Two selection factors play **the** most important roles in choosing **the** optimal **satellite** mission scenarios: (i) **the** performance **of** **the** mission in retrieving **the** geophysical signals, and (ii) technical and stability issues connected with **the** mission. **From** a technical viewpoint, **the** **missions** are chosen by **the** altitude not less than 290 km, while **from** **the** 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 **future** **satellite** **missions** 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 **the** **gravity** **field** (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 **of** **the** 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 **orbit** **of** 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:

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In 2002 **the** **satellite** pair **Gravity** Recovery And Climate Experiment (GRACE) was launched. **The** GRACE mission aimed for improvement **of** **the** **gravity** **field** determination using tracking between **the** two low satellites (ll-SST), i.e. a tracking pair **of** satellites in Low **Earth** **Orbit** (LEO) (Figure 1.1). **The** mission was capable **of** measuring **the** **time**-**variable** **gravity** **field**, and increased spatial resolution **of** mass transport measurement inside **the** **Earth** system. **The** original GRACE mission was designed for five years performance in **orbit**. However, **the** **satellite** mission provided **gravity** **field** measurements beyond **the** designated lifetime up to **the** **time** **of** writing this statement, but **the** batteries failure may shorten lifetime **of** **the** 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 **gravity** **field** 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 **of** **the** GRACE **satellite** mission is **the** set **of** inter-**satellite** range-rate measurement. **The** GPS receivers on **the** satellites allow for precise **orbit** determination **of** **the** satellites as well as precise **time**-tagging **of** **the** 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 **of** **the** satellites (Touboul et al., 1999).

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One basic research **field** **of** geodesy or **Earth** system science is to develop and apply new methodologies and algorithms for **gravity** **field** modeling, in particular based on data **from** **the** dedicated **satellite** **gravity** **missions** Challenging Minisatellite Payload (CHAMP), **Gravity** Recovery and Climate Experiment (GRACE), **Gravity** **field** 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 **gravity** **field** models based on **the** invariant theory. Compared to traditional methods, where these GGs are affected by attitude errors, Invariants **of** **the** Gravitational Gradient Tensor (IGGT) in combination with least squares adjustment avoid **the** problem **of** inaccurate rotation matrices. **The** application **of** **the** first tensor invariant (**the** trace **of** **the** gravitational tensor) in **gravity** **field** determination yields **the** trivial solution while **the** observation equation **of** **the** third one (**the** determinant **of** **the** 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 **of** **the** **gravity** **field** model’s spherical harmonic coefficients. For this specific application **of** **the** 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 **of** **the** **design** matrix like applied in **the** **time**-wise and direct approaches.

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Received: 10 August 2019; Accepted: 6 September 2019; Published: 11 September 2019 Abstract: **Time**-**variable** **gravity** **field** models derived **from** observations **of** **the** **Gravity** 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 **of** **the** GRACE Science Data System, has reprocessed **the** complete GRACE mission and released an improved GFZ GRACE RL06 monthly **gravity** **field** **time** 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 **of** **the** RL06 **gravity** **field** 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 **of** **the** **gravity** **field** coefficients are more realistic. These findings are confirmed as well by independent validation **of** **the** 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 **the** **Earth** system.

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1 Introduction
When concerned with **the** determinants **of** **the** 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 **of** **the** **gravity** 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 **of** **the** size **of** **the** two economies and their distance, borrowing an analogy **from** physics that has since named **the** relation: **gravity**. While **the** theoretical underpinnings **of** **gravity** **of** 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.

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six **gravity** **field** **time** series was considered. If only ITG 2010 was considered, **the** convergence decreased such that a reliable estimation **of** **the** 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 **of** **the** covariance components is not completely understood up to now. Further theoretical and experimental investigations are necessary. Two reasons might cause **the** insufficient convergence **of** **the** 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 **of** **the** 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 **of** **the** 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 **of** **the** 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 **the** **time** derivatives **of** **the** polar motion are unavailable, they have to be approximated. If **the** **time** derivative **of** **the** current epoch is approximated by a difference quotient, **the** current epoch depends on measurements **of** **the** previous and **the** next epoch. Thus, **the** measurements **of** **the** 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 **of** **the** 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).

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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. [41] **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 **of** **the** 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.

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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 **of** **gravity** 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 **from** **the** 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 **from** **the** 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 **of** **the** 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 **of** **the** spatial and temporal patterns (as provided by StICA) likely mitigates **the** clustered behavior. However, it is worth mentioning that after application **of** StICA, both **of** **the** 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.

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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 **gravity** **field** 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 **of** **the** 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 **from** **the** copyright holder. To view a copy **of** this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.

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These uncertainties arise in part **from** an inability to observe **gravity** wave dynamics in sufficient detail to constrain key dynamical aspects **of** **the** parameteriza- tions ( Alexander et al. 2010 ). **Satellite** remote sensors, for example, suffer similar resolution constraints to global models, resolving only longer-wavelength com- ponents **of** **the** **gravity** 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 **of** **gravity** wave generation, propagation and breakdown through deep layers **of** **the** 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 **time** **of** year; designing near-real-**time** flight- planning strategies to locate, intercept, and observe specific aspects **of** **gravity** 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.

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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 **of** **the** system are included in **the** guidance such that **the** feedforward control brings **the** system in a state which has only small deviations **from** **the** required state given by **the** guidance. With these small deviation it is possible to **design** a linear feedback controller.

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This world is a curious place. **The** physics that governs humans’ everyday lives, and also most **of** **the** 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 **of** **the** 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.

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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 **of** **the** different scale factors and, in turn, **of** **the** 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.

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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 **of** **the** small **satellite** **missions** 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 **of** **the** data. **The** other aspect for bad telemetry is **the** attitude control being used for **the** pico satellites **missions**. Most **of** **the** 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 **the** **satellite** 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.

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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.

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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 **of** **the** acquired data. **The** level 1 processor corrects **the** hyperspectral image for systematic effects **of** **the** 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 **of** **the** 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.

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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 **the** **Earth** and other terrestrial planets, and **the** size-frequency and relative velocity distribution **of** **the** impactors. In **the** wider scientific context, many **of** these models are based on **the** orbital evolution **of** **the** 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 **from** **the** 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 **the** **Earth**-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 **the** **Earth**-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 **of** **the** 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. [3]

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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 **of** **the** semiclassical **S**-matrix in a **time**-dependent exter- nal metric, such as de Sitter. **The** reason is **the** eternal nature **of** **the** 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-**time** **of** **the** 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 **of** **the** approximate toy model which we possess. Despite its simplicity, this model allows us to capture **the** key essence **of** **the** 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 **of** **the** fact that **the** coherent state |N i itself has a finite lifetime. Still, an effective

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In **the** second case, **the** related point mass method was called PM-FRE, in which **the** number and **the** positions **of** **the** RBFs were completely or partly unknown. A search process was developed for finding **the** RBFs, in which **the** magnitudes and positions **of** **the** 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 **of** **the** 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 **of** **the** magnitudes **of** all found RBFs based on **the** whole input data was carried out while keeping **the** positions **of** **the** RBFs fixed. This led to **the** two-step approach **of** PM-FRE, which is one **of** **the** major innovations **of** this thesis.

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