Since the year 2000, some periodic investigations have been performed in the Lilla Edet region to monitor and possibly determine the landslide of the area with GPS measurements. The responsible consultant has conducted this project by setting up some stable stations for GPS receivers in the risky areas of Lilla Edet and measured the independent baselines amongst the stations according to their observation plan. Here, we optimise the existing surveying network and determine the optimal configuration of the observation plan based on different criteria.We aim to optimise the current network to become sensitive to detect 5 mm possible displacements in each net point. The network quality criteria of precision, reliability and cost are used as object functions to perform single, bi and multiobjective optimisation models. It has been shown in the results that the singleobjective model of reliability, which is constrained to the precision, provides much higher precision than the defined criterion by preserving almost all of the observations. However, in this study, the multiobjective model can fulfil all the mentioned quality criteria of the network by 17% less measurements than the original observation plan, meaning 17%of saving time, cost and effort in the project.
An optimal design of a geodetic network can fulfill the requested precision and reliability of the network, and decrease the expenses of its execution by removing unnecessary observations. The role of an optimal design is highlighted in deformation monitoring network due to the repeatability of these networks. The core design problem is how to define precision and reliability criteria. This paper proposes a solution, where the precision criterion is defined based on the precision of deformation parameters, i. e. precision of strain and differential rotations. A strain analysis can be performed to obtain some information about the possible deformation of a deformable object. In this study, we split an area into a number of threedimensional finite elements with the help of the Delaunay triangulation and performed the strain analysis on each element. According to the obtained precision of deformation parameters in each element, the precision criterion of displacement detection at each network point is then determined. The developed criterion is implemented to optimize the observations from the Global Positioning System (GPS) in Skåne monitoring network in Sweden. The network was established in 1989 and straddled the Tornquist zone, which is one of the most active faults in southern Sweden. The numerical results show that 17 out of all 21 possible GPS baseline observations are sufficient to detect minimum 3 mm displacement at each network point. © 2018 Walter de Gruyter GmbH, Berlin/Boston.
In order to detect the geohazards, different deformation monitoring networks are usually established. It is of importance to design an optimal monitoring network to fulfil the requested precision and reliability of the network. Generally, the same observation plan is considered during different time intervals (epochs of observation). Here, we investigate the case that instrumental improvements in sense of precision are used in two successive epochs. As a case study, we perform the optimisation procedure on a GPS monitoring network around the Lilla Edet village in the southwest of Sweden. The network was designed for studying possible displacements caused by landslides. The numerical results show that the optimisation procedure yields an observation plan with significantly fewer baselines in the latter epoch, which leads to saving time and cost in the project. The precision improvement in the second epoch is tested in several steps for the Lilla Edet network. For instance, assuming two times better observation precision in the second epoch decreases the number of baselines from 215 in the first epoch to 143 in the second one.
One of the GOCE satellite mission goals is to study the Earth's interior structure including its crustal thickness. A gravimetricisostatic Moho model, based on the Vening MeineszMoritz (VMM) theory and GOCE gradiometric data, is determined beneath Iran's continental shelf and surrounding seas. The terrestrial gravimetric data of Iran are also used in a nonlinear inversion for a recoveringMoho model applying the VMM model. The newlycomputed Moho models are compared with the Moho data taken from CRUST2.0. The rootmeansquare (RMS) of differences between the CRUST2.0 Moho model and the recovered model from GOCE and that from the terrestrial gravimetric data are 3.8 km and 4.6 km, respectively.
In the gravimetric approach to determine the Moho depth an isostatic hypothesis can be used. The Vening Meinesz–Moritz isostatic hypothesis is the recent theory for such a purpose. Here, this theory is further developed so that the satellite gravity gradiometry (SGG) data are used for recovering the Moho depth through a nonlinear integral inversion procedure. The kernels of its forward and inverse problems show that the inversion should be done in a larger area by 5° than the desired one to reduce the effect of the spatial truncation error of the integral formula. Our numerical study shows that the effect of this error on the recovered Moho depths can reach 6 km in Persia and it is very significant. The iterative Tikhonov regularization in a combination with either generalized cross validation or quasioptimal criterion of estimating the regularization parameter seems to be suitable and the solution is semiconvergent up to the third iteration. Also the Moho depth recovered from the simulated SGG data will be more or less the same as that obtained from the terrestrial gravimetric data with a root mean square error of 2 km and they are statistically consistent.
Today, the recent global Earth's gravity model, EGM08, is successfully utilised for different purposes in geosciences. Here, EGM08 is used to compute a geoid model for Fennoscandia and since it is restricted to degree and order 2160, the higher frequencies of the geoid, or the truncation bias, is recovered directly from terrestrial gravity anomalies using a simple formula. The total topographic and atmospheric effects are computed and added to the derived geoid as well. A very simple EGM08based nonintegral geoid estimator is developed and applied for computing the geoid of Fennoscandia. The outcome of the estimator is compared with the Global Positioning System (GPS)/levelling data of Sweden, Denmark, Finland and Norway. Numerical results show the successful performance of the presented estimator as the geoid become closer to GPS/levelling data than the one computed solely with EGM08. This study will show that considering the truncation bias of EGM08 will reduce the root mean square error (RMSE) of the differences between the geoid and GPS/levelling data by about 1.3 cm and the additive topographic and atmospheric corrections by 1 cm further. It is shown that the correlations among the data have no significant influence on the estimated geoid.
The Moho surface can be determined according to isostatic theories and the recent Vening MeineszMoritz (VMM) theory of isostasy has been successful for this purpose. In this paper, we will study this method from a theoretical prospective and try to find its connection to the AiryHeiskanen (AH) and Vening Meinesz original theories. We develop Jeffrey’s inverse solution to isostasy according to the recent developments of the VMM method and compare both methods in similar situations. We will show that they are generalisations of the AH model in a global and continuous domain. In the VMM spherical harmonic solution for Moho depth, the mean Moho depth contributes only to the zerodegree term of the series, whilst in Jeffrey’s solution it contributes to all frequencies. We improve the VMM spherical harmonic series further so that the mean Moho can contribute to all frequencies of the solution. This modification makes the VMM global solution superior to the Jeffrey one, but in a global scale, the difference between both solutions is less than 3 km. Both solutions are asymptoticallyconvergent and we present two methods to obtain smooth solutions for Moho from them.
The idea of this paper is to present estimators for combining terrestrial gravity data with Earth gravity models (EGMs) and produce a highquality source of the Earth's gravity field data through all wavelengths. To do so, integral and pointwise estimators are mathematically developed, based on the spectral combination theory, in such a way that they combine terrestrial data with one and/or two Earth gravity models. The integral estimators are developed so that they become biased or unbiased to a priori information. For testing the quality of the estimators, their global mean square errors (MSEs) are generated using an Earth gravity model08 model and one of the recent products of the gravity field and steadystate ocean circulation explorer (GOCE) mission. Numerical results show that the integral estimators have smaller global root mean square errors (RMSEs) than the pointwise ones but they are not efficient practically. The integral estimator of the biased type is the most suited due to its smallest global root mean square error comparing to the rest of the estimators. Due largely to the omission errors of Earth gravity models the pointwise estimators are not sensitive to the Earth gravity model commission error; therefore, the use of highdegree Earth gravity models is very influential for reduction of their root mean square errors. Also it is shown that the use of the ocean circulation explorer Earth gravity model does not significantly reduce the root mean square errors of the presented estimators in the presence of Earth gravity model08. All estimators are applied in the region of Fennoscandia and a cap size of 2° for numerical integration and a maximum degree of 2500 for generation of bandlimited kernels are found suitable for the integral estimators.
The idea of this paper is to refine the terrestrial gravimetric data with the Earth's gravity models (EGMs) and produce a high quality source of gravity data. For this purpose, biased and unbiased integral estimators are presented. These estimators are used to refine gravimetric data over Fennoscandia with the ITGGRACE2010s and GO_CONS_GCF_2_DIR_R2 EGMs, which are the recent products of the gravity field and climate experiment (GRACE) and the gravity field and steadystate ocean circulation explorer (GOCE) satellite missions. Numerical results show that the biased integral estimator has smaller global root mean square error (RMSE) than the unbiased one. Also a simple strategy is presented to downweight the lowfrequencies the terrestrial data in spectral combination. The gravity anomalies, computed by EGM08, are compared to the refined anomalies for evaluation purpose. In the case of using a cap size of 1° for integration the EGM08 gravity anomalies are more correlated with the refined ones. Also the bandlimited kernels can simply be generated to maximum degree of the used EGMs for both estimators. Comparisons of the combined anomalies and those of EGM08 show insignificant differences between the biased and unbiased estimators in practice. However, the biased estimator seems to be proper one for gravity data refinement due to its smaller global RMSE.
Today, most of the aircrafts are navigated by global navigation satellite systems (GNSSs). Landing is a dangerous phase of a flight especially when an airport runway is not clearly seen from the aircrafts. In such cases, GNSSs are useful for a safe landing under the circumstances that healthy signals, free of any interference, reach to GNSSs receiver antennas mounted on the aircrafts. This shows the importance of establishing GNSS interference localisation security networks around airports. Designing a good configuration for the points with GNSS antennas at for receiving interference signals is important for a successful localisation of the interference device. Here, the timedifference of the arrivals of an interference signal to such points or anchor nodes (ANs), are used as observables, and a security network with four ANs is optimally designed along the runways of the Arlanda airport to reduce the dilution of precision (DOP) of the network. Our study showed that by such an optimisation, the maximum DOP value can reduce by 50% meaning a significant increase in the probability of a successful interference device localisation.
This new book deals with, as its titles says, the analysis and process of gravity field of the Earth. The book is mathematically oriented, and the authors have explained how Mathematics is used for local analysis of the gravity field. The basic and required background knowledge, needed for understanding the book, are given in early chapters as well as appendices. They are used for explaining gravity field analysis from measurement surveys, corrections, and processes to direct and inverse problems in exploration Geophysics with interesting exercises, numerical and real examples.
The satellite gradiometry data (SGD) can be used for studying the crustal structure in addition to the Earth’s gravity field. This paper will show how this type of data is related to the Moho discontinuity or the boundary between the Earth’s crust and mantle. Here, the Vening MeineszMoritz (VMM) theory of isostasy is used and its mathematical formulae are modified to use the SGD instead of the Earth gravity models. A linear integral equation with a wellbehaving kernel is presented by approximating the Moho depth formula derived based on the VMM theory. The error of this approximation is less than 300 m in Iran as the study area. Furthermore, this paper shows that the contribution of the higher degree harmonics than 215 is less than 1% with respect to the total signal of Moho undulations. This means that the use of SGD is meaningful as they sense the harmonics of the Earth’s gravity field to this degree. Two methods of onestep and twostep are proposed for Moho determination and applied in Iran. It is shown that to reduce the effect of spatial truncation error of the integral formulae of both methods the central area should be smaller by 6^{◦} than the inversion area. Numerical studies show that the twostep approach is superior to the other one and the root mean squared error of differences between the Moho model recovered by an Earth gravity model and SGD is about 1.5 km in Iran.
Elastic thickness (Te) is one of mechanical properties of the Earth's lithosphere. The lithosphere is assumed to be a thin elastic shell, which is bended under the topographic, bathymetric and sediment loads on. The flexure of this elastic shell depends on its thickness or Te. Those shells having larger Te flex less. In this paper, a forward computational method is presented based on the Vening Meinesz–Moritz (VMM) and flexural theories of isostasy. Two Moho flexure models are determined using these theories, considering effects of surface and subsurface loads. Different values are selected for Te in the flexural method to see by which one, the closest Moho flexure to that of the VMM is achieved. The effects of topographic/bathymetric, sediments and crustal crystalline masses, and laterally variable upper mantle density, Young's modulus and Poisson's ratio are considered in whole computational process. Our mathematical derivations are based on spherical harmonics, which can be used to estimate Te at any single point, meaning that there is no edge effect in the method. However, the Te map needs to be filtered to remove noise at some points. A median filter with a window size of 5° × 5° and overlap of 4° works well for this purpose. The method is applied to estimate Te over South America using the data of CRUST1.0 and a global gravity model.
Subcrustal stress induced by mantle convection can be determined by the Earth's gravitational potential. In this study, the spherical harmonic expansion of the simplified Navier–Stokes equation is developed further so satellite gradiometry data (SGD) can be used to determine the subcrustal stress. To do so, we present two methods for producing the stress components or an equivalent function thereof, the socalled S function, from which the stress components can be computed numerically. First, some integral estimators are presented to integrate the SGD and deliver the stress components and/or the S function. Second, integral equations are constructed for inversion of the SGD to the aforementioned quantities. The kernel functions of the integrals of both approaches are plotted and interpreted. The behaviour of the integral kernels is dependent on the signal and noise spectra in the first approach whilst it does not depend on extra information in the second method. It is shown that recovering the stress from the vertical–vertical gradients, using the integral estimators presented, is suitable, but when using the integral equations the vertical–vertical gradients are recommended for recovering the S function and the vertical–horizontal gradients for the stress components. This study is theoretical and numerical results using synthetic or real data are not given.
Different gravitational force models are used for determining the satellites’ orbits. The satellite gravity gradiometry (SGG) data contain this gravitational information and the satellite accelerations can be determined from them. In this study, we present that amongst the elements of the gravitational tensor in the local northoriented frame, all of the elements are suitable for this purpose except T_{xy}. Three integral formulae with the same kernel function are presented for recovering the accelerations from the SGG data. The kernel of these integrals is wellbehaving which means that the contribution of the farzone data is not very significant to their integration results; but this contribution is also dependent on the type of the data being integrated. Our numerical studies show that the standard deviations of the differences between the accelerations recovered from T_{zz}, T_{xz} and T_{yz} and those computed by an existing Earth´s gravity model reduce by increasing the cap size of integration. However, their root mean squared errors increase for recovering T_{y}from T_{yz}. Larger cap sizes than 5 is recommended for recovering T_{x} and T_{z}_{ }but smaller ones for T_{y}.
The spherical harmonic expressions of the horizontal subcrustal stress components induced by the mantle convection are convergent only to low degrees. In this paper, we use the method of stress (S) function with numerical differentiation and present a formula for determining the degree of convergence from the mean Moho depth. We found that for the global mean Moho depth, 23 km, this convergence degree is 622 and for Iran, 35 km, it is 372. Also, three methods are developed and applied for computing the subcrustal stress, (1) direct integration with a spectral kernel limited up to the degree of convergence, (2) integral inversion with a kernel having closedform formula without any frequency limit, and (3) solving an integral equation with limited spectral kernel to the convergence degree. The second method has no divergence problem and its kernel function is well behaving so that the system of equations from which the S function is determined is stable, and no regularisation is needed to solve it. It should be noted that for using this method the resolution of the recovery should be higher than 0.5° × 0.5°, otherwise the recovered S function and correspondingly the stress components will have smaller magnitude than those derived from the other two methods. Our numerical studies for stress recovery in Iran and its surrounding areas show that the methods, which use the limited spectral kernels to the convergence degree, deliver consistent results to that of the spherical harmonic expansion.
Global models of the Earth gravity field and topographic/bathymetric data can be used for the gravimetric determination of the Moho discontinuity based on the Vening MeineszMoritz theory. In this paper, we mathematically develop this method in such a way that the local data can be used for Moho modelling. Two integral formulae are presented, one for integrating the data and one for their inversion. The kernels of both integrals are wellbehaving meaning that the contribution of farzone quantities being integrated are not very significant in the results. Both of these methods are applied for computing the Moho model of Iran and their results are compared to the Moho model determined based on the global models. Consistency of the computed Moho models from the simulated data and the global models verifies the correctness of both approaches. The presented methods are consistent even for the case of using real data. Numerical results show that the minimum value of the Moho models derived by the simulated data and global models are about 31 km, whilst those derived from the real data are about 3 km smaller. Similarly, the mean value of Moho depths derived from real data is about 1 km smaller than that from the global models.
The sublithospheric stress due to mantle convection can be computed from gravity data and propagated through the lithosphere by solving the boundaryvalue problem of elasticity for the Earth's lithosphere. In this case, a full tensor of stress can be computed at any point inside this elastic layer. Here, we present mathematical foundations for recovering such a tensor from gravitational tensor measured at satellite altitudes. The mathematical relations will be much simpler in this way than the case of using gravity data as no derivative of spherical harmonics or Legendre polynomials is involved in the expressions. Here, new relations between the spherical harmonic coefficients of the stress and gravitational tensor elements are presented. Thereafter integral equations are established from them to recover the elements of stress tensor from those of the gravitational tensor. The integrals have no closedform kernels, but they are easy to invert and their spatial truncation errors are reducible. The integral equations are used to invert the real data of the gravity field and steadystate ocean circulation explorer (GOCE) mission, in November 2009, over the South American plate and its surroundings to recover the stress tensor at a depth of 35 km. The recovered stress fields are in good agreement with the tectonic and geological features of the area.
The traditional expressions of the gravitational vector (GV) and the gravitational gradient tensor (GGT) have complicated forms depending on the first and the secondorder derivatives of associated Legendre functions (ALF), and also singular terms when approaching the poles. This article presents alternative expressions for the GV and GGT, which are independent of the derivatives, and are also nonsingular. By using such expressions, it suffices to compute the ALF to two additional degrees and orders, instead of computing the first and the second derivatives of all the ALF. Therefore, the formulas are suitable for computer programming. Matlab software as well as an output of a numerical computation around the North Pole is also presented based on the derived formulas.
So far the recent Earth's gravity model, EGM08, has been successfully applied for different geophysical and geodetic purposes. In this paper, we show that the computation of geoid and gravity anomaly on the reference ellipsoid is of essential importance but error propagation of EGM08 on this surface is not successful due to downward continuation of the errors. Also we illustrate that some artefacts appear in the computed geoid and gravity anomaly to lower degree and order than 2190. This means that the role of higher degree harmonics than 2160 is to remove these artefacts from the results. Consequently, EGM08 must be always used to degree and order 2190 to avoid the numerical problems. © 2013 Elsevier B.V.
Different approximations are used in Moho modelling based on isostatic theories. The wellknown approximation is considering a plate shell model for isostatic equilibrium, which is an oversimplified assumption for the Earth’s crust. Considering a spherical shellmodel, as used in the Vening MeineszMoritz (VMM) theory, is a more realistic assumption, but it suffers from different types of mathematical approximations. In this paper, the idea is to investigate such approximations and present their magnitudes and locations all over the globe. Furthermore, we show that the mathematical model of Moho depth according to the VMM principle can be simplified to that of the plate shell model after four approximations. Linearisation of the binomial term involving the topographic/bathymetric heights is sufficient as long as their spherical harmonic expansion is limited to degree and order 180. The impact of the higher order terms is less than 2 km. The Taylor expansion of the binomial term involving the Moho depth (T) up to second order with the assumption of T2 = TT0, T0 is the mean compensation depth, improves this approximation further by up to 4 km over continents. This approximation has a significant role in Moho modelling over continents; otherwise, loss of frequency occurs in the Moho solution. On the other hand, the linear approximation performs better over oceans and considering higher order terms creates unrealistic frequencies reaching to a magnitude of 5 km in the Moho solution. Involving gravity data according to the VMM principle influences the Moho depth significantly up to 15 km in some areas.
Estimation of variance in an ordinary adjustment model is straightforward, but if the model becomes unstable or illconditioned its solution and the variance of the solution will be very sensitive to the errors of observations. This sensitivity can be controlled by stabilizing methods but the results will be distorted due to stabilization. In this paper, stabilizing an unstable condition model using Tikhonov regularization, the estimations of variance of unit weight and variance components are investigated. It will be theoretically proved that the estimator of variance or variance components has not the minimum variance property when the model is stabilized, but unbiased estimation of variance is possible. A simple numerical example is provided to show the performance of the theory.

The subcrustal stress components due to mantle convection have a direct relation with the spherical harmonic coefficients of the Earth's disturbing potential like those of the Moho model, developed by the Vening–Meinesz–Moritz theory. In this paper, the relation between the stress components and the global and local models of Moho is mathematically developed in three different ways. Here, we present the S function (S) with a numerical differentiation approach to generate the stress components and we show that its spherical harmonic series is convergent to a degree of about 600 based on a mean global Moho depth of 23 km. An integral approach is developed for integration of a local Moho model for the stress recovery, but the kernels of this integral are not likely to be convergent and should be generated by their spectral forms to a limited degree. Another method is developed based on integral inversion, which is free of any mathematical problem and suitable for recovering S from an existing model of Moho. Our numerical presentation shows that the stress has a good agreement with the tectonic boundaries and the places at which the curvature of the Moho surface changes.
The Gravity field and steadystate Ocean Circulation Explorer (GOCE) mission is dedicated to recover spherical harmonic coefficients of the Earth's gravity field to degree and order of about 250 using its satellite gradiometric data. Since these data are contaminated with coloured noise, therefore, their inversion will not be straightforward. Unsuccessful modelling of this noise will lead to biases in the harmonic coefficients presented in the Earth's gravity models (EGMs). In this study, five of the recent EGMs of GOCE such as two direct, two timewise and one spacewise solution are used to degree and order 240 and their reliability is investigated with respect to EGM08 which is assumed as a reliable EGM. The detected unreliable coefficients and their errors are replaced by the corresponding ones from EGM08 as a combination strategy. A condition adjustment model is organised for each two corresponding coefficients of GOCE EGMs and EGM08; and errors of the GOCE EGMs are calibrated based on a scaling factor, obtained from a posteriori variance factor. When the factor is less than 2.5 it will be multiplied to the error otherwise the error of EGM08 coefficient will be considered as the calibrated one. At the end, a simple geoid estimator is presented which considers the EGMs and their errors and its outcomes are compared with the corresponding geoid heights derived from the Global Positioning System (GPS) and the levelling data (GPS/levelling data), over Fennoscandia. This comparison shows that some of the combinedcalibrated GOCE EGMs are closer to the GPS/levelling data than the original ones.
Gravity and topographic/bathymetric data are used for gravimetric modelling of Moho discontinuity by hydrostatic or flexural theories of the isostasy. Here, two hydrostatic models, based on the Vening MeineszMoritz (VMM) principle, and two based on the loading theories and flexural isostasy are compared over Tibet Plateau. It is shown that the Moho models generated based on the VMM theory and flexural isostasy have very good agreements if the mean compensation depth and the mean elastic thickness are selected properly. However, the model computed based on the flexural isostasy is smoother. A more rigorous flexural model, which considers the membrane stress and curvature of the lithosphere, is used to model the Moho surface over the study area. It is shown that the difference between the Moho models, derived by considering and ignoring these parameters, is not significant. By combination of the flexural and VMM hydrostatic models new mathematical formulae for crustal gravity anomalies are provided and it is shown that the crustal gravity anomalies produced by them are also equivalent.
A smart city is a city equipped with many sensors communicating with each other for different purposes. Cybersecurity and signal security are important in such cities, especially for airports and harbours. Any signal interference or attack on the navigation of autonomous vehicles and aircraft may lead to catastrophes and risks in people's lives. Therefore, it is of tremendous importance to develop wireless security networks for the localisation of any radio frequency interferer in smart cities. Time of arrival, angle of arrival, timedifference of arrivals, received signal strength and received signal strength difference (RSSD) are known observables used for the localisation of a signal interferer. Localisation means to estimate the coordinates of an interferer from some established monitoring stations and sensors receiving such measurements from an interferer. The main goal of this study is to optimise the geometric configuration of the monitoring stations using a desired dilution of precision and/or variancecovariance matrix (VCM) for the transmitter's location based on the RSSD. The required mathematical models are developed and applied to the Arlanda international airport of Sweden. Our numerical tests show that the same configuration is achieved based on dilution of precision and VCM criteria when the resolution of design is lower than 20 m in the presence of the same constraints. The choice of the pathloss exponent in the mathematical models of the RSSDs is not important for such low resolutions. Finally, optimisation based on the VCM is recommended because of its larger redundancy and flexibility in selecting different desired variances and covariances for the coordinates of the transmitter.
Precise navigation, as a method for guiding vehicles from one point to another, is an important subject these days especially in navigation of aircraft. Global navigation satellite systems (GNSSs) are capable tools for such a purpose. Any intentional or unintentional interference in satellite signals may cause risks of deadly accidents. Therefore, it is tremendously important to control airports or harbors and locate any existing radio frequency interference device. This localization can be done based on measuring time of arrival (TOA), angle of arrival (AOA), or time difference of arrival (TDOA) of signals from the device to sensors or receivers at some basepoints. In this article, a method is proposed based on these arrivals for optimizing the configuration created by these basepoints from a large grid of points covering a control area. Furthermore, a simulation test was performed to verify the theory, and after that a control network was designed and optimized for the international Landvetter Airport of Sweden. Our simulation studies show that when the AOA is used, our optimization is more robust with respect to the control grid resolution. In addition, optimization based on the TDOA improves the coverage over the control area with a significant reduction of error of control points, but because of the special geometric shape of the Landvetter Airport, such an optimization was not successful.
Satellite gravimetry and the solid Earth: mathematical foundations presents the theories behind satellite gravimetry data and their connections to solid Earth. It covers the theory of satellite gravimetry and data analysis, presenting it in a way that is accessible across geophysical disciplines. Through a discussion of satellite measurements and the mathematical concepts behind them, the book shows how various satellite measurements, such as satellite orbit, acceleration, vector gravimetry, gravity gradiometry, and integral energy methods can contribute to an understanding of the gravity field and solid Earth geophysics. Bridging the gap between geodesy and geophysics, this book is a valuable resource for researchers and students studying gravity, gravimetry and a variety of geophysical and Earth Science fields.
The Earth’s gravity potential can be determined from its secondorder partial derivatives using the spherical gradiometric boundaryvalue problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased and unbiasedtypes are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biasedtype integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of farzone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250 km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral
The Earth gravity field is a signature of the Earth’s mass heterogeneities and structures and applied in Geodesy and Geophysics for different purposes. One of the main goals of Geodesy is to determine the physical shape of the Earth, geoid, as a reference for heights, but Geophysics aims to understand the Earth’s interior. In this chapter, the general principles of geoid determination using the wellknown methods of RemoveComputeRestore, StokesHelmert and leastsquares modification of Stokes’ formula with additive corrections are shortly discussed. Later, some Geophysical applications like modelling the Mohorovičić discontinuity and density contrast between crust and uppermantle, elastic thickness, ocean depth, sediment and ice thicknesses, sublithospheric and lithospheric stress, Earthquakes and epicentres, postglacial rebound, groundwater storage are discussed. The goal of this chapter is to briefly present the roll of gravity in these subjects.
The satellite gravity gradiometry (SGG) data can be used for local modelling of the Earth's gravity field. In this study, the SGG data in the local northoriented and orbital frames are inverted to the gravity anomaly at sea level using the secondorder partial derivatives of the extended Stokes formula. The emphasis is on the spatial truncation error and the kernel behaviour of the integral formulas in the aforementioned frames. The paper will show that only the diagonal elements of gravitational tensor at satellite level are suitable for recovering the gravity anomaly at sea level. Numerical studies show that the gravity anomaly can be recovered in Fennoscandia with an accuracy of about 6 mGal directly from onorbit SGG data.
Surveying Engineering Education (SEE) programmes are often of three years and the students learn how to model the shape of the Earth's surface by specific instruments, applying mathematics and computer software. They are inherently practicallyoriented and majority of their courses contain simulated practical assignments close to the job market. Workintegrated learning (WIL), which is an approach to intentionally involve learners with practical works so that the learn the concepts by using them, is a suitable for SEEs. Different WIL models such as simulated assignment, collaborative learning with help of industry, and cooperative education, are applicable in SEE, which can increase the students' selfconfidence, motivation, academic performance, and employability potential in addition to prepare them for working environments. Here, the focus is on the cooperative education in SEE, which is done outside universities. Literatures about WIL lacks attention to SEEs, there is a need for more researches with focus on the stateoftheactual in this subject rather to see challenges in the work placement of students in businessoriented private sectors. Our literature study and interviews of three graduates from three subsequent graduation years and two students having experience in work placement showed clear supports of the governmental organisations. However, some students experienced difficulties in private companies. The job has been stressful for them and they were sometimes used like labours. Making clear agreements with private companies, clarifying the roles and missions of the students and companies, covering the costs and compensating time are important factors, which need to be considered. Inviting active engineers from companies for performing simulated assignments close to reality at campus will be very helpful for preparing the students for work placement for practical parts of the courses. It is also recommended that cooperative education needs to be performed with a full supervision of university.
In the traditional method of optimal design of displacement monitoring networks a higher precision, times better than the desired accuracy of displacements, is considered for the net points in such a way that the accuracy of the detected displacements meets the desired one. However, in this paper, we develop an alternative method by considering the total number of observations in two epochs without such a simple assumption and we call it twoepoch optimisation. This method is developed based on the GaussHelmert adjustment model and the variances of the observations are estimated instead of the weights to optimise the observation plan. This method can deliver the same results as the traditional one, but with less required observations in each epoch
One of the problems in the singleobjective optimisation models (SOOMs) for optimising geodetic networks is the contradiction of the controlling constraints, which may lead to their violation or infeasibility in the optimisation process. One way to solve this problem is to use a biobjective optimisation model (BOOM) instead of SOOMs. In this paper, we will use the BOOM of precision and reliability and investigate the influence of the controlling constraints in a twodimensional simulated network. Our studies show that the unconstrained BOOM is a good model, which almost fulfils our precision and reliability demands of the network. This model is also economical as more observables are removed from the plan whilst adding the controlling constraints leads to including more observables, which have no significant role
The tensor of gravitation is traceless as the gravitational field of the Earth is harmonic outside the Earth's surface. Therefore, summation of the 2ndorder horizontal derivatives on its diagonal components should be equal to the radial one but with the opposite sign. The gravity field can be recovered locally from either of them, or even their combination. Here, we use the inorbit diagonal components of the gravitational tensor measured by the gravity field and steady stateocean circulation explorer (GOCE) mission for recovering gravity anomaly with a resolution of 1°×1° at sea level in Ethiopia. In order to solve the system of equations, derived after discretisation of integral equations, the Tikhonov regularisation is applied and the bias of thi sregularisation is estimated and removed from the estimated gravity anomalies. The errors of the anomalies are estimated and their significance of recovery from these diagonal components is investigated. Statistically, the difference between the recovered anomalies from each scenario isnot significant comparing to their errors. However, their joint inversion of the diagonal components improved the solution by about 1 mGal. Furthermore, the inversion processes arebetter stabilised when using errors of the input data compared with its exclusion, but at the penalty of degradation in accuracy of the estimates.
In this study, we develop three estimators to optimally combine seismic and gravimetric models of Moho surface. The first estimator combines them by their special harmonic coefficients; the second one uses the spherical harmonic coefficients of the seismic model and use integral formula for the gravimetric one. The kernel of the integral terms of this estimator shows that a cap size of 20^{◦} is required for the integration, but since this integral is presented to combine the low frequencies of the gravimetric model, a low resolution model is enough for the integration. The third estimator uses the gravity anomaly and converts its low frequencies to those of the gravimetric Moho model, meanwhile combining them with those of seismic one. This integral requires an integration domain of 30^{◦} for the gravity anomalies but since the maximum degree of this kernel is limited to a specific degree, the use of its spectral form is recommended. The kernel of the integral involving the gravity anomalies, developed for recovering high frequencies of Moho, is written in a closedfrom formula and its singularity is investigated. This kernel is wellbehaving and decreases fast, meaning that it is suitable for recovering the high frequencies of Moho surface.
The gravimetric model of the Moho discontinuity is usually derived based on isostatic adjustment theories considering floating crust on the viscous mantle. In computation of such a model some a priori information about the density contrast between the crust and mantle and the mean Moho depth are required. Due to our poor knowledge about them they are assumed unrealistically constant. In this paper, our idea is to improve a computed gravimetric Moho model, by the Vening MeineszMoritz theory, using the seismic model in Fennoscandia and estimate the error of each model through a combined adjustment with variance component estimation process. Corrective surfaces of bilinear, biquadratic, bicubic and multiquadric radial based function are used to model the discrepancies between the models and estimating the errors of the models. Numerical studies show that in the case of using the bilinear surface negative variance components were come out, the biquadratic can model the difference better and delivers errors of 2.7 km and 1.5 km for the gravimetric and seismic models, respectively. These errors are 2.1 km and 1.6 km in the case of using the bicubic surface and 1 km and 1.5 km when the multiquadric radial base function is used. The combined gravimetric models will be computed based on the estimated errors and each corrective surface.
The NKG2015 geoid model covers the Nordic and Baltic countries and has been computed based on the leastsquares modification of Stokes’ formula with additive corrections method. New and precise terrestrial, airborne and shipborne gravimetric measurements, the recent global gravity model of the gravity field and steadystate ocean circulation explorer (GOCE) and detailed digital terrain models over each territory have been used for computing this new geoid model. Some estimates for the error of this model have been roughly presented by comparing it with the global navigation satellite system(GNSS) data over each country. In this paper, our goal is to have a closer look at the relative error of this model by performing some statistical tests and finding the proper corrective surface for absorbing the systematic errors over each country. Our main assumption is realisticity of the errors of GNSS/levelling data and we will investigate its consequences in estimating the error of the geoid model. Our results show that the 4parameter corrective surface is suitable for modelling the systematic trends of the differences between the gravimetric and GNSS geoid heights in Sweden, Denmark and Finland, but a filtered discrepancies by a confidence interval of 95% should be used for Sweden. A 7aparameter model is suitable for the filtered discrepancies with the confidence interval of 95% in Norway. Based on the selected corrective surface and our newly developed regional iterative variance estimator, the confidence interval for the error of NKG2015 geoid model in Sweden, Denmark and Norway yielded 06.5 mm, 1.85.2 mm, 14.817.7 mm, respectively with a confidence level of 95%. We could not estimate the geoid error in Finland because the given error of the GNSS/levelling heights is significantly larger than the size of residuals. Based on the selected corrective surfaces and our presented local variance estimator, the average error of geoid becomes 3.6, 2.4, 8.8 and 5.8 mm with a confidence interval of 68%, respectively, over Sweden, Denmark, Norway and Finland.
In this paper, three independent Earth gravity models (EGMs) ofGO_CONS_GCF_2_TIM_R4, AIUBGRACE03S and ULux_CHAMP2013s are combined to degree and order 120. The geoid models of these EGMs are computed and compared with the Global Positioning System (GPS) and levelling data over Fennoscandia. We found that the simple mean of these geoid models is closer to the GPS/levelling data than their weighted mean. This means that errors of the EGMs are not properly estimated as they are used in the weighted mean solution. We develop a method based on solving a nonlinear condition adjustment model to calibrate the errors so that the result of weighted mean becomes the same as that of the simple mean. Numerical results show slight changes in the errors of GRACE03S but large ones in those of GO_CONS_GCF_2_TIM_R4 and ULux_CHAMP2013s. Furthermore, the weighted mean solution considering the calibrated errors and some additional constraints is better than GOCO03S to degree and order 120 over Fennoscandia.
One of the major issues associated with a regional Moho recovery from the gravity or gravitygradient data is the optimal choice of the mean compensation depth (i.e., the mean Moho depth) for a certain area of study, typically for orogens characterised by large Moho depth variations. In case of selecting a small value of the mean compensation depth, the pattern of deep Moho structure might not be reproduced realistically. Moreover, the definition of the mean compensation depth in existing isostatic models affects only lowdegrees of the Moho spectrum. To overcome this problem, in this study we reformulate the Sjöberg and Jeffrey’s methods of solving the VeningMeinesz isostatic problem so that the mean compensation depth contributes to the whole Moho spectrum. Both solutions are then defined for the vertical gravity gradient, allowing estimating the Moho depth from the GOCE satellite gravitygradiometry data. Moreover, gravimetric solutions provide realistic results only when a priori information on the crust and upper mantle structure is known (usually from seismic surveys) with a relatively good accuracy. To investigate this aspect, we formulate our gravimetric solutions for a variable Moho density contrast to account for variable density of the uppermost mantle below the Moho interface, while taking into consideration also density variations within the sediments and consolidated crust down to the Moho interface. The developed theoretical models are applied to estimate the Moho depth from GOCE data at the regional study area of the Iranian tectonic block, including also parts of surrounding tectonic features. Our results indicate that the regional Moho depth differences between Sjöberg and Jeffrey’s solutions, reaching up to about 3 km, are caused by a smoothing effect of Sjöberg’s method. The validation of our results further shows a relatively good agreement with regional seismic studies over most of the continental crust, but large discrepancies are detected under the Oman Sea and the Makran subduction zone. We explain these discrepancies by a low quality of seismic data offshore.
Temporal variations in the Earth’s gravity field can be used for monitoring of lithospheric deformations. The network of continuously operating gravity stations is required for this purpose but a global coverage by such network is currently extremely sparse. Temporal variations in longwavelength part of the Earth’s gravity field have been, however, observed by two satellite missions, namely the Gravity Recovery And Climate Experiment (GRACE) and the GRACE FollowOn (GRACEFO). These satellite gravity observations can be used to study longwavelength deformations of the lithosphere. Consequently, considering the lithosphere as a spherical elastic shell and solving the partial differential equation of elasticity for it, the stress, strain and displacement inside the lithosphere can be estimated. The lower boundary of this shell is assumed to be stressed by mantle convection, which has a direct relation to the Earth’s gravity field according to Runcorn’s theory. Changes in gravity field lead to changes in the sublithospheric stress and the stress propagated throughout the lithosphere. In this study, we develop mathematical models in spherical coordinates for describing the stress propagation from the sublithosphere through the lithosphere. We then organize a system of observation equations for finding a special solution to the boundaryvalue problem of elasticity in the way that provides a stable solution. In contrast, models presented in previously published studies are illposed. Furthermore, we use constants of the solution determined from the boundary stresses to determine the strain and displacements leading to these stresses, while in previous studies only the stress has been considered according to rheological properties of the lithosphere. We demonstrate a practical applicability of this theoretical model to estimate the stress–strain redistribution caused by the SarePol Zahab 2018 earthquake in Iran by using the GRACEFO monthly solutions.
The spatial truncation error (STE) is a significant systematic error in the integral inversion of satellite gradiometric and orbital data to gravity anomalies at sea level. In order to reduce the effect of STE, a larger area than the desired one is considered in the inversion process, but the anomalies located in its central part are selected as the final results. The STE influences the variance of the results as well because the residual vector, which is contaminated with STE, is used for its estimation. The situation is even more complicated in variance component estimation because of its iterative nature. In this paper, we present a strategy to reduce the effect of STE on the a posteriori variance factor and the variance components for inversion of satellite orbital and gradiometric data to gravity anomalies at sea level. The idea is to define two windowing matrices for reducing this error from the estimated residuals and anomalies. Our simulation studies over Fennoscandia show that the differences between the 0.5°×0.5°0.5°×0.5° gravity anomalies obtained from orbital data and an existing gravity model have standard deviation (STD) and root mean squared error (RMSE) of 10.9 and 12.1 mGal, respectively, and those obtained from gradiometric data have 7.9 and 10.1 in the same units. In the case that they are combined using windowed variance components the STD and RMSE become 6.1 and 8.4 mGal. Also, the mean value of the estimated RMSE after using the windowed variances is in agreement with the RMSE of the differences between the estimated anomalies and those obtained from the gravity model.
The orbital elements of a low Earth orbiting satellite and their velocities can be used for local determination of gravity anomaly. The important issue is to find direct relations among the anomalies and these parameters. Here, a primary theoretical study is presented for this purpose. The Gaussian equations of motion of a satellite are used to develop integral formulas for recovering the gravity anomalies. The behaviour of kernels of the integrals are investigated for a twomonth simulated orbit similar to that of the Gravity field and steadystate ocean circulation explorer (GOCE) mission over Fennoscandia. Numerical investigations show that the integral formulas have neither isotropic nor wellbehaved kernels. In such a case, gravity anomaly recovery is not successful due to large spatial truncation error of the integral formulas. Reformulation of the problem by combining the orbital elements and their velocities leads to an integral with a wellbehaved kernel which is suitable for our purpose. Also based on these combinations some general relations among the orbital elements and their velocities are obtained which can be used for validation of orbital parameters and their velocities
All quantities, which are measured in the gravity field of the Earth are affected by the field, therefore, there should be correlations amongst them. Here, we focus on some gravimetricallydetermined quantities like deflections of vertical, deflections of Moho, verticalhorizontal gravity gradients and the shear sublithospheric stress components due to mantle convection. We show that how these quantities are related to each other mathematically so that one of them can be written in term of another. This somehow proves the presence of the mentioned correlations theoretically. Also, we generate the maps of these quantities over the IndoPak and surrounding areas and show how similar they are. Thereafter, they are explained and interpreted geologically. Our investigations show that the maps of these quantities are in good agreements with topographic and geological features. The map of the verticalhorizontal gravity gradients shows more detailed information of the gravity field due to signal amplification at high degrees, that of Moho deflection shows subsurface features due to reduction of the effect of topographic masses. The map of the shear sublithospheric stress components is much smoother than the gradients, as expected, and has good agreement with the collisional and subduction zones as well.
Relationship amongst gravity gradients, deflection of vertical, Moho deflection and the stresses derived by mantle convectionsa case study over IndoPak and surroundings. Available from: https://www.researchgate.net/publication/292538682_Relationship_amongst_gravity_gradients_deflection_of_vertical_Moho_deflection_and_the_stresses_derived_by_mantle_convectionsa_case_study_over_IndoPak_and_surroundings [accessed Feb 1, 2016].
Seismic data are primarily used in studies of the Earth's inner structure. Since large partsof the world are not yet sufficiently covered by seismic surveys, products from the Earth's satellite observation systems have more often been used for this purpose in recent years. In this study we use the gravitygradient data derived from the Gravity field and steadystate Ocean Circulation Explorer (GOCE), the elevation data from the Shuttle Radar Topography Mission (SRTM) and other global datasets to determine the Moho density contrast at the study area which comprises most of the Eurasian plate (including parts of surrounding continental and oceanic tectonic plates). A regional Moho recovery is realized by solving the Vening MeineszMoritz's (VMM) inverse problem of isostasy and a seismic crustal model is applied to constrain the gravimetric solution. Our results reveal that the Moho density contrast reaches minima along the midoceanic rift zones and maxima under the continental crust. This spatial pattern closely agrees with that seen in the CRUST1.0 seismic crustal model as well as in the KTH1.0 gravimetricseismic Moho model. However, these results differ considerably from some previously published gravimetric studies. In particular, we demonstrate thatt here is no significant spatial correlation between the Moho density contrast and Moho deepening under major orogens of Himalaya and Tibet. In fact, the Moho density contrast under most of the continental crustal structure is typically much more uniform.
In this research, a modified form of Vening MeineszMoritz (VMM) theory of isostasy for the secondorder radial derivative of gravitational potential, measured from the Gravity field and steadystate Ocean Circulation Explorer (GOCE), is developed for local Moho depth recovery. An integral equation is organised for inverting the GOCE data to compute a Moho model in combination with topographic/bathymetric heights of SRTM30, sediment and consolidated crystalline basement and the laterallyvarying density contrast model of CRUST1.0. A Moho model from EGM2008 to degree and order 180 is also computed based on the same principle for the purpose of comparison. In addition, we compare both of them with the 3 available seismic Moho models; two global and one regional over the IndoPak region. Numerical results show that our GOCEbased Moho model is closer to the all seismic models than that of EGM2008. The model is closest to the regional one with a standard deviation of 5.5 km and a root mean squares error of 7.8 km, which is 2.3 km smaller than the corresponding one based on EGM2008.
Sublithospheric stresses can be estimated by analysis of gravity field measurements. Depending on the measured gravimetric quantity, different methods can be employed to estimate those sublithospheric stresses. Here, we further develop the Runcorn's theory for estimation of mantle stresses (1967) such that a Moho model and full topographic information are used to recover the function from which the stress can be computed by taking derivatives northwards and eastwards. We develop new integral equations for such a purpose and recover this function by solving those integral equations locally over the IndoPak (IndiaPakistan) region from (1) a gravimetric Moho model computed from the SRTM (Shuttle Radar Topography Mission) and the Earth gravity model EGM2008, (2) SRTM and the seismic Moho model of CRUST1.0 and (3) data and measurements of the GOCE (Gravity field and steadystate Ocean Circulation Explorer) mission. Finally, we perform a joint inversion of seismic and GOCE data for the same purpose. The numerical results show that the use of a seismic Moho model recovers information about the stress field which is not seen in the results derived from a gravimetric Moho model. A combination of the seismic Moho model, SRTM and GOCE yields a better stress field than that of either the seismic and/or gravimetric data alone. The magnitudes of the sublithospheric stress are computed from the shear stress components over the area and good agreement is seen between the recovered combined stress field, the regional tectonic boundaries and the seismicity of the World Stress Map 2008 database.
Monitoring deformation of manmade structures is very important to prevent them from a risk of collapse and save lives. Such a process is also used for monitoring change in historical objects, which are deforming continuously with time. An example of this is the Vasa warship, which was under water for about 300 years. The ship was raised from the bottom of the sea and is kept in the Vasa museum in Stockholm. A geodetic network with points on the museum building and the ship's body has been established and measured for 12 years for monitoring the ship's deformation. The coordinate time series of each point on the ship and their uncertainties have been estimated epochwisely. In this paper, our goal is to statistically analyse the ship's hull movements. By fitting a quadratic polynomial to the coordinate time series of each point of the hull, its acceleration and velocity are estimated. In addition, their significance is tested by comparing them with their respective estimated errors after the fitting. Our numerical investigations show that the backside of the ship, having highest elevation and slope, has moved vertically faster than the other places by a velocity and an acceleration of about 2 mm/year and 0.1 mm/year2, respectively and this part of the ship is the weakest with a higher risk of collapse. The central parts of the ship are more stable as the ship hull is almost vertical and closer to the floor. Generally, the hull is moving towards its port and downwards
The dedicated satellite mission GOCE will sense various small mass variations along its path around the Earth. Here we study the effect of the Earth's topography and atmosphere on GOCE data. The effects depend on the magnitude of topographic height, and they will therefore vary by region. As the effect of the atmosphere and topography must be removed from the total gravity anomaly prior to geoid determinations, these effects should also be removed to simplify the downward continuation of the GOCE data to the sea level. The main goal of this article is to investigate the direct topographic and atmospheric effects in a rough region like Iran. Maps of the direct effects and their statistics are presented and discussed. Numerical results show maximum direct topographic and atmospheric effects on the GOCE data can reach 2.64 E and 5.53 mE, respectively, when the satellite flies over Iran. The indirect effect of the atmospheric and topographic masses are also formulated and presented.