GRAVITY-FIELD recovery of the EARTH using reconstruction of spherical harmonic coefficients up to specified degree and order requires proper data sampling based on Shannon-Nyquist rate. Since, these coefficients are globally significant, the sampling must be done uniformly on the EARTH, which it takes much time and expense to collect and process data. Many studies have been done in the FIELD of sampling analysis of spherical harmonics [1, 2]. Sneeuw [2] showed a lack of Nyquist sampling rate can cause aliasing of second type in GRAVITY-FIELD modeling. Recently based on Compressive Sensing (CS) theorem the sampling rate can be substantially reduced and a signal can be approximated in sparse sense with fewer sampled data that has main role in reconstruction. In this case, the desired signal can be reconstructed, using only some base functions, which are most strongly correlated with the problem. Therefore, based on this strategy, the base functions posing the best solution to the problem will be selected and the sampling rate for regional GRAVITY FIELD modeling will be decreased significantly. When we say a signal is m-sparse, it means that there are at most nonzero components in the signal. In this case, only m coefficients of the signal have large magnitude, and others are zero, or have very small values. Here, the desired signal can be reconstructed with its large components without loss of more information. The zero-norm of a vector which is defined as, specifies the sparsity-level of a signal. Sparse approximation has been discussed in many studies [4, 5, 6, 7, 8, 9]. The basic idea proposed by Mallat and Zhang [4] is called matching pursuit (MP), which is an iterative sparse approximation method to reconstruct a signal under specified conditions by replacing a complex sparse problem with a simple optimized solution. Pati et al. [6] modified this algorithm into orthogonal matching pursuit (OMP), which is used for non-orthogonal dictionaries and converges faster than MP. The regularized orthogonal matching pursuit (ROMP) algorithm popularized by Needell and Vershynin [8] is an iterative sparse approximation method where at each iteration m nonzero components of unknown parameters that most closely resemble the properties of the desired signal are selected. Needell and Tropp [9] refined the ROMP algorithm with compressive sampling matching pursuit (CoSaMP), which identifies locations of the large energy of a signal at each iteration. All these algorithms try to find column vectors in the design matrix that most strongly correlate with the desired signal. It is also assumed that the design matrix is well-posed and prior knowledge of the sparsity-level of the signal is clear. Usually, in practical application an ill-posed problem may be encountered, also the sparsity-level of the signal is not exactly clear, which make it difficult to use conventional iterative methods of CS. In this paper we present a new dynamic algorithm called Stabilized Orthogonal Matching Pursuit (SOMP) for GRAVITY-FIELD recovery of the EARTH using sparse approximation of geopotential spherical harmonic coefficients, which is compatible with the ill-posed problem and can determine the sparsity-level of the signal, properly. Numerical result of the calculated spherical harmonics coefficients up to degree and order 36 shows that the algorithm is able to reconstruct the EARTH's GRAVITY-FIELD with precision in mind the number of samples is 50% lower than the Nyquist rate.

In this paper a method for computation of mean GRAVITY value within the EARTH between the computation point and the geoid needed for precise orthometric height computations is presented.The method presented is based on following steps: (1) computation of the global and regional GRAVITY effects using ellipsoidal harmonic expansion to degree and order 360 plus the centrifugal acceleration. (2) Computation of the gravitational effect of terrain masses within the radius of 55km around the computational point applying Newton integral in the equal area map projection. (3) Computation of the GRAVITY at two points on the surface of the EARTH and on the geoid using steps 1-2, computing mean and standard deviation of the computed GRAVITY values and increasing the number of points within the EARTH to meet the predefined standard deviation for the computation of mean GRAVITY within the EARTH. (4) Deriving the mean GRAVITY within the EARTH from the steps 1-3. The proposed method for the computation of mean GRAVITY within the EARTH is checked against the observed GRAVITY values within the EARTH in an exploration borehole. The test computations are made in the following two modes: (a) Computation of GRAVITY values at the observation points in the borehole and comparison of the computed values with the observed values (b) Computing mean GRAVITY within the EARTH using the proposed method and the observed GRAVITY values. According to the test computations at a depth of 474.7 m computed GRAVITY differs from the observed GRAVITY by 10.768 mGal and the computed mean GRAVITY from the observed mean GRAVITY by 5.56 mGal.

Global warming is one of the problems of human civilization and decarbonization policy is the main solution to this problem. In this work, we propose an alternative method of using the GRAVITY assist by the asteroids to increase the orbital distance of the EARTH from the Sun. We can manipulate the orbit of asteroids in the asteroid belt by solar sailing and propulsion engines to guide them towards the Mars orbit and a gravitational scattering can put asteroids in a favorable direction to provide an energy loss scattering from the EARTH. The result would be increasing the orbital distance of the EARTH and consequently cooling down the EARTH's temperature. We calculate the increase in the orbital distance of the EARTH for each scattering. The time scale for lowering the orbit of an asteroid for a gravitational impact is $70$ years. For the consecutive of scatterings of asteroids, we investigate the feasibility of performing this project.

Detailed GRAVITY FIELD modeling requires ground, sea, airborne and satellite GRAVITY observations. Sea GRAVITY observations have been always suffering from the low accuracy and noises.  On the other hand satellite altimetry provides accurately sea level variations. In this paper possibility of GRAVITY FIELD modeling at sea areas, especially at islands, using satellite altimetry is studied. A method for this purpose is devised which algorithmically can be explained as follows: (1) Obtain Mean Sea Level (MSL) from satellite altimetry. (2) Obtain Sea Surface Topography (SST) from oceanographic sea current and studies. (3) Compute the geoid by removing SST from MSL. (4) Apply inverse ellipsoidal Bruns formula and compute the GRAVITY potential at the reference ellipsoid. (5) Remove gravitational effect of ellipsoidal harmonic expansion to degree and order 360 and the centrifugal acceleration. (6) Remove terrain effect using Newton integral over the terrain masses at the radius of 55 km around the computational points. (7) Upward continue the harmonic gravitational potential derived in step 6 using ellipsoidal Abel-Poisson integral. (8) Restore the effects removed in steps 5 - 6 at the computational point. This procedure is applied for GRAVITY FIELD modeling at Geshm Island and its surrounding area. The details of the method and results of the case study are presented in this paper.

The possibility of modeling GRAVITY FIELD of the EARTH along the leveling lines using the GRAVITY observations at the leveling benchmarks has been studied. As the case study, GRAVITY observations along the first order-leveling network of Iran has been considered. The mathematical models that have been developed for this purpose are polynomials of degree 4 and degree 8. The efficiency of the aforementioned models has been compared with the ellipsoidal harmonic expansion to degree and order 20, 180, 360 and the Somigliana-Pizzeti GRAVITY FIELD. The test computations have proven that a polynomial model of degree 4 can be considered as the best choice for the GRAVITY FIELD modeling along the leveling lines. Based on the case study of GRAVITY FIELD modeling along the leveling lines of Iran, the polynomial model of degree 4 can provide 20 mGal accuracy which is quite enough to provide the accuracy level of the GRAVITY data needed for the precise leveling based on the current demands. Therefore, by applying such a model the need for the GRAVITY observations along the leveling lines will be reduced and as such great reduction in the expense of the precise leveling observations will be achieved.

GRAVITY gradient tensor is a matrix containing the second order derivatives of the EARTH’s GRAVITY FIELD, which has numerous applications in geodesy and geophysics. To date, much effort has been done for estimating GRAVITY gradient tensor with reasonable accuracy. This quantity can be estimated via using various methods, and one of these methods is applying finite-difference method to GRAVITY observations. Finite-difference method can estimate GRAVITY gradient tensor directly by using the mathematical concept of gradient, regardless of extra assumptions. This ability of finite-difference method, from theoretical point of view, provides the possibility of accurate estimation of GRAVITY gradient tensor without considering additional assumptions to the problem. This study tends to introduce and evaluate Finite difference method for estimating the gradient tensor and present formulae for determining GRAVITY gradient tensor from land-based GRAVITY observations. In this paper, the proposed equations are numerically tested by means of using a global GRAVITY model of the EARTH. Global GRAVITY model of the EARTH (EGM 2008) is a geopotential model of the EARTH consisting of spherical harmonic coefficients up to degree 2190 and order 2159. There are numerous uses for these high degree potential coefficient models. One of these uses is modeling and estimating GRAVITY gradient tensor.Finally, GRAVITY gradient tensor is estimated by the proposed method for 6350 GRAVITY stations located in Costal Fars region in a northern part of the Persian Gulf, between the latitudes from 26.5 N to 27.27 N and longitudes from 53.41 E to 55.58 E. The target area is about 10000 square kilometers. About 8500 square kilometers of the study area is located in moderate mountainous regions, and about 1500 square kilometers is located in flat coastal areas. The altitudinal distribution and spatial distribution of GRAVITY in study area are shown in figure 1 and 2 respectively. Numerical experiments of this study demonstrate the ability of this method in GRAVITY gradient tensor estimation with acceptable accuracy. For example, numerical experiments showed that the proposed method can estimate diagonal components of GRAVITY gradient tensor (second order derivatives of the EARTH’s GRAVITY FIELD in east, north and vertical directions) with the accuracy values of 12.46, 34.49 and 454.82 Eotvos respectively. The spatial distribution of the GRAVITY gradient tensor components obtained from finite difference method in study area are shown in figure 3.Finally, according to the theoretical concepts discussed in this paper, It can be said because the finite difference method using from derivative and difference concepts directly for estimating GRAVITY gradient tensor, it is expected that this method provide accurate estimation of GRAVITY gradient tensor, As this is happen in the simulation conducted. However the accuracy of this method is very dependent on distances between sampling stations and by reducing distance between the stations, the accuracy of proposed method will be increased. The numerical results of this study also showed that the proposed method can provide accurately estimate of GRAVITY gradient tensor components In some stations surrounded by suitable spatial distribution of gravitational observations.

Many different methods for GRAVITY FIELD modelling have been investigated, among which, the harmonic expansion has been widely used due to harmonic nature of GRAVITY potential FIELD that satisfies Laplace equation in an empty space. This method, however, cannot reach to a high resolution in a GRAVITY FIELD, and suffers from omitting the high frequency GRAVITY signals and therefore it is not appropriate for local GRAVITY FIELD modelling. To overcome this drawback and recover high frequency features of GRAVITY FIELD, appropriate basis functions with local support should be used. One of the methods for local GRAVITY FIELD modeling based on local harmonic function is spherical cap harmonic analysis. In this method, the Dirikhlet boundary value problem for Laplace equation is solved for boundary conditions on the surface of a spherical cap which results in Eigen expansion of the solution in terms of the associated Legendre function of noninteger degree and integer order. Another method that can be used for local GRAVITY FIELD modeling is rectangular harmonic analysis. In this method, Laplace equation is solved in a local Cartesian coordinate system and boundary conditions are applied on a plane area which. In this approach, trigonometric functions are used as basis functions. In this study, the problem of local GRAVITY FIELD modeling based on both spherical cap, and rectangular harmonic expansion is investigated. Also, a numerical study is conducted to show the performance of each method for local GRAVITY FIELD modeling. To do so the observations of vector airborne gravimetry in the northwest of Tanzania in Highland region are used to derive the coefficients of each model. The low-frequency part of observed GRAVITY FIELD is removed from the data using EGM2008 geo-potential model, and the resulting residual GRAVITY FIELD is considered for local modelling. Since the governing equations for determination of the coefficients suffer from an ill-conditioning problem, it is necessary to apply some regularization schemes to find the optimum solution. Here, the Tikhonov regularization method is utilized to obtain the regular solution. In this study, the edge effect for each model is also analyzed. To show this effect, the results of models are compared with the observations of GRAVITY at some control points distributed both within the study area and its margin. It should be noted that the maximum degree of expansion in harmonic series, plays an important role in appropriate fitting of local GRAVITY FIELD models to the GRAVITY data and it has significant effects on the computational task of determining the coefficients of each model. For this purpose, local GRAVITY FIELD modelling is calculated with different value of maximum degree of expansion and then regarding to the result (accuracy of local GRAVITY model by comparing with control points), appropriate value of maximum degree of expansion for each model is determined. Finally the results of two models are compared to each other to show the performance of each models in local GRAVITY FIELD modeling. The results of this study reveal that ASHA has the ability to model local GRAVITY with accuracy of about 1 mGal, and RHA method in the best situation can just achieve to a 3 mGal accuracy, although the convergence rate in RHA model is faster than ASHA model. Also by comparing the edge effect on each models, it is seen that the edge effect in two models and in all directions occurred but in a Z direction of RHA model that are more significant than the other directions in two models and one may conclude that the edge effect of RHA are much larger than that of ASHA. Finally, the result obtained shows that ASHA model can have better results for local GRAVITY modelling.

Edge detection and edge enhancement techniques play an essential role in interpreting potential FIELD data. This paper describes the application of various edge detection techniques to GRAVITY data in order to delineate the edges of subsurface structures. The edge detection methods comprise analytic signal, total horizontal derivative (THDR), theta angle, tilt angle, hyperbolic of tilt angle (HTA), normalised total horizontal gradient (TDX) and normalised horizontal derivative (NTHD). The results showed that almost all filters delineated edges of anomalies successfully. However, the capability of these filters in edge detection decreased as the depth of sources increased. Of the edge enhancement filters, normalized standard deviation filter provided much better results in delineating deeper sources. The edge detection techniques were further applied on a real GRAVITY data from the Gheshm sedimentary basin in the Persian Gulf in Iran. All filters specified a northeast-southwest structural trend. The THDR better outlined the structural morphology and trend. Moreover, it indicated the salt plugs much better than other filters. Analytic signal and THDR successfully enhanced the edges of the shorter wavelength residual structures. Normalized standard deviation (NSTD), TDX and hyperbolic of tilt angle (HTA) filters highlighted the likely fault pattern and lineaments, with a dominant northeast-southwest structural trend. This case study shows that the edge detection techniques provides valuable information for geologists and petroleum engineers to outline the horizontal location of geological sources including salt plugs and stand out buried faults, contacts and other tectonic and geological features.

The modification of laws of physics at short intervals is an important result of the theory of quantum GRAVITY. For instance, commutative relations of standard quantum mechanics change on scales of length-called Planck length. It should be noted that these changes can be neglected at low energy levels but they are considerable only at high energy levels such as the initial universe. In this regard, the principle of uncertainty of standard quantum mechanics is changed with modified relations of uncertainty including a visible minimum of Planck order. Early moments of the universe, which included the inflation period, was a period with noticeable effects of quantum GRAVITY due to the high energy level, and as such, the effects can be studied during this period. To do this, characteristics of the inflation period can be examined according to initial parameters of the universe such as the initial fluctuations in the formation of the universe structure and the spectral index. On the other hand, vector cosmology models have been taken into consideration by researchers. These models include an action in which a vector FIELD (in addition to the scalar FIELD) is included to investigate effects of violation of the Lorentz invariance in observations. The present paper investigated effects of quantum GRAVITY (with effects on non-commutative geometry and generalization of the uncertainty principle) on parameters of a vector cosmological model. The vector model was used as this scenario had acceptable adaptation to parameters of cosmology after inflation (e. g. the transition from the Phantom boundary, etc. ) (Nozari and Sadatian, 2009). Furthermore, the present study could test this vector model for determining parameters of the inflation period based on effects of quantum GRAVITY. According to calculations in the present paper, we concluded that, first: the density of scalar perturbations decreased in the vector model based on effects of quantum GRAVITY (the reduction of standard model was more considerable), and second: due to the ignorance of effects quantum GRAVITY, the scalar spectral index parameter remained invariant as observations indicate, but due to large enough gravitational effects (depending on amount of β), the spectral index parameter is not maintained its invariance scale. According to obtained modification in the present study, the quantum GRAVITY can be tested for the density of scalar perturbation (which can be measured by observing the spectrum of cosmic microwave background radiation). In order to compare our results with other studies, we can refer to (Zhu et al, 2014) where they examined the spectral index in accordance with high-order correction mechanism. It also indicated that a single asymmetric approximation does not lead to a considerable error value for the spectral index, and the invariance scale is maintained. Furthermore, the paper (Hamber and Sunny Yu, 2019) found the same results for invariance scale of the spectral index according to the Wilson normalization analysis method. Therefore there was no need to have common assumptions in the inflation period. Finally, it should be noted that despite a great number of studies on effects of quantum GRAVITY, the reviewed model of this paper considers a state in which the effects can be investigated at all stages of the universe evolution from inflation till now.