Reviews of Geophysics

A spectral assessment review of current satellite-only and combined Earth gravity models

Authors

  • Dimitrios Tsoulis,

    Corresponding author
    1. Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Thessaloniki, Greece
    • Corresponding author: Dimitrios Tsoulis, Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Thessaloniki, Greece. (tsoulis@auth.gr; http://users.auth.gr/tsoulis)

    Search for more papers by this author
  • Konstantinos Patlakis

    1. Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Thessaloniki, Greece
    Search for more papers by this author

Abstract

[1] The realization over the last decade of dedicated gravity field satellite missions enabled the production of a series of new satellite-only and combined models for the Earth's gravity field. Using different sensors, measurement techniques, and algorithmic procedures, the final product in each case is a set of spherical harmonic coefficients representing the series expansion of the gravitational potential up to a certain maximum degree, depending on the mission characteristics and the range of the available data. The present review performs a detailed quantified analysis of a representative selection of currently available CHAMP (Challenging Minisatellite Payload), GRACE (Gravity Recovery and Climate Experiment), GOCE (Gravity Field and Steady-State Ocean Circulation Explorer), and combined Earth gravity models. In this comparative analysis, we also include the so-called topographic/isostatic gravity models that represent the contribution of global digital elevation maps for the topography and ocean bathymetry. Applying a range of available spatial and spectral accuracy and assessment measures, such as correlation per degree and order, smoothing per degree and order, signal-to-noise ratio, gain, degree variances, and error degree variances, one gains a quantified “peek” inside the quality of these models spanning over their whole spectrum. The applied error and assessment measures are defined both in an absolute and relative manner with respect to other similar models or some reference Earth gravity models. Furthermore, the nature of the performed analysis (degree-wise, order-wise, and cumulative) permits the identification of distinct spectral bandwidths in these models, enables the quantification of some standard features of the observed field, such as its “long wavelength”, “short wavelength”, or “very high frequency part”, and specifies the attenuation of the gravity signal with increasing altitude from the Earth's surface. An examination of the assessment quantities reveals certain bandwidths of these models with characteristic statistical features. A band-limited synthesis of these bandwidths in the space domain quantifies the corresponding contributions in terms of selected gravity field functionals, including second-order derivatives at GOCE altitude.

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 inline image and inline image 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 m2 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 inline image and inline image 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 inline image and inline image, 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 inline image and inline image [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.

2 THEORETICAL BACKGROUND

[14] The basic reference quantity is the gravitational potential. Knowledge of the potential permits the computation of other gravity functionals, either on the Earth's surface or at a given satellite altitude. Some of these functionals are even linked directly to certain observations obtained from the new missions at satellite altitude. For the mathematical formulation of the gravitational potential, different alternatives exist. When the geometry and consistency of a given source is given in a rigorous manner, then usually analytical procedures can be applied, producing a closed analytical expression for the integral expression of the potential at the given computation point. However, the exact analytical solutions are not always possible, when for example the geometry implies an integration that cannot be solved analytically. Thus, numerical or even hybrid solutions can be elaborated in order to compute the gravitational potential of the corresponding mass distribution. When instead of simple one-, two-, or three-dimensional distributions, we deal with the total masses of the Earth the mathematical tool that has been established for the representation of the gravitational potential due to these masses at an arbitrary space point is the following series expansion [Kaula, 1966]

display math(1)

[15] Equation (1) is known as the development of the potential in spherical harmonics and is used extensively in connection with global gravity field modeling, satellite methods in gravity field determination, and satellite dynamics [Kaula, 1966; Heiskanen and Moritz, 1967]. The first term equals the potential of a homogeneous sphere and expresses the representation of the Earth in a central field approximation. The equation is expressed in terms of spherical coordinates; thus, the position of the computation point is defined by its radial distance r, co-latitude θ, and longitude λ. inline image are the fully normalized associated Legendre functions, with l and m denoting degree and order, respectively. GM expresses the product of Newton's gravitational constant G times the total mass of the Earth M, inline image and inline image are the fully normalized spherical harmonic coefficients, and R denotes an approximate value for the Earth's radius. The first summation runs through degrees up to the upper truncation limit of the expression at a specific degree L called the maximum degree (l = Lmax) of the geopotential model. The second summation sums up the orders, where for each l, it holds m ≤ l. Equation (1) identifies with a Fourier series defined in the range − π ≤ θ ≤ π, 0 ≤ λ ≤ 2π and is also referred to as spherical harmonic synthesis. The latter formulation implies that given the numerical values of the potential harmonic coefficients inline image and inline image up to a certain Lmax, the corresponding potential can be synthetically computed at any arbitrary field point. Indeed, apart from inline image and inline image all other quantities in (1) express geometrical definitions on the sphere and can be computed numerically without any other prior information. inline image and inline image are the fundamental parameters in this procedure. They contain the dynamics of the process expressing the gravity-related information in this expression. Their computation is linked to the inverse form of equation (1), i.e., an inverse Fourier expression in the same range of parameters, known as spherical harmonic analysis. The general expression referring to the gravitational potential of the Earth's total mass distribution reads [Tsoulis, 2001]

display math(2)

where M is the Earth's mass in a spherical approximation, inline image, inline image is a mean density value of the Earth's masses, e.g., 5500 kg m−3, and Ylm denotes the abbreviation

display math(3)

[16] The basic asset of equation (2) is the prerequisite that the actual density distribution in the Earth's interior ρ(Q) is known. This permits the integration in the radial direction r, which leads to expressions defined on the sphere, whose numerical computation can be performed in terms of spherical harmonic analysis. Equation (2) is linked with the potential of the Earth's masses and is the basis for the evaluation of topographic/isostatic gravity models using as input data the geometry and consistency of the crustal masses.

[17] In the most general case of the other satellite-only or combined gravity models, however, coefficients inline image and inline image are obtained through a different procedure than the direct integration expression (2). The primary observations (e.g., range, range rates, accelerometry, and gradiometry) enter the corresponding observation model, which defines the measurement principle (e.g., satellite-to-satellite-tracking and gradiometry) and describes the dynamics of the system by means of additional parameters, such as nongravitational effects, initial state parameters, gravity effects of third bodies, atmospheric parameters, and Earth rotation parameters. After linearization of the observation model with respect to inline image and inline image, the coefficients together with their error estimates are computed by means of a standard adjustment procedure.

[18] The implementation of these procedures (direct numerical integration followed by spherical harmonic analysis and adjustment procedure applied on an extended observational model) leads to the production of a set of inline image and inline image, i.e., the production of an Earth gravity model up to the corresponding Lmax. These procedures are tedious; they depend strongly on the details of the observational model or the way the analysis algorithm is implemented numerically on the sphere and can lead to different solutions, even when the same amount of primary observations is used.

[19] The International Center for Global Earth Models in Potsdam (http://icgem.gfz-potsdam.de/ICGEM/) collects and archives a thorough list of existing gravity field models in terms of their coefficients inline image and inline image and the corresponding error estimates. Additional information accompanying these data include the adopted value GM for the gravitational constant times the Earth total mass, which is used when creating the model, the adopted Earth's equatorial radius value R, the information whether the included errors are the formal or calibrated ones after some kind of calibration, a degree dependent scaling being the most typical one, and the tide system to which the coefficient inline image refers. These additional data permit the necessary numerical rescaling between different models. Thus, in many cases the coefficients of two different geopotential models are not given with respect to the same values of GM and R. A scale factor should be applied in this case rescaling the spherical harmonic coefficients so that they refer to the same values of GM and R. Such a scale factor is given by [Lemoine et al., 1998]

display math(4)

where inline image are the spherical harmonic coefficients of the first model inline image; GM(1), R(1), and GM(2), R(2) the values associated with the first and second models, respectively; and inline image the scaled spherical harmonic coefficients of the first model fully compliant with the second model.

[20] A further important issue is the permanent tide system to which harmonic coefficients of a single model refer. There are three systems in common use: Mean, Zero and Tide-free. In each system, the contributions to the potential, deformation of the crust, and best fitting ellipsoid to the geoid have their own definitions. In order to compare different geopotential models, it is mandatory that they refer to the same permanent tide system. The effect of changing a permanent tide system is seen only in the inline image term. The corresponding corrections are given by [Smith, 1998]

display math(5)
display math(6)

where a is the semi-major axis of a reference ellipsoid, γ a latitude dependent normal gravity value, and k the (fundamentally unknowable) zero frequency Love number. Different values for k have been adopted for different models, e.g., for Earth Gravitational Model EGM96 k = 0.3. Also, the way the inline image term of a specific model is converted to a different tide system varies between different models. For example, in EGM96 and EGM2008, the corresponding correction is given by [Lemoine et al., 1998]

display math(7)

3 TOPOGRAPHIC/ISOSTATIC GRAVITY MODELS

[21] The geodetic interest in isostasy was triggered by the release of global digital elevation models and the numerical ability to perform heavy computational tasks, such as the spherical harmonic analysis algorithm of a global elevation dataset. In essence, these models deal with the gravitational potential due to the topographic masses, whose upper boundary surface is described geometrically by the values of the global digital elevation model. The lower boundary surface which limits the lower boundary of the integration of the topographic masses has to do with the assumption regarding the way the crustal masses are in an isostatic equilibrium with the underlying thicker masses of the mantle. The spherical harmonic analysis of the potential defined by the direct integration of the topographic masses between the aforementioned boundary surfaces using some crustal density information or assumption leads to a set of potential harmonic coefficients that cannot be regarded as an equivalent to an Earth gravity model. They express the power of the topographic masses, and they define the spectrum of the so-called compensated topography. If the same integration is performed with respect to some kind of density contrast, then the resulting set of coefficients define a so-called topographic/isostatic model. The latter is linked usually to the isostatic hypothesis that was used for the geometrical description of the crustal masses. A formal description of an Airy/Heiskanen type topographic/isostatic potential has been given by Sünkel [1985], who defined it as the potential of all mass disturbances δρ relative to an ideal crustal layer of uniform density ρο and thickness D, superimposed upon underlying material of equally uniform density ρm. Pavlis and Rapp [1990] on the other hand define the topographic/isostatic potential as the difference between the potential induced by an Earth, which would behave as prescribed by a certain isostatic hypothesis minus a reference potential induced by a simplified Earth model consisting of a crustal layer of uniform density ρc and thickness D superimposed on a denser layer of uniform density ρm. The two definitions are equivalent, the latter referring to a difference between two potentials and the former to a potential induced by differences in mass, both leading to an identical result.

[22] The key element in the evaluation of these models is the definition of the lower boundary surface between crust and mantle. The geometry of the Moho boundary layer defines the lower boundary of the integration of the disturbing masses in the radial direction and results from the application of an isostatic mechanism that describes the hydrostatic equilibrium between the lighter crust floating over the denser mantle. In the case of the Airy/Heiskanen isostatic model for example, equation (2) becomes [Sünkel, 1986]

display math(8)

with AT and AC denoting the surface topography and the compensation part defined, respectively, as

display math(9)

and

display math(10)

[23] In the last two expressions, ρcr and Δρ stand for the constant density of the crust and the constant density contrast between crust and mantle, respectively, and D is the thickness of the crust at zero elevation, also known as compensation depth, i.e., the thickness of the crust beyond which the compensation mechanism for root or anti-root thickness t according to Airy/Heiskanen begins to apply. Replacing ocean depths by equivalent rock topography and taking the convergence of the verticals into account, the coefficients of the isostatically compensated topography, i.e., the topographic/isostatic gravity model according to Airy/Heiskanen, are obtained as the difference between the uncompensated topography and the compensation part according to Rummel et al. [1988]

display math(11)

with the separate definitions being, respectively,

display math(12)

and

display math(13)

[24] The notation hilm is used to describe spherical harmonic expressions of the elevation function h denoting topography and bathymetry information on a global equidistant grid. They read

display math(14)
display math(15)
display math(16)

[25] The numerical computation of equations (14)(16) is performed with the tools of spherical harmonic analysis. Thereby, certain computational issues have to be properly addressed. Since the elevation data of a global digital elevation database express mean values valid over the corresponding geographical compartment, the formulation of the discrete case of spherical harmonic analysis has to be modified accordingly by replacing the terms sin and cos with their integral definitions over individual compartments, with λ varying between the two neighboring longitude values λk − 1 and λk [Albertella and Sacerdote, 1995; Tsoulis, 1999]. Also, when using mean values in the discrete case of spherical harmonic analysis, the integrals of the associated Legendre functions over the corresponding compartment latitude values θk − 1 and θk have to be evaluated, following a closed iteration algorithm [Gerstl, 1980; Tsoulis, 1999]. Last but not least, the numerical implementation of equations of the type (14)–(16) is linked to the loss of orthogonality of the Legendre functions due to the latitude sampling. Unlike the discretization in the longitude direction where the orthogonality is preserved, the θ-sampling destroys the validity of the orthogonality property of the Legendre functions. The different numerical ways to retrieve the loss of orthogonality in the θ-direction, e.g., by inserting θ-dependent weights to each parallel θi, lead to different solutions in the final adjustment step for the computation of the potential harmonic coefficients, categorized as exact or approximate solutions [Sneeuw, 1994; Tsoulis, 1999]. However, the fulfillment of the orthogonality property is not a strict requirement in the process of an evaluation of a new gravity model. For example, EGM2008 relies on a block-diagonal approach for the high orders, which makes the orthogonality property obsolete.

[26] The Airy/Heiskanen model is used as the standard isostatic hypothesis behind topographic/isostatic gravity models. The use of the Pratt/Hayford mechanism leads to a model with almost no compensating effect with respect to the spectrum of the global elevation model for the lower degrees and an energy damping or compensating feature for degrees above 60 with increasing compensating effect for increasing degrees [Tsoulis, 2001, 2005]. The incorporation of the geometry and consistency information of global crustal database CRUST 2.0 [Bassin et al., 2000], providing a global layered structure of the crust in terms of geometry and density of seven distinct crustal layers from the surface topography down to the Moho boundary, led to the computation of a further topographic/isostatic gravity model, where a modification to the standard Airy/Heiskanen compensation mechanism had to be introduced [Tsoulis, 2004]. As the interpretation of topographic/isostatic models is strongly related to density structures in the Earth's interior, one has to ascertain the incorporation of all available data sources and methodologies to improve the information content of such models. In this respect, Sünkel [1985] applied a Vening-Meinesz isostatic model for his derivation of a topographic/isostatic model. Vening-Meinesz [1931] argued that the compensation had to have a more regional character, introducing a smoothing of the compensation surface. Sünkel investigated the optimal choice of the compensation smoothing operator providing a very useful link to current purely flexural isostatic models [Watts, 2001]. Kaban et al. [1999] have used a Bouguer admittance function in calculating the gravity and geoid signal of their global isostatic model. The derivation of adapted admittance functions, which take the flexural strength of the lithosphere better into account, could provide further developments in topographic/isostatic analysis.

[27] Topographic/isostatic models provide a useful tool in gravity field modeling. Although they do not contain local analysis, e.g., local and regional crustal and lithospheric data, which would provide a more realistic description of the actual isostatic mass balance in the area under consideration, they fulfill a number of criteria that make them unique in gravity field interpretation. Free air gravity anomalies represent a characteristic example. They are defined as the difference between observed gravity g and normal gravity γo defined at a reference ellipsoid, if one neglects the existence of masses between these two points. The formal definition reads using scalar notation [Torge, 2001]

display math(17)

with

display math(18)

the vertical gradient of g, expressing the natural decrease of the gravity signal with increasing distance from the Earth's surface, and H the orthometric height of the field point measured along the curved plumb line of the Earth's gravity field showing elevation with respect to the geoid [Hackney and Featherstone, 2003].

[28] Although globally, or even regionally, free air gravity anomalies are relatively small and smooth, they become rugged and correlated to local mass distributions on the short scales. The need to smooth these anomalies, which is a requirement to geoid determination, can be performed efficiently by using isostatic models, without any loss of physical information that any low-pass filtering would cause. Furthermore, isostatic models are easy to compute and their indirect effect is negligibly small. Thus, they enable nicely the identification of topographic patterns, such as mountains and ocean ridges, when compared globally with free air anomalies and smooth the observed field efficiently when applied regionally [Göttl and Rummel, 2009].

[29] Topographic/isostatic potential coefficients do not contain contributions from deeper mass inhomogeneities. They reflect the contribution of mass disturbances that are situated at the crust and are therefore more profoundly linked to the high-frequency part of the observed field. Their low-frequency coefficients are in principle not realistic and are modeled by other means, when isostatic models are combined with other gravity information to construct synthetic or reference Earth gravity models. Fitting the degree variance spectra of a topographic/isostatic potential and a global geopotential model, or filtering the potential coefficients with a combination of a low-pass filter for the geopotential model and a high-pass filter for the topographic/isostatic potential coefficients, leads to a merging of the topographic/isostatic model with some given geopotential model in order to support the construction of a so-called synthetic Earth model [Haagmans, 2000; Claessens, 2003].

4 ABSOLUTE SPECTRAL ASSESSMENT TOOLS

[30] The most common way to represent the spherical harmonic coefficients of a given gravity model is by means of the so-called triangular representation, where the coefficients are ordered in the form of a triangle, whose vertical axis displays the degree l and the horizontal axis the order m. Typical quantities that are displayed in this triangular format are the coefficients of a model, their errors, or the coefficient differences with respect to an existing gravity field model leading to the so-called 2-D spectrum, 2-D error spectrum, and 2-D difference spectrum, respectively. Thus, the triangular representation apart from a first quick look of a model offers a means of spectral evaluation by revealing possible zonal, sectorial, or tesseral characteristics, especially in the case of the error spectrum, and permitting an interpretation of the dynamical characteristics of the model in question. Apart from expressing a complete qualitative spectral image of one model, this kind of representation provides important quantitative information as well. An additional tool, especially for comparison of the spherical harmonic signal, is the representation of the error and difference spectra by means of more condensed 1-D quantities.

[31] The most common absolute measure of a model is its degree variance or spectral power, defined as [Rapp, 1982, 1986]

display math(19)

and denoting the total signal power of a certain degree l. Degree variances express the power spectrum of thecoefficients of one model, not providing any information about its accuracy. The error degree variance on the other hand expresses the total error power at a given degree as follows:

display math(20)

where inline image and inline image are the standard deviations of coefficients inline image and inline image, respectively.

[32] It is common practice to compute instead of degree variances and error degree variances the square root or the weighted square root (degree variance divided by the number of coefficients, 2 l + 1) of these quantities, the so-called degree RMS spectrum and error degree RMS spectrum. Both degree variance and degree RMS express rough assessment measures and are representative for a single σlm only in the case of an isotropic error spectrum.

[33] The above quantities can be also considered for a given gravity functional, such as geoid heights, gravity anomalies, or gravity gradients. A simple multiplication with the appropriate eigenvalue leads to the error measure for the corresponding functional. These eigenvalues are summarized from the so-called Meissl's scheme, which provides the spectral linkage between different functionals of the disturbing potential at different altitudes from the Earth's surface [Rummel and van Gelderen, 1995]. Thus, the mean Earth radius R scales the dimensionless degree variance into a geoid degree variance, the eigenvalue GM(l − 1)R− 2 scales the dimensionless degree variance into a gravity anomaly degree variance, and the eigenvalue GM(l + 1)(l + 2)R− 3 scales the dimensionless degree variance into a radial gravity gradient degree variance.

[34] The weakness of the error degree variance to express the total error power has to do with the nonisotropy of the coefficients' error spectrum. Nonisotropy implies here that the error spectrum depends also on the order m. Thus, the need arises to introduce another error measure which exhibits the total power of a given order m. The order variance and error order variance are given, respectively, by

display math(21)

and

display math(22)

[35] Another measure for the relative quantification of gravity field models is the difference degree and order variance computed, respectively, by

display math(23)
display math(24)

where inline image and inline image are the differences between the two sets of coefficients.

[36] As equation (24) clearly demonstrates, when the models are of different maximum solvable degree and order, the difference order variance does not show the real order difference but the difference between the first model and the truncated version of the second one. For comparison of geopotential models of different maximum degree and thus different spatial resolution on the Earth's surface, the commission error should be considered, in a certain bandwidth. The commission error can be expressed in terms of various gravity field functionals as

display math(25)

with L the degree up to which the cumulative error or difference is computed, inline image the gravity field functional, λl the eigenvalue for the specific functional, and cl the error degree variance or difference degree variance.

[37] Introducing an identical relation depending on the order m, one should rescale the coefficients multiplying them by the eigenvalue λl before computing the order degree variance and finally the cumulative error by order as

display math(26)

where M is the order up to which the cumulative error or difference is computed, inline image denotes the gravity field functional, and cm the error order variance or difference order variance.

[38] It should also be noted that the coefficients of all models are affected by omission (or equally aliasing) errors, especially towards their highest corresponding degree, obtained by the fact that every model is defined only up to a certain maximum degree. Thus, all spectral comparisons between models of different solvable degrees are also affected by this error, especially the differences defined close to the common spectral range of the models.

5 RELATIVE SPECTRAL ASSESSMENT TOOLS

[39] When different gravity models are available, it is useful to obtain a more detailed insight to their spectral characteristics. This can be performed by means of relative spectral assessment tools that refer to two different global geopotential solutions. One of these tools that can be used for the comparison of two sets of potential harmonic coefficients is the correlation by degree and by order given, respectively, by

display math(27)

and

display math(28)

where inline image and inline image denote the fully normalized spherical harmonic coefficients of the two expansions, symbolically expressed here as models A and B, and inline image are the respective degree or order variances.

[40] The correlation per degree provides a numerical quantification of a direct comparison between the two models, though it cannot reflect entirely the agreement or disagreement between them. Even if a high correlation by degree exists, the two models may still differ by a dominant scale factor. Therefore, the definition of additional complementary assessment measures is necessary. A better quantification and understanding of the correlations offered by equations (27) and (28) is obtained from the so-called smoothing per degree or per order given, respectively, by the expressions

display math(29)

and

display math(30)

[41] This defines a quantity expressing the degree of smoothing one obtains if model A is subtracted from B [Tscherning 1985]. An additional quantity of special interest for comparative analysis of two models obtained from an identical evaluation algorithm and primary observations is the percentage difference by degree or by order, defined as

display math(31)

and

display math(32)

[42] In addition to these spectral quantities, two more relative error measures are considered here, the signal-to-noise ratio (SNR), which expresses the ratio between the signal and the error spectrum of a single model, and Gain, which is defined as the ratio between a model's error spectrum and another error spectrum, serving as the a priori model. The 10-base logarithm of these relative measures represents the number of significant digits and the gain in significant digits, respectively, [Sneeuw, 2000]

display math(33)

and

display math(34)

inline image in equation (33) denotes inline image or inline image, and inline image stands for inline image or inline image. The exponent new in (34) expresses the model which is evaluated and exponent old the a priori model, e.g., EGM2008.

[43] Both SNR and Gain are mainly used as one-dimensional measures (Figure 1); i.e., when expressed per degree, they are given, respectively, by

display math(35)

and

display math(36)
Figure 1.

Graphical explanation of the error measures (left) gain and (right) signal-to-noise ratio.

[44] The signal-to-noise ratio determines the maximum solvable degree, i.e., the degree up to which the specific model has full power and the signal is stronger than noise. Thus, a model has significant power up to degree l when SNRl = 1. In a 1-D representation of significant digits, that specific degree up to which the model has significant power is defined at that point in the graph where the curve crosses the zero x axis. In a same way, Gain determines the degree up to which the accuracy of the coefficients' computation is better than that of an a priori model. Thus, Gain determines degree bandwidths and not a maximum degree, in which the accuracy of the coefficient's computation is better than that of an a priori model.

6 ASSESSMENT SURVEY

[45] The available Earth gravity models can be grouped into three categories. The satellite-only models are those obtained from the analysis of the detected orbit perturbations, various sensor (accelerometer, star sensor, or gradiometer), and tracking observation data of the CHAMP, GRACE, or GOCE satellites gathered over finite time periods. The merged analysis of satellite-only, surface gravity, and altimetry data leads to the computation of the combined gravity models. In this category, one finds also the so-called reference gravity models or Earth Gravity Models, including models EGM96 [Lemoine et al., 1998] and EGM2008 [Pavlis et al., 2012]. Finally, the topographic/isostatic models, which were described above, are linked solely with the availability and analysis of a global digital elevation database accompanied with an assumption about isostasy between crust and mantle. They form an independent category of Earth gravity models that can reach also high and very high degrees of expansion. In the following, the assessment tools that were presented in the previous sections are implemented aiming for a thorough analysis of a selection of the most characteristic gravity models of all three categories. Apart from the evaluation of the spectral assessment quantities, a systematic spatial representation is performed of some gravitational functionals both for the entire spectrum and for special limited spectral bandwidths that emerge after the implementation of the spectral assessment tools.

[46] Table 1 lists all models considered in the present survey. Their selection was performed so that the recent state-of-the-art models of the three satellite missions are included. Also, an effort was made so that the different groups working on the evaluation of the models are being represented in this listing. All models are expressed in the tide-free system, and the harmonic coefficients of each model have been properly rescaled so that they refer to the GM and R values of the reference model GRS80 (Geodetic Reference System 1980).

Table 1. Geopotential Models Used in Present Studya
ModelYearDegreeDataSource
  1. aS denotes satellite, G gravity, and A altimetry data.
TUM-2S200460S(Champ)

Wermuth et al. [2004]

AIUB-CHAMP03S2010100S(Champ)

Jäggi et al. [2010]

EIGEN-CHAMP05S2010150S(Champ)

Flechtner et al. [2010]

AIUB-GRACE03S2011160S(Grace)

Jäggi et al. [2010]

GGM03S2008180S(Grace)

Tapley et al. [2007]

EIGEN-GL05S2008150S(Grace,Lageos)

Förste et al. [2008]

ITG-Grace03s2007180S(Grace)

Mayer-Gürr et al. [2010a]

ITG-Grace2010s2010180S(Grace)

Mayer-Gürr et al. [2010b]

GGM03C2009360S(Grace),G,A

Tapley et al. [2007]

EIGEN-GL05C2008360S(Grace,Lageos),G,A

Förste et al. [2008]

EIGEN-51C2010359S(Grace,Champ),G,A

Bruinsma et al. [2010]

GOCE-spw2010210S(GOCE)

Migliaccio et al. [2010]

GOCE-dir (r1)2010240S(GOCE)

Bruinsma et al. [2010]

GOCE-dir (r2)2011240S(GOCE)

Bruinsma et al. [2011]

Pail et al. [2011b]

GOCE-tim (r1)2010224S(GOCE)

Pail et al. [2010a]

GOCE-tim (r2)2011250S(GOCE)Pail et al. [2011a] Pail et al. [2011b]
GOCO01S2010224S(GOCE,Grace)

Pail et al. [2010b]

GOCO02S2011250S(GOCE,Grace)

Goiginger et al. [2011]

EGM200820082159S(Grace),G,A

Pavlis et al. [2012]

T/I20052160DTM2002

Pavlis et al. [2005]

[47] The combined model EGM2008 [Pavlis et al., 2012] is used as a common reference model for all spectral comparisons which are performed in the sequel. EGM2008 is the result of combining the ITG-Grace03s (ITG—Institute of Theoretical Geodesy, University of Bonn) gravitational model [Mayer-Gürr et al., 2010a] with a global 5 arc-minute equiangular grid of free air gravity anomalies. The latter is formed by merging terrestrial, altimetry-derived, and airborne gravity data. In addition, EGM2008 incorporates the gravity information obtained by global topography in the frame of the topographic/isostatic formulation in order to improve the spectral content over areas with low resolution of gravity data. Thus, EGM2008 expresses currently the best knowledge of the actual field in terms of direct gravity observations and high-frequency contributions due to the topography. For this reason, it is selected as the reference model for all performed comparisons. Figure 2 presents both the full coefficients and calibrated error spectrum of EGM2008 and their truncated image up to degree l = 360 in order to emphasize its long wavelength features. Coefficients and their standard deviations present a near isotropic behavior beyond degree 100. This feature should be assigned to a combination of reasons related with the data and the computational strategy of EGM2008. These reasons include but are not restricted to (1) the incorporation of accurate and detailed gravimetric data forming a global equidistant grid, (2) the utilization of dense global elevation information which permits to increase computationally the spectral resolution of certain areas to the uniform global high resolution of the model, and (3) the fact that the least-squares estimation procedure which is applied to the global grid of gravity anomalies for the evaluation of the model's coefficients is based on a Block-Diagonal approximation of the normal equations [Pavlis et al., 2012]. Also from these plots, an almost stable uncertainty in the computation of the EGM2008 coefficients beyond degree 720 is apparent. Zonal harmonics up to degree 60 seem to have been computed with a better accuracy than the rest of the EGM2008 spectrum, while the uncertainty decreases proportional with the increasing degree l.

Figure 2.

(left) 2-D spectra and (right) 2-D error spectra of EGM2008, (first row) full spectrum version and (second row) truncated up to degree 360.

[48] The proportional decrease of the coefficient errors with increasing degree l is also apparent in Figure 3, where the square roots of degree and order variances are plotted. Model “T/I” expresses the full up to degree and order 2160 topographic/isostatic model, which was obtained from an Airy/Heiskanen isostatic mechanism applied to the 2 min × 2 min global terrain and bathymetry model DTM2002 (Digital Terrain Model 2002) [Saleh and Pavlis, 2003]. “Topo” denotes the spectrum of the uncompensated topography for the same terrain model. The compensation effect of the Airy/Heiskanen model is evident for the lower degrees, where the power of the T/I model is significantly less than that of the uncompensated topography. With increasing degree, this compensation effect is lost and the two spectra (T/I and “Topo”) converge to the same power. However, for the higher degree range, both T/I and “Topo” expose significantly less power than EGM2008.

Figure 3.

(left) Square root degree and (right) order variances of EGM2008 and T/I.

[49] A quantified spectral comparison between EGM2008 and T/I can be performed by means of the available correlation tools. Figure 4 presents the obtained correlations per degree and per order. It can be seen that the coefficient sets are characterized by a high degree of correlation between degrees 720 and 1800. The two correlation patterns for degree and order are almost the same. However, it is interesting to observe the differences in correlation of the very last orders in the second row of Figure 4, demonstrating the amplitude of the effect of the aforementioned omission errors. The smoothing coefficients by degree and by order, which are also shown in Figure 4, confirm the aforementioned spectral bandwidth of agreement between EGM2008 and T/I. Finally, the small peaks occurring at order 720 for the correlation and smoothing coefficients per order (lower panels of Figure 4) and around degree 900 for the correlation and smoothing coefficients per degree (upper panels of Figure 4) should be related with the fact that in the development of EGM2008, a forward modeling approach has been implemented in order to compute so-called “fill-in” gravity anomalies for degrees beyond 720 [Pavlis et al., 2012].

Figure 4.

Correlation and smoothing coefficients per degree and per order between EGM2008 and T/I.

[50] The transfer of the full spectra in the space domain permits a quantification of the comparisons in terms of selected gravity field functionals. Figure 5 presents the full difference spectrum between EGM2008 and T/I expressed in geoid heights, gravity anomalies, and second-order radial derivatives of the disturbing potential both at the Earth's surface and at a GOCE simulated altitude of 250 km. A close study of these plots—e.g., of the areas over which large discrepancies occur combined with the lateral numerical variance of the obtained fields, i.e., the identification of neighboring differences with opposing sign—leads to the conclusion that the obtained patters follow the overall variability and general pattern structure of the observed EGM2008 signal. The T/I coefficients represent the high-frequency part of the observed geopotential and when they are subtracted from the reference field EGM2008 produce a residual field which mirrors the long to medium wavelengths of the field. This information is already inherent in the small degree range; thus, the variability and overall structure of the residual field coincides with that of EGM2008. This observation gains in significance, since EGM2008 uses terrain-related information in the recovery of its coefficients above degree 720. The T/I model on the other hand expresses purely the gravity field content which is produced by the global crustal masses, floating over the denser material of the mantle. This contribution belongs in terms of gravity field theory to the high-frequency part of the real field and depends on the actual geometrical relief of the used global elevation model and the numerical assumptions regarding the density of the crustal masses. Although the relation between the two coefficient sets is not linear, the residual field between EGM2008 and T/I roughly removes the terrain-related information of EGM2008, providing a picture of the larger (long to medium) wavelength features of the gravity field. This becomes evident in all subplots of Figure 5 but especially in the upper panel of the figure where the residual field is expressed in terms of geoid heights and gravity anomalies. On the other hand, the second-order radial derivatives of the residual field computed at the Earth's surface (lower left subplot), though topography-free, still manage to capture some characteristic regional medium wavelength features, such as the Andes, the Himalayas, or the boundary between the Pacific and Eurasian plates. The gravity signal attenuation is apparent in the lower right subplot of this figure which presents second-order radial derivatives at a typical GOCE altitude 2 orders of magnitude smaller than those computed at the Earth's surface.

Figure 5.

Geoid heights, gravity anomalies, and second-order radial derivative of T at h = 0 and second-order radial derivative of T at h = 250km (lower right panel) differences between EGM2008 and T/I.

6.1 CHAMP-Only Models

[51] The CHAMP-only models that were selected for the present survey include TUM-2S, AIUB-CHAMP03S, and EIGEN-CHAMP05S. TUM-2S is a gravity field model produced at the TUM (Technical University of Munich), which is based on 2 years of CHAMP GPS orbit tracking and accelerometry data from the time interval 11 March 2002 to 10 March 2004, computed using the energy balance approach. The latter is a technique which leads to gravity field determination by means of a two step approach. First, a high-precision orbit is computed from the GPS observations. Then, the resulting position and velocity coordinates are used as in situ observations for gravity field recovery by means of the so-called energy balance equations, a formulation steaming from the definition of the energy conservation principle for the satellite's motion. These energy balance observations are formed as a time series along the orbital track, and the whole procedure is based solely on the satellite's tracking data, being independent from any prior gravity information [Visser et al., 2003; Gerlach et al., 2003]. AIUB-CHAMP03S is a global gravity field model produced at the AIUB (Astronomical Institute, University of Bern), which is computed from GPS satellite-to-satellite tracking data from the time interval 2002 to 2009. This model has been computed using the celestial mechanics approach, which is a term used to describe the numerical integration of the variational equations without applying any regularization to the data [Beutler et al., 2010a; 2010b]. Finally, model EIGEN-CHAMP05S is produced at the GeoForschungsZentrum (GFZ) Potsdam and has been computed from CHAMP data out of the period October 2002 through September 2008. It should be stressed that the common underlying theory and methodology used for the evaluation of the GFZ, AIUB, and CSR (Center for Space Research at the University of Texas at Austin) models is the numerical integration of the equations of motion and variational equations. The celectial mechanics approach, which is merely a technical term describing these procedures, is very efficient in terms of organizing the necessary computations which lead to the design matrix and the corresponding normal equations. Although differences in the estimation of the different parameters exist, all implementations should lead to the same results when applied to the same data.

[52] Figure 6 presents the full spectra of the three CHAMP-only models in terms of their coefficients inline image and inline image and their corresponding standard deviations. These full error spectra reveal the nonisotropic behavior of the CHAMP models with the exception of model TUM-2S, which is nearly isotropic in the whole spectral range. The reason for this behavior in the TUM-2S spectrum is the fact that a uniform weighting has been applied and orbit information has been converted to in situ energy balance observations. Thus, the resulting error spectrum is not realistic. In other words, the obtained isotropy can be regarded as an artifact of the selected methodology. For both AIUB-CHAMP03S and EIGEN-CHAMP05S, sectorial harmonics seem to have been computed with a better accuracy than the other harmonics and it is obvious that the uncertainty in computation increases proportional to degree l. The spectra of these two models appear similar, because the same underlying method was applied. It is also important to note that in the case of AIUB and GFZ models, a starting or a priori gravity model such as EGM96 [Lemoine et al., 1998] or GRIM5 (GRIM—Groupe de Recherches Géodésie Spatiale Toulouse (GRGS) and German geodetic research Institute Munich) [Biancale et al., 2000; Gruber et al., 2000] was used. Thus, these models are not perfectly CHAMP-only. As they are based on data from many satellites as well as terrestrial and satellite altimetry data, orbital resonances of many other satellites are already well-represented in these gravity field solutions. The TUM-2S model on the other hand is a model computed from scratch. Furthermore, the characteristic pattern for EIGEN-CHAMP05S appearing in the bandwidth beyond degree l = 70 should be probably due to Kaula's a priori information which is used for the evaluation of the model at this spectral range. Kaula's a priori information, mentioned also as a Kaula type of regularization in gradiometric analysis, is based on the so-called Kaula's rule of thumb, an empirical degree variance model providing a rough estimate for the course of the gravity field spectrum. Kaula's empirical rule emerged from autocovariance analysis of terrestrial gravimetry data and has the form inline image [Kaula, 1966]. It implies that for a given degree, the quadratic mean over all orders of the harmonic coefficients of the Earth's gravity field decreases approximately according to 10− 5 × l− 2, l denoting degree. Kaula's rule expresses in other words the long wavelength behavior of the gravity field, according to which the short wavelengths are of much smaller power that the longer wavelengths.

Figure 6.

(left) 2-D spectra and (right) 2-D error spectra of the considered CHAMP-only models.

[53] Figure 7 displays the power spectrum and the respective error degree variances of the three CHAMP-only models. A comparison with EGM2008 in terms of difference degree variances takes also place. The considered quantities are again geoid heights, gravity anomalies, and second-order radial derivative of the disturbing potential. The evaluations take place at the Earth's surface and at satellite altitude (h = 250 km).

Figure 7.

Square root degree variances, error variances, and difference variances with respect to EGM2008 of the three considered CHAMP-only models in terms of different gravity functionals.

[54] It is clear that at least in terms of signal power, all models present an almost identical variation up to degree 60, whereas their errors fluctuate in a different way along the whole common spectral range. AIUB-CHAMP03S is more sensitive than the other two CHAMP-only models in the whole spectrum, with TUM-2S having in a comparative sense the worst performance in the common spectral range. However, it should be noted that the AIUB-CHAMP03S and TUM-2S errors are the formal ones in contrary to model EIGEN-CHAMP05S, which was distributed with calibrated errors. This could be the reason why the degree error variance of EIGEN-CHAMP05S does not increase for degrees beyond l = 70. A different pattern of fluctuation is observed with the difference variances with respect to EGM2008 (gray-scaled curves). This demonstrates that the errors of EIGEN-CHAMP05S were not properly calibrated. A similar conclusion cannot be drawn for the other models of this figure, since they are given in terms of formal errors. At the same time, it is interesting to observe the smoothness in the difference degree variances expressed in terms of the other three gravity field functionals against the one for geoid heights up to degree 30.

[55] The additive effects of individual degrees to the numerical evaluation of some gravity functional produce the so-called degree-wise accumulated or cumulative curves. Figure 8 presents these curves in terms of geoid heights and gravity anomalies for the considered CHAMP-only models. Commission errors and accumulated difference amplitudes per degree with respect to EGM2008 are also given in the right column of this figure. The differences are computed with respect to the truncated versions of EGM2008 up to the corresponding maximum degree of the CHAMP-only models. Tables 2 and 3 present these computations in numerical detail and in a band-limited fashion including also computations for the radial gradient of the disturbing potential at h = 0 and h = 250 km.

Figure 8.

Cumulative errors by degree of different CHAMP-only models and cumulative differences with respect to truncated versions of EGM2008 in terms of geoid heights and gravity anomalies.

Table 2. Cumulative Errors of CHAMP-Only Models in Terms of Different Gravity Field Functionals
Degree (up to) 102030506090100120150
TUM-2SN (cm)0.170.531.275.8311.85----
Δg (mGal)0.0020.0130.0480.3890.967----
Trr (E)0.0000.0000.0020.0300.090----
inline image (mE)0.0250.1960.7374.7049.714----
AIUB CHAMP03SN (cm)0.010.020.060.430.988.8517.39--
Δg (mGal)0.0000.0010.0030.0290.0811.1282.478--
Trr (E)0.0000.0000.0000.0020.0080.1570.383--
inline image (mE)0.0010.0080.0390.3500.8025.5729.416--
EIGEN CHAMP05SN (cm)0.030.140.453.127.2347.5962.8783.4399.33
Δg (mGal)0.0000.0030.0180.2130.6005.8758.39112.47416.677
Trr (E)0.0000.0000.0010.0170.0560.7981.2322.0703.186
inline image (mE)0.0040.0540.2752.5505.94031.11937.66443.43445.279
Table 3. Cumulative Differences of CHAMP-Only Models With Respect to EGM2008 in Terms of Different Gravity Field Functionals
Degree (up to) 102030506090100120150
TUM-2SN (cm)2.002.723.578.8317.06----
Δg (mGal)1.3680.0170.0460.1010.556----
Trr (E)0.0000.0010.0050.0430.127----
inline image (mE)0.2270.6971.5506.76913.799----
AIUB CHAMP03SN (cm)0.270.310.360.781.5110.8920.84--
Δg (mGal)0.0010.0040.0080.0470.1201.3752.957--
Trr (E)0.0000.0000.0000.0040.0110.1900.456--
inline image (mE)0.0170.0590.1210.5791.2096.92811.363--
EIGEN CHAMP05SN (cm)0.320.460.631.653.2025.1236.7255.1170.20
Δg (mGal)0.0030.0080.0190.1060.2583.1695.0388.57512.418
Trr (E)0.0000.0000.0010.0080.0240.4370.7551.4642.460
inline image (mE)0.0350.1220.2921.2822.59415.99921.10726.53028.414

[56] The overall differences in the spectral domain between the AIUB-CHAMP03S and the truncated to degree 150 EGM2008 model amount to 20.8 cm in terms of geoid undulations, 3 mGal in terms of gravity anomalies, 0.46 Eötvös in terms of second-order radial derivative of the disturbing potential on the Earth's surface, and 11.4 mE at satellite altitude. For EIGEN-CHAMP05S, the corresponding differences accumulate to 70.2 cm, 12.4 mGal, 2.46 Eötvös, and 28.41 mE, respectively, and for TUM-2S 17.1 cm, 1.4 mGal, 0.13 Eötvös, and 13.8 mE. It is important to underline that these values are indicative for the performance of the individual models relative to the corresponding spectral bandwidth of EGM2008. Thus, they do not represent accuracy estimates in an absolute sense, as the models are of different maximum degree. For example, the bigger cumulative value of EIGEN-CHAMP05S in terms of geoid undulations (70.2 cm) compared with that of TUM-2S (17.1 cm) does not imply that TUM-2S is more accurate in terms of geoid than EIGEN-CHAMP05S. The cumulative curves in Figure 8 should be evaluated individually. It is interesting to note that the error magnitude in terms of gravity gradient at satellite altitude for the TUM-2S model is higher than that of AIUB-CHAMP03S. Furthermore, these computations quantify the overall imported error when computing the gravity field functionals based solely on CHAMP-only models.

[57] The order-wise accumulated computations in terms of the same functionals, which are not shown here, reveal the major contribution of lower orders to the overall error in computing the corresponding functional. It is clear that cumulative errors and differences by order cannot be considered to compare different models because of the different number of spherical harmonic coefficients for the individual orders, depending on the maximum degree of resolution of each model. On the other hand, these measures can serve complementary to the cumulative curves by degree and complete the information extracted from each model.

[58] The signal-to-noise ratio and gain for the three CHAMP-only models with respect to EGM2008 expressed both in one and two dimensions are presented in Figures 9 and 10, respectively. Since the coefficient errors are sometimes calibrated and sometimes they are not, it is important to mention that SNR expresses a not always calibrated quantity. Figure 9 displays the significant digits, i.e., the 10-base logarithm of signal-to-noise ratio, for the three CHAMP-only models—TUM-2S, AIUB-CHAMP03S, and EIGEN-CHAMP05S. The first two of these models seem to have a strong signal along the whole spectral range unlike the third one for which the noise dominates the signal in the bandwidth starting roughly from degree l = 75 and onwards.

Figure 9.

Significant digits for the three considered CHAMP-only models.

Figure 10.

Gain in significant digits of the three CHAMP-only models with respect to EGM2008.

[59] Compared to EGM2008, the CHAMP models have almost no gain in accuracy except from the very first degrees, as Figure 10 nicely demonstrates. With the exception of TUM-2S which shows no gain over EGM2008 for the whole spectral range, the AIUB-CHAMP03S and EIGEN-CHAMP05S models present gain until degrees 30 and 10, respectively. It is clear that near-sectorial coefficients tend to have a small gain with respect to EGM2008 and that near-zonal coefficients tend to have the biggest difference in computational accuracy, with EGM2008 being better in almost two 7orders of magnitude. Finally, Figure 10 shows that model TUM-2S is not isotropic when compared to EGM2008, proving that the isotropy observed in Figure 6 was indeed an artifact.

[60] Figures 11 and 12 display the obtained correlation and smoothing coefficients by degree, respectively, between the three CHAMP-only models and various reference models. We notice an almost perfect coincidence between all three CHAMP models up to degree l = 45. The correlation remains high up to degree l = 80 among the CHAMP models and all other reference models. Beyond this limit, the coefficients of the CHAMP models begin to vary considerably with respect to the specific reference model. The correlation coefficients have a completely different behavior when the topographic/isostatic model is used as a reference. It can be seen in this case that the coefficient sets are characterized by a high degree of correlation between degrees l = 10 and l = 80. In order to obtain a more detailed numerical insight of these comparisons, the reader is referred to the entries of Table 4.

Figure 11.

Correlation coefficients per degree between the three CHAMP-only models and various reference models.

Figure 12.

Smoothing coefficients per degree between the three CHAMP-only models and various reference models.

Table 4. Correlation (in Percentage) and Smoothing per Degree for the Three CHAMP-Only Models With Respect to Different Reference Models
  EGM2008T/IITG-Grace2010sEIGEN-GL05CGOCE-spwGOCE-dir (r2)GOCO02S
 DegreeCor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.
TUM-2S2100.000.00−57.930.10100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.050.07100.000.00100.000.00100.000.00100.000.00100.000.00
2099.990.0065.870.0699.990.0099.990.0099.990.0099.990.0099.990.00
3099.980.0072.130.0599.980.0099.980.0099.980.0099.980.0099.980.00
4099.860.0370.930.0599.860.0399.860.0399.860.0399.860.0399.860.03
5099.230.1766.410.0699.220.1799.220.1799.230.1799.230.1799.220.17
6095.471.0774.280.0595.481.0795.481.0795.501.0795.471.0895.481.07
AIUB-CHAMP03S2100.000.00−57.930.10100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.060.07100.000.00100.000.00100.000.00100.000.00100.000.00
20100.000.0065.820.06100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0071.940.05100.000.00100.000.00100.000.00100.000.00100.000.00
40100.000.0071.270.05100.000.00100.000.00100.000.00100.000.00100.000.00
50100.000.0067.850.06100.000.00100.000.0099.990.00100.000.00100.000.00
6099.970.0175.170.0599.970.0199.970.0199.960.0199.960.0199.970.01
7099.710.0669.540.0599.710.0699.710.0699.710.0699.710.0699.710.06
8098.950.2168.960.0699.020.2099.020.2099.000.2099.030.1999.020.20
9095.110.9657.730.0795.260.9395.230.9395.330.9195.260.9395.270.93
10083.203.0857.500.0783.413.0483.213.0883.443.0483.383.0583.403.05
EIGEN-CHAMP05S2100.000.00−57.930.01100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.040.01100.000.00100.000.00100.000.00100.000.00100.000.00
20100.000.0065.800.01100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0071.870.00100.000.00100.000.00100.000.00100.000.00100.000.00
40100.000.0071.120.01100.000.00100.000.00100.000.00100.000.00100.000.00
5099.970.0067.610.0199.970.0099.970.0099.960.0099.970.0099.970.00
6099.810.0075.040.0099.810.0099.810.0099.810.0099.810.0099.810.00
7098.950.0268.160.0198.950.0298.950.0298.940.0298.960.0298.950.02
8094.240.1568.180.0194.300.1594.310.1594.230.1594.300.1594.300.15
9074.571.0053.470.0174.980.9874.920.9874.780.9874.950.9875.010.98
10054.562.1633.920.0155.042.1654.872.1054.702.1555.152.1655.042.16
12037.564.7419.630.0238.834.5338.644.4339.174.4839.094.5238.784.53
1502.282.032.540.012.322.014.971.922.611.992.712.003.321.99

[61] In addition to correlation and smoothing per degree, correlation and smoothing per order are given in Figures 13 and 14, respectively. The CHAMP-only models refer to the same models that were used in the correlation per degree computations. It can be seen that there is a strong correlation between TUM-2S and AIUB-CHAMP03S, and all the used reference models at the whole spectrum range. EIGEN-CHAMP05S possesses a different behavior with a decay in correlation from order m = 30 and onwards. As far as the correlation by order of the CHAMP-only models with respect to the T/I model is concerned, it can be observed that high correlation occurs at the same bandwidths as in the correlation by degree plots. Finally, the smoothing coefficients by degree and by order confirm the spectral bandwidths in which the corresponding correlations in degree and order were established by the respective computations.

Figure 13.

Correlation coefficients per order between the three CHAMP-only models and various reference models.

Figure 14.

Smoothing coefficients per order between the three CHAMP-only models and various reference models.

[62] Figure 15 shows the representation of TUM-2S and TUM-2S minus EGM2008 truncated at degree l = 60, in terms of geoid heights, gravity anomalies, and second-order radial derivative of the disturbing potential at the Earth's surface and at satellite altitude. Figures 16 and 17 display the representation of the two other CHAMP-only models and their differences to EGM2008 truncated at the corresponding maximum degree of each model, in terms of the same gravitational functionals. A main comment drawn from the comparisons presented in these three plots is that TUM-2S model has a limited resolution ability due to its relatively low maximum degree of expansion (lmax = 60) and EIGEN-CHAMP05S model is problematic over polar regions. Furthermore, the residual fields of the three CHAMP-only models with respect to EGM2008 exhibit large discrepancies over the polar areas, probably due to the already mentioned poor resolution of the near-zonal coefficients for all the three CHAMP-only models.

Figure 15.

Geoid heights, gravity anomalies, second-order radial derivative of the disturbing potential at h = 0, and second-order radial derivative at h = 250km for TUM-2S (left column) and differences with respect to EGM2008 truncated at the maximum degree of TUM-2S model (right column).

Figure 16.

Geoid heights, gravity anomalies, second-order radial derivative of the disturbing potential at h = 0, and second-order radial derivative at h = 250km for AIUB-CHAMP03S (left column) and differences with respect to EGM2008 truncated at the maximum degree of AIUB-CHAMP03S model (right column).

Figure 17.

Geoid heights, gravity anomalies, second-order radial derivative of the disturbing potential at h = 0, and second-order radial derivative at h = 250km for EIGEN-CHAMP05 (left column) and differences with respect to EGM2008 truncated at the maximum degree of EIGEN-CHAMP05S model (right column).

[63] GRACE Gravity Model is a global gravity model provided by CSR, thus applying also the underlying theoretical concept of the celestial mechanics approach, which is determined from 47 months of GRACE K-band intersatellite range-rate data, GPS tracking, and GRACE accelerometry from January 2003 to December 2006 (January 2004 data were excluded). Model EIGEN-GL05S was provided jointly by GeoForschungsZentrum Potsdam and Groupe de Recherches Géodésie Spatiale Toulouse and was computed from GRACE data observed during August 2002 until January 2007, and LAGEOS data was taken from January 2006 to December 2006. ITG-Grace03s is a static field computed from GRACE K-band range-rate measurements observed from September 2002 to April 2007, and ITG-Grace2010s is a static field calculated from GRACE-only data from August 2001 to August 2008. The evaluation of the ITG models is based on the determination of short arcs of the satellite orbit. Thereby, the methodology includes an integrated processing of both the high-precision relative measurements provided by the intersatellite K-band ranging system with the comparatively inaccurate GPS observations [Mayer-Gürr, 2008].

[64] The full spectra of the five GRACE-only models in terms of their coefficients and their standard deviations reveal a near isotropic behavior for of the models, in the whole spectral range, with the errors increasing proportional to increasing degree l. Figure 18 shows the full coefficients and error spectra of model ITG-Grace2010s. When compared with the coefficients of EGM2008, the GRACE-only coefficients produce differences which are almost identical to the corresponding error spectra of the models, thus expressing the fact that the GRACE-only coefficients, which are considered here, are properly calibrated. ITG-Grace03s differences with respect to EGM2008 are quite small in the degree range up to l = 60, probably due to the use of the GRACE-only model in the computation of EGM2008. The other five models present an almost identical behavior with the differences in the zonal coefficients of the low degrees being small and fluctuating up to specific orders.

Figure 18.

2-D spectrum and 2-D error spectrum of ITG-Grace2010s.

[65] Figure 19 shows the signal and error amplitudes per degree of the five GRACE-only gravity field solutions and the difference degree amplitudes between these models and EGM2008. All of the models have almost identical spectral power for the low degrees and resolve fully the gravity field up to their maximum degree, with the GGM03S signal curve in particular showing an abrupt increase from degree l ≈ 150. It also should be noted that GGM03S and EIGEN-GL05S errors are calibrated and present a same spectral behavior through their common spectral range. In addition, one can outline the identical amplitudes in degree differences up to l = 120 for all models except ITG-Grace03s, which is the GRACE model used in the compilation of EGM2008. The lower panel of Figure 19 depicts the error and difference degree variances of the previous CHAMP-only models. It is important to remark that the two groups of satellite-only solutions are of noncomparable accuracy, with the corresponding error variance curves being different to each other to a factor of 0.01 over the whole degree range. This verifies the a priori theoretical knowledge of the accuracy profiles between the high-low (CHAMP) and the low-low (GRACE) Satellite-to-Satellite Tracking principles, according to which the latter performs much better over the whole degree range in terms of error variance curves to a factor of 10−2 [ESA, 1999].

Figure 19.

Square root degree variances, error variances, and difference variances with respect to EGM2008 of different (first row) GRACE-only and (second row) CHAMP-only models.

[66] Figure 20 presents cumulative errors by degree and differences with respect to EGM2008 expressed in terms of radial gradient of the disturbing potential and computed on the Earth's surface and at satellite altitude. It is interesting to see the almost linear trend in the errors curves and that there are differences of at least 1 order of magnitude between the models with calibrated errors and those with formal ones. In addition, we observe the obvious increase in cumulative differences beyond degree l = 50 and the different behavior of model EIGEN-GL05S at satellite altitude with its maximum value becoming larger than that of ITG-Grace2010s. Tables 5 and 6 provide detailed cumulative errors and differences with respect to EGM2008 up to selected degrees in terms of different gravity field functionals.

Figure 20.

Cumulative errors by degree of different GRACE models and cumulative differences with respect to truncated versions of EGM2008 in terms of second-order radial derivative of the disturbing potential on the Earth's surface and at satellite altitude.

Table 5. Cumulative Errors of GRACE Models in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120130140150160180
AIUB GRACE03SN (cm)0.000.000.000.010.010.080.160.571.112.154.208.20-
Δg (mGal)0.0000.0000.0000.0000.0010.0100.0220.0990.2090.4380.9221.923-
Trr (E)0.0000.0000.0000.0000.0000.0010.0030.0180.0420.0950.2140.476-
inline image (mE)0.0000.0000.0010.0050.0100.0520.0850.2150.3370.5220.8041.228-
GGM03SN (cm)0.050.050.050.080.110.571.013.225.7510.2618.3332.74104.48
Δg (mGal)0.0000.0000.0010.0040.0080.0710.1420.5511.0732.0753.9887.62727.553
Trr (E)0.0000.0000.0000.0000.0010.0100.0220.1020.2140.4470.9211.8807.650
inline image (mE)0.0030.0070.0140.0460.0820.3630.5581.2391.7992.5773.6495.1129.780
EIGEN GL05SN (cm)0.040.050.050.070.100.661.214.248.1615.7828.43--
Δg (mGal)0.0000.0000.0010.0030.0080.0820.1710.7311.5353.2126.196--
Trr (E)0.0000.0000.0000.0000.0010.0110.0260.1350.3080.6951.432--
inline image (mE)0.0030.0070.0120.0380.0760.4180.6661.6042.4893.8425.572--
ITG Grace03sN (cm)0.000.000.010.020.030.170.311.071.993.706.8812.7238.56
Δg (mGal)0.0000.0000.0000.0010.0020.0210.0440.1840.3740.7521.5022.97310.112
Trr (E)0.0000.0000.0000.0000.0000.0030.0070.0340.0750.1620.3480.7352.793
inline image (mE)0.0000.0010.0030.0120.0220.1070.1700.4070.6140.9141.3431.9483.685
ITG Grace2010sN (cm)0.000.000.000.010.010.060.120.400.751.412.675.1319.86
Δg (mGal)0.0000.0000.0000.0000.0010.0080.0170.0690.1410.2870.5851.2035.270
Trr (E)0.0000.0000.0000.0000.0000.0010.0030.0130.0280.0620.1360.2981.470
inline image (mE)0.0000.0010.0020.0050.0090.0410.0650.1530.2310.3470.5180.7751.772
Table 6. Cumulative Differences of GRACE Models With Respect to EGM2008 in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120130140150160180
AIUB GRACE03SN (cm)0.420.420.420.420.432.383.666.127.729.5412.4318.56-
Δg (mGal)0.0010.0010.0010.0030.0080.3010.5050.9771.3301.7702.5154.147-
Trr (E)0.0000.0000.0000.0000.0010.0420.0760.1700.2510.3590.5540.998-
inline image (mE)0.0110.0150.0200.0420.0861.4682.0492.7823.0773.3133.5824.021-
GGM03SN (cm)0.080.090.090.100.162.393.696.389.0112.6919.3430.48112.17
Δg (mGal)0.0000.0010.0010.0030.0110.3050.5121.0271.5992.4574.0856.96929.622
Trr (E)0.0000.0000.0000.0000.0010.0420.0770.1800.3090.5140.9261.6988.235
inline image (mE)0.0040.0100.0160.0420.1041.5012.0832.8753.3723.8924.6285.57710.642
EIGEN GL05SN (cm)0.130.130.130.140.182.443.796.9010.2213.9419.44--
Δg (mGal)0.0000.0010.0010.0030.0100.3120.5261.1211.8372.6964.059--
Trr (E)0.0000.0000.0000.0000.0010.0430.0790.1980.3580.5620.911--
inline image (mE)0.0050.0100.0170.0440.0981.5302.1313.0463.6914.2374.826--
ITG Grace03sN (cm)0.070.070.070.070.102.333.636.077.558.9710.9414.3137.62
Δg (mGal)0.0000.0000.0000.0010.0060.2990.5050.9691.2971.6422.1603.0879.772
Trr (E)0.0000.0000.0000.0000.0010.0420.0760.1690.2440.3290.4670.7252.693
inline image (mE)0.0020.0020.0020.0160.0601.4582.0442.7653.0363.2203.3943.6104.650
ITG Grace2010sN (cm)0.090.100.100.110.142.353.635.917.158.099.1210.6222.57
Δg (mGal)0.0000.0010.0010.0030.0080.3010.5040.9391.2161.4471.7262.1605.742
Trr (E)0.0000.0000.0000.0000.0010.0420.0760.1630.2260.2840.3600.4871.569
inline image (mE)0.0040.0100.0160.0390.0831.4672.0442.7232.9483.0653.1503.2313.634

[67] The signal-to-noise ratio and gain computations with respect to EGM2008 both in one and two dimensions have been performed for all five GRACE-only models. AIUB-GRACE03S appears with a qualitative signal along the whole spectral range, unlike EIGEN-GL05S in which the noise dominates the signal in the bandwidth from degree l = 140 and onwards. In the other three models—GGM03S, ITG-Grace03s, and ITG-Grace2010s—the noise equals in magnitude the signal in the last degrees bandwidth. Compared to EGM2008, the GRACE-only models have a gain in accuracy for the long to medium wavelengths. GGM03S and EIGEN-GL05S present an identical trend with almost no gain in the near-zonal coefficients and a small gain in the tesseral coefficients up to degree l = 110. The AIUB-GRACE03S model has an overall gain over EGM2008 except for the degrees beyond l = 140, and ITG GRACE show an isotropic gain structure over EGM2008, which fades gradually from degree l = 140 and onwards. Figure 21 displays significant digits, i.e., the 10-base logarithm of signal-to-noise ratio, and Figure 22 presents the gain in significant digits for all five GRACE-only models. These figures include also the two-dimensional representation of signal-to-noise ratio and gain representative only for ITG-Grace2010s.

Figure 21.

Significant digits for the five GRACE-only models.

Figure 22.

Gain in significant digits for the five GRACE models with respect to EGM2008.

[68] Correlation by degree computations between the GRACE-only models and various reference models are presented in Figure 23. We notice an almost perfect coincidence between the five GRACE-only models in terms of correlations with the different reference models almost up to degree l = 120. The correlation remains high up to the same degree among the different GRACE models and all the other models that are used as reference except for the computed correlation for EIGEN-CHAMP05S, which starts to decay from degree l = 75. The correlation coefficients have a totally different structure, when the topographic/isostatic model is used as reference, an observation made for the CHAMP-only models as well. The different coefficient sets are characterized by a striking degree of correlation especially in the degree bandwidth between l = 10 and l = 140. Table 7 summarizes a selection of detailed numerical overview of the results obtained for the correlation computations.

Figure 23.

Correlation coefficients per degree between the five GRACE models and various reference models.

Table 7. Correlation (in Percentage) and Smoothing per Degree for Three Selected GRACE Models With Respect to Different Reference Fields
  EGM2008T/IEIGEN-GL05CGOCE-spwGOCE-tim (r2)GOCE-dir (r2)GOCO02S
 DegreeCor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.
EIGEN-GL05S2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0071.930.48100.000.00100.000.0099.980.00100.000.00100.000.00
50100.000.0067.810.55100.000.00100.000.0099.910.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
8099.940.0069.820.56100.000.0099.990.0099.860.0099.990.00100.000.00
10099.590.0169.170.5599.930.0099.940.0099.860.0099.950.0099.950.00
12098.460.0367.600.5797.640.0599.350.0199.310.0299.380.0199.410.01
15051.411.2037.180.9447.601.2450.911.1849.901.2149.901.2250.591.20
GGM03S2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0071.930.48100.000.00100.000.0099.980.00100.000.00100.000.00
50100.000.0067.810.55100.000.00100.000.0099.910.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
8099.940.0069.840.56100.000.0099.980.0099.860.0099.990.00100.000.00
10099.600.0169.030.5699.910.0099.940.0099.860.0099.960.0099.970.00
12098.850.0267.400.5797.840.0499.620.0199.580.0199.650.0199.670.01
15074.680.4457.390.7072.290.4877.400.4077.170.4176.820.4177.280.40
16046.750.7830.660.9147.550.7746.770.7847.670.7747.580.7848.270.77
18011.910.999.920.9910.520.9910.430.9912.760.9811.470.9913.100.98
ITG-Grace2010s2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0071.930.48100.000.00100.000.0099.980.00100.000.00100.000.00
50100.000.0067.810.55100.000.00100.000.0099.910.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
8099.940.0069.890.56100.000.0099.990.0099.860.00100.000.00100.000.00
10099.640.0169.160.5699.920.0099.980.0099.890.0099.990.00100.000.00
12099.290.0167.800.5798.240.0399.910.0099.870.0099.960.0099.980.00
15097.030.0667.810.5695.010.1098.450.0398.770.0298.810.0299.050.02
16092.640.1470.480.5389.610.2092.680.1493.830.1293.930.1294.760.10
18061.300.6346.120.8159.490.6562.850.6163.210.6161.660.6264.830.58

[69] In addition to the correlation per degree, correlation per order is computed as well. Figure 24 collects the correlation coefficients per order computations for the GRACE-only models using the same reference models that were used in the correlation per degree computations. It is evident that there is a strong correlation between ITG-Grace2010s and all reference models up to the order m = 150. All the other GRACE-only models show a degradation in correlation starting from order m = 100, with model's GGM03S correlation coefficients with respect to various reference models being scattered from order m = 60, with possible explanations including the characteristic geometry of the GRACE ground track patterns and the truncation of the monthly temporal gravity field solutions at degree and order 60. Furthermore, the correlation by order plots of the GRACE-only models with respect to the T/I model shows that a high correlation occurs at the order bandwidth between m = 10 and m = 100.

Figure 24.

Correlation coefficients per order between the five GRACE models and various reference models.

[70] The correlation bandwidths that were retrieved above can be confirmed by the computations of smoothing coefficients by degree and by order, given in Figures 25 and 26, respectively.

Figure 25.

Smoothing coefficients per degree between the five GRACE models and various reference models.

Figure 26.

Smoothing coefficients per order between the five GRACE models and various reference models.

[71] A rigorous spatial representation for the entire spectrum of the GRACE-only models has been performed of the same group of gravity field functionals as the ones used already. One of the main outcomes refers to the typical error stripe pattern usually present in the GRACE solutions, which is significantly reduced in the representations of EIGEN-GL05S and ITG-Grace2010s models. In addition, a strong noise pattern is observed in the GGM03S gravity anomalies and radial gradient of the disturbing potential on the Earth's surface, which relates to the higher global standard deviations of these functionals compared to other GRACE-only models. This pattern almost disappears when the bandwidth l = 121 to l = 180 is excluded from the computations. Considering the residual fields of the GRACE-only models with respect to EGM2008, truncated at the corresponding maximum degree of the GRACE-only field, one notices that the patterns of large differences over the Amazon basin, Central Africa, and Central Asia coincide with the gaps in terrestrial data used for the compilation of the EGM2008 solution. Furthermore, the difference between EIGEN-GL05S and EGM2008 reveals that there is a problem with the resolution of EIGEN-GL05S in polar regions, while the residual fields between the ITG GRACE models and EGM2008 reflect low signal power in the degree bandwidth l ≈ 150 to l = 180. When this bandwidth is excluded, the noise patterns are reduced significantly. Figure 27 presents representatively for all GRACE models the full spectrum forward computations of geoid heights, gravity anomalies, and second-order radial derivatives of the disturbing potential on the Earth's surface and at 250 km for model ITG-Grace2010s and its difference spectrum with respect to EGM2008.

Figure 27.

Geoid heights, gravity anomalies, second order radial derivative of the disturbing potential at h=0 (3rd row) and at h=250km (4th row) for ITG-Grace2010s (left column) and differences with respect to EGM2008 truncated at the maximum degree of the ITG-Grace2010s model (right column).

6.2 Combined Models

[72] The combined models that are considered here include GGM03C, EIGEN-GL05C, and EIGEN-51C. GGM03C is a gravity model based on the combination of GRACE gravity field information from model GGM03S with land and ocean gravity data. EIGEN-GL05C is produced by the GeoForschungsZentrum Potsdam and is computed by combining GRACE and LAGEOS data from model EIGEN-GL05S with gravimetric and altimetry surface data. EIGEN-51C is a gravity model computed from CHAMP and GRACE observations from the period October 2002 to September 2008 and global gravity anomaly data.

[73] The coefficients spectra of all models are almost isotropic beyond degree l = 100. Up to this degree limit, the error spectra show strong resemblance to the GRACE-only model spectra revealing the degree bandwidth of the combined models where use of GRACE information has been made. Figure 28 presents the full spectra in terms of coefficients and their standard deviations for model EIGEN-51C.

Figure 28.

2-D spectrum and 2-D error spectrum of combined model EIGEN-51C.

[74] The incorporation of satellite-only information in the compilation of these models is apparent from their error spectra. For example, the 2-D error spectrum of EIGEN-GL05C presents the same noise pattern, as the one appearing in GRACE-only model EIGEN-GL05S in the spectral bandwidth between degree and order 73 to degree and order 90, which reveals the use of the specific GRACE-only model information up to degree and order 90 for the construction of EIGEN-GL05C.

[75] Degree variances and error degree variances of the three combined models expressed in terms of the common group of gravity field functionals are displayed in Figure 29. It is clear that in terms of signal power and errors, all models present an almost identical variation along the whole harmonic spectrum. More specifically, GGM03C dominates the bandwidth from degree l = 60 to degree l = 150, while beyond degree l = 150, the EIGEN-GL05C model is the dominant one. Furthermore, considering the degree variance differences of the combined models with respect to EGM2008, the identical behavior for the whole spectral range of the combined models becomes apparent, except for EIGEN-51C for which the differences with respect to EGM2008 present a discontinuity of one order of magnitude at degree l = 110. Moreover, the signal and difference degree amplitudes shown in Figure 29 demonstrate the overall power decay, or equally smoothing effect of the gravity field with increasing altitude from the Earth's surface.

Figure 29.

Square root degree variances, error variances, and difference variances with respect to EGM2008 of different combined models in terms of selected gravity field functionals.

[76] Figure 30 shows the power spectrum and the respective error order variances and difference order variances of the three combined models with respect to EGM2008. It is obvious, as it is the case in the satellite only models as well, that the order-relevant errors have an almost constant value from the very first until the higher orders where a decay of almost 2 orders of magnitude occurs. The same smoothness through the whole order range is present in the difference curves of the models with respect to EGM2008.

Figure 30.

Square root order variances, error variances, and difference variances with respect to EGM2008 of the three considered combined models.

[77] Figure 31 exhibits cumulative errors and differences with respect to EGM2008 in terms of geoid heights and gravity anomalies. The dominant presence of GRACE information up to degree l = 120 is apparent, with the contribution to the overall error being quite small beyond this degree threshold. EIGEN-51C differences with respect to EGM2008 are of 1 order of magnitude smaller than the differences of the other combined models to EGM2008 from in the degree range l = 110 to l = 359. This level of agreement between EIGEN-51C and EGM2008 should be attributed to the fact that EGM2008 information is already embedded in the development of EIGEN-51C. In more detail, EIGEN-51C incorporates the DNSC08GRA global gravity anomaly data set, a global marine gravity field based on radar altimetry data and EGM2008 [Andersen et al., 2010]. Tables 8 and 9 provide a more detailed numerical insight regarding these cumulative computations.

Figure 31.

Cumulative errors by degree of different combined models and cumulative differences with respect to truncated versions of EGM2008 in terms of geoid heights and gravity anomalies.

Table 8. Cumulative Errors of the Three Combined Models in Terms of Different Gravity Field Functionals
Degree (up to) 1050100120150180210250359/360
GGM03CN (cm)0.050.080.822.135.608.3210.1211.7814.42
Δg (mGal)0.0000.0040.1150.3591.1531.9412.5963.3585.112
Trr (E)0.0000.0000.0170.0650.2550.4900.7301.0732.173
inline image (mE)0.0030.0450.4630.8601.4061.5591.5841.5881.588
EIGEN GL05CN (cm)0.040.071.624.548.119.9311.0111.8712.99
Δg (mGal)0.0000.0030.2320.7661.5892.1502.5753.0143.858
Trr (E)0.0000.0000.0360.1390.3370.5120.6780.8901.465
inline image (mE)0.0030.0380.8731.8082.3822.4672.4782.4792.479
EIGEN-51CN (cm)0.010.051.174.117.859.8111.0712.2213.96
Δg (mGal)0.0000.0030.1650.7031.5532.1502.6423.2104.475
Trr (E)0.0000.0000.0250.1290.3320.5170.7070.9751.809
inline image (mE)0.0010.0380.6481.5782.2122.3062.3192.3212.321
Table 9. Cumulative Differences of the Combined Models With Respect to EGM2008 in Terms of Different Gravity Field Functionals
Degree (up to) 1050100120150180210250359/360
GGM03CN (cm)0.090.123.596.1210.4412.8214.2315.4817.72
Δg (mGal)0.0000.0050.4970.9802.0112.7483.3113.9465.663
Trr (E)0.0000.0000.0750.1710.4240.6540.8731.1782.342
inline image (mE)0.0050.0662.0352.7823.3903.4933.5073.5083.509
EIGEN GL05CN (cm)0.130.143.686.9111.8514.2415.6816.8818.97
Δg (mGal)0.0000.0040.5101.1272.2893.0343.6104.2255.858
Trr (E)0.0000.0000.0770.2000.4820.7160.9421.2412.364
inline image (mE)0.0050.0442.0843.0293.7623.8663.8793.8813.881
EIGEN-51CN (cm)0.080.103.635.215.525.786.126.699.53
Δg (mGal)0.0000.0040.5030.8000.8830.9841.1561.5053.629
Trr (E)0.0000.0000.0760.1340.1570.1950.2740.4511.751
inline image (mE)0.0050.0472.0492.5402.5812.5872.5892.5892.589

[78] The computation of order-wise accumulated curves in terms of geoid heights, gravity anomalies, and radial gradient of the disturbing potential at h = 0 and h = 250 km reveals, as it is the case for the satellite only models, the major contribution of the lower orders to the overall error.

[79] The signal-to-noise ratio and gain in significant digits with respect to EGM2008 are shown for the three combined gravity field solutions in terms of one-dimensional representations in Figures 32 and 33, respectively. Two-dimensional plots of signal-to-noise ratio and gain with respect to EGM2008 representatively for EIGEN-51C are also included. The significant digits presented in Figure 32 demonstrate almost identical patterns for the three combined models. Furthermore, it is clear that the signal is dominant over noise for the entire range of the spherical harmonic spectrum while a strong signal-to-noise ratio is present mostly at the long to medium wavelengths and up to degree l = 110. In Figure 33, the gain of the combined models with respect to EGM2008 is displayed. All three models have an identical behavior with gain in the tesseral coefficients of the low degrees, almost no gain in the near-zonal coefficients of the low degrees and no gain with respect to EGM2008 for the bandwidth beyond degree l = 110.

Figure 32.

Significant digits for the three combined models.

Figure 33.

Gain in significant digits for the three combined models with respect to EGM2008.

[80] Correlation coefficients with reference to various models are presented in Figure 34. An almost perfect coincidence can be documented between the two EIGEN models along the whole range of the spherical harmonic spectrum, while the correlation for GGM03C starts to decay beyond the degree l = 300. In general, a strong correlation is apparent up to degree l = 110 of all three combined models against almost all the other models used as reference. From this degree onwards, the correlation loses slightly in magnitude but remains extremely high, almost 70%, for the EGM2008 reference field. A different correlation pattern is observed for the reference field of EIGEN-CHAMP05S, where the correlation decreases rapidly from degree 75 and onwards, even for model EIGEN-51C which contains also CHAMP information. Furthermore, when the reference is model T/I, the coefficient sets are characterized by a high degree of correlation between degrees l = 10 and l = 330. Table 10 provides a more thorough numerical insight to the above correlations.

Figure 34.

Correlation coefficients per degree between the three combined models and various reference models.

Table 10. Correlation (in Percentage) and Smoothing per Degree for Combined Models With Respect to Different Reference Fields
  EGM2008T/IAIUB-GRACE03SGOCE-spwGOCE-tim (r2)GOCE-dir (r2)GOCO02S
 DegreeCor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.
GGM03C2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
10099.660.0169.090.5699.940.0099.920.0099.850.0099.940.0099.950.00
15096.960.0764.460.5987.100.2495.670.0995.950.0896.050.0896.120.08
16096.110.0868.840.5366.180.5694.710.1195.050.1194.720.1295.060.11
21096.060.0975.740.44--39.551.0482.650.3381.690.3782.430.33
24094.570.1274.700.45----41.520.8641.741.3841.230.85
25094.680.1275.550.44----32.110.92--32.050.91
30089.510.2271.730.49----------
36062.730.7450.460.78----------
EIGEN-GL05C2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
10099.630.0168.640.5699.910.0099.900.0099.820.0099.920.0099.920.00
15096.380.0865.240.5887.350.2495.250.1095.630.0995.750.0995.810.09
16096.090.0868.150.5466.430.5694.660.1195.050.1194.820.1195.080.11
21095.980.0974.850.45--41.610.9882.330.3381.370.3782.120.33
24094.160.1372.960.47----44.220.8344.411.3243.810.82
25094.830.1173.970.46----33.530.91--33.490.90
30089.200.2770.260.51----------
36075.172.0260.021.32----------
EIGEN-51C2100.000.00−57.931.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.0067.070.70100.000.00100.000.00100.000.00100.000.00100.000.00
60100.000.0075.070.47100.000.00100.000.0099.860.00100.000.00100.000.00
10099.660.0169.070.5699.980.0099.970.0099.880.0099.980.0099.990.00
15099.890.0068.550.5588.690.2197.570.0597.840.0497.840.0498.010.04
16099.830.0074.070.4868.450.5397.070.0697.500.0597.340.0697.640.05
21099.500.0179.550.41--41.680.9484.430.2983.190.3284.110.29
24098.770.0280.190.39----45.670.8046.151.1745.120.80
25098.560.0380.580.40----35.250.88--35.120.88
30094.330.1177.140.42----------
35973.720.4860.120.64----------

[81] Figure 35 provides the corresponding computations for correlation per order using the same reference fields. All combined models show a decline in correlation per order with respect to the various reference models from the order m = 120. This feature is however not consistent and follows a rather chaotic pattern. Moreover, it is of special interest to observe the almost zero or negative correlation in a specific bandwidth consisting of the last orders, where no dependency on the reference model is visible when one approaches the maximum degree and order of the expansion, a feature primarily connected with the effect of the omission error.

Figure 35.

Correlation coefficients per order between the three combined models and various reference models.

[82] Correlation coefficients by themselves cannot imply an agreement or disagreement of the above models, as they may differ by a scale factor. The computation of the degree of smoothing is a further assessment measure that offers a complementary tool to the conclusions drawn by the correlation computations. Figures 36 and 37 present the computations for the combined and reference models of the smoothing per degree and order, respectively. From the results displayed in these figures, one can deduce easily that the smoothing coefficients affirm the correlation results.

Figure 36.

Smoothing coefficients per degree between the three combined models and various reference models.

Figure 37.

Smoothing coefficients per order between the three combined models and various reference models.

[83] A rigorous spatial computation of all gravity field functionals has been performed for the entire spectrum of all selected combined models and the residual fields with respect to EGM2008. Large discrepancies appear in regions where the terrestrial gravity data are known to be of low accuracy, such as South America, Africa, Central Asia, and Antarctica. That leads to the remark that GRACE information contained in the combined solutions contributes to the improvement of gravity field resolution over these areas. Figure 38 demonstrates that the best agreement to EGM2008 is obtained for the combined model EIGEN-51C, as it was already expected from the previous analysis remarks.

Figure 38.

Geoid heights, gravity anomalies, and second-order radial derivative of the disturbing potential at h = 0 and at h = 250km for EIGEN-51C (left column) and differences with respect to EGM2008 truncated at the maximum degree of EIGEN-51C (right column).

6.3 GOCE Models

[84] With the CHAMP and GRACE-only models being mathematically derived through the high-low and low-low Satellite-to-Satellite-Tracking configurations with the fundamental satellite observations being the satellite positions, the availability of gradiometry data from GOCE enabled a first direct measurement of gravity field functionals, namely, second-order derivatives of the gravitational potential, from satellite orbit. In the period 2010–2011, the ICGEM released seven GOCE-based models that are examined here. The names given to these models abbreviate the computational method used in their evaluation. These models include GOCE-only models as well as GOCE-based models where GOCE derived gravity field information is merged with other available satellite-only or combined models.

[85] GOCE-spw model is computed using the space-wise approach with GOCE data from the period November 2009 to 11 January 2010, using EGM2008 as a priori information for the error calibration in order to enhance the validity of the long wavelengths of the solution. The space-wise approach treats the observables as functions of spatial coordinates, applying to the observation model in the space domain a direct integration technique or a least-squares collocation method to estimate the spherical harmonic coefficients of the gravity field through a standard spherical harmonic analysis of data on a sphere [Rummel et al., 1993]. The signature of the noise spectrum of the accelerometers, which affects the separation of gravitational and nongravitational accelerations derived from the orbit and thus the validity of the estimation results, can be modeled by embedding a Wiener filter into an iterative scheme along the orbit [Migliaccio et al., 2004; Reguzzoni, 2003].

[86] GOCE-dir (r1) model is a gravity field solution computed from GOCE data from November 2009 to 11 January 2010, using the so-called direct approach, which is equivalent to the variational equations or celestial mechanics approach, for the representation of the orbit perturbations part. This approach is based on the least-squares solution of the inverse problem. The partial derivatives of the unknowns, i.e., the spherical harmonic coefficients, are computed and the normal equations are generated based on daily arcs. Then, these daily normal equations are stacked for the entire period, and the resulting normal matrix is inverted using Cholesky decomposition. Due to its definition, the direct approach, also referred to as brute-force method, is linked to extensive demands in terms of CPU speed and available computer memory [Pail et al., 2011b]. EIGEN-GL05C combined model is used as background gravity model, and EIGEN-51C is used as additional information for the regularization applied to overcome the polar gaps. It should be noted that (r1) denotes the release of the model for the specific computation approach, e.g., GOCE-dir (r1) is the first released GOCE-based model, using the direct approach philosophy. GOCE-dir (r2) model is based on GOCE data from November 2009 to June 2010, using ITG-Grace2010s GRACE-only model up to degree and order 150 as a background model and a priori information for a regularization applied to the polar gaps problem.

[87] GOCE-tim (r1) is the first gravity field solution that relies exclusively on GOCE data. It is based on the data obtained in the period from November 2009 to 11 January 2010 and is computed using the time-wise approach. This methodology takes advantage of the incoming time flow of data, treats them as a time series along the orbit, performs a Fourier transform of the observation equations, an analysis step in which the link between the Fourier and the spherical harmonic coefficients is explicitly known, and estimates the spherical harmonic coefficients by exploiting the full normal equation matrix linked to the considered GOCE observations [Pail et al., 2011b]. The time-wise approach permits the use of a definition surface other than that of a sphere, leading to hybrid or semi-analytical approaches, which comprise the characteristics of both the time- and space-wise approaches [Sneeuw, 2003]. No a priori gravity field information is applied to the evaluation of model GOCE-tim (r1), while a Kaula regularization is applied to all zonal and near-zonal coefficients as well to coefficients beyond degree l = 170. The former treats the regularization required to deal with the polar gap problem, which affects the specific coefficients according to the rule of thumb defined by Sneeuw and van Gelderen [1997], while the latter aims to improve the signal-to-noise ratio in the very high degrees [Pail et al., 2011b]. Different types of regularizations are typical in gradiometric analysis. The determination of the gravity field from satellite gradiometry data is an inverse or ill-posed problem requiring regularization. A standard praxis in the computation procedure of the relevant GOCE models is by adding a priori information. A Kaula type of regularization is a common practice in GOCE gradiometric analysis [Bouman and Koop, 1998; Ditmar et al., 2003]. GOCE-tim (r2) is a gravity model based on GOCE data from November 2009 to 5 July 2010. A Kaula regularization is applied to the coefficients of degrees higher than 180, in order to improve the signal-to-noise ratio in this spectral range.

[88] GOCO01S, the first product of GOCO (Gravity Observation COmbination) project, is based on a combination of the normal equations of a GOCE-based model using a data span from November 2009 to 31 December 2009 and the GRACE-only model ITG-Grace2010s. In order to improve the signal-to-noise ratio in the higher degrees, a Kaula regularization to the coefficients with l > 170 is applied. GOCO02S is based on the combination of GOCE data comprising of 8 months Satellite Gravity Gradiometry and 12 months of GOCE high-low satellite-to-satellite tracking data, GRACE-only gravity information in terms of the ITG-Grace2010s model, 8 years of CHAMP data, and various Satellite Laser Ranging data taken from five satellites spanning over 5 years. A Kaula regularization was applied to the coefficients above degree 180.

[89] The two-dimensional signal and error spectra of the GOCE-based models offer a first quick look to their spectral characteristics. GOCE-spw does not perform well in terms of accuracy at the coefficients above degree l = 170, at zonal coefficients as well as coefficients of orders m = 16, 32, and 48, providing a corresponding striped pattern up to degree l = 80. In addition, a small uncertainty in the computation of the low degrees coefficients is noted, probably due to the contributions of EGM2008. GOCE-dir (r1) 2-D error plot shows that sectorial coefficients beyond degree and order 60, and all the coefficients above degree 180 have been computed comparatively with the worst accuracy. A low accuracy for the low degree coefficients up to l = 10 is also demonstrated. Furthermore, in the patterns of the GOCE-dir (r2) error spectrum, one relates to the GRACE information that was used in the model's computation. Sectorial coefficients have been computed with a worse accuracy than in the GOCE-dir (r1) solution, while there is an improvement in the computation of tesseral harmonics from degrees l = 40 to l ~ 150.

[90] GOCE-tim (r1) performs poorly in terms of accuracy at the zonal coefficients, as one should expect, due to the inclination of the GOCE satellite, which provides no coverage over the poles. As for the GOCE-tim (r2) error spectrum, it is almost identical to that of GOCE-tim(r1), with a better recovery of the sectorial harmonics. Finally, the error spectra of the two GOCO models, GOCO01S and GOCO02S, demonstrate the dominant presence of GRACE information until degree l = 170 and l = 180, respectively, with the Kaula regularization beyond these degree thresholds being also visible. Figure 39 displays the coefficients and error spectra of models GOCE-spw, GOCE-dir (r2), and GOCO02S.

Figure 39.

2-D coefficients and error spectra of GOCE-spw, GOCE-dir (r2), and GOCO02S.

[91] Figure 40 displays the power spectrum and the respective error degree variances and difference degree variance with respect to EGM2008 for all considered GOCE-based geopotential models. First of all, the quite different behavior of each model is due to the different computation approaches that have been used for their production. Both GOCE-tim models show an almost identical profile through the whole common spectral range, whereas the GOCE-dir models' performance starts to deviate from each other from degree l = 120 and onwards. It should also be noted that up to degree 120, the GOCE-spw solution seems to be the one with better performance among the five GOCE-only models. It is important to mention that the term GOCE-only refers to the models that use GOCE observations irrespective of the fact whether any other gravity field a priori information was used or not. From all aforementioned models, only GOCE-tim models can be considered as GOCE-only in the strict sense, as they do not use any prior information other than GOCE data. As it is expected, both GOCO models are more accurate up to degree l = 150 demonstrating the dominant GRACE information at this spectral range.

Figure 40.

Square root degree variances, error variances (dashed lines), and difference variances with respect to EGM2008 of different GOCE models.

[92] From the computations of the coefficient differences with respect to EGM2008 included in the same figure, it can be also deduced that the GOCE-only models show an improvement compared to EGM2008 at certain spectral ranges, more specifically l = 2–180 for GOCE-tim (r1), l = 2–200 for GOCE-tim (r2), l = 80–240 for GOCE-dir (r2), and l = 80–180 for GOCE-spw and GOCE-dir (r1) models. Furthermore, both GOCO models show an improvement with respect to EGM2008 over the whole spherical harmonic spectrum.

[93] The power spectrum order variances demonstrate a steep decrease in power of one order of magnitude in the spectral range per order. The corresponding error order variances and difference order variances with respect to EGM2008 demonstrate an almost constant value for the errors with increasing order from the very first up to the higher orders of the expansion.

[94] Degree variances, error variances, and difference variance have been also considered in terms of different gravity field functionals. Figure 41 presents second-order derivatives of the disturbing potential on the Earth's surface and at a GOCE satellite altitude (h = 250 km). Figure 42 includes cumulative errors and differences with respect to EGM2008 in terms of geoid heights for all GOCE-based models. Compared to the previous analysis of the GRACE-only models, it can be deduced that GOCE models perform better from degree l = 120 and onwards. Moreover, the cumulative curves of the GOCO models show the dominant presence of GRACE information up to degree l = 150 and the small contribution of GOCE to the overall error. In order to obtain a more detailed insight the results of Tables 11-14 are included where cumulative errors and differences with respect to EGM2008 are displayed in terms of all gravity field functionals used through the present study.

Figure 41.

Square root degree variances, error variances and difference variances with respect to EGM2008 of different GOCE models in terms of radial gradient of the disturbing potential on the Earth's surface and at a GOCE altitude.

Figure 42.

Cumulative errors by degree of different GOCE models and cumulative differences with respect to EGM2008 in terms of geoid heights.

Table 11. Cumulative Errors of GOCE-Only Models in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120150180210224240250
GOCE-spwN (cm)0.070.190.360.771.011.631.822.263.486.6315.10---
Δg (mGal)0.0010.0050.0130.0460.0710.1650.2030.3050.6401.6004.447---
Trr (E)0.0000.0000.0010.0030.0060.0200.0270.0500.1370.4241.395---
inline image (mE)0.0010.0700.1960.5830.7921.1911.2601.3611.4721.5721.677---
GOCE-dir (r1)N (cm)0.250.320.521.081.311.922.112.523.364.796.647.538.51-
Δg (mGal)0.0020.0050.0170.0620.0880.1850.2250.3240.5701.0471.7392.1022.537-
Trr (E)0.0000.0000.0010.0050.0070.0220.0300.0520.1170.2650.5130.6580.843-
inline image (mE)0.0240.0830.2630.7951.0081.4011.4731.5651.6391.6771.6901.6931.694-
GOCE-dir (r2)N (cm)0.620.771.171.842.132.953.173.544.014.575.847.079.12-
Δg (mGal)0.0040.0120.0370.0950.1320.2700.3190.4150.5690.7991.3701.9102.813-
Trr (E)0.0000.0000.0020.0070.0110.0320.0420.0640.1070.1860.4010.6140.984-
inline image (mE)0.0560.1890.5641.2531.5392.0962.1812.2672.3072.3182.3242.3292.328-
GOCE-tim (r1)N (cm)0.250.591.022.072.633.904.164.645.688.0913.6116.88--
Δg (mGal)0.0030.0130.0350.1200.1820.3770.4310.5550.8851.7343.7565.022--
Trr (E)0.0000.0000.0020.0090.0160.0450.0560.0840.1760.4431.1511.635--
inline image (mE)0.0350.2030.5431.5432.0432.8822.9763.0813.1643.2153.2523.261--
GOCE-tim (r2)N (cm)0.200.470.801.622.083.173.393.784.505.769.2311.990.200.47
Δg (mGal)0.0020.0100.0270.0940.1450.3110.3570.4550.6851.1552.4873.5705.0085.896
Trr (E)0.0000.0000.0010.0070.0130.0370.0460.0690.1330.2860.7611.1741.7562.138
inline image (mE)0.0280.1620.4231.2061.6152.3372.4172.5032.5622.5872.6062.6132.6172.619
Table 12. Cumulative Differences of GOCE Models With Respect to EGM2008 in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120150180210224240250
GOCE-spwN (cm)0.040.170.390.811.022.783.966.279.4712.4618.84---
Δg (mGal)00.0040.0140.0470.070.3330.530.9821.772.7255.111---
Trr (E)0.0000.0000.0010.0030.0060.0450.0790.1700.3670.6691.539---
inline image (mE)0.0060.0640.2230.6050.7941.8342.3392.9963.4173.5223.575---
GOCE-dir (r1)N (cm)0.340.350.370.480.562.493.776.038.099.039.539.699.86-
Δg (mGal)0.0010.0030.0050.020.0320.3110.5170.9511.4581.7751.9912.0752.178-
Trr (E)0.0000.0000.0000.0020.0030.0430.0780.1640.2920.3970.4870.5280.583-
inline image (mE)0.0190.0390.0710.2560.3541.5702.1392.8143.0943.1273.1313.1313.131-
GOCE-dir (r2)N (cm)0.30.310.320.420.52.443.716.039.0111.6615.4517.7921.17-
Δg (mGal)0.0010.0020.0040.0180.0290.3070.510.9561.6832.5323.9734.9516.429-
Trr (E)0.0000.0000.0000.0010.0030.0430.0770.1660.3470.6171.1501.5442.171-
inline image (mE)0.0180.0290.0530.2280.3201.5372.1012.7883.1953.2903.3233.3303.335-
GOCE-tim (r1)N (cm)0.691.392.344.665.767.848.511012.3414.7919.1221.64--
Δg (mGal)0.0060.030.080.270.3930.7250.8721.2471.9492.8544.655.754--
Trr (E)0.0000.0010.0040.0200.0340.0850.1140.1980.3850.6821.3511.794--
inline image (mE)0.0860.4641.2343.4554.4475.8136.0586.3916.6086.6646.6876.692--
GOCE-tim (r2)N (cm)0.751.362.234.475.577.858.5610.0312.1813.9216.418.2620.720.75
Δg (mGal)0.0060.0280.0750.2580.3810.7420.8931.2581.9032.5623.6684.5475.7350.006
Trr (E)0.0000.0010.0040.0190.0320.0880.1170.1990.3720.5921.0231.3871.9012.213
inline image (mE)0.0860.4331.1543.3064.2995.7946.0566.3866.5896.6296.6416.6446.6456.646
Table 13. Cumulative Errors of the Two GOCO Models in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120150180210224240250
GOCO01SN (cm)0.000.000.000.010.010.060.100.311.455.1312.0415.63--
Δg (mGal)0.0000.0000.0000.0000.0010.0070.0140.0520.3121.3223.5704.883--
Trr (E)0.0000.0000.0000.0000.0000.0010.0020.0100.0720.3601.1171.611--
inline image (mE)0.0000.0010.0020.0050.0090.0380.0570.1190.2990.5550.7370.775--
GOCO02SN (cm)0.000.000.000.010.010.050.080.241.043.207.7310.7614.3416.38
Δg (mGal)0.0000.0000.0000.0000.0010.0060.0110.0410.2220.8202.3043.4014.8025.652
Trr (E)0.0000.0000.0000.0000.0000.0010.0020.0080.0510.2230.7241.1341.6962.060
inline image (mE)0.0000.0010.0010.0050.0080.0310.0460.0940.2200.3640.4750.5080.5290.536
Table 14. Cumulative Differences of the Two GOCO Models With Respect to EGM2008 in Terms of Different Gravity Field Functionals
Degree (up to)102030506090100120150180210224240250
GOCO01SN (cm)0.140.140.140.150.172.353.635.918.8711.7216.8519.65--
Δg (mGal)0.0000.0010.0010.0030.0080.3000.5030.9391.6592.5724.4835.612--
Trr (E)0.0000.0000.0000.0000.0010.0420.0760.1630.3430.6331.3271.773  
inline image (mE)0.0050.0100.0160.0390.0831.4662.0422.7213.1253.2263.2723.282  
GOCO02SN (cm)0.100.100.100.110.142.353.625.928.8611.0614.0716.2018.9220.40
Δg (mGal)0.0000.0010.0010.0030.0080.3000.5030.9411.6562.3643.5424.4445.6506.342
Trr (E)0.0000.0000.0000.0000.0010.0420.0760.1630.3410.5681.0121.3791.8932.204
inline image (mE)0.0040.0100.0160.0390.0841.4662.0412.7233.1283.2083.2323.2383.2423.243

[95] The corresponding order-wise accumulated computations outline the major contribution of the lower orders to the overall error in computation of the gravity field functionals. Furthermore, models based on the same computational strategy present an identical cumulative profile along the entire spherical harmonic spectrum per order.

[96] The signal-to-noise ratio and gain computations with respect to EGM2008 have been performed both in one and two dimensions for all GOCE-based GOCE-spw, and both GOCE-dir models seem to have a strong signal along the entire spectral range. On the other hand, in the GOCE-tim and GOCO models, the noise dominates over the signal beyond degree and order 210.

[97] Figure 43 displays the gain of all GOCE-based models with respect to EGM2008. All GOCE-only models present almost identical patterns with gain for the tesseral coefficients from degree l ≈ 70 to degree l ≈ 120–150, while GOCO models have a gain over EGM2008 up to degree l = 150. The above results are also visible in the one-dimensional representation of gain in Figure 44. Gain in significant digits with respect to EGM2008 is observed for the GOCE-only models from degree 60–80 to degree 140–180 and for GOCO models from degree 2 to degree 150–170.

Figure 43.

Gain in significant digits for GOCE models with respect to EGM2008.

Figure 44.

1-D signal-to-noise ratio and gain representations for GOCE models with respect to EGM2008.

[98] Correlation by degree between the GOCE-based models and various reference models are presented in Figure 45. We notice an almost perfect coincidence between all GOCE models almost up to degree l = 160. Furthermore, a high degree of correlation is observed up to degree l = 120 between the GOCE models and the combined solutions EIGEN-51C and EGM2008.

Figure 45.

Correlation coefficients per degree between GOCE models and various reference models.

[99] It should be also noted that a perfect correlation of the GOCE-dir (r1) solution to these reference models up to its maximum degree exists. Concerning the correlation of the GOCE models with respect to the T/I model, this proves to be rather high until degree l = 200, with GOCE-dir (r1) being highly correlated up to degree l = 240. Table 15 augments these observations with more detailed numerical results over specific orders.

Table 15. Correlation (in Percentage) and Smoothing per Degree for GOCE Models With Respect to EGM2008 and T/I
  GOCE-spwGOCE-tim (r1)GOCE-tim (r2)GOCE-dir (r1)GOCE-dir (r2)GOCO01SGOCO02S
 DegreeCor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.Cor.Sm.
EGM20082100.000.00100.000.00100.000.00100.000.00100.000.00100.000.00100.000.00
10100.000.00100.000.00100.000.00100.000.00100.000.00100.000.00100.000.00
30100.000.0099.970.0099.980.00100.000.00100.000.00100.000.00100.000.00
60100.000.0099.810.0099.860.00100.000.00100.000.00100.000.00100.000.00
7099.980.0099.750.0099.790.0099.990.0099.990.0099.990.0099.990.00
8099.930.0099.810.0099.810.0099.940.0099.940.0099.940.0099.940.00
10099.610.0199.550.0199.540.0199.610.0199.640.0199.640.0199.640.01
12099.210.0299.180.0299.200.0299.390.0199.270.0299.290.0199.280.01
15097.700.0597.670.0597.970.0498.950.0297.980.0498.040.0498.150.04
18093.490.1393.740.1396.520.0799.090.0295.800.0894.260.1296.610.07
20074.490.5278.670.5490.650.2099.040.0288.860.2177.990.5590.420.21
21042.011.5867.321.0884.880.3899.220.0283.480.3668.531.0784.630.40
224--42.072.3466.140.9798.770.0259.890.7843.202.3266.081.05
240----44.432.5298.630.0345.111.01--43.922.89
250----35.584.36------35.465.11
T/I2−57.931.00−57.931.00−57.931.00−57.931.00−57.931.00−57.931.00−57.931.00
1067.070.7067.120.7067.090.7067.060.7067.070.7067.070.7067.070.70
3071.880.4871.680.4971.750.4971.930.4871.920.4871.930.4871.930.48
6075.070.4774.700.4774.690.4775.050.4775.090.4775.070.4775.080.47
7069.430.5270.390.5170.210.5169.500.5269.470.5269.460.5269.470.52
8069.890.5670.050.5570.130.5569.810.5669.890.5669.900.5669.900.56
10069.180.5569.100.5669.020.5669.220.5569.170.5669.140.5669.140.56
12067.920.5767.520.5767.710.5767.580.5767.850.5767.870.5767.920.57
15067.530.5667.880.5668.300.5568.480.5568.060.5568.390.5568.600.55
18073.440.5074.650.4877.740.4578.350.4476.320.4774.400.4877.390.46
20062.530.6164.650.5872.420.4977.600.4470.460.5364.140.5972.490.48
21039.140.9959.370.7171.420.4979.900.4169.720.5259.780.7171.280.49
224--32.581.4252.480.8278.420.4147.800.7934.001.4052.640.84
240----35.821.4280.140.3934.850.93--35.561.55
250----30.261.99------30.272.24

[100] In addition, correlation per order is shown in Figure 46. With the exception of correlation to EIGEN-CHAMP05S model that starts decay from order 30, all the other estimated coefficients show strong resemblance to correlation coefficients by degree. It is of interest to observe the behavior of correlation coefficients at the last orders range, with the very last orders exhibiting an almost perfect correlation. The above results are confirmed by the computation of smoothing coefficients by degree and by order shown in Figures 47 and 48, respectively.

Figure 46.

Correlation coefficients per order between GOCE models and various reference models.

Figure 47.

Smoothing coefficients per degree between GOCE models and various reference models.

Figure 48.

Smoothing coefficients per order between GOCE models and various reference models.

[101] A thorough spatial representation of all gravity field functionals used up to this point has been performed for all GOCE-based models using their entire spectrum and the residual fields with respect to EGM2008. Figure 49 displays for all considered models the results obtained for model GOCO02S. First of all, an identical distribution of functional values is observed globally and for all models with almost the same mean value, with the exception of the GOCE-tim models, which produce a higher mean value probably due to the problematic coverage of the polar regions. Furthermore, large discrepancies appear in regions where the terrestrial gravity data are known to be of low accuracy, such as South America, Africa, Central Asia, and Antarctica. That leads to the conclusion that GOCE data contribute to the improvement of gravity field resolution over these areas, even for Antarctica where the data coverage is not optimal. The best agreement to EGM2008 is obtained for the GOCE-dir (r1) model, due to the use of EIGEN-51C as a background model. Finally, large differences are retrieved from the difference between GOCE-tim (r1) and EGM2008 over the region south to Australia. This feature is a known problem which is related with the cross-track gravity gradients observed by GOCE. In more detail, a peculiar pattern for these gradients is visible close to the magnetic poles, which is related to a drift in the differential scale factor of the gradiometer [Frommknecht et al., 2011; Rispens and Bouman, 2011]. These scale factors, and other instrumental correction parameters, are determined roughly every 2 months and applied as corrections to the observations [Frommknecht et al., 2011]. This leakage related to the cross-track gravity gradients indicates that model GOCE-tim (r1) is affected by this anomaly, to such a level that it can be observed from the computed radial gradients as well. Following releases of GOCE-tim models include an improvement due to this effect [Bouman and Fuchs, 2012].

Figure 49.

Geoid heights, gravity anomalies, and second-order radial derivative of the disturbing potential at h = 0 and at h = 250km for GOCO02S (left column) and differences with respect to EGM2008 truncated at the maximum degree of GOCO02S model (right column).

7 CONCLUDING REMARKS

[102] Since the realization of the first dedicated satellite gravity field missions and the release of the respective gravity models, great effort has been put on devising means of identifying known or reproducible features of the Earth's gravity field. The availability over the last years of global digital databases with information regarding the geometry and density distribution of distinct layers of the Earth's crust permits a rigorous modeling of the implied gravity signal and thus enables the numerical investigation of correlations that might exist between these contributions and the respective quantities that may be obtained from limited bandwidths of the new gravity field models. 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 satellite altitude, satellite-only global geopotential models can be recovered only up to a certain resolution. Moreover, they also represent time-wise averaged values of the geopotential. Although assessed for internal accuracy, the actual performance of the derived models in terms of interpreting the static part of the gravity field remains largely unknown. Thus, besides some general remarks of the type “…they capture the medium-to-long wavelength part of the observed gravity field…”, a detailed and justified interpretation of the information content of the new models regarding the static gravity field component is an open challenge in current gravity field research. However, due to the nature of the gravity field regarding the nonuniqueness of the buried sources that define the observed signal, the actual performance of many models depends on the different comprehensive validation and assessment studies that are available. It is the scope of the present survey to offer a detailed documentation of the spectral and spatial performance of some currently representative models, with the aspiration that it could assist ongoing and future validation and calibration work that is done by many groups [e.g., Lemoine et al., 1998; Gruber et al., 2011; Pavlis et al., 2012].

[103] The increasing number of new satellite-only and combined Earth gravity models in terms of sets of potential harmonic coefficients up to a truncated maximum degree and order that are becoming available either from the sole analysis of satellite tracking, accelerometry, or gradiometry data, or from the combination of satellite data with terrestrial information and global digital elevation models for the Earth's topography/bathymetry and crustal structure, sets new challenges in the field of gravity field modeling, interpretation, and analysis. A central challenge is to define means of interpretation, modeling, and analysis which can be applied both independently and combined for the purpose of assessing and interpreting a given gravity model. The spectrum of every model contains contributions from all the underlying sources in the Earth's interior; thus, the task of analysis and interpretation could include, but not be restricted to, band-limited approaches for the characterization and understanding of the distinct spectral bandwidths of the model. The existence of digital databases for the layered representation of the crust down to the crust-mantle boundary permits in this context the forward computation of the individual contributions of these masses to arbitrary points in space, thus also at satellite altitude [Tsoulis et al., 2011].

[104] The investigation of the level of agreement between different gravity models is a task that provides a solid background for interpretation and modeling purposes. It can be carried out both in the spectral domain, based on the information content offered by the coefficients and their errors and in the space domain through standard tools of expressing these coefficients in terms of selected gravity field functionals. As many of the updated versions of the new satellite only or combined models that are released by the same groups 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. The spectral assessment quantities that were considered in the present survey both in terms of the original coefficients and their accuracies and their degree- and order-wise expression to some selected gravity field functional quantify the level of agreement between different CHAMP-only, GRACE-only, combined, and GOCE models, defining the specific bandwidths where the respective correlations take place. This first-level analysis provides a useful interface for the actual scientific exploitation of these models in terms of (1) identifying sources of the observed field in the Earth's interior and (2) linking the satellite-borne gravity field information with the much higher spectral and spatial resolution of a topographic/isostatic model.

[105] As our knowledge of the observed gravity field is defined to a large extent by the available gravity models, it is important to consult the available validation studies as guides that quantify the different models and promote their special features. The present survey attempts to perform this task. The combined Earth Gravity Model EGM2008 and a topographic/isostatic gravity model based on an Airy/Heiskanen isostatic hypothesis were used as reference models in this procedure. The information gained from this analysis can be grouped with respect to each model belonging in the same class of models and with respect to the two reference models.

[106] CHAMP models present a similar pattern of consistency in terms of correlation per degree and order with EGM2008 as the other groups of models. In their spectral range, all CHAMP models show a very good agreement with EGM2008. In this class model, EIGEN-CHAMP05S clearly stands out both as far as its inconsistency with EGM2008 and T/I, a fact that has to be taken into account in conjunction with its increased resolution up to degree 150.

[107] The inclusion of the ITG-Grace03s model in the evaluation of EGM2008 is apparent in the study of the GRACE-only models. However, it is ITG-Grace2010s that shows the most profound agreement amongst GRACE-only models with both EGM2008 and T/I. In general, GRACE models are much more consistent with EGM2008 and T/I than CHAMP models.

[108] The spectral comparisons of the combined models are the least scattered amongst all classes of models. This is not surprising as the combined models include information from both reference models as well as the satellite-only models against which they are compared here.

[109] Finally, GOCE models present the most satisfactory results amongst the satellite-only models if one compares the numerical values and variation pattern of the different assessment quantities with the increased resolution, compared with the other two classes of models. In this class of models, the performance of GOCE-dir(r1) stands out proving an astonishing agreement with EGM2008 and T/I in the whole of its spectral range.

[110] It is clear that elaborate validation procedures with regional or local independent sources of field or model data are necessary for an assessment that could produce outcomes with geophysical relevance, as it is being done in the frame of diverse current validation studies [e.g., Gruber et al., 2011]. However, the present survey demonstrates the effect that intrinsic methodological decisions have on the quality and overall spectral characteristics of the obtained model. It is remarkable how all the different models, despite of their vast differences in terms of measurement principle and methodological evaluation, succeed in recovering the greatest part of the observed signal and the fundamental signatures of the real field, as the spatial representations demonstrated. The clear qualitative and quantitative leap that is offered from the GOCE models proves that the resources invested in new, direct, and more accurate satellite observations are now returning high quality information on the Earth's gravity field. In this respect, efforts on innovative observation techniques both in terms of single principles (e.g., interferometric observations) and combination of different techniques in the frame of new satellite constellations should be encouraged and considered. There is still a great distance to go from the current GOCE models to a satellite-based global gravity field monitoring system with a spatial resolution similar to that of terrestrial Earth Gravity Models.

Acknowledgments

[111] The International Center for Global Earth Models (ICGEM) at the Helmholtz Center, GeoForschungsZentrum Potsdam (GFZ-Potsdam) maintains a freely accessible database of almost all models used in the present survey at http://icgem.gfz-potsdam.de/ICGEM. Financial support through European Space Agency (ESA) contract 22316/09/NL/CBI is highly appreciated. The resp Editor Mark Moldwin and an anonymous reviewer are thanked sincerely for their constructive criticism and useful comments.

Ancillary