Two selection factors play the most important roles in choosing the optimal satellite mission scenarios: (i) the performance ofthe mission in retrieving the geophysical signals, and (ii) technical and stability issues connected with the mission. From a technical viewpoint, themissions are chosen by the altitude not less than 290 km, while fromthe view of geodetic sensitivity an orbit height not larger than 320 km is preferable. That is a trade-off between higher sensitivity to short wavelength phenomena by lower altitude and a shorter mission life time due to a larger atmospheric drag force. This decision is due to the expectation that futuresatellitemissions will benefit from drag-free technology like GOCE which allows the mission to fly at lower altitudes (Marchetti et al., 2008; St Rock et al., 2006; Wiese et al., 2011b). Furthermore, an intersatellite distance of 100 km of an inline formation equipped with laser interferometry is chosen as a trade-off between instrument performance and rel- ative accuracy in determining short wavelengths features in thegravityfield (Wiese et al., 2009). The stability problem with Pendulum and Cartwheel formations as well as the laser in- terferometry pointing issue limit the choices to inline formations and conservative Pendulum formations with small opening angle (GFO). However, due to the higher performance ofthe GFO formation compared to the inline configuration, the GFO would be a favorite scenario for a single pair satellite mission. The scenario is chosen on a repeat orbitof β/α = 507/32 which shows a good performance for 6-day recovery (Table 6.1). For dual satellite pairs, two different formation scenarios are selected:
In 2002 thesatellite pair Gravity Recovery And Climate Experiment (GRACE) was launched. The GRACE mission aimed for improvement ofthegravityfield determination using tracking between the two low satellites (ll-SST), i.e. a tracking pair of satellites in Low EarthOrbit (LEO) (Figure 1.1). The mission was capable of measuring thetime-variablegravityfield, and increased spatial resolution of mass transport measurement inside theEarth system. The original GRACE mission was designed for five years performance in orbit. However, thesatellite mission provided gravityfield measurements beyond the designated lifetime up to thetimeof writing this statement, but the batteries failure may shorten lifetime ofthe mission at any time. Moreover, the mission may run out fuel and changes its orbit to lower altitudes due to atmospheric drag forces. The launch of GRACE provided unprecedented improvement in determining the Earth’s gravityfield and the data are vastly used for geo- physical purposes, among them for hydrological, glaciology and atmospheric studies. The GRACE mission consists of two identical satellites in near-polar orbit (by inclination of 89 ◦ ) that are separated in along-track direction by approximately 220 km. The mission altitude is approximately 500 km, but due to lack of altitude control, the satellites’ orbit continually decays by atmospheric drag forces. A K-Band microwave ranging system is employed to mea- sure the distance change between the two satellites at level of few tenths of micron/second. The main observable ofthe GRACE satellite mission is the set of inter-satellite range-rate measurement. The GPS receivers on the satellites allow for precise orbit determination ofthe satellites as well as precise time-tagging ofthe inter-satellite range-rate measurements (Tap- ley et al., 2004). Moreover, each spacecraft is equipped with a high precision accelerometer to measure and remove the effect of all non-conservative forces like atmospheric drag, solar radiation pressure, Earth radiation pressure which allows to isolate the gravitational motion ofthe satellites (Touboul et al., 1999).
One basic research fieldof geodesy or Earth system science is to develop and apply new methodologies and algorithms for gravityfield modeling, in particular based on data fromthe dedicated satellitegravitymissions Challenging Minisatellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE), Gravityfield and steady-state Ocean Circulation Explorer (GOCE) and Gravity Recovery and Climate Experiment-Follow-on (GRACE-FO) as well as combined with ground gravity data (e.g., air-shipborne and terrestrial measurements). In this thesis, I investigated how to use GOCE Gravitational Gradients (GGs) to build global gravityfield models based on the invariant theory. Compared to traditional methods, where these GGs are affected by attitude errors, Invariants ofthe Gravitational Gradient Tensor (IGGT) in combination with least squares adjustment avoid the problem of inaccurate rotation matrices. The application ofthe first tensor invariant (the trace ofthe gravitational tensor) in gravityfield determination yields the trivial solution while the observation equation ofthe third one (the determinant ofthe gravitational tensor) is more complicated with a correspondingly larger linearization error. Therefore, the second IGGT approach is studied in this thesis which is a quadratic function ofthegravityfield model’s spherical harmonic coefficients. For this specific application ofthe IGGT I derived mathematical and stochastic models (parameterization, linearization and weight determination). This was done by a Taylor expansion to get linearized observation equations for the least squares method and also to show that the Lagrange remainder i.e., the linearization error can be ignored if the used a-priori model (e.g., EIGEN-5C) was sufficiently accurate. I also deduced the stochastic model i.e., determined the weighting equation from an adopted law of measurement error propagation for the non-uniform accurate GOCE GGs. As the GOCE GGs were measured in a band-limited manner, a forward and backward finite impulse response band-pass filter was applied to the data, which could also eliminate filter caused phase change. In this way, it avoided filtering both, the observations and columns ofthedesign matrix like applied in thetime-wise and direct approaches.
Received: 10 August 2019; Accepted: 6 September 2019; Published: 11 September 2019 Abstract: Time-variablegravityfield models derived from observations oftheGravity Recovery and Climate Experiment (GRACE) mission, whose science operations phase ended in June 2017 after more than 15 years, enabled a multitude of studies of Earth’s surface mass transport processes and climate change. The German Research Centre for Geosciences (GFZ), routinely processing such monthly gravity fields as part ofthe GRACE Science Data System, has reprocessed the complete GRACE mission and released an improved GFZ GRACE RL06 monthly gravityfieldtime series. This study provides an insight into the processing strategy of GFZ RL06 which has been considerably changed with respect to previous GFZ GRACE releases, and modifications relative to the precursor GFZ RL05a are described. The quality ofthe RL06 gravityfield models is analyzed and discussed both in the spectral and spatial domain in comparison to the RL05a time series. All results indicate significant improvements of about 40% in terms of reduced noise. It is also shown that the GFZ RL06 time series is a step forward in terms of consistency, and that errors ofthegravityfield coefficients are more realistic. These findings are confirmed as well by independent validation ofthe monthly GRACE models, as done in this work by means of ocean bottom pressure in situ observations and orbit tests with the GOCE satellite. Thus, the GFZ GRACE RL06 time series allows for a better quantification of mass changes in theEarth system.
When concerned with the determinants ofthe volume of flows of goods, trade economists often have to resort to aggregate trade figures, by country, or sometimes state and province. This makes an aggregation of its determinants equally necessary. This article, building on earlier work by Head and Mayer (2009), sets out to provide an aggregation of trade costs that is derived from a very general representation ofthegravity equation, while remaining agnostic to its micro-foundation. I apply the method to compute time-varying distances using nighttime satellite imagery. Using these theory-consistent distances, the elasticity of trade with respect to distance can be estimated in the within-dimension of a panel, allowing to control for time-invariant unobserved country pair characteristics. Further, the use of these distances produces the noteworthy results of significantly lower estimates of coefficients for variables that are correlated with distance. Most notable is an up to 50 % decrease in the estimated effect of borders on trade, i.e. the net cost of crossing a border. In its earliest and simplest form, Tinbergen et al. (1962) described the volume of trade flows between countries as a function ofthe size ofthe two economies and their distance, borrowing an analogy from physics that has since named the relation: gravity. While the theoretical underpinnings ofgravityof international trade have since received drastic improvements with Anderson (1979), Anderson and van Wincoop (2003) and others, the employed distance measures have seen surprisingly little attention.
six gravityfieldtime series was considered. If only ITG 2010 was considered, the convergence decreased such that a reliable estimation ofthe covariance component were not possible. Therefore it was concluded that the partial redundancies might play a major role. Though the estimation of covariance components is described on a theoretical basis in literature, no further numerical examples for the estimation of co- variance components are available so far. The reason for the deteriorated convergence ofthe covariance components is not completely understood up to now. Further theoretical and experimental investigations are necessary. Two reasons might cause the insufficient convergence ofthe covariance components: First the SMCTE might not estimate the traces with sufficient accuracy. Finding tighter bounds for the trace estimation of asymmetric matrices (Equation 3.57) might clarify the effect ofthe SMCTE. The second, more likely reason for inaccurate covariance components might be an ill-conditioned matrix S (Equa- tion 3.42), which is probably caused by too small partial redundancies ofthe associated observations. The proposed linear least-squares solver is proven to be an adequate approach for the mutual validation. However, other approaches also might be suitable. The Kalman and Bayes filter are highly efficient, recursive analysis methods. Two steps are performed in each filter epoch. First, based on previous epochs and a mathematical model, the filters predict the state vector, containing the unknown parameters, for the current epoch. In the second step, the observations ofthe current epoch are taken into account and are used to improve the predicted state vector. Then, the improved state vector is the basis for the next prediction. However, as thetime derivatives ofthe polar motion are unavailable, they have to be approximated. If thetime derivative ofthe current epoch is approximated by a difference quotient, the current epoch depends on measurements ofthe previous and the next epoch. Thus, the measurements ofthe epochs are not independent on each other and three different residuals occur for the same observation in three consecutive epochs. If the temporal correlations are stretched over several epochs, the application of recursive filters is additionally complicated. However, despite ofthe difficulties due to the epoch dependencies, the Kalman filter is currently investigated, in order to estimate Love numbers (personal communication S. Kirschner and F. Seitz, September 2012).
What we have seen in this chapter is that Yang Mills theory simplifies in the large N limit. In particular we have seen that at leading order in 1/N only planar diagrams contribute and expectation values of gauge invariant operators factorize. Furthermore we have seen, that it is natural to think of Yang Mills theory at large N as some kind of noncritical string theory, where glueballs behave like closed strings, while mesons behave like open strings. Furthermore we have seen that in this picture the mass of baryons is consistent with interpreting them as D0-branes, although here one should note, that the more natural interpretation is in terms of a soliton of mesons, which can be seen in chiral peturbation theory as a skyrmion. A similar situation arises in Type I string theory, here we have a 5-brane, as well as open strings in the bulk. The type-I 5-brane can be either seen as an open string soliton or as a D-Brane. So in string theories with bulk open string there is no sharp distinction between D-Branes and open string solitons.  The open string sector gives a SO(32) gauge theory. We can take the instanton solution of 4 dimensional Yang-Mills theory and lift it to 10 di- mensions, by simply making it extend into the other 6 directions, this gives us a six dimensional object. In this language this object is just a soliton ofthe open string sector, however the instanton has a scale modulus. If we shrink the instanton to zero size it can actually be understood as the 5-brane, so we see, that the question of whether a given object should be understood as a brane or a soliton is somewhat ambiguous if there is an open string sector and which description should be used depends on the problem.
In a recent study, Boergens et al. (2014) applied the Spatio-temporal ICA (StICA) and the Temporal ICA techniques, which were formulated based on the entropy criterion (Hyvärinen, 1999b) to identify patterns ofgravity changes over North America and the African continent. Their results indicated a slightly better separation performance of StICA compared to those of Temporal ICA. The StICA technique is slightly different fromthe Spatial ICA and Temporal ICA introduced in this thesis. StICA searches for the patterns in the data set that contain small dependences in space and time. Therefore, the StICA-derived patterns are not strictly indepen- dent with compared to the ICs derived fromthe introduced Spatial ICA and Temporal ICA techniques. In practice, however, the results of StICA might be easier to interpret since in reality there exist small dependences between different spatial, as well as between different temporal source signals. For example, one can consider that two regions exhibit similar TWS changes but with slightly different time latencies. Thus, their temporal changes would be statistically correlated. An application ofthe Temporal ICA to separate TWS changes over these two areas results to a clustered behavior as it was shown in the simulation study of Section 5.1.3. One might argue that in such cases a trade off between the mutual independence ofthe spatial and temporal patterns (as provided by StICA) likely mitigates the clustered behavior. However, it is worth mentioning that after application of StICA, both ofthe StICA-derived spatial and tempo- ral components are not anymore orthogonal. Furthermore, StICA maximizes the independence of sources over space and time, without necessarily producing independence in either space or time. Therefore, in case of comparable outcomes, we recommend the use of either Temporal or Spatial ICA due to the mentioned computational and statistical benefits.
Funding Open Access funding provided by Universität Bern. Data Availability Swarm related data products are provided by ESA (http://swarm -diss.eo.esa.int). Auxiliary products used for precise orbit determination are provided by the IGS via anonymous ftp (http://ftp. aiub.unibe .ch/). The GRACE gravityfield solutions are provided via ICGEM (http://icgem .gfz-potsd am.de/home). For precise orbit deter- mination and gravity determination the development version 5.3 ofthe Bernese GNSS software was used, which is not publicly available. Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly fromthe copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
These uncertainties arise in part from an inability to observe gravity wave dynamics in sufficient detail to constrain key dynamical aspects ofthe parameteriza- tions ( Alexander et al. 2010 ). Satellite remote sensors, for example, suffer similar resolution constraints to global models, resolving only longer-wavelength com- ponents ofthegravity wave spectrum ( Wu et al. 2006 ). These gaps motivated a Deep Propagating Gravity Wave Experiment (DEEPWAVE; Fritts et al. 2016 ) to acquire the most intensive observations to date ofgravity wave generation, propagation and breakdown through deep layers ofthe atmosphere (see Fig. 2 of Fritts et al. 2016 ), using instruments on the National Science Foundation (NSF)/National Center for Atmospheric Research (NCAR) Gulfstream V research aircraft (NGV; Laursen et al. 2006 ). Yet this very lack of observational knowledge about gravity waves that spurred DEEPWAVE also compli- cated logistical planning for an NGV-based gravity wave measurement campaign: for example, identifying the best site and timeof year; designing near-real-time flight- planning strategies to locate, intercept, and observe specific aspects ofgravity wave dynamics; and assessing whether executed flights achieved their requisite science goals. Stratospheric gravity waves observed by infrared nadir sensors, such as the Atmospheric Infrared Sounder (AIRS) on NASA’s Aqua satellite, proved pivotal in these and other areas. This paper describes that work, focusing in particular on new and innovative uses of operational near-real-time radiances, used successfully for the first time during DEEPWAVE, which could find future uses in field campaigns and other applications.
For position and pointing a control system was designed. The overall structure for both controllers (position and attitude) is a feedback controller together with feedforward as shown in figure 3. The feedforward control imposes the predicted forces and torques on the plant, which are needed to follow the trajectory given by the guidance. In an undisturbed ideal case the system then follows the trajectory provided by the guidance with a zero control error such that the feedback control loop has no effect. Nonlinearities ofthe system are included in the guidance such that the feedforward control brings the system in a state which has only small deviations fromthe required state given by the guidance. With these small deviation it is possible to design a linear feedback controller.
The striking improvements in long- to medium-wavelengths gravityfield recovery achieved with GPS-CHAMP and GPS-GRACE high-low and GRACE K-band range low-low satellite-to-satellite tracking prompted us to combine thesatellite data with surface data from altimetry over the oceans and gravimetry over the continents to generate a new, high resolution global gravityfield model: EIGEN-CG01C. The model is complete to degree/order 360 in terms of spherical harmonics and resolves half-wavelengths of 55 km in the geoid and gravity anomaly fields. A special band-limited combination method has been applied in order to preserve the high accuracy fromthesatellite data in the lower frequency band ofthe geopotential and to allow for a smooth transition to the high- frequency band, dominated by the surface data. Compared to pre-CHAMP/GRACE global high- resolution gravityfield models, the accuracy was improved by one order of magnitude to 4 cm and 0.5 mgal in terms of geoid heights and gravity anomalies, respectively, at a spatial resolution of 200 km half-wavelength. The overall accuracy at degree/order 360 is estimated to be 20 cm and 5 mgal, respectively, and benefits significantly from recently released new gravity anomaly compilations over the polar regions. In general, the accuracy over the oceans is better than over the continents reflecting the higher quality ofthe available surface data.
This world is a curious place. The physics that governs humans’ everyday lives, and also most ofthe stuﬀ that we can build, is extremely well de- scribed by the quantum theory of electrons and nuclei interacting through Maxwell’s laws. In spite ofthe simplicity of this theory, the world displays a stunning variety of diﬀerent phenomena. Among these are some that ini- tially surprise even students of physics, not to mention their initial discov- erers. One example is superconductivity, the phenomenon in which rather simple materials like mercury or lead, at low temperatures, lose any resistiv- ity [Onn11] and gain the ability to sustain currents when no external voltage is applied [Onn14], while expelling magnetic fields [MO33]. A satisfactory theoretical explanation was only given much later [GL50; BCS57]. Another example is the fractional quantum Hall eﬀect [TSG82], in which a material acts as if it contained particles with a fraction of an electron charge, even though no such elementary particles are actually there [Lau83]. In these ex- amples, the eﬀective theory of electrons is still completely valid. The key reason that such striking phenomena may occur is the conspiration of many particles in conjunction with the laws of quantum mechanics - they are col- lective quantum eﬀects.
not correspond to the desired coupled quintessence scenario: indeed, these are exactly the terms responsible for the fifth force, originating in (3.22) and leading to an effective gravitational force as in Eq. (3.26). In other words, we point out that the standard spherical collapse, as used for example in [Nunes and Mota, 2006] does not include the main ingredient of coupled quintessence. A fifth attractive force acting between CDM particles and mediated by the cosmon is absent, although densities are indeed coupled to each other as in (3.28) - (3.30). The reason for this can be seen as follows: spherical col- lapse is by construction based on gravitational dynamics and cannot account for other external forces unless appropriately modified. The dynamics in the spherical collapse models are governed by the usual Friedmann equa- tions, which are particular formulations of Einstein’s field equations. Hence, only gravitational forces determine the evolution ofthe different scale factors and, in turn, ofthe density contrast. We note that, though in the limit of small couplings the difference can be small, for strongly coupled scenarios a completely different evolution is obtained. This is simply connected to the fact that for small couplings gravity is still the crucial ingredient to fuel the collapse.
A 21 days simulation cycle shows an increase in the number of access as soon as all the ground stations are active. The number of access increases by one extra pass on an average per day. Most important factor is the redundancy in receiving data. Most ofthe small satellitemissions especially CubeSats work on low bit rates. The probability of receiving bad telemetry is high therefore the communication protocols do need to be well scripted to attain the telemetry data in a best possible way. The Peruvian Satellite Network cleans up all the constraints by having redundant stations for the reception ofthe data. The other aspect for bad telemetry is the attitude control being used for the pico satellites missions. Most ofthe CubeSats use the permanent magnets to control its attitude in space. The latest development in CubeSats is the use of reaction wheels or magneto- torques. Still the usage of such attitude systems is yet to be verified for full effectiveness. The attitude system plays an important role in supporting thesatellite to communicate with the ground station. Even though the permanent magnets are used to drag the tumbling in 2 axes, it still tumbles in all 3 axes, which leads to loss of contact and bad telemetry.
difficult to calibrate with any degree of confidence.
In the present paper, we show that economists already possess a toolkit for im- proving on both approaches: structural-gravity modelling. Structural gravity models are now common in international trade, where they are used to study the observed pattern of economic interactions across space and to assess the impact of trade-policy changes. They have provided simple microfoundations to explain why certain types of data − such as trade, migration or commuting flows − exhibit “gravity” patterns. There exist well-understood empirical approaches for estimating the impact of geo- graphy on interactions consistently with these models. Moreover, such models share convenient properties that make it easy to analyse the welfare impact of barriers that restrict interactions across space.
At the ground receiving station processing starts with the collection of data from different sources, e.g. the hyperspectral instrument, star sensors, GPS and housekeeping data. The transcription processor derives additional information, e.g. the quality ofthe acquired data. The level 1 processor corrects the hyperspectral image for systematic effects ofthe focal plane detector matrix, e.g. radiometric non-uniformities, and converts the system corrected data to physical at-sensor radiance values based on the currently valid calibration values. The spectral and radiometric in-flight calibration is based on dark current measurements performed for each data take as well as by utilization of a full aperture diffuser plate and further calibration equipment, e.g. internal light sources. The level 2- geo processor creates orthoimages based on Direct Georeferencing techniques implementing a line-of-sight model, which uses on-board measurements for orbit and attitude determinations as well as the sensor look direction vectors based on the currently valid geometric calibration values. Furthermore it is foreseen to automatically extract ground control points from existing reference data sets of superior quality (e.g. the Image2006 database with about 10-20 m absolute geometric accuracy or Image2009 database to be generated or USGS ETM+ land cover dataset) by image matching techniques to improve the geometric accuracy better than one pixel size (Müller, R. et al., 2008). The geometric in- flight calibration is based on data takes combined with ground control points. Terrain displacements are taken into account by a global digital elevation model (e.g. derived from SRTM-C/X band, Tandem-X or ASTER). The level 2-atm processor performs atmospheric and haze correction ofthe images by estimating the aerosol optical thickness and the columnar water vapour separately for land and water surfaces. The model uses the radiative transfer equation and takes the date, the sensors’ spectral response functions as well as view and solar geometry into account to convert physical at-sensor radiance values to surface reflectance values. In order to ensure the spectral, radiometric, and geometric accuracy of all EnMAP products they are periodically validated within time series and with data from other sources, e.g. field measurements.
With respect to SSSBs, the innermost solar system still is largely uncharted territory. Information on the orbital parameters and approximate size of previously unknown IEOs and other objects passing through this region is critical for the evaluation of SSSB distribution models. These models are used primarily for two important purposes: In the planetary defence context, they serve to determine the overall risk and frequency of impacts on theEarth and other terrestrial planets, and the size-frequency and relative velocity distribution ofthe impactors. In the wider scientific context, many of these models are based on the orbital evolution ofthe solar system as a whole, and modelled SSSB populations serve as sets of test particles that as a whole record and statistically image the integrated influence of various gravitational and non-gravitational effects over time. To determine the relative strength of these effects and their variation over time, observed and modelled populations can be compared at varied parameter settings, and before and after correction for observational biases which may also be determined in the process. The energy-frequency distribution of impactors is also used to determine the age of solid surfaces in the solar system, expanding the relative dating of planetary surfaces fromthe size-frequency distribution of craters alone. For absolute dating, a reference is required which can only be provided by returned samples that are dated by isotope clocks, such as the Apollo and Luna Moon rocks. The orbital and size-frequency distribution of impactors varies across the solar system. For example, IEOs can presently not reach objects beyond theEarth-Moon-system, and Aten asteroids can not reach the surfaces of main belt asteroids, although both may well migrate over time due to long-term perturbations to hit or become part of either group of solar system bodies. These localized SSSB population differences have to be modelled to determine the absolute age of planetary surfaces outside theEarth-Moon system as long as local surface samples remain unavailable. The high number of observed and modelled bodies enables sound statistical results. Each body adds seven or more parameters to the database; its orbit parameters, estimated size, and occasionally its shape and other physical properties. For example, the model population by Bottke and Morbidelli used in the evaluation ofthe AsteroidFinder’s performance contains 57649 virtual objects, including 1190 virtual IEOs, down to a limiting absolute magnitude H = 23.0, corresponding to a diameter of about 100 m at an albedo of 0.15. 
Finally, we note that establishing the connection between the semiclassical and the quantum S-matrix evolutions sheds new light on the standard difficulties of defining in- and out-states ofthe semiclassical S-matrix in a time-dependent exter- nal metric, such as de Sitter. The reason is the eternal nature ofthe background metric. As we have seen, in the quantum language this eternity translates to the approximation in which the initial and final states of gravitons can be taken as the same undisturbed coherent state |N i. But for finite N , this approximation is good only for a finite time: For finite N , the coherent state cannot be eternal. As we shall see, precisely because of backreaction to it, the coherent state has a charac- teristic lifetime, which defines the quantum break-timeofthe system. This time scales as N . Consequently, the coherent state can be treated as truly eternal only in the limit (2.131), i.e. for infinite N and zero coupling. This makes the whole story self-consistent, at least at the level ofthe approximate toy model which we possess. Despite its simplicity, this model allows us to capture the key essence ofthe semiclassical problem as well as of its quantum resolution. In short, we do not need to worry about defining final S-matrix states on top of de Sitter in the light ofthe fact that the coherent state |N i itself has a finite lifetime. Still, an effective
In the second case, the related point mass method was called PM-FRE, in which the number and the positions ofthe RBFs were completely or partly unknown. A search process was developed for finding the RBFs, in which the magnitudes and positions ofthe RBFs were estimated simultaneously by solving a series of small-scale nonlinear problems. This search process aimed at minimizing the residuals between the predicted and observed gravity values (i.e., data misfit). Before starting it, several model factors (e.g., initial depth and depth limits, optimization direction, etc.) had to be defined properly so that a good approximation can be guaranteed. They were all numerically investi- gated and discussed. Due to the depth limits on the selected point mass RBFs, the nonlinear problem to be solved in the search process was bound-constrained. Consequently, the choice of a suitable optimization algorithm was necessary. Among the four tested iteration algorithms (i.e., LM, NLCG, L-BFGS, and L-BFGS-B), the L-BFGS-B algorithm proved to be the most proper one. The search process was usually terminated by satisfying a defined data misfit, or by satisfying a given number of point mass RBFs, which is defined based on the number of observations or by testing different choices. Sometimes, the criterion for stopping the search process was realized by considering the data misfit as a function ofthe number of RBFs. In this case, if the data misfit decreased very slowly, the search can be stopped accordingly. After the search process, a set of point mass RBFs with known positions and magnitudes are obtained. Because the point mass RBFs were selected and estimated individually, a readjustment ofthe magnitudes of all found RBFs based on the whole input data was carried out while keeping the positions ofthe RBFs fixed. This led to the two-step approach of PM-FRE, which is one ofthe major innovations of this thesis.