Papers by Johannes Bouman
The Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the European Space Agency... more The Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the European Space Agency's (ESA) satellite gravity mission to determine the Earth's mean gravity field. GOCE delivers gravity gradients, a new type of satellite data. We study how these data can improve modeling of the Earth's lithosphere. We discuss the use of the original GOCE gravity gradients versus the use of gravity gradients in grids at satellite altitude or close the Earth's surface and conclude that grids are easier to handle than the original data because one does not have to deal with very different error characteristics of the different gradients, given in a rotating frame at varying heights. The downward continuation to the surface enhances signal and better reflects the near-surface geology. But this does not outweigh the amplification of noise and omission errors, which is why we recommend using the field at mean satellite altitude for lithospheric modeling. The NorthEast Atlantic region is ideal to analyze the additional value of GOCE gravity gradients because it is a well-studied region in terms of regional geophysics. We calculated the gradient sensitivity for crustal depth slices using a 3D lithospheric model. This reveals that especially interfaces with large density contrasts have a distinct signal in the gravity gradients, but that they are quite insensitive to intra-crustal density sources, which can have quite a large effect on surface gravity data. We also show that the satellite gradients have a depth sensitivity well suited to study the upper mantle density structure, making them complementary to gravity and seismic tomography. In the underexplored Rub' al-Khali area the GOCE vertical gradient was used to invert for crustal thickness. The updated Moho model gives a good fit to four of the six gradients and independent depths from seismic stations. The Moho model was used to update the heat flow model and source rock maturity maps, which are generally consistent with known source rock maturity trends in the surrounding regions. GOCE gradients are therefore useful to map crustal thickness and deep regional structures for frontier areas. In combination with other data, heat flow can be modeled which is essential for basin maturity evaluation.
GOCE will be the first satellite ever to measure the second order derivatives of the Earth’s grav... more GOCE will be the first satellite ever to measure the second order derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors and/or outliers have been removed to the extent possible from the data. It is necessary to detect as many outliers as possible in
the data pre-processing because undetected outliers may lead to erroneous results when the data are further processed, for example in the recovery of a gravity field model. Outliers in the
GOCE gravity gradients, as they are likely to occur in the real observations, will be searched for and detected in the processing step preceding gravity field analysis. As the diagonal gravity gradients are the main gradient observables for GOCE, three methods are discussed to detect outliers in these gradients. The first is the tracelessness condition, that is, the sum of the diagonal gradients has to be zero. The second method compares GOCE gravity gradients with model or filtered gradients. Finally, along track interpolation of gravity gradient anomalies is discussed. Since the difference between an interpolated value and a measured value is large when outliers are present, along track interpolation is known to be
suitable for outlier detection. The advantages and disadvantages of each method are discussed and it is shown that the final outlier detection algorithm, which is a combination of the three methods, is able to detect almost all outliers while the number of falsly detected outliers remains small.
The Gravity field and steady state Ocean Circulation Explorer (GOCE) is the European Space Agency... more The Gravity field and steady state Ocean Circulation Explorer (GOCE) is the European Space Agency’s mission that combines GPS tracking and gravity gradiometry to determine the Earth’s mean gravity field with unprecedented, global accuracy with a
spatial resolution down to 80 km. This resolution makes GOCE gravity gradient data in particular useful for lithospheric scale modeling. However, the relation between coordinates in a model frame and at satellite altitude is not straightforward, and most geophysical modeling programs require a planar approximation, which may not be appropriate for satellite data. We derive the exact relation between the model reference
frame, in which gradients from lithospheric modeling are given, and the local north-oriented frame in which GOCE gradients at 255 km altitude are given. We generated gradients from a GOCE gravity field model and assessed whether the orientation
differences between local north-oriented frame and model reference frame are relevant. In addition, we assessed the same for airborne gradiometry at an altitude of 5 km because
these data are complementary to GOCE. We find that if the regional area has a longitude extension of 5° , the errors stay below 10%. For larger areas the standard deviation of the systematic errors may be 40% of the signal standard deviation. Comparing topographic mass reduction in planar and spherical approximation, one sees significant long wavelength differences in terms of gravity gradients or gradient-tensor invariants. The maximum error is up to 1 E at satellite altitude compared with maximum signal amplitude of 3 E. Planar approximation is therefore not accurate enough for topographic
mass reduction of GOCE gravity gradients.
EOS paper together with Reiner Rummel and Rune Floberghagen
The vertical gradients of gravity anomaly and gravity disturbance can be related to horizontal fi... more The vertical gradients of gravity anomaly and gravity disturbance can be related to horizontal first derivatives of deflection of the vertical or second derivatives of geoidal undulations. These are simplified relations of which different variations have found application in satellite altimetry with the implicit assumption that the neglected terms -using remove-restore - are sufficiently small. In this paper, the different simplified relations are rigorously connected and the neglected terms are made explicit. The main neglected terms are a curvilinear term that accounts for the difference between second derivatives in a Cartesian system and on a spherical surface, and a small circle term that stems from the difference between second derivatives on a great and small circle. The neglected terms were compared with the dynamic ocean topography (DOT) and the requirements on the GOCE gravity gradients. In addition, the signal root-mean-square (RMS) of the neglected terms and vertical gravity gradient were compared, and the effect of a remove-restore procedure was studied. These analyses show that both neglected terms have the same order of magnitude as the DOT gradient signal and may be above the GOCE requirements, and should be accounted for when combining altimetry derived and GOCE measured gradients. The signal RMS of both neglected terms is in general small when compared with the signal RMS of the vertical gravity gradient, but they may introduce gradient errors above the spherical approximation error. Remove-restore with gravity field models reduces the errors in the vertical gravity gradient, but it
appears that errors above the spherical approximation error cannot be avoided at individual locations. When computing the vertical gradient of gravity anomaly from satellite altimeter data using deflections of the vertical, the small circle term is readily available and can be included. The direct computation of the vertical gradient of gravity disturbance from satellite altimeter data is more difficult than the computation of the vertical gradient of gravity anomaly because in the former case the curvilinear term is needed, which is not readily available.
The GOCE mission, a part of ESA’s Living Planet Programme, aims at improved gravity field modelli... more The GOCE mission, a part of ESA’s Living Planet Programme, aims at improved gravity field modelling at high spatial resolution. On-board GOCE a gradiometer, in combination with a scientific on-board GPS receiver, measures the earth’s gravity field with unprecedented accuracy. These measurements have been used to compute GOCE gravity field solutions and combined GOCE/GRACE solutions. The main difference between the solutions is how they incorporate the required a priori information, which consists either of existing gravity field
models or Kaula’s rule for the signal variances of the gravity field. We assessed four series of models by comparing the gravity gradients they predict with the measured GOCE gradients. The analysis of the gravity gradients fits may reveal differences between the different solutions that can be attributed to the solution strategy, assuming that the measurement errors are homogeneous. We compared the GOCE gradients with existing state-of-the-art global gravity
field models and conclude that the gradient errors are indeed globally homogeneous, with the exception of the cross-track gradient especially south of Australia. Furthermore, we find
that the use of existing global gravity field models as a priori information should be avoided because this may increase the gradient residuals in spatial and spectral domain. Finally, we
may conclude that the GOCE and GRACE data are compatible and complementary.
International Association of Geodesy Symposia, 2001
ABSTRACT The expected high resolution and precision of a global gravity field model derived from ... more ABSTRACT The expected high resolution and precision of a global gravity field model derived from satellite gradiometric observations is unprecedented compared to nowadays satellite-only models. However, a dedicated gravity field mission will most certainly fly in a non-polar (sun-synchronous) orbit, such that small polar regions will not be covered with observations. The resulting inhomogeneous global data coverage, together with the downward continuation problem, leads to unstable global solutions and regularization is necessary. Regularization gives rise to a bias in the solution, mainly in the polar areas although in other regions as well. Undoubtedly, the combination with gravity related measurements in the polar areas, like airborne gravimetry, will improve the stability of the solution. Consequently, the bias is reduced and the quality is likely to be better. Open questions are, for example, how accurate gravity anomalies must be, what spatial sampling is required, and how large the area with observations should be. Moreover, it is unknown whether measurements in one polar area only (e.g. North Pole) is sufficient. In order to answer these questions, a gravity field solution from gradiometry-only will be compared with a solution from gradiometry combined with several airborne gravimetric scenarios. Special attention is given to the quality improvement and bias reduction relative to the gradiometry-only solution. The coefficients of a spherical harmonic series are the unknowns and their errors are propagated to, for example, geoid heights.
International Association of Geodesy Symposia, 2005
GOCE will be the first satellite ever to measure the second derivatives of the Earth's gravitatio... more GOCE will be the first satellite ever to measure the second derivatives of the Earth's gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment. ¨ © !
GOCE will be the first ever satellite mission measuring gravity gradients using a dedicated instr... more GOCE will be the first ever satellite mission measuring gravity gradients using a dedicated instrument called a gradiometer. High resolution gravity field recovery will be possible from these gradients under the condition that systematic errors have been removed to the extent possible from the data and the gravity gradients accuracy has been assessed. High resolution and high accuracy terrestrial gravity data could possibly be used for these purposes. As an example, we will show how Canadian gravity data can be used for the external calibration of the gravity gradients, that is, correcting the gradients for systematic effects, and how the gravity gradient error can be assessed with these terrestrial data. Our focus will be on the long wavelength biases present in the terrestrial data, the differences in information content between gravity data at the Earth's surface and gravity gradients at satellite altitude, and how these problems can possibly be solved by filtering both data ...
Journal of Geodesy, 2005
The satellite missions CHAMP, GRACE, and GOCE mark the beginning of a new era in gravity field de... more The satellite missions CHAMP, GRACE, and GOCE mark the beginning of a new era in gravity field determination and modeling. They provide unique models of the global stationary gravity field and its variation in time. Due to inevitable measurement errors, sophisticated pre-processing steps have to be applied before further use of the satellite measurements. In the framework of the GOCE mission, this includes outlier detection, absolute calibration and validation of the SGG (satellite gravity gradiometry) measurements, and removal of temporal effects. In general, outliers are defined as observations that appear to be inconsistent with the remainder of the data set. One goal is to evaluate the effect of additive, innovative and bulk outliers on the estimates of the spherical harmonic coefficients. It can be shown that even a small number of undetected outliers (< 0:2% of all data points) can have an adverse effect on the coefficient estimates. Consequently, concepts for the identification and removal of outliers have to be developed. Novel outlier detection algorithms are derived and statistical methods are presented that may be used for this purpose. The methods aim at high outlier identification rates as well as small failure rates. A combined algorithm, based on wavelets and a statistical method, shows best performance with an identification rate of about 99%. To further reduce the influence of undetected outliers, an outlier detection algorithm is implemented inside the gravity field solver (the Quick-Look Gravity Field Analysis tool was used). This results in spherical harmonic coefficient estimates that are of similar quality to those obtained without outliers in the input data.
Journal of Geodesy, 2011
GOCE is ESA's gravity field mission and the first satellite ever that measures gravitational grad... more GOCE is ESA's gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth's gravitational potential. The goal is to determine the Earth's mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V X X , V Y Y , V Z Z and V X Z are much more accurate than V XY and V Y Z , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10 −3 level. Furthermore, we found that the error of V X X and V Y Y is approximately at the level of the requirement on the gravitational gradient trace, whereas the V Z Z error is a factor of 2-3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2-35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.
Journal of Geodesy, 2009
One of the products derived from the gravity field and steady-state ocean circulation explorer (G... more One of the products derived from the gravity field and steady-state ocean circulation explorer (GOCE) observations are the gravity gradients. These gravity gradients are provided in the gradiometer reference frame (GRF) and are calibrated in-flight using satellite shaking and star sensor data. To use these gravity gradients for application in Earth scienes and gravity field analysis, additional preprocessing needs to be done, including corrections for temporal gravity field signals to isolate the static gravity field part, screening for outliers, calibration by comparison with existing external gravity field information and error assessment. The temporal gravity gradient corrections consist of tidal and nontidal corrections. These are all generally below the gravity gradient error level, which is predicted to show a 1/ f behaviour
Journal of Geodesy, 2004
It is the aim of the GOCE mission to determine a model of the Earth's gravity field with high acc... more It is the aim of the GOCE mission to determine a model of the Earth's gravity field with high accuracy and resolution. For this purpose, gravity gradients will be measured in combination with highlow satellite-to-satellite tracking. The gravity gradients are derived from pair-wise differenced accelerations as determined by the six three-axes accelerometers that form the GOCE gradiometer. Since the measured accelerations suffer from errors of a random and systematic nature, the gravity gradients may suffer from random and systematic errors as well. Systematic errors are, for example, a scale factor and a bias. The common accelerations of the paired accelerometers also are contaminated with such errors. The common accelerations are used in the drag-free control of the satellite and are important for the separation of the gravitational and non-gravitational forces in the gravity field determination. The checking of the gravity gradients and the common accelerations against independent data (i.e. external to the GOCE satellite) in order to free the observations as well as is possible from systematic errors is called external calibration. The possibilities and limitations of using terrestrial gravity data and global gravity models for external calibration of the gravity gradients are reviewed. It turns out that the determination of a gravity gradient scale factor and bias using just the accurate knowledge of the central term and the flattening (J 2 ) of the Earth's gravity field is not good enough. When global gravity field models are used for the calibration, higher degrees and orders should be taken into account as well. With today's existing global models it seems to be possible to remove the greater part of the systematic errors of the GOCE gradients. A gravity gradient bias can accurately be recovered using terrestrial gravity data in a regional approach with least squares collocation. However, since regional data are used it may not be possible to determine calibration parameters valid for the whole (global) gravity gradient data set. Nevertheless, regional terrestrial gravity data could be used to validate the measured and calibrated gravity gradients. In addition, a possible use of GOCE high-low satellite-to-satellite tracking data to calibrate the common accelerations is explored; it is shown that this approach fails. If more accurate gravity field information becomes available then such a calibration may become feasible.
Geophysical Journal International, 2011
ESA's GOCE mission aims at improved global and regional gravity field information with high spati... more ESA's GOCE mission aims at improved global and regional gravity field information with high spatial resolution by measuring gravity gradients. A local analysis of the GOCE gravity gradient tensor may benefit from a rotation from the gradiometer reference frame to local reference frames such as the local north oriented reference frame. As the GOCE gravity gradients include accurate and less accurate measured gradients, the point-wise tensor rotation of the GOCE-only measurements may suffer from the projection of the errors of the less accurate gravity gradients onto the accurate gravity gradients. In addition, the GOCE gravity gradients have high accuracy in the measurement bandwidth but low accuracy below, and tensor rotation may cause leakage of the large error below the measurement bandwidth to the measurement bandwidth. Degradation of the rotated gravity gradients is circumvented by replacing the less accurate tensor components, as well as the signal below the measurement bandwidth of the accurate gravity gradients, with model signal. The combination of GOCE and model gravity gradients is performed by determining the effective measurement bandwidth (EMB), that is, the bandwidth in which the integrated signal-to-noise ratio of the GOCE gravity gradients is maximized. We find that the determination of the EMB is relatively independent of the reference gravity field model that is used. The lower bound of the EMB is well below the pre-mission specifications for the four accurate gravity gradients. In addition, we assess how much GOCE contributes to the gravity gradient signal in local frames and how much the model. For the radial gravity gradient the relative GOCE contribution is 98 per cent on average, whereas this is 65-97 per cent for the other gravity gradients. These numbers strongly depend on the local frame under consideration and on the geographical position.
earth.esa.int
The availability of gravity data from recent satellite missions (CHAMP, GRACE) and the future GOC... more The availability of gravity data from recent satellite missions (CHAMP, GRACE) and the future GOCE mission will improve our current knowledge of the Earth's geoid. An accurate description of the geoid is essential for a proper interpretation of altimetric sea level data, which can help us to better understand the ocean circulation. This study evaluates the sensitivity of an estimated ocean circulation to the definition of the geoid, using a global ocean circulation model (OGCM) and an advanced data assimilation technique.
Advances in Space Research, 2008
Advances in Geosciences, 2003
GOCE will be the first satellite gravity mission measuring gravity gradients in space using a ded... more GOCE will be the first satellite gravity mission measuring gravity gradients in space using a dedicated instrument called a gradiometer. High resolution gravity field recovery will be possible from these gradients. Such a recovery requires a proper description of the gravity gradient errors, where the a priori error model is for example based on end-to-end instrument simulations. One way to test the error model against real data, i.e. to see if the a priori model really describes the actual error, is to compare along track interpolated gradients with the measured gradients. The difference between the interpolated and measured gravity gradients is caused by, among others, the interpolation error and the measurement errors. The idea is that if the interpolation error is small enough, then the differences should be predicted reasonably well by the error model. This paper discusses a simulation study where the gravity gradient errors are generated with an end-to-end instrument simulator. The measurement error will be compared with the interpolation error and we will assess the latter as a function of the sampling interval.
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Papers by Johannes Bouman
the data pre-processing because undetected outliers may lead to erroneous results when the data are further processed, for example in the recovery of a gravity field model. Outliers in the
GOCE gravity gradients, as they are likely to occur in the real observations, will be searched for and detected in the processing step preceding gravity field analysis. As the diagonal gravity gradients are the main gradient observables for GOCE, three methods are discussed to detect outliers in these gradients. The first is the tracelessness condition, that is, the sum of the diagonal gradients has to be zero. The second method compares GOCE gravity gradients with model or filtered gradients. Finally, along track interpolation of gravity gradient anomalies is discussed. Since the difference between an interpolated value and a measured value is large when outliers are present, along track interpolation is known to be
suitable for outlier detection. The advantages and disadvantages of each method are discussed and it is shown that the final outlier detection algorithm, which is a combination of the three methods, is able to detect almost all outliers while the number of falsly detected outliers remains small.
spatial resolution down to 80 km. This resolution makes GOCE gravity gradient data in particular useful for lithospheric scale modeling. However, the relation between coordinates in a model frame and at satellite altitude is not straightforward, and most geophysical modeling programs require a planar approximation, which may not be appropriate for satellite data. We derive the exact relation between the model reference
frame, in which gradients from lithospheric modeling are given, and the local north-oriented frame in which GOCE gradients at 255 km altitude are given. We generated gradients from a GOCE gravity field model and assessed whether the orientation
differences between local north-oriented frame and model reference frame are relevant. In addition, we assessed the same for airborne gradiometry at an altitude of 5 km because
these data are complementary to GOCE. We find that if the regional area has a longitude extension of 5° , the errors stay below 10%. For larger areas the standard deviation of the systematic errors may be 40% of the signal standard deviation. Comparing topographic mass reduction in planar and spherical approximation, one sees significant long wavelength differences in terms of gravity gradients or gradient-tensor invariants. The maximum error is up to 1 E at satellite altitude compared with maximum signal amplitude of 3 E. Planar approximation is therefore not accurate enough for topographic
mass reduction of GOCE gravity gradients.
appears that errors above the spherical approximation error cannot be avoided at individual locations. When computing the vertical gradient of gravity anomaly from satellite altimeter data using deflections of the vertical, the small circle term is readily available and can be included. The direct computation of the vertical gradient of gravity disturbance from satellite altimeter data is more difficult than the computation of the vertical gradient of gravity anomaly because in the former case the curvilinear term is needed, which is not readily available.
models or Kaula’s rule for the signal variances of the gravity field. We assessed four series of models by comparing the gravity gradients they predict with the measured GOCE gradients. The analysis of the gravity gradients fits may reveal differences between the different solutions that can be attributed to the solution strategy, assuming that the measurement errors are homogeneous. We compared the GOCE gradients with existing state-of-the-art global gravity
field models and conclude that the gradient errors are indeed globally homogeneous, with the exception of the cross-track gradient especially south of Australia. Furthermore, we find
that the use of existing global gravity field models as a priori information should be avoided because this may increase the gradient residuals in spatial and spectral domain. Finally, we
may conclude that the GOCE and GRACE data are compatible and complementary.
the data pre-processing because undetected outliers may lead to erroneous results when the data are further processed, for example in the recovery of a gravity field model. Outliers in the
GOCE gravity gradients, as they are likely to occur in the real observations, will be searched for and detected in the processing step preceding gravity field analysis. As the diagonal gravity gradients are the main gradient observables for GOCE, three methods are discussed to detect outliers in these gradients. The first is the tracelessness condition, that is, the sum of the diagonal gradients has to be zero. The second method compares GOCE gravity gradients with model or filtered gradients. Finally, along track interpolation of gravity gradient anomalies is discussed. Since the difference between an interpolated value and a measured value is large when outliers are present, along track interpolation is known to be
suitable for outlier detection. The advantages and disadvantages of each method are discussed and it is shown that the final outlier detection algorithm, which is a combination of the three methods, is able to detect almost all outliers while the number of falsly detected outliers remains small.
spatial resolution down to 80 km. This resolution makes GOCE gravity gradient data in particular useful for lithospheric scale modeling. However, the relation between coordinates in a model frame and at satellite altitude is not straightforward, and most geophysical modeling programs require a planar approximation, which may not be appropriate for satellite data. We derive the exact relation between the model reference
frame, in which gradients from lithospheric modeling are given, and the local north-oriented frame in which GOCE gradients at 255 km altitude are given. We generated gradients from a GOCE gravity field model and assessed whether the orientation
differences between local north-oriented frame and model reference frame are relevant. In addition, we assessed the same for airborne gradiometry at an altitude of 5 km because
these data are complementary to GOCE. We find that if the regional area has a longitude extension of 5° , the errors stay below 10%. For larger areas the standard deviation of the systematic errors may be 40% of the signal standard deviation. Comparing topographic mass reduction in planar and spherical approximation, one sees significant long wavelength differences in terms of gravity gradients or gradient-tensor invariants. The maximum error is up to 1 E at satellite altitude compared with maximum signal amplitude of 3 E. Planar approximation is therefore not accurate enough for topographic
mass reduction of GOCE gravity gradients.
appears that errors above the spherical approximation error cannot be avoided at individual locations. When computing the vertical gradient of gravity anomaly from satellite altimeter data using deflections of the vertical, the small circle term is readily available and can be included. The direct computation of the vertical gradient of gravity disturbance from satellite altimeter data is more difficult than the computation of the vertical gradient of gravity anomaly because in the former case the curvilinear term is needed, which is not readily available.
models or Kaula’s rule for the signal variances of the gravity field. We assessed four series of models by comparing the gravity gradients they predict with the measured GOCE gradients. The analysis of the gravity gradients fits may reveal differences between the different solutions that can be attributed to the solution strategy, assuming that the measurement errors are homogeneous. We compared the GOCE gradients with existing state-of-the-art global gravity
field models and conclude that the gradient errors are indeed globally homogeneous, with the exception of the cross-track gradient especially south of Australia. Furthermore, we find
that the use of existing global gravity field models as a priori information should be avoided because this may increase the gradient residuals in spatial and spectral domain. Finally, we
may conclude that the GOCE and GRACE data are compatible and complementary.