## 1 INTRODUCTION

[2] The launch of satellite CHAMP on 15 July 2000 initiated a new era in gravity field research [*Reigber et al*., 2002]. The dedicated gravity satellite missions that followed—GRACE [*Tapley et al*., 2004] and GOCE [*Drinkwater et al*., 2007; *Rummel et al*., 2011], both launched on 17 March of the years 2002 and 2009, respectively—were motivated by the retrieval of data with almost global coverage and homogeneous accuracy, and the production of satellite-only solutions of the gravity field with unprecedented spatial and temporal (for the case of GRACE) resolution and accuracy. From the primary observations of these satellites (range and range-rate measurements provided by Global Positioning System (GPS) receivers and dedicated instruments measuring intersatellite distances, accelerometry, and gradiometry data), different assessment strategies including high-low and low-low Satellite-to-Satellite-Tracking and Satellite Gravity Gradiometry lead to the determination of global models for the geopotential in the form of fully normalized spherical harmonic coefficients and of degree *l* and order *m* that represent numerical constants in a series representation of the geopotential up to a certain maximum degree and order. Although this paper focuses on spherical harmonic expansions, it is important to mention that other efficient gravity field representations exist, including mass concentrations [e.g., *Luthcke et al*., 2008] or regional basis functions [e.g., *Han et al*., 2008].

[3] The gravity field of the Earth plays a fundamental role in geodesy. The figure of the Earth is constantly formed under the influence of gravity and most geodetic observations refer to gravity [*Torge*, 2001]. A proper definition of the mathematical model describing almost any geodetic observing technique requires knowledge of the gravity field. In addition to that, analysis of the observed gravity signal either on the Earth's surface or at some altitude provides information that can be exploited for the study of the structure of the hidden sources in the Earth's interior that produce the observed field. This can be done either in the frame of the forward or the inverse problem in potential field theory, thus linking geodesy with geophysics [*Blakely*, 1996].

[4] Applying Newton's law of gravitation to the entire Earth masses, one can obtain the gravitational attraction at a single field point as an integral expression summing the individual contributions of the infinitesimal mass elements into which Earth is subdivided. If we place a unit mass at the attracted field point, the gravitational attraction obtains dimensions of acceleration and is called gravitational acceleration or simply gravity when we want to refer to its magnitude. In SI, the unit of gravity is m s^{−2} and the unit of gravity gradient components is s^{−2}. The units usually used in geodesy and geophysics are the milli-Gal, defined as 1 mGal = 10^{−3}, Gal = 10^{−5} m s^{−2}, and named after Galileo (1 Gal = 1 cm s^{−2}); for the gravity gradient, the Eötvös unit named after the Hungarian geophysicist Loránd Eötvös is frequently used, 1 E = 10^{−9} s^{−2}, or its subdivision milli-Eötvös, defined as 1 mE = 10^{−12} s^{−2} [*Torge*, 1989]. The gravitational potential is the scalar function which is linked to the gravitational attraction with a gradient operator, the gravity gradients are thus defined as the second order derivatives of the potential. The gravitational potential is also defined through an integral expression over the entire Earth masses and expresses the energy inherited at one point due to its specific location with respect to the attracting source. Its unit in SI is m^{2} s^{−2}. Both gravitational attraction and potential are functions of the position and consistency of the attracting sources. Thus, their analytical computation for arbitrary space points would require the knowledge of the exact position and density of all masses in the Earth's interior. As this information remains unknown, alternative approximations are sought that can describe the observed field adequately in terms of accuracy and spatial resolution.

[5] The mathematical tool that permits a numerical approximation of the gravitational potential and subsequently all of its functionals is its expression in a spherical harmonic series expansion. The obtained spherical harmonic coefficients and constitute a single determination of the gravity field and permit the determination of gravity in terms of potential, acceleration, and gravity gradients everywhere in space. The spherical harmonic formulation is a global and uniform approach and provides a very flexible tool for gravity field modeling with a vast range of applications.

[6] For the determination of a single gravity field solution in the spherical harmonic formulation, i.e., for the evaluation of a set of spherical harmonic coefficients and , a global grid of gravity-related observations is necessary. Moreover, if a homogeneous solution in terms of quality is envisaged, the aforementioned grid should be equidistant and of a uniform quality as well. In the case of an equidistant grid that fully covers the globe and is of homogeneous quality, the orthogonality property of the Legendre functions, which is crucial in the discrete formulation of spherical harmonic computations, is preserved [*Sneeuw*, 1994; *Tsoulis*, 1999]. However, this orthogonality is not always required and for most of the spherical harmonic gravity models not achieved. The observations defining the global grid may either be potential or gravity values, which may be used to evaluate the coefficients numerically in the frame of an inverse Fourier transform algorithm characterized in this particular case as spherical harmonic analysis [*Sneeuw*, 1994]. The global grid may be assembled either by terrestrial or by satellite measurements. The availability of GPS and the possibility to compute high-precision real-time orbits by placing a GPS receiver on board a typical Earth orbiting satellite initiated the implementation of so-called gravity field dedicated satellite missions, such as CHAMP, GRACE, and GOCE, with CHAMP providing unprecedented data to magnetic field and atmospheric profiling as well. The advantages of the satellite against the terrestrial observation techniques include global coverage, uniform accuracy, measurements free from local disturbances, repetition in data coverage which permits the study of time-variable gravity effects, and many others. On the other hand, though satellites provide more or less homogeneous data sets that are clearly superior in terms of uniform quality to terrestrial observations, the actually obtained quality of satellite data is not uniform and there are regional dependencies, e.g., geographical dependence of GPS coverage, magnetic equator effects, and reduced quality in polar areas.

[7] The CHAMP satellite realizes the so-called high-low Satellite-to-Satellite Tracking method (high-low SST), where the low Earth orbiter is the CHAMP satellite and the satellite revolving in high orbits is the constellation of GPS satellites. The difference in geocentric position between the two defines for that instance the intersatellite distance defined in the so-called line-of-sight linking the CHAMP with the corresponding GPS satellite. Due to the continuous tracking of the GPS receiver on-board CHAMP, this distance is obtained as a time series. The subsequent derivatives of this series with respect to time define the relative motion and thus the acceleration of the CHAMP satellite with respect to the GPS satellite. In this indirect manner, one obtains accelerations at satellite altitude that can be related with the spherical harmonic expression of the gravity field as observations in a linearized system of equations where the unknowns are the coefficients and [*Reigber et al*., 2002]. After its placement in orbit at about 454 km altitude and an inclination of about 87.3°, the satellite gradually reached an altitude of 320 km in March 2009 providing unprecedented tracking, atmospheric, and magnetic data for a straight decade until its re-entry in September 2010.

[8] The satellite mission GRACE consists of a twin satellite constellation that emerged from copying the main CHAMP satellite excluding its extendable arm into two duplicates and adding an intersatellite Laser measurement device between the two satellites. With all of the gravity-related CHAMP instrumentation on board, the GRACE system realizes a simultaneous implementation of two measurement principles, namely, high-low SST for each one of the satellite pair with the GPS constellation and low-low SST with the two GRACE satellites being the two satellites in a low orbit defining the principle. In the latter observation technique, the subsequent time derivative of the intersatellite distances define the gravity gradient component along the line-of-sight direction. Thus, apart from the typical high-low SST observables, the GRACE constellation defines in addition a 1-axis gradiometer in orbit [*Tapley et al*., 2004]. The initial orbit characteristics were two identical spacecrafts flying about 220 km apart in an almost polar orbit with an inclination of 89°, at an altitude of 500 km. Since the mission has started, orbit maneuvers are executed in order to keep the satellite separation between 170 and 220 km. As of 19 November 2012, the two satellites were at an altitude of 440 km and their separation reached 262 km.

[9] The satellite GOCE applies the principle of satellite gradiometry. Having a 3-axis gradiometer as its basic measurement sensor, the satellite materializes the only direct gravity measurement principle from an orbiting satellite, that of gravity gradiometry. Instead of translating range and range rates into accelerations, the principle of gravity gradiometry is based on the measurement of the forces applied in order to keep the test masses of the six accelerometers in place. The test masses are placed in pairs on the edges of three arms with lengths measured with an accuracy of 10^{−7} m. The control forces to keep them in place are measured with a precision of the order of 10^{−12} m s^{−2} in terms of acceleration, thus providing, when combined in pairs, direct observations of all elements of the gravity gradient tensor at the corresponding Gradiometer Reference Frame. The analysis of these measurements leads to the determination of a gravity field which is characterized by the highest spatial resolution and accuracy for its complete spectral range than any other satellite-only gravity model [*Rummel et al*., 2011]. The GOCE satellite was injected into a sun-synchronous, near circular orbit with an inclination of 96.7° at an altitude of about 280 km, from where the satellite fell gradually to its operational altitude of around 260 km. From that level, it has been decided to lower the level of the satellite to 235 km, an altitude the satellite has reached in February 2013.

[10] Another satellite system used as complementary source of direct measurements in the development of model EIGEN-GL05 (European Improved Gravity model of the Earth by New techniques—GRACE/LAGEOS based on GFZ release 05 time series) [*Förste et al*., 2008] is LAGEOS (Laser Geodynamics Satellites). The LAGEOS satellites are passive vehicles covered with retro-reflectors designed to reflect laser beams transmitted from ground stations. By measuring the time between transmission of the beam and reception of the reflected signal from the satellite, stations can precisely measure the distance between themselves and the satellite according to the Satellite Laser Ranging measurement principle [*Seeber*, 2003]. These distances can be used to calculate station positions to within 1–3 cm. Long-term data sets can be used to monitor the motion of the Earth's tectonic plates, determine the Earth's gravity field in terms of a set of spherical harmonic coefficients, measure the wobble in the Earth's axis of rotation, and measure the length of the day. The two LAGEOS satellites orbit at altitudes around 5800 km with inclinations of 109.8° and 52.6°, respectively.

[11] Due to problems related mainly with the sampling of the data both in space and time and the attenuation of the observed gravity field signal with increasing altitude, satellite-only geopotential models can be recovered only up to a certain resolution. In order to increase this resolution, the inclusion of other data and algorithmic procedures are necessary. Thus, the combination of data from the satellite gravity missions with altimetry data, global elevation models, and surface gravity measurements leads to the derivation of so-called combined models for the Earth's gravity field. These models are complete up to a much higher degree and order than satellite-only models. However, they include heterogeneous sources of the observed field, possibly affected by other signals, e.g., the correlation of altimetry data with ocean dynamics. In this respect, an additional category of gravity field models are the topographic/isostatic models, which represent a gravity field determination based solely on the geometric information of a global elevation model for continents and bathymetry expressing the variations of the outer crustal boundary with air and water, under some hypothesis for the isostatic mass balance between crust and mantle. Since their maximum degree is constrained only by the spatial resolution of the global elevation model, topographic/isostatic models can reach computationally high and very high frequency bandwidths of the observed gravity field. This fact makes them an exceptional tool for gravity field analysis and interpretation.

[12] The objective of this review is to perform a thorough spectral and spatial assessment of these different new gravity models. For example, as some of these models are obtained through an identical analysis procedure and using the same primary data types (with the only difference being the time span of the involved satellite observations and consequently the maximum degree and order of the evaluated harmonic coefficients), it would be useful to be able to identify any possible spectral correlations between the respective coefficient sets, and if such a correlation exists, to define the specific spectral bandwidth where it occurs. In order to obtain a first quantification of their spectral characteristics, a selection of spectral quantities was computed for all of the considered models, such as correlation coefficient per degree, smoothing per degree, and percentage difference by degree. Furthermore, typical geodetic measures in computing different geopotential functionals, such as Root Mean Square (RMS) geoid undulation and gravity anomaly difference curves, have been also evaluated for the same models, using the combined model EGM2008 as the reference model [*Pavlis et al*., 2012]. The comparisons quantify the level of agreement between different CHAMP-only, GRACE-only, combined, and GOCE models, defining the specific bandwidths where the corresponding correlations take place.

[13] After presenting the theoretical background of the series representation of the gravity field, we provide an insight to the definition and basic assets of topographic/isostatic gravity models, which serve in the present survey as the independent source of assessment for all of the examined satellite-only and combined models. The following section provides the definition of all spectral and spatial assessment measures that are considered here. The comparisons are grouped into four categories presented in the corresponding sections, namely, for CHAMP-only, GRACE-only, combined, and GOCE models. The final section provides a synthesis of the results of the assessment pointing out the main findings of the comparisons and highlighting future possibilities in gravity field modeling, validation and interpretation.