The Earth’s gravitational potential can be expressed by the well-known spherical harmonic expansion. The computationaltime of summing up this expansion is an important practical issue which can be reduced by an efficientnumerical algorithm. This paper proposes such a method for block-wise synthesizing the anomaly quantities onthe Earth surface using vectorization.Fully-vectorization means transformation of the summations to the simple matrix and vector products. It is not apractical for the matrices with large dimensions. Here a semi-vectorization algorithm is proposed to avoid workingwith large vectors and matrices. It speeds up the computations by using one loop for the summation either ondegrees or on orders. The former is a good option to synthesize the anomaly quantities on the Earth surfaceconsidering a digital elevation model (DEM). This approach is more efficient than the two-step method whichcomputes the quantities on the reference ellipsoid and continues them upward to the Earth surface. The algorithmhas been coded in MATLAB which synthesizes a global grid of 50 x 50 (corresponding 9 million points) of gravityanomaly or geoid height using a geopotential model to degree 360 in 10000 seconds by an ordinary computer with2G RAM.

During recent years there has been a great focus on the climate changes within the media. More or less every day more newspaper articles are presented about the global warming issue and the effect on us human race. Climate models predict higher temperatures and more rain in the northern part of Europe. It is also predicted that the weather will become more extreme e.g. it will rain a lot during longer periods than has been the norm. If these predictions are correct, the amount of water that is going to be transported away in streams and rivers will increase and so also will the subsoil water level. The latter increases the risk for landslides in areas with fine grained soils. An early warning system that is able to alert people before a landslide take place would be of great interest.

The purpose of this work is to develop a complete real-time displacement monitoring system based on observations from several GPS-receivers that could be used as an early warning system. Due to the complex correlation structure of the traditionally used double differences, an alternative method based on undifferenced observations is used. Theoretically this approach shows some advantages and simplifies the correlative structure of observables compared to the double differenced method. A complete model for the undifferenced approach is presented in this thesis including its software implementation.

A displacement detection system includes not only the positioning algorithms, but also methods to detect if any displacement occurs. There are many methods available to discriminate displacements, which are used in the traditional control of manufacturing processes. Several of these methods are compared in this thesis, such as the Shewhart chart, different Weighted Moving Average (WMA) charts and the CUmulative SUMmation (CUSUM). Practical tests show that it is possible to detect an abrupt shift on sub centimetre level at the same epoch as the shift occurs. Smaller shifts are also detectable with the applied approach but with a slightly longer detection time.

 The fennoscandia post glacial uplift rate of today, the time averaged vertical velocity, is usually determined by mareographs and repeated leveling. We consider situations where parts of the levelling not are observed only once. We consider here methods that handle such situations which has relevance to Norway, and we have applied the methods to southern Norway.

We have tested the different methods known to us, which have been used up to now. Least squares collocation has been treated, and both variance and covariance function for collocation with zero-degree polynomial trend surface. But still, collocation is one of the best, if not the best method.

Some new methods have been introduced. Especially the method we have called the geometric filter is promising and clearly an alternative to collocation.

Other types of measurement have been discussed. Along the Norwegian coast there are grooves in rocks, old water marks made mostly in the last century, which have been remeasured recently. This gives us reliable values for the land uplift, but with varying degree of accuracy.

We also considered geological data to see if they can be utilized for the determination of land uplift of today. They can, but are all located along the coastline, where they are least needed, and very few are now available.   

The main purpose of this study is to compute a gravimetric geoid model of Greece using the least squares modification method developed at KTH. In regional gravimetric geoid determination, the modified Stokes' formula that combines local terrestrial data with a global geopotential model is often used nowadays. In this study, the optimum modification of Stokes' formula, introduced by Sjöberg (2003), is employed so that the expected mean square error (MSE) of the combined geoid height is minimized. According to this stochastic method, the geoid height is first computed from modified Stokes' formula using surface gravity data and a global geopotential model (GGM). The precise geoid height is then obtained by adding the topographic, downward continuation, atmospheric and ellipsoidal corrections to the approximate geoid height. In this study the downward continuation correction was not considered for the precise geoid height computations due to a limited DEM. The dataset used for the computations, consisted of terrestrial gravimetric measurements, a DEM model and GPS/Levelling data for the Greek region. Three global geopotential models (EGM96, EIGEN-GRACE02S, EIGEN-GL04C) were tested for choosing the best GGM to be combined into the final solution. Regarding the evaluation and refinement of the terrestrial gravity measurements, the cross-validation technique has been used for detection of outliers. The new Greek gravimetric geoid model was evaluated with 18 GPS/Levelling points of the Greek geodetic network. After using a 7-parameter model to fit the geoid model to the GPS/Levelling data, the agreement between the absolute geoid heights derived from the gravimetric method and the GPS/Levelling data, was estimated to 27 cm while the agreement for the relative geoid heights after the fitting, to 0.9 ppm. In an optimal case study, considering the accuracies of the ellipsoidal and orthometric heights as σh≈±10 cm and σH≈±20 cm respectively, the RMS fit of the model with the GPS/Levelling data was estimated to σN≈±15 cm. The geoid model computed in this study was also compared with some previous Greek geoid models, yielding better external accuracy than them.

The Polar Regions are not covered by satellite gravity gradiometry data if the orbital inclination of the satellite is not equal to 90 degrees. This paper investigates the feasibility of determining gravity anomaly (at sea level) by inversion of satellite gravity gradiometry data in these regions. Inversion of each element of tensor of gravitation as well as their joint inversion are investigated. Numerical studies show that gravity anomaly can be recovered with an error of 3 mGal in the north polar gap and 5 mGal in south polar gaps in the presence of 1 mE white noise in the satellite data. These errors can be reduced to 1 mGal and 3 mGal, respectively, by removing the regularization bias from the recovered gravity anomalies.

The traditional expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics have complicated forms involved with first- and second-order derivatives of spherical harmonics and also singular terms. In this paper we present alternative expressions for these quantities, which are simpler and contain no singular terms. The presented formulas are useful for those disciplines of geosciences which are involved with potential theory, tensor spherical harmonics and second-order derivatives of spherical harmonic series in the local northoriented frame. A simple numerical test on the solution of the gradiometric boundary value problems presents the correctness of these new expressions and ability of the solutions to continue the gravity gradients from satellite level down to sea level using spherical harmonics.

The satellite gravity gradiometric data are influenced by laterally varying density in topographic masses, while in most of studies a constant density for the masses was considered. This assumption causes an error in estimating the topographic effect. This paper theoretically and numerically investigates the methods of Sjoberg as well as Novak and Grafarend to consider the laterally varying density for topographic masses in formulation of topographic potential in spherical harmonics.

In mathematical modeling of the topographic and atmospheric potentials in spherical harmonics, the topographic heights can binomially be expanded a certain order, usually to the third order. Some studies have been done on the effect of each order on geoid and gravity anomaly. However similar study on the satellite gravity gradiometric data is missed yet. This paper will investigate this matter globally. It presents that the contribution of the second- and third-order topographic terms is within 0.08 E and 2 mE, respectively on satellite gravity gradiometric data at 250 km level. Also the contribution of these terms is within 0.5 mE and 0.08 mE for the atmospheric effect.

Errors of estimated parameters in an adjustment process should be scaled according to the size of the estimated residuals or misclosures. After computing a quasigeoid (geoid), its biases and tilts, due to existence of systematic errors in the terrestrial data, are removed by fitting a corrective surface to the misclosures of the differences between the GNSS/levelling data and the quasi-geoid (geoid). Variance component estimation can be used to re-scale or calibrate the error of the GNSS/levelling data and the quasi-geoid (geoid) model. This paper uses this method to calibrate the errors of the recent quasi-geoid model, the GNSS and the normal heights of Sweden. Different stochastic models are investigated in this study and based on a 7-parameter corrective surface model and a three-variance component stochastic model, the calibrated error of the quasi-geoid and the normal heights are 6 mm and 5 mm, respectively and the re-scaled error of the GNSS heights is 18 mm.

Computational time is an important matter in numerical aspects and it depends on the algorithm and computer that is used. An inappropriate algorithm can increase computation time and cost. The main goal of this paper is to present a vectorization algorithm to speed up the global gradiometric synthesis and analysis. The paper discusses details of this technique and its very high capabilities. Numerical computations show that the global gradiometric synthesis with 0.5 degrees x 0.5 degrees resolution can be done in a few minutes (6 minutes) by vectorization, which is considerable less compared to several hours (9 hours) by an inappropriate algorithm. The global gradiometric analysis of representation by spherical harmonics up to degree and order of 360, can be performed within one hour using vectorization, but if an inconvenient algorithm is used it can be delayed more than 1 day. Here we present the vectorization technique to gradiometric synthesis and analysis, but it can also be used in many other computational aspects and disciplines.

Satellite gravity gradiometry is a technique to determine a precise high-resolution geopotential model based on spatial-differential accelerometry. The satellite gradiometric data plays an important role in this respect and they must be validated before doing any computation. One way of validating such a data is to use the second-order partial derivatives of the extended Stokes formula to generate the gradients at satellite level, from terrestrial gravimetric data. A global coverage of the terrestrial data is required to perform the integration, but having such coverage is neither practical nor reasonable, and the integrals should be modified. The integrals’ kernel is not isotropic (except for second-order radial derivative) and modification of such integrals will not be easy task. Here, general integral estimators for vertical-vertical, vertical-horizontal and horizontal-horizontal gradients are presented, based on combination of the gradients, so that that the estimators become modifiable. Least-squares modification minimizes not only the truncation error of the integral, but the errors of global gravitational model and the terrestrial data. Elements of the system of equations, from which the modification parameters based on biased, unbiased and optimum least-squares modification is derived, are mathematically formulated.

The gravity anomalies at sea level can be used to validate the satellite gravity gradiometry data. Validation of such a data is important prior to downward continuation because of amplification of the data errors through this process. In this paper the second-order radial derivative of the extended Stokes' formula is employed and the emphasis is on least-squares modification of this formula to generate the second-order radial gradient at satellite level. Two methods in this respect are proposed: (a) modifying the second-order radial derivative of extended Stokes' formula directly, and (b) modifying extended Stokes' formula prior to taking the second-order radial derivative. Numerical studies show that the former method works well but the latter is very sensitive to the proper choice of the cap size of integration and degree of modification.

Solution of the gradiometric boundary value problems leads to three integral formulas. If we are satisfied with obtaining a smooth solution for the Earth's gravity field, we can use the formulas in regional gravity field modelling. In such a case, satellite gradiometric data are integrated on a sphere at satellite level and continued downward to the disturbing potential (geoid) at sea level simultaneously. This paper investigates the gravity field modelling from a full tensor of gravity at satellite level. It studies the truncation bias of the integrals as well as the filtering of noise of data. Numerical studies show that by integrating T (zz) with 1 mE noise and in a cap size of 7A degrees, the geoid can be recovered with an error of 12 cm after the filtering process. Similarly, the errors of the recovered geoids from T (xz,yz) and T (xx-yy, 2xy) are 13 and 21 cm, respectively.

The gravity gradiometric data are affected by the topographic and atmospheric masses. In order to fulfill Laplace-Poisson’s equation and to simplify the downward continuation process, these effects should be removed from the data. However, if the analytical downward continuation is considered, the gravity gradients can be continued downward disregarding such effects but the result will be biased. The topographic and atmospheric biases can be expressed in terms of spherical harmonics and studying these biases gives some ideas about analytical downward continuation of these quantities to sea level. In formulation of harmonic coefficients of the topographic and atmospheric biases, a truncated binomial expansion of topographic height is used. In this paper, we show that the harmonics are convergent to the third term of this binomial expansion. The harmonics of the biases on Vzz are convergent to the first term and they are convergent in Vxy for all the terms. The harmonics of the other components of the gravity gradient tensor are convergent to the second terms, while the third terms are only symptotically convergent. This means that in terrestrial and airborne gradiometry the biases should be computed just to the second order term, while in satellite gravity gradiometry, e.g. GOCE, the third term can also be considered.

Optimal estimation of geopotential coefficients is an important aspect of gravitational field recovery using satellite gravity gradiometry. The combination of gradiometric data and the use of tensor spherical harmonics is useful in this field. Here, we present a new strategy for combining different spectral solutions of the gradiometric boundary value problem by defining and formulating degree-order variance components and using the condition adjustment model. Numerical results show that the spectral combination of considering one degree-order variance component for each type of observation yields better results than the case where one degree-order variance component is estimated for each integral solution of the gradiometric boundary value problem. In this study, the estimates of the variance components are not considered in the standard way; rather, these components are mainly used to absorb the discretization error of the integral solutions. This method is capable of combining integrals in geosciences disciplines.

The topographic and atmospheric masses influence the satellite gravity gradiometry data, and it is necessary to remove these effects as precise as possible to make the computational space harmonic and simplify the downward continuation of such data. The topographic effects have been formulated based on constant density assumption for the topographic masses. However in this paper we formulate and study the effect of lateral density variation of crustal and topographic masses on the satellite gravity gradiometry data. Numerical studies over Fennoscandia and Iran show that the lateral density variation effect of the crust on GOCE data can reach to 1.5 E in Fennoscandia and 1 E in Iran. The maximum effect of lateral density variation of topography is 0.1 E and 0.05 E in Iran and Fennoscandia, respectively.

The lack of satellite gravity gradiometric data, due to inclined orbit, in the Polar Regions influences the geopotential coefficients obtained from the solutions of gradiometric boundary value problems. This paper investigates the polar gaps effect on these solutions and it presents that the near zero-, first- and second-order geopotential coefficients are weakly determined by the vertical-vertical, vertical-horizontal and horizontal solutions, respectively. Also it shows that the vertical-horizontal solution is more sensitive to the lack of data than the other solutions.

An ill-posed problem which involves heterogonous data can yield good results if the weight of observations is properly introduced into the adjustment model. Variance component estimation can be used in this respect to update and improve the weights based on the results of the adjustment. The variance component estimation will not be as simple as that is in an ordinary adjustment problem, because the result of the solution of an ill-posed problem contains a bias due to stabilizing the adjustment model. This paper investigates the variance component estimation in those ill-posed problems solved by the truncation singular value decomposition. The biases of the variance components are analyzed and the biased-corrected and the biased-corrected non-negative estimators of the variance components are developed. The derivations show that in order to estimate unbiased variance components, it suffices to estimate and remove the bias from the estimated residuals.

In solution of gradiometric boundary value problem in space a regular grid of satellite gravity gradiometry data is required. This grid is considered on a sphere with radius of the mean Earth sphere and altitude of satellite. However, the gravitational gradients are measured by a gradiometer mounted on GOCE satellite and orbital perturbations of the satellite influence GOCE observations as well. In this study we present that these effects are about 2 E on GOCE data. Also numerical studies on the gravitational gradients in orbital frame show that the perturbations of co-latitude are more significant than that of inclination. The effect of perturbed inclination is less than -9 mE while the effect of perturbed co-latitude is within -173 mE in one day revolution of GOCE.

The harmonic synthesis and analysis of the elements of gravitational tensor can be done in few minutes if a suitable programming algorithm is used. Vectorization is an efficient technique for such processes, but the size of matrices will increase when the resolution of synthesis or analysis is high; say higher than 0.5 degrees x 0.5 degrees. Here, we present a technique to manage the computer memory and computational time by excluding one computational loop from the matrix products and we call this method semi-vectorization. Based on this technique, we synthesize the gravitational tensor using the EGM96 geopotential model and after that we analyze the tensor for recovering the geopotential coefficients. MATLAB codes are provided which are able to analyze 224 millions gradiometric data, corresponding to a global grid of 2.5' x 2.5' on a sphere in 1,093 s by a personal computer with 2 Gb RAM.

The main point of this paper is to evaluate the perturbations in orbital elements of a low Earth orbiting satellite. The outcome of a numerical orbit integration process is the position and velocity vectors of satellite in an inertial coordinate system. The velocity and position vectors are converted into the corresponding orbital elements. Perturbations in a satellite motion affect the orbital elements in the sense of Keplerian motion. In this paper after introducing the perturbing forces acting on a satellite, the method of converting the position and velocity into the orbital elements is presented, and finally the perturbationsin orbital elements of the low Earth orbiting satellite of CHAMP are evaluated. The numerical results show that, disregarding the geopotential perturbing forces, the air drag is the most predominant among other perturbing forces: rotational deformation, solar radiation, third body effect, solid Earth tide, oceantide, and general relativity arranged by their magnitude respectively.

Estimated variance components may come out as negative numbers without physical meaning. One way out of this problem is to use non-negative methods. Different approaches have been presented for the solution. Sjöberg presented a method of Best Quadratic Unbiased Non-Negative Estimator (BQUNE) in the Gauss-Helmert model. This estimator does not exist in the general case. Here we present the Modified BQUNE (MBQUNE) obtained by a simple transformation from the misclosures used in the BQUE to residuals. In the Gauss-Markov adjustment model the BQUNE and MBQUNE are identical, and they differ in condition and Gauss-Helmert models only by a simple transformation. If the observations are composed of independent/disjunctive groups the MBQUNE exists in any adjustment model and it carries all the properties of the BQUNE (when it exists). The presented variance component models are tested numerically in some simple examples. It is shown that the MBQUNE works well for disjunctive groups of observations.

Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes' (geopotential) coefficients. This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects, the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to the Earth's surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is harmonic in the exterior to sea level. This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed.

Atmospheric masses play an important role in precise downward continuation and validation of satellite gravity gradiometry data. In this paper we present two alternative ways to formulate the atmospheric potential. Two density models for the atmosphere are proposed and used to formulate the external and internal atmospheric potentials in spherical harmonics. Based on the derived harmonic coefficients, the direct atmospheric effects oil the satellite gravity gradiometry data are investigated and presented ill the orbital frame over Fennoscandia. The formulas of the indirect atmospheric effects oil gravity anomaly and geoid (downward continued quantities) are also derived using the proposed density models. The numerical results show that the atmospheric effect can only be significant for precise validation or inversion of the GOCE gradiometric data at the mE level.

The problem of handling outliers in a deformation monitoring network is of special importance, because the existence of outliers may lead to false deformation parameters. One of the approaches to detect the outliers is to use robust estimators. In this case the network points are computed by such a robust method, implying that the adjustment result is resisting systematic observation errors, and, in particular, it is insensitive to gross errors and even blunders. Since there are different approaches to robust estimation, the resulting estimated networks may differ. In this article, different robust estimation methods, such as the M-estimation of Huber, the "Danish", and the L-1-norm estimation methods, are reviewed and compared with the standard least squares method to view their potentials to detect outliers in the Tehran Milad tower deformation network. The numerical studies show that the L-1-norm is able to detect and down-weight the outliers best, so it is selected as the favourable approach, but there is a lack of uniqueness. For comparison, Baarda's method "data snooping" can achieve similar results when the outlier magnitude of an outlier is large enough to be detected; but robust methods are faster than the sequential data snooping process.

In this thesis a system for aided inertial navigation (aided INS or AINS) has been testedthrough a number of kinematic experiments. In the experiments, data was collected andpost-processed by different methods. The system was built up by an IMU (inertial measurementunit) aided by a GPS receiver and an odometer. To fuse and filter the sensor dataa Kalman filter from the AINS Matlab Toolbox has been used.The following equipment was used• iNAV-RQH; IMU from iMAR• System 4000SSE; GPS receiver from Trimble• Correvit L-CE; odometer from Corsys-DatronThe following methods (and combinations of these) were used on the acquired data• INS with Zero velocity updates (ZUPT)• INS under non holonomic constraints• INS integrated with GPS• INS integrated with odometerThe work resulted in a number of trajectories; some of these could be compared with areference trajectory, in other cases this was not possible due to the nature of the experiment.Also there are no guidelines or standards on how to test a system in order to compare itwith other systems.Over all, good results were achieved with the system. The AINS toolbox worked outwell and it was indicated that the odometer could be used as an important aiding source inan aided inertial navigation system.

Today there are different methods to automatically generate 3D models from laser data, whichgive varying results. The two methods used in this study are, a fully automatic methoddeveloped by the Swedish Defence Research Agency, FOI, and a semiautomatic methodwithin the program Terra Scan which is developed by Terrasolid, a Finnish company.The report describes the way these methods have been implemented, and it evaluates theoutput quality of each model set created from specified criteria.In order to be able to verify the quality, a statistical model was created with the purpose ofusing it for other situations as well in the future.The modelling methods gave different results and both had their strengths and theirweaknesses, which makes them applicable for different tasks. The strength in FOI’s method isthat it could recreate even more advanced buildings and still get a good visual impression, e.g.Stockholm’s central station.One advantage with the method used in Terra Scan is that it has good capabilities and tools toedit the models, which makes it possible to vary the quality depending on the purpose.In order to generate the best result this method is not supposed to be used in an automaticway.

This article describes an algorithm for polynomial interpolation of GPS satellite coordinates and its implementation in MATLAB. The algorithm is intended for realtime processing software and computes the position and velocity of GPS satellites from both broadcast and precise ephemerides. Tests with different orders of polynomials, and with different time spans used for polynomial fitting, show suitable settings with respect to the required interpolation precision.

Using GPS technology in motion detection and deformation measurements are getting more common. Different concepts are used; GPS as a stand-alone system or together with other surveying methods. A lot of research is done in this field and several types of applications have been developed. In this work only GPS receivers are used and the focus is on centimetre level motions over short periods of time. Because the created horizontal motion is known the accuracy of the by GPS detected motion can be computed for different settings and in different environments. A brief overview of the limitations and error sources in GPS surveying is also given.The motions are monitored with a commercial software, Motion Tracker, from Trimble and with On-the-Fly processed data files from Trimble Total Control that are run in a Matlab program in order to get RMS values for the detected motions and plots showing the created true motion and the by GPS measured motion. In addition the procedure to write a batch file in order to run GNNET-RTK, a software from GEO++, is investigated. This software package makes it possible to simultaneously do real-time processing of several reference and rover stations.When the conditions for GPS surveying are good, the quality of detected motions show promising results. Millimetre accuracy is achieved even if only GPS sensors are used and the epochs are as short as one minute. In built up areas with a lot of obstructions of the sky the error sources and limitations of GPS, e.g. multipath, has a significant impact on the results and longer observation times are needed. In such areas this work indicates that for demanding high accuracy applications GPS needs other types of sensors as complement, to be able to provide stable reliable solutions with millimetre accuracy.

Repeated gravity measurements were carried out from 1991 until 1999 at sites SE of Vatnajokull, Iceland, to estimate the mass flow and deformation accompanying the shrinking of the ice cap. Published GPS data show an uplift of about 13 +/- A 5 mm/a near the ice margin. A gravity decrease of -2 +/- A 1 mu Gal/a relative to the Hofn base station, was observed for the same sites. Control measurements at the Hofn station showed a gravity decrease of -2 +/- A 0.5 A mu Gal/a relative to the station RVIK 5473 at Reykjavik (about 250 km from Hofn). This is compatible, as a Bouguer effect, with a 10 +/- A 3 mm/a uplift rate of the IGS point at Hofn and an uplift rate of similar to 20 mm/a near the ice margin. Although the derived gravity change rates at individual sites have large uncertainties, the ensemble of the rates varies systematically and significantly with distance from the ice. The relationship between gravity and elevation changes and the shrinking ice mass is modelled as response to the loading history. The GPS data can be explained by 1-D modelling (i.e., an earth model with a 15-km thick elastic lithosphere and a 7 center dot 10(17) Pa center dot s asthenosphere viscosity), but not the gravity data. Based on 2-D modelling, the gravity data favour a low-viscosity plume in the form of a cylinder of 80 km radius and 10(17) to 10(18) Pa center dot s viscosity below a 6 km-thick elastic lid, embedded in a layered PREM-type earth, although the elevation data are less well explained by this model. Strain-porosity-hydrology effects are likely to enhance the magnitude of the gravity changes, but need verification by drilling. More accurate data may resolve the discrepancies or suggest improved models.

Nowadays, Global Geopotential Models (GGMs) are used as a routine stage in the procedures to compute a gravimetric geoid. The GGMs based geoidal height also can be used for GPS/levelling and navigation purposes in developing countries which do not have accurate gravimetric geoid models. Also, the GGM based gravity anomaly including the digital elevation model can be used in evaluation and outlier detections schemes of the ground gravity anomaly data. Further, the deflection of vertical and gravity gradients components from the GGMs can be used for different geodetic and geophysical interpretation purposes. However, still a complete and user-friendly software package is not available for universities and geosciences communities. In this article, first we review the procedure for determination of the basic gravity field and gradient components from the GGMs, then general MATLAB based software is presented and applied as a sample case study for determination of gravity components based on the most recent EIGEN-GL04C GRACE model in Sweden.

The modification of Stokes' formula allows the user to compensate the lack of a global coverage of gravity data by a combination of terrestrial gravity and a global geopotential model. The minimization of the errors of truncation gravity data and potential coefficients could be treated in a least-squares sense as is the basic ingredient in the Royal Institute of Technology (KTH) approach as proposed by Sjoberg in 1984. This article presents the results from a joint project between KTH and the National Land Survey of Sweden, whose main purpose is to evaluate the KTH approach numerically and to compute a gravimetric geoid model for Sweden. The new geoid model (KTH06) was computed based on the least-squares modification of Stokes' formula, the GRACE global geopotential model, a high-resolution digital terrain model and the NKG gravity anomaly database. The KTH06 was fitted to 1162 GPS/levelling points by a 7-parameter transformation, yielding an all-over fit of 19 mm and 0.17 ppm. The fit is even smaller than the estimated internal accuracy for the geoid model (28 mm). If we assume that the accuracy of the GPS and levelling heights are 10 mm and 5 mm, respectively, it follows that the accuracy of the expected gravimetric geoid heights are of the order of 11 mm. Also, we found a significant expected difference between the KTH06 and NKG2004 models in rough topographic areas (up to 36 cm). As the major ground data and global geopotential model were almost same in the two models, we believe that there are different reasons that come into play for interpreting the discrepancies between them, as the method for eliminating outliers from the gravity database, the interpolated denser gravity observations using the high-resolution digital elevation model before Stokes' integration, the potential of the LSM kernel, which matches the errors of the terrestrial gravity data, GGM and the truncation error in an optimum way, and the effect of applying more precise correction terms in the KTH approach compared to the remove-compute-restore method. It is concluded that the least-squares modification method with additive corrections is a very promising alternative for geoid computation.

A new geoid model for Iran (IRG04) is computed based on the method of least squares modification of Stokes formula based on the most recent gravity anomaly database, SRTM high resolution Digital Elevation Model (DEM) and GRACE GGM02 Global Geopotential Model. In order to define a new height datum for Iran, we attempt to combine this high resolution gravimetric geoid model with GPS/levelling data by using a corrective surface idea. The corrective surface is constructed based on 224 GPS/levelling points and then evaluated with 35 independent points. Different interpolation techniques were tested for the creation of the corrective surface; among them the Kriging method gives the minimum RMS and noise level versus the suggesting that GPS/levelling data. The RMS of fitting the new combined geoid model versus GPS/levelling data is 0.09 in, it is near 4 times better accuracy compared with the original gravimetric geoid model. Comparing the later model, the new surface should be more convenient and useful in definition of the new height datum, specifically in engineering and GPS/levelling projects.

Only with satellites it is possible to cover the entire Earth densely with gravity field related measurements of uniform quality within a short period of time. However, due to the altitude of the satellite orbits, the signals of individual local masses are strongly damped. Based on the approach of Petrovskaya and Vershkov we determine the gravity gradient tensor directly from the spherical harmonic coefficients of the recent EIGEN-GL04C combined model of the GRACE satellite mission. Satellite gradiometry can be used as a complementary tool to gravity and geoid information in interpreting the general geophysical and geodynamical features of the Earth. Due to the high altitude of the satellite, the effects of the topography and the internal masses of the Earth are strongly damped. However, the gradiometer data, which are nothing else than the second order spatial derivatives of the gravity potential, efficiently counteract signal attenuation at the low and medium frequencies.

In this article we review the procedure for estimating the gravity gradient components directly from spherical harmonics coefficients. Then we apply this method as a case study for the interpretation of possible geophysical or geodynamical patterns in Iran. We found strong correlations between the cross-components of the gravity gradient tensor and the components of the deflection of vertical, and we show that this result agrees with theory. Also, strong correlations of the gravity anomaly, geoid model and a digital elevation model were found with the diagonal elements of the gradient tensor.

The application of geodetic techniques to study crustal deformations associated with the geodynamic activity of a region is considered as a fundamental tool in seismic risk mitigation and in earthquake prediction research. In principle, the crustal deformation analysis is a purely geodetic problem as it concerns alteration of the Earth shape, so that deformations of the crust are directly connected with geodetic observables. The Tornquist zone across Skane in southern Sweden is a classical fault zone that separates the Precambrium gneisses of the Baltic shield in the north from Phanerozoic Europe to the south. In this region, a Global Positioning Network (GPS) was established to study possible crustal motions. The aim of this article is to improve on previous study in to estimate the possible crustal strains and dilation parameters by a finite element analysis. Results show that the areas with maximum shear strain and dilation are located exactly in the active fault zones and their intersections. However, further observations in a dense network as well as integration with geological and geophysical data are needed to fully explore the recent crustal motions over the Tornquist zone.

A number of regional gravimetric geoid models have recently been determined for the Iran area, and a common problem is to select the best model, e.g. for engineering applications. A related problem is that in order to improve the local geoid models, the selection of the best Global Geopotential Model (GGM) model for the region is essential, to be used in a combined solution from GGM and local gravimetric data. We discuss these problems by taking advantage of 260 GPS/levelling points as an external tool for validation of different global and local geoid models in the absolute and relative senses. By using relative comparisons of the height differences between precise levelling and GPS/geoid models we avoid possible unknown systematic effects between the different types of observables.

The study shows that the combination of the newly released GRACE model (GGM02C) with EGM96 geoid model fits the GPS/levelling data in Iran with the best absolute and relative accuracy among the GGMs. Among the local geoid models, the newly gravimetric geoid model IRG04 agrees considerably better with GPS/levelling than any of the other recent local geoid models. Its rms fit with GPS/levelling is 55 cm. Hence, we strongly recommend the use of this new model in any surveying engineering or GPS/levelling projects in the area.

Any errors in digital elevation models (DEMs) will introduce errors directly in gravity anomalies and geoid models when used in interpolating Bouguer gravity anomalies. Errors are also propagated into the geoid model by the topographic and downward continuation (DWC) corrections in the application of Stokes's formula. The effects of these errors are assessed by the evaluation of the absolute accuracy of nine independent DEMs for the Iran region. It is shown that the improvement in using the high-resolution Shuttle Radar Topography Mission (SRTM) data versus previously available DEMs in gridding of gravity anomalies, terrain corrections and DWC effects for the geoid model are significant. Based on the Iranian GPS/levelling network data, we estimate the absolute vertical accuracy of the SRTM in Iran to be 6.5 m, which is much better than the estimated global accuracy of the SRTM (say 16 m). Hence, this DEM has a comparable accuracy to a current photogrammetric high-resolution DEM of Iran under development. We also found very large differences between the GLOBE and SRTM models on the range of -750 to 550 m. This difference causes an error in the range of -160 to 140 mGal in interpolating surface gravity anomalies and -60 to 60 mGal in simple Bouguer anomaly correction terms. In the view of geoid heights, we found large differences between the use of GLOBE and SRTM DEMs, in the range of -1.1 to 1 m for the study area. The terrain correction of the geoid model at selected GPS/levelling points only differs by 3 cm for these two DEMs.

Iran is one of the most active regions in the world with respect,to earthquakes and tectonic motions in the lithosphere. In order to study the impact of the geoid model in detecting plate tectonic boundaries and in the establishment of an accurate height datum for future geodynamic observations, a new combined gravimetric geoid model for Iran (IRG04C) was computed by the method of least squares modification of Stokes formula based on the most recent gravity anomaly database, SRTM high resolution Digital Elevation Model (DEM), GRACE GGM02 Global Geopotential Model and GPS/levelling data. The RMS fit of the new geoid model versus GPS/levelling data is 9 cm, which is a 10 times better fit compared to the most recent published gravimetric geoid model in the area. An integrated approach, combining gravity, geoid and seismology data as well as a digital elevation model, was used to find possible correlations between these parameters. Our investigation indicates that all earthquakes with magnitude over 6.0 in the Richter scale are located in areas with a geoid slope exceeding 7.5%. The study shows a significant correlation between the horizontal gradient of the geoid and plate tectonic activities.

This thesis is an accuracy research of detail surveying with networkRTK. Another objective of the thesis concerns some proposed controlmethods and to estimate if they are practically useful, and if the theoreticalassumptions of accuracy levels can be adjusted to standardisedtolerances for such control methods.The control methods are “revisit with network RTK” and “separatecontrol with conventional technique”, both in plane and height. Therevisit can be processed alone, or integrated with the productionmeasurements. Conventional technique involves a totalstation or alevelling instrument. The authors consider the control methods to bepractically useful.The modification of the theoretical assumptions of accuracy levelshas been based on an extensive material of measurements, calculations,analyses and considerations. This thesis achieved a standarderror in plane of 10 mm and in height of 15 mm (without the standarderror of the geoid model), which is comparable with similarstudies carried out lately. However, the conditions in this study areconsidered to be very favourable. To achieve the accuracy in plane, atripod of some type is necessary to minimize the influence from thecentering standard errors.Our modified accuracy levels can in the future be adjusted to standardisedtolerances, but more studies at different locations andunder other conditions are then necessary.