomalies after a number of such earthquakes. For instance, in the regions of the subduction earthquakes, the growth of the positive gravity anomaly above the oceanic trench was revealed after two events with magnitudes M_w=8.5 in the Sumatra region (the Nias earthquake of March 2005 and the Bengkulu event of September 2007 near the southern termination of Sumatra Island), after the earthquake with M_w=8.5 on Hokkaido in September 2007, a doublet Simushir earthquake with the magnitudes M_w = 8.3 and 8.1 in the Kuriles in November 2006 and January 2007, and after the earthquake off the Samoa Island in September 2009 (M_w=8.1). The steady changes in the gravity field have also been recorded after the earthquake in the Sichuan region (May 2008, M_w = 8.0) and after the doublet event with magnitudes 8.6 and 8.2, which occurred in the Wharton Basin of the Indian Ocean on April 11, 2012. The detailed analysis of the growth of the positive anomaly in gravity after the Simushir earthquake of November 2006 is presented. The growth started a few months after the event synchronously with the seismic activation on the downdip extension of the coseismically ruptured fault plane zone. The data demonstrating the increasing depth of the aftershocks since March 2007 and the approximately simultaneous change in the direction and average velocity of the horizontal surface displacements at the sites of the regional GPS network indicate that this earthquake induced postseismic displacements in a huge area extending to depths below 100 km. The total displacement since the beginning of the growth of the gravity anomaly up to July 2012 is estimated at 3.0 m in the upper part of the plate’s contact and 1.5 m in the lower part up to a depth of 100 km. With allowance for the size of the region captured by the deformations, the released total energy is equivalent to the earthquake with the magnitude M_w = 8.5. In our opinion, the growth of the gravity anomaly in these regions indicates a large-scale aseismic creep over the areas much more extensive than the source zone of the earthquake. These processes have not been previously revealed by the ground-based techniques. Hence, the time series of the GRACE gravity models are an important source of the new data about the locations and evolution of the locked segments of the subduction zones and their seismic potential.     

Formation of large-scale disturbances in the upper atmosphere caused by acoustic gravity wave sources on the Earth’s surface

The results of a model study of the acoustic gravity wave (AGW) propagation from the Earth’s surface to the upper atmospheric altitudes have been considered. Numerical calculations have been performed using a nonhydrostatic model of the atmosphere, which takes into account nonlinear and dissipative processes originating when waves propagate upward. The model source of atmospheric disturbances has been specified in an area localized on the Earth’s surface. The disturbance source frequency spectrum includes harmonics at frequencies of 0.5ωg-1.5ωg (ωg is the Brunt-Väisälä frequency near the Earth’s surface). The calculations indicated that AGW propagation and dissipation over the source result in the fact that the region of large-scale spatial disturbances of the upper atmosphere mean state is formed at ～200 km altitudes. This region substantially affects AGW propagation and results in waveguide propagation of AGWs with periods shorter than the Väisälä-Brunt period at the altitude of a disturbed atmosphere. The dissipation of AGWs propagating in such a waveguide results in a waveguide horizontal expansion. The extension of the disturbed region of the mean state of the upper atmosphere and, consequently, the waveguide length can reach ～1000 km, if the AGW ground source operates for ～1 h. The physical mechanism by which large-scale disturbances are formed in the upper atmosphere, based on the propagation and dissipation of AGWs with periods shorter than the Väisälä-Brunt period in the upper atmosphere, explains why these disturbances are rapidly generated and localized above AGW sources located on the Earth’s surface or in the lower atmosphere.     

Harmonic sources of the main geomagnetic field in the Earth’s core

The large-scale harmonic magnetic-convective sources of the main geomagnetic field in the Earth’s core have been determined for the first time. The determination is based on a complete system of eigenfunctions of the magnetic diffusion equation in a homogeneously conducting sphere, which is surrounded by an insulator. The sources of the main geomagnetic field observed, which is responsible for the distribution of the electric currents generating this field in the core, are expressed in terms of large-scale eigenfunctions. In this case, the dipole sources are directly related to the observed geomagnetic dipole, whereas the quadrupole sources are related to the quadrupole, etc. The time variations in the obtained sources are responsible for individual spatiotemporal features in the generation or suppression of each Gaussian component of the observed geomagnetic field. When the commonly accepted observational international geomagnetic reference field (IGRF) models were used to partially reveal these time variations, it became possible to specify the estimate of the Earth’s core conductivity and determine the minimum period that can separate us from the commencement of further inversion or excursion.     

Effect of the magnetic field on measurements of the electron density and temperature by cylindrical probes in the Earth’s ionosphere

In the 1960s and 1970s, quite simply produced cylindrical Langmuir probes were used in the USSR both on satellites (Kosmos-378, Intercosmos-2, -4, -8, -10, -19) and to measure the electron density and temperature on vertical launched rockets (Vertical’-4, -6, -10) within the Intercosmos program. These measurements were first made at middle latitudes. With increasing inclination of the orbits of launched satellites (satellites had no stabilization), falling sections were sometimes observed on probe characteristics in the electron saturation region. The Intercosmos-Bulgaria-1300 satellite, which was stabilized along three axes and was equipped with a cylindrical probe whose longitudinal axis was always directed downward to the Earth, was launched in 1981. This satellite allowed definite conclusions on the effect of the geomagnetic field on the form of the probe characteristic and, hence, on the determination of the electron density and temperature. Probe characteristics with falling sections are presented. These measurements are compared with those performed in a laboratory plasma. The appearance of negative sections on the probe characteristics is shown to be due to the effect of the geomagnetic field. The degree of this effect depends both on the electron density and temperature and on the probe voltage.     

The effect of wind on the gravity wave propagation in the Earth’s ionosphere

The effect of ionospheric wind on the gravity wave propagation is studied. These waves arise in the ionosphere due to intensification of their sources near the Earth’s surface during enhanced seismic activity. The influence of the wind on these waves is connected with the Ampere’s force that produces the ion-drag force acting on the atmosphere. This results in the occurrence of the discrete wave spectrum the maximum of which increases in proportion to the numbers of the natural scale. Furthermore, these waves are amplified during propagation from the source region in the direction perpendicular to the wind direction. These peculiarities of the gravity waves can be used for monitoring of seismic activity based on the ionosphere sounding.     

Perturbation of the magnetic field in the Earth’s magnetosphere due to plateau creation in the radial distribution of plasma pressure

The nonlinear perturbation of a dipole field by a system of transverse currents, which arises due to the radial pressure distribution when the pressure is almost independent of the radial distance, is analyzed. This distribution of pressure was observed in the experiment. The radial dependences of the magnetic field depression, transverse current density, and volumes of magnetic flux tubes have been obtained at different values of the plasma parameter via nonlinear simulation. It is shown that a dependence of the volume of magnetic flux tube on the radial distance can change and a region of the negative gradient of volume can appear at some plasma-parameter values.     

Spatial and temporal variation of gravity field in the capital region

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Estimation of the temperature in the Earth’s interior from surface measurements of the electromagnetic field

The possibility of contactless remote estimation of the temperature in the Earth’s interior from surface magnetotelluric (MT) measurements is examined. The neuronet analysis of MT and temperature measurements in the Bishkek geodynamic research area (the Northern Tien Shan) showed that a contactless electromagnetic geothermometer can in principle be realized. An optimal method including MT measurements and treatment of available thermograms is developed. The method minimizes uncertainties of the remote temperature estimation. The use of six to eight thermograms for calibration of electromagnetic data is shown to provide a 12% relative error of prediction, and a priori geological information available for the region under study can reduce this error. Areas of practical application of a contactless electromagnetic geothermometer are outlined.     

Reflectionless acoustic gravity waves in the Earth’s atmosphere

The vertical wave propagation in an inhomogeneous compressible atmosphere is studied in the framework of a linear theory. Under specific conditions imposed on atmospheric parameters, solutions can be found in the form of travelling waves with variable amplitudes and wave numbers that do not reflect in the atmosphere in spite of its strong inhomogeneity. Model representations for the sound speed have been found, for which waves can propagate in the atmosphere without reflection. A wave energy flux retains these reflectionless profiles, which confirms that energy can be transferred to high altitudes. The number of these model representations is fairly large, which makes it possible to approximate real vertical distributions of the sound speed in the Earth??s atmosphere using piecewise reflectionless profiles. The Earth??s standard atmosphere is shown to be well approximated by four reflectionless profiles with weak jumps in the sound speed gradient. It has been established that the Earth??s standard atmosphere is almost completely transparent for the considered vertical acoustic waves in a wide range of frequencies, which is confirmed by observational data and conclusions derived using numerical solutions of original equations.     

Intensity of geomagnetic field in the Precambrian and evolution of the Earth’s deep interior

Reliable data on the paleointensity of the geomagnetic field can become an important source of information both about the mechanisms of generation of the field at present and in the past, and about the internal structure of the Earth, especially the structure and evolution of its core. Unfortunately, the reliability of these data remains a serious problem of paleomagnetic research because of the limitations of experimental methods, and the complexity and diversity of rocks and their magnetic carriers. This is true even for relatively “young” Phanerozoic rocks, but investigation of Precambrian rocks is associated with many additional difficulties. As a consequence, our current knowledge of paleointensity, especially in the Precambrian period, is still very limited. The data limitations do not preclude attempts to use the currently available paleointensity results to analyze the evolution and characteristics of the Earth’s internal structure, such as the age of the Earth’s solid inner core or thermal conductivity in the liquid core. However, such attempts require considerable caution in handling data. In particular, it has now been reliably established that some results on the Precambrian paleointensity overestimate the true paleofield strength. When the paleointensity overestimates are excluded from consideration, the range of the field strength changes in the Precambrian does not exceed the range of its variation in the Phanerozoic. This result calls into question recent assertions that the Earth’s inner core formed in the Mesoproterozoic, about 1.3 billion years ago, triggering a statistically significant increase in the long-term average field strength. Instead, our analysis has shown that the quantity and quality of the currently available data on the Precambrian paleointensity are insufficient to estimate the age of the solid inner core and, therefore, cannot be useful for solving the problem of the thermal conductivity of the Earth’s core. The data are consistent with very young or very “old” inner core ages and, correspondingly, with high or low values of core thermal conductivity.     

Determination of the Earth＇s free core nutation parameters by using tidal gravity data

The Earth＇s free core nutation （FCN） is a retrograde eigenrnode which is attributed to the interaction between the solid mantle and the liquid core of the rotational elliptical Earth. This mode appears as an eigenmode of nearly diurnal free wobble （NDFW） in a terrestrial reference frame with a period of about one day （XU et al, 2001）. Therefore, the NDFW will lead to an obvious resonance enhancement in the diurnal tidal gravity observations, especially those of the tidal waves with frequencies closed to its eigenfrequency such as P1, K1, ψ1 and Ф1. The FCN resonance parameters can be retrieved accurately by high-precision tidal gravity observations, especially those recorded with the superconducting gravimeters （SG）. The Global Geodynamics Project （GGP） organized by IUGG took it as an important content for determining the FCN resonance parameters by using gravity data. However, the results are affected by many factors such as station location, background noise, the selection of the tide-generating potential tables, ocean tide models, data processing techniques and so on. In our study, the FCN parameters will be retrieved by using the SG observations at Wuhan, and the effects of the choices of various tide-generating potential tables, oceanic models and weight functions on the estimation of the FCN parameters will be discussed in detail,     

Spectral expressions for modelling the gravitational field of the Earth’s crust density structure

We derive expressions for computing the gravitational field (potential and its radial derivative) generated by an arbitrary homogeneous or laterally varying density contrast layer with a variable depth and thickness based on methods for a spherical harmonic analysis and synthesis of gravity field. The newly derived expressions are utilised in the gravimetric forward modelling of major known density structures within the Earth’s crust (excluding the ocean density contrast) beneath the geoid surface. The gravitational field quantities due to the sediments and crust components density contrasts, shown in numerical examples, are computed using the 2 × 2 arc-deg discrete data from the global crustal model CRUST2.0. These density contrasts are defined relative to the adopted value of the reference crustal density of 2670 kgm⁻³. All computations are realised globally on a 1 × 1 arc-deg geographical grid at the Earth’s surface. The maxima of the gravitational signal due to the sediments density contrast are mainly along continental shelf regions with the largest sedimentary deposits. The corresponding maxima due to the consolidated crust components density contrast are over areas of the largest continental crustal thickness with variable geological structure.