Rotation of GOCE gravity gradients to local frames

Geophys. J. Int. (2011) 187, 743–753 doi: 10.1111/j.1365-246X.2011.05162.x

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Rotation of GOCE gravity gradients to local frames

Martin J. Fuchs and Johannes BoumanDeutsches Geodatisches Forschungsinstitut (DGFI), Alfons-Goppel-Str. 11, 80539 Munich, Germany.E-mail: [email protected]

Accepted 2011 July 18. Received 2011 July 18; in original form 2011 February 18

S U M M A R YESA’s GOCE mission aims at improved global and regional gravity field information withhigh spatial resolution by measuring gravity gradients. A local analysis of the GOCE gravitygradient tensor may benefit from a rotation from the gradiometer reference frame to localreference frames such as the local north oriented reference frame. As the GOCE gravitygradients include accurate and less accurate measured gradients, the point-wise tensor rotationof the GOCE-only measurements may suffer from the projection of the errors of the lessaccurate gravity gradients onto the accurate gravity gradients. In addition, the GOCE gravitygradients have high accuracy in the measurement bandwidth but low accuracy below, andtensor rotation may cause leakage of the large error below the measurement bandwidth tothe measurement bandwidth. Degradation of the rotated gravity gradients is circumvented byreplacing the less accurate tensor components, as well as the signal below the measurementbandwidth of the accurate gravity gradients, with model signal. The combination of GOCE andmodel 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 gravitygradients is maximized. We find that the determination of the EMB is relatively independentof the reference gravity field model that is used. The lower bound of the EMB is well belowthe pre-mission specifications for the four accurate gravity gradients. In addition, we assesshow much GOCE contributes to the gravity gradient signal in local frames and how muchthe model. For the radial gravity gradient the relative GOCE contribution is 98 per cent onaverage, whereas this is 65–97 per cent for the other gravity gradients. These numbers stronglydepend on the local frame under consideration and on the geographical position.

Key words: Time-series analysis; Spatial analysis; Satellite geodesy; Space geodetic surveys.

1 I N T RO D U C T I O N

The objective of ESA’s GOCE mission is to determine the Earth’smean gravity field with an accuracy of 1–2 cm in terms of geoidheight and 1 mGal in terms of gravity anomalies at 100 km res-olution (ESA 1999). GOCE carries an Electrostatic Gravity Gra-diometer and a GPS receiver to achieve this mission goal. Thegradient data are used to derive the short to medium wavelengthgravity field information, whereas the GPS tracking data allow ac-curate determination of the GOCE orbit and are used to derivethe medium to long wavelength gravity field information. The gra-diometer consists of six highly precise three axes accelerometers.The acceleration differences combined with information from thestar sensors (STRs) allow separating the gravity gradients from an-gular rates and angular accelerations once the measurements arecalibrated (Cesare 2008; Cesare & Catastini 2008). The Level 1bgravity gradients require further scientific pre-processing such asthe application of temporal corrections, data screening and external

calibration (Bouman et al. 2009). These Level 2 gradients are usedin gravity field analysis, (Pail et al. 2011) and may directly be ap-plied, for example, (Pedersen & Rasmussen 1990; Mikhailov et al.2007).

A difficulty in the direct application of the GOCE gradients maybe that they are given in the Gradiometer Reference Frame (GRF),which is an instrumental frame approximately Earth pointing, butwith continuous rotations around all three axes. Hence the GRFis only indirectly related to the Earth (details are given in Section2). It may therefore be advantageous to rotate the gravity gradienttensor to a geographical reference frame such as the Local NorthOriented Frame (LNOF) also called the North-West-Up frame. Acomplicating factor is thereby that four of the GOCE gradients inthe GRF have high accuracy (V XX, V YY, V ZZ and V XZ) whereasthe other gradients (V XY and V YZ) are less accurately determined.A direct rotation of the gradient tensor from GRF to other localframes would project part of the larger V XY and V YZ errors ontothe other gradients in the rotated frame (Muller 2003). In addition,

C© 2011 The Authors 743Geophysical Journal International C© 2011 RAS

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744 M. J. Fuchs and J. Bouman

the gradients are most accurate in the Measurement Bandwidth(MB) between 0.5 mHz and 0.1 Hz, while the errors increase forlow frequencies. This low frequency error may leak to the MB dueto tensor rotation (Bouman 2007).

Bouman (2007) therefore studied methods to circumvent thegravity gradient degradation caused by projection and leakage. Onemethod is to predict by Least-Squares Collocation (LSC) all sixgradients in the rotated frame (Tscherning 1993) using the fouraccurate gravity gradients in the GRF. Because of the high compu-tational costs and because the signal covariance function needs to beregionally scaled, spatial blocks of typically 6◦ × 6◦ are used for theLSC prediction. As the scaling of the signal covariance may differbetween neighbouring blocks, the rotated gradients may be dis-continuous at block boundaries (Bouman 2007). Another methodtherefore is to substitute the less accurate GOCE gradients withmore accurate model gradients and to perform a direct, point-wisetensor rotation. The computational costs are modest and a gradientdata set of for example, one month can be uniformly rotated. Oneof the disadvantages of the point-wise method may be that modeland measured components are mixed and the result may be biasedtowards the model input. The signal below the MB for all gradientsis replaced by model signal, for example from a GRACE-basedgravity field model, which is expected to be more accurate at thesewavelengths, see for example, (ESA 1999).

In this paper, we address two questions concerning the GOCEgradient point-wise tensor rotation. First, the replacement of theGOCE gradient signal below the MB is to a certain extent arbitrary.In the MB GOCE is expected to be more accurate than state-of-the-art gravity field information, whereas the latter is expected to bemore accurate than the GOCE information at longer wavelengths.The GOCE gradients are combined with model gradients by band-pass filtering the former and by low-pass filtering the latter with thecomplement of the bandpass filter. The question is therefore howto choose the filter cut-on and cut-off frequencies. We present dif-ferent methods to do so in an optimal way. Secondly, the gradientsin the rotated local frame will be a combination of original GOCEgradients and model gradients. It is therefore useful to be able toassess how much gravity field information in the rotated gradientsstems from GOCE and how much from the model. We present amethod to estimate the model content, illustrated with real GOCEdata for different relevant local reference frames.

The paper is organized as follows. Section 2 discusses the localreference frames that play a role in the GOCE gradient tensor ro-tation. Our algorithm for the tensor rotation is presented in Section3. Different methods to choose the optimal cut-on frequency arediscussed in Section 4 and the assessment of the model content inthe rotated gradients is the topic of Section 5. Finally, Section 6contains the conclusions.

2 L O C A L R E F E R E N C E F R A M E S

The GOCE gravity gradients are given in the GRF, which is aninstrument related frame. Since many geophysical applications usegeographically oriented frames, it is convenient to rotate the gravitygradients from the GRF to other local frames. Three nominal localreference frames are being used for the rotation: the Local OrbitalReference Frame (LORF) in two different fashions and the LNOF.The definitions of these reference frames are given below, see also(Gruber et al. 2010; Rummel et al. 2011).

The LORF is a reference frame that corotates with the satellite.Two slightly different definitions of the LORF exist, see Fig. 1.

Figure 1. LORF-v and LORF-r.

In the first definition, the X -axis is parallel to the orbital velocityvector, in the second definition the Z-axis is in parallel with theradial direction. In both cases, the origin of the LORF is locatedat the satellite centre of mass (CoM). Note that the origin of theGRF does not coincide with the satellite CoM—the offset along theX -axis is about 2 cm (negligible small for this analysis).

The LORF-v is oriented with respect to the satellite velocityvector:

(1) The X -axis is parallel to the instantaneous direction of theorbital velocity vector (v) with the same sign as this vector.

(2) The Y -axis is parallel to the instantaneous direction of theorbital angular momentum (n), with the same sign as n. The vectorsv and n are orthogonal by definition, since n = r × v, where r is thevector from the Earth centre to the satellite CoM.

(3) The Z-axis is parallel to v × n with the same sign as v × nand complements the right-handed frame. The Z-axis points almostradial outward, as the GOCE orbit is near circular.

The LORF-r is oriented with respect to the orbital radial vector:

(1) The Z-axis is parallel to the instantaneous direction of theradial vector r from the geocentre to the satellite CoM with thesame sign as this vector.

(2) The LORF-r and LORF-v Y -axes are equal.(3) The X -axis complements the right-handed frame with X =

Z × Y . The X -axis is almost parallel to the instantaneous directionof the orbital velocity vector, as the GOCE orbit is near circular.

The LNOF is a right-handed North-West-Up frame with the X -axis pointing North, the Y -axis pointing West and the Z-axis Up:

(1) The Z-axis is defined as the vector from the geocentre to theorigin pointing radial outward (equal to Z-axis LORF-r).

(2) The Y -axis is parallel to the normal vector to the plane of thegeocentric meridian of the satellite CoM, pointing westward.

(3) The X -axis is parallel to the normal vector to the plane definedby the Y - and Z-axes.

The GRF is kept as good as possible parallel to LORF-v bykeeping the nose of the satellite as close as possible to the instan-taneous velocity direction using the magnetic torquers. GOCE hasbeen launched in the summer configuration with the Z-axis point-ing towards the Earth. We therefore have approximately X LORF =X GRF, Y LORF = −Y GRF, and ZLORF = −ZGRF. Fig. 2 displays the yaw,roll and pitch angles between the GRF and LORF-v and LNOF forascending tracks with a binning of 1◦ × 1◦. The time-period for

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Geophysical Journal International C© 2011 RAS

Rotation of GOCE gradients to local frames 745

Figure 2. Yaw, roll and pitch angles (from top to bottom) for ascending tracks to rotate from GRF to LORF-v (left-hand side) or LNOF (right-hand side).Angles are in degrees. In the column on the left the diagonal gradients affected by the rotation are indicated.

this analysis covers GOCE data from 2009 October 31 until 2010January 11. The maximum variations for yaw (Z), roll (X) and pitch(Y) for GRF to LORF-v are 6.7◦, 2.2◦ and 2.0◦, respectively andfor GRF to LNOF 84.7◦, 2.1◦ and 0.6◦, respectively. The maximumvariations for the GRF to LORF-r rotations are almost identical tothe GRF to LORF-v variations. The roll and pitch angles for LORF-r and LNOF are about equal because both systems are aligned withthe radial vector (coming from the geocentre). The subsequent inplane rotation around the yaw axis then differs as the LORF-r isapproximately aligned with the satellite’s velocity vector and theLNOF with the geographical meridian. The variations for the roll,pitch and yaw angles between GRF and LORF-v differ, because themoments of inertia around the three axes differ (almost equal forY and Z, much smaller for X) and because the attitude control bythe magnetic torquers depends on the magnetic field. In particular,there is no or little control around the Z-axis (yaw) close to themagnetic poles where the magnetic field lines are vertical and no orlittle control around the X -axis (roll) close to the magnetic equatorwhere the magnetic field lines are horizontal and roughly in flightdirection.

In the so-called GOCE space-wise gravity field solution the ob-servation equations are set up in the LORF-v. To overcome thedifferences between the LORF-v and the GRF an iterative schemeis applied in which an intermediate space-wise solution is used to re-duce the observations. Numerical tests have shown that the iterativescheme converges (Migliaccio et al. 2004a,b).

3 P O I N T - W I S E RO TAT I O N

The rotation of the gravitational tensor from one Cartesian frame toanother is given by

Vab = RVi j RT (1)

with Vij and Vab the tensor before and after rotation, and R therotation matrix that connects the two reference frames. The gravitygradients in the rotated frame are therefore a linear combinationof the gradients in the original frame. Because four of the GOCEgradients in the GRF have high accuracy (V XX, V YY, V ZZ and V XZ)and two (V XY and V YZ) are less accurately determined, a directrotation of the gradient tensor from GRF to other local frames wouldproject part of the larger V XY and V YZ errors onto the other gradientsin the rotated frame (Muller 2003). In addition, the gradients aremost accurate in the MB between 5 mHz and 0.1 Hz, while theerrors increase as 1/f for low frequencies. This low frequency errormay leak to the MB due to tensor rotation (Bouman 2007).

Model values replace the less accurate gradients as well as thesignal below the MB of the accurate gradients, to circumvent thegradient contamination. Two gradient tensor matrices can be set up,one with model values Uij and one with GOCE measurements Vij :

V Gi j =

⎛⎜⎝

VX X 0 VX Z

0 VY Y 0

VX Z 0 VZ Z

⎞⎟⎠ ; U G

i j =

⎛⎜⎝

UX X UXY UX Z

UXY UY Y UY Z

UX Z UY Z UZ Z

⎞⎟⎠ ,

(2)

C© 2011 The Authors, GJI, 187, 743–753



where the superscript G signifies that the gradients are given inthe GRF, and the Vij are filtered to keep the MB (indicated with abar), whereas some of the Uij are low-pass filtered (indicated with atilde). The cut-on frequency fl determines the lower limit of the MBand the upper limit for the model pass band (PB). Combining thecomponents of the measurement tensor V G

i j and the model tensor

U Gi j leads to a complete gravity field tensor that includes model and

GOCE measured gradients:

V Gi j = V G

i j + U Gi j . (3)

The model gravity gradients can be computed using for examplea GRACE-based global gravity field model or a GOCE-only gravityfield model. Alternatively, one might use a GOCE-only model tocompute UXY and UY Z , and a GRACE-based model to compute UXY

and UY Z . Thus all gradient information in the MB—either gradientmeasurements or gradient model values—would come from GOCE(which is desirable) and all gradient information below the MBwould come from a GRACE-based model. The combined gravitytensor can be rotated to any reference frame using a point-wisetensor rotation, eq. (1), without severe degradation of the gravitygradient accuracy in the MB.

4 E F F E C T I V E M E A S U R E M E N TB A N DW I D T H

Two parameters that need to be set are the filter cut-on and cut-offfrequency to combine the GOCE gradients with model gradients.The GOCE gradients are bandpass filtered because it is expectedthat GOCE will be the most accurate source for shorter wavelengths,with gradient noise increase above 0.1 Hz. On the other hand, themodel gradients that are computed with state-of-the-art global grav-ity field models are expected to be more accurate than GOCE atlonger wavelengths, mainly because of the gravity field informationprovided by GRACE. These model gradients are therefore low-passfiltered using the complement of the bandpass filter. It is expectedthat there is a certain (maybe narrow) frequency band in which boththe GOCE and model gradients have about equal accuracy.

There are at least two different philosophies to choose the ‘opti-mal’ cut-on frequency fl for the combination of GOCE and modelgradients. The first is to choose fl such that it falls in the frequencyband in which the GOCE and model gradients have about equal ac-curacy. The combined gravity gradients in the GRF would then be

the best gradients available. A different philosophy is to maximizethe information delivered by GOCE, that is, to determine the effec-tive MB, where the gradient errors are minimal and behave ideallyas white noise. Compared with the best-combined gradients, thesegradients may be less accurate at intermediate wavelengths. Nev-ertheless, these GOCE-only gradients in the effective MB provideaccurate gravity field information, independent of other data sets.We therefore adopted the second philosophy.

Although the MB of the GOCE gradients is defined to be between5 mHz and 0.1 Hz, this does not necessarily mean that the fl of theeffective MB is 5 mHz or that the cut-off frequency is 0.1 Hz. Theactual GOCE gradient error may differ from the specifications—asit does (Bouman et al. 2011)—and fl and cut-off frequency fu (ufor upper) should therefore be based on an analysis of real data.We propose here a method to choose fl and fu based on signal-to-noise estimation. The aim is to obtain the maximum integratedsignal-to-noise ratio (SNR) of the gradients inside the MB.

4.1 Upper limit of the effective MB

The upper limit fu of the effective MB (the cut-off frequency) isdetermined by the frequency at which the gradient signal, of aspecific gradient, equals the estimated gradient error. Figs 3 and4 show the spectral densities (SDs) of the gradient signals anderrors for the data period 2009 November 1 until 2010 January11. The gradient signals (in red) were computed with the GOCEtime-wise model (Pail et al. 2011) and the gradient error SDs forthe diagonal gradients (in green) are derived in the computationof a quick-look gravity field model (Mayerhofer et al. 2010). ForV XZ no error SD is available because this component is not usedin the quick-look gravity field processing. The gradient differences(in blue) between time-wise and measured gradients were filteredto the MB and provide a reasonable error estimate compared withthe available estimated quick-look error SDs. The magenta lineindicates the lower end of the specified MB (5 mHz), the cyan lineindicates the upper end of the specified MB (0.1 Hz), and the blackline is the frequency at which the SNR = 1. The exact values fu aregiven in Table 1. The spherical harmonic (SH) degree in Table 1is computed as fu· 5400, which is the approximate time in secondsneeded for one orbital revolution (exact value 5383.18 ± 0.30 s).The spatial resolution is computed as 20.000 km L–1, with L theSH degree.

Figure 3. Spectral densities for VXX and VYY for the GRF. In red the SDs for the gradient signal computed with the GOCE time-wise model, in green theestimated gradient error SD and in blue the SD of the difference between time-wise and measured gradients filtered to the MB. The magenta line indicates thelower end of the specified MB (5 mHz), the cyan line indicates the upper end of the specified MB (0.1 Hz) and the black is the frequency at which the SNR = 1.

C© 2011 The Authors, GJI, 187, 743–753



Figure 4. Spectral densities for VZZ and VXZ for the GRF. In red the SDs for the gradient signal computed with the GOCE time-wise model, in green theestimated gradient error SD (not available for VXZ ) and in blue the SD of the difference between time-wise and measured gradients filtered to the MB. Themagenta line indicates the lower end of the specified MB (5 mHz), the cyan line indicates the upper end of the specified MB (0.1 Hz) and the black is thefrequency at which the SNR = 1.

Table 1. Upper limits of the effective measurement bandwidth for the accu-rate gravity gradients in the GRF for different SNRs. The resolution (Res.)is rounded to 5 km.

SNR = 1 SNR = 1/9

Gravity fu SH Res. fu SH Res.gradient (mHz) degree (km) (mHz) degree (km)

VXX 32.4 175 115 41.0 221 90VYY 23.3 126 160 32.3 174 115VZZ 29.7 160 125 38.9 209 95VXZ 29.7 160 125 38.8 209 95

Averaging measurements leads to a reduction of the random er-rors. It makes therefore sense to consider not only SNR = 1 butalso other values, for example, SNR = 1/9 which corresponds to10 per cent signal versus 90 per cent noise. The correspondingvalues for the upper frequency fu of the effective MB are given inTable 1. As an example, one has on average 100 gradients in 1◦ ×1◦ blocks after about 2.5 months of GOCE mission duration. Av-eraging these values to obtain block mean values reduces the noiseby approximately a factor of 10 and the SNR of the block meanvalues is about 1. The upper frequency of the effective MB for thedifferent gravity gradients corresponds to a SH degree of roughly200–220, which agrees well with the maximum SH degree of thefirst 3 GOCE-only gravity field models, which are based on roughly2.5 months of GOCE data: L is 210, 224 and 240 for the so-calledspace-wise, time-wise and direct models, respectively (Pail et al.2011).

4.2 Lower limit of the effective MB

The optimal lower limit fl of the effective MB is determined byintegration of the signal and noise energy from the upper limit fu toa variable lower limit fl. Because the error of the accurate gravitygradients is flat in the MB and increases rapidly below the MB as1/f, and because the gradient signal tends to increase steadily butslowly with increasing frequency, see Figs 3 and 4, we expect that theratio of the integrated gradient signal and gradient error increasesin the effective MB and decreases below the MB. In other words,we expect that there is a maximum integrated SNR at a certaincut-on frequency. This fl defines the lower limit of the effective

measurement bandwidth (EMB). Thus the integrated SNR is

SNR( fl ) = S( fl )

N ( fl )=

∫ flfu

∣∣Vi j ( f )∣∣2

d f∫ fl

fu

∣∣V εi j ( f )

∣∣2d f

, (4)

where Vij(f ) is the Fourier transformed gradient signal and V εi j ( f )

the Fourier transformed gradient error. The signal energy can becomputed from model data (shown in red in Figs 3 and 4) or GOCEmeasurements, whereas the noise energy can be computed fromthe difference between GOCE gradients and gradients from globalgravity field models (shown in blue in Figs 3 and 4) or from theestimated gradient error SDs (shown in green in Figs 3 and 4) givenby the gravity field quick-look estimation process (Mayerhofer et al.2010).

The SNR (4) is computed in the spectral domain. Alternatively,the SNR can be computed in the time domain as

SNR ( fl ) =1n

∑nk=1

∣∣V Gi j (tk)

∣∣2

1n

∑nk=1

∣∣V Gi j (tk) − U G

i j (tk)∣∣2

, (5)

where V Gi j and U G

i j are the bandpass filtered GOCE and modelgradient time-series respectively, given in the GRF, with the MBfrom fl to fu, and tk the discrete times of the measurements.

The ratios (4) and (5) are in principle the same, but differ inthe practical realization and the spectral behaviour with respect todata gaps or outliers. For the SD computation in (4) Welch’s methodincluding a Kaiser window has been used. Computing (5) an eighth-order Butterworth filter has been used to establish optimal flatnessinside the MB. For the time domain approach (5) bad flagged datahas been excluded from the analysis, where for (4) in general dataperiods have been chosen to establish continuous measurement datawith almost no outliers. Also note that the time domain equivalentof the estimated error SDs is not available and that therefore theSNR (5) can only be computed with GOCE—model differences.

4.3 Global analysis to determine the cut-on frequency

The SNR can be evaluated either globally or regionally, which isdiscussed in this section and Section 4.4, respectively. The errorsin the GOCE gradients are not only derived from the given errorSDs but also estimated by taking the difference between GOCEand model gradients. The latter are computed with existing global

C© 2011 The Authors, GJI, 187, 743–753



Table 2. Optimal cut-on frequency fl in mHz derived by the time domain residual method, the spectral method and the error SD method for different referencemodels. A cut-off frequency fu at SNR = 1 was used to compute fl .

EGM96 EIGEN-5C EGM2008 ITG-GRACE GOCE-QL GOCE-TIM

Tim

e

Spectral

SD

Tim

e

Spectral

SD

Tim

e

Spectral

SD

Tim

e

Spectral

SD

Tim

e

Spectral

SD

Tim

e

Spectral

SD

VXX 1.5 1.5 1.5 3.6 3.7 3.7 3.9 3.9 3.7 4.0 4.1 4.1 4.1 4.1 4.1 3.9 3.9 3.9VYY 1.4 1.4 1.3 3.5 3.2 3.2 3.5 3.5 3.5 3.7 3.6 3.6 3.7 3.6 3.7 3.7 3.7 3.7VZZ 1.1 1.2 1.1 2.8 2.8 2.8 3.0 2.8 2.8 3.1 3.0 3.0 3.1 3.0 3.0 3.1 3.1 3.1VXZ 0.8 0.8 – 3.1 3.1 – 3.3 3.3 – 3.4 3.4 – 3.4 3.4 – 3.3 3.3 –

gravity field models and the errors in these models are thereforeimportant as well. To assess the influence of these model errors, weperformed the global analysis using a suit of different gravity fieldmodels: EGM96 (Lemoine et al. 1998), EIGEN-5C (Forste et al.2008), EGM2008 (Pavlis et al. 2008), ITG-GRACE2010 (Mayer-Gurr et al. 2010) and the GOCE quick-look and time-wise models.

The residuals along the GOCE orbit have been used to maximizethe integrated SNR in the spectral (4) and time domain (5). Due tothe difference in the methods and in the models small deviationsbetween the optimal fl may occur. Table 2 shows the cut-on frequen-cies for the four accurate GOCE gravity gradients using differentmethods and gravity field models. The cut-off frequency that wasused is that at which the SNR = 1, see Table 1.

In general there is a good agreement between the fl derived withdifferent methods and different models. Differences between meth-ods and models for a specific gravity gradient are less than 0.5mHz, which in terms of SH degree corresponds to l = 3 or less.The fl computed with EGM96 are an exception, which is discussedbelow. In any case, the lower end of the effective MB seems to be 3–4 mHz for all four accurate gravity gradients. In terms of SH degreethis corresponds to l = 16–22 which is well below the lower boundof the specified MB of 5 mHz (l = 27). Note that in the currentrelease of the GOCE data (2011 July) V YY may also include non-gravitational signal, which yields increased residuals with respect tothe reference gravity fields (Bouman et al. 2011; Bouman & Fuchs2011).

When the EGM96 gravity field model is used for the analysis, theoptimal fl is significantly smaller compared with the other models.Since the release of EGM96 much more precise measurements havebecome available and therefore gravity gradient differences, com-pared with today’s state-of-the-art gravity field models, are largeinside the MB, whereas the total gradient signal energy does notdiffer much from that of recent gravity field models. Therefore theweight in the resulting ratio behaves crucially different. Dependenton the spectral error of the reference model the residuals are domi-nated by signal energy and increase towards lower frequencies. Thusthe model biases in EGM96 may lead to smaller fl.

We also derived the optimal fl using an upper integration fre-quency fu that corresponds to SNR = 1/9, see Table 1. The fl are allequal or somewhat smaller than those given in Table 2, with maxi-mum differences of 0.2 mHz for all nominal gravity gradients. Ourmethod is therefore robust against small changes of the upper MBlimit, which is attributed to the small signal energy at the upper MBlimit compared with the lower MB limit and the steep intersectionof the noise floor. Furthermore, the fl were derived for different filterorders. Also in this case they do not change very significantly.

It is clear from the error SDs shown in Figs 3 and 4 that the gravitygradients suffer from systematic errors at 1, 2, etc. CPR (cycles-per-revolution), and the lower end of the effective MB can therefore notbe chosen arbitrarily low. This is illustrated in Fig. 5 that displays

the binned residuals between the GOCE and GOCE time-wise V ZZ

gravity gradients filtered with cut-on frequencies of 2 and 4 mHz forthe time-period 2009 November 8 till 2010 January 11. This time-period has been chosen because here the precise gravity gradientcomponents (V XX, V YY, V ZZ and V XZ) are largely free of outliersand data gaps. Whereas the 4 mHz lower bound leads to a ratherhomogeneous residual structure, stronger oscillations are visible forthe 2 mHz lower bound. In this case systematic errors at 12 CPRstart to dominate the residuals (1 CPR = 1/5400 Hz, 12/5400 =2.22 mHz).

4.4 Regional analysis to determine the cut-on frequency

The maximum of the integrated SNR that determines the optimalfl cannot only be evaluated globally but also regionally if the timedomain method (5) is used. We selected two test areas where on theone hand terrestrial gravity data that went into EGM2008 is believedto be of good quality and second where regional residuals (GOCEminus model) are more or less homogeneously distributed. One ofthe test areas is located in central Europe, defined from 37.5◦ to55◦N and 0◦ to 17◦E, the second test area is located at the southernhemisphere covering Australia from 40◦ to 10◦S and 110◦ to 160◦E.Fig. 6 displays the binned V ZZ differences between EGM2008 andGOCE inside the European test area for a cut-on frequency of4 mHz. The data period for the local analysis covers 2009 November8 until 2010 January 11, flagged data have been excluded from theanalysis.

Table 3 shows that for all gradients the optimal cut-on frequencycorresponds well with the fl determined with the global method,except for VXX where the difference in the European test area is sig-nificantly larger as compared with Table 2. The reason is probablythat the orbit height is lower for the Northern hemisphere comparedwith the Southern hemisphere, which causes systematic errors at 1,2,. . ., CPR that are correlated with the signal to be larger for theNorthern hemisphere. Again the fl are below the GOCE specifica-tion of 5 mHz, which corresponds to a SH degree of 27 and a spatialresolution of about 740 km. Note: while the global analysis also in-cludes anomalous VYY tracks close to the magnetic poles (Boumanet al. 2011; Bouman & Fuchs 2011), the local analysis is almost freeof this error. In addition, the resulting derived absolute value of theSNR is much larger as compared with the global analysis becausethe differences EGM2008-GOCE are small in the chosen areas.

4.5 Summary

The effective MB is determined by an upper frequency at whichthe gravity gradient SNR equals, for example, 1 or 1/9, and by alower frequency at which the integrated SNR in the effective MBis maximized. To choose a cut-on frequency is a trade-off between

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Figure 5. Binned residuals for GOCE time-wise in GRF for VZZ in 2◦ grid cells with respect to different fl . Spatial representation for different cut-off-frequencies(top panel) 2 mHz and (bottom panel) 4 mHz in V ZZ for 6 million samples. The averages over latitude are shown in the right column.

Figure 6. Binned GOCE residuals in the GRF for EGM2008 in a centralEuropean test area for VZZ with a cut-on frequency of 4 mHz (bin size of0.5◦).

signal bandwidth and measurement noise and we can use differ-ent methods to do so. Consequently, the fl determination dependsamong others on the quality of the reference gravity field models.Nevertheless, there is in general a good agreement between the fl

derived with different methods and different models. If the modelerrors are relatively large, such as for EGM96, then the estimatedfl are not reliable in contrast to fl determined with GRACE- andGOCE-based gravity field models. The optimal fl seem to be rela-tively insensitive to the choice of the cut-off frequency fc and alsonot very sensitive to the filter order that is used to filter the gravitygradients.

Table 3. Results for the optimal cut-on frequency fl in the GRF derivedby the time domain residual method; Central European and Australian testareas. Spatial resolution rounded to 5 km.

Central-Europe Australia

Gravity fl SH Resolution fl SH Resolutiongradient (mHz) degree (km) (mHz) degree (km)

VXX 4.9 26 755 4.3 23 860VYY 3.5 19 1060 3.4 18 1090VZZ 3.6 19 1030 3.0 16 1240VXZ 4.0 22 930 3.5 19 1060

All obtained cut-on frequencies are between 3 and 4 mHz whichis below the GOCE specification of 5 mHz. The obtained cut-on fre-quencies show small deviations for the different gravity gradients,which can be explained by the different signal and noise contribu-tions.

5 M O D E L C O N T E N T I N T H E RO TAT E DG R AV I T Y G R A D I E N T S

Once the EMBs of the four accurate GOCE gravity gradients VXX ,VYY , VZZ and VXZ have been determined, the gradients can be filteredand complemented with gradient model information. Using alsomodel information to replace GOCE VXY and VYZ in the GRF, thecomplete tensor (3) can be rotated to other reference frames using(1). Unavoidably, the gradients in the rotated frame are a mix ofthe measured and modelled gradients, which means that the rotatedgradients will be biased towards the global gravity field model thatwas used as input. This is not necessarily a problem when the lessaccurate gradients are replaced by model values computed from aGOCE-only gravity field model. Note, however, that all GOCE-onlymodels include some kind of a priori information either in the formof Kaula’s rule or in form of existing global gravity field models.

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Bouman & Fuchs (2011) discuss how this a priori information mayaffect the GOCE-only solutions.

One important quantity to assess the quality of the rotated gra-dients is the model content present in the gradients after rotation.Here we study and analyse the model content in the spectral andtime domain for the LNOF and for the LORF-r/v as defined in Sec-tion 2. To evaluate the model content present in these frames themodel and measured gradients are separately rotated and combinedafterwards:

V Li j = R

(U G

i j + V Gi j

)RT = RU G

i j RT + RV Gi j RT

= U Li j + V L

i j , (6)

where V Li j are the rotated gravity gradients in LORF-r/v or LNOF.

To simplify the analysis we chose a cut-on frequency of 4 mHz forall gravity gradients.

Fig. 7 shows the spectral densities (SDs) of the VXX , VYY and VZZ

gravity gradients after rotation to the LORF-v for the data periodof 2009 October 31 until 2010 January 11. The GOCE and modelgradients as well as their differences are shown using the gradienttensors (6). For the model gradients the GOCE time-wise model hasbeen used. The contribution of the GOCE gradients below the MB

of course quickly decreases. The amount of model information inthe rotated gradients varies for the different gravity gradients, butis relatively small as the rotations from GRF to LORF are small.The rotation around the Z-axis is much larger when rotating fromGRF to LNOF, and therefore also the model contribution for VXX

and VYY . This is shown in Fig. 8, from which it becomes clear thatalso in this case the model contribution to VZZ remains small.

The relative model content present in the rotated frames can bequantified by taking the ratio of the total model signal and GOCEgravity gradient signal inside the MB:

RL =∣∣U L

i j

∣∣∣∣U L

i j

∣∣ + ∣∣V Li j

∣∣ × 100. (7)

The evaluation of RL in each measurement point allows a geograph-ical representation.

Figs 9 and 10 show the relative model content for VXX , VYY , VZZ ,VXY , VXZ and VYZ in the LORF-v and LNOF, respectively for thetime period 2009 October 31 until 2010 January 11 with a spatialbinning of 1◦ × 1◦. Because the yaw rotation angle is the largestangle going from GRF to LORF-r/v or LNOF, the rotation takesplace specifically in the XY -plane. The model contribution of UXY

Figure 7. SD plot of GOCE and model gravity gradients in the LORF-v. From left to right-hand side: VXX , VYY and VZZ . GOCE gradients filtered to effectiveMB (blue); Model gradients (cyan); Model gradients filtered to effective MB (green); GOCE gradients minus model gradients (purple). The lower limit of theMB has been set to 4.0 mHz (solid black line).

Figure 8. SD plot of GOCE and model gravity gradients in the LNOF. From left to right-hand side: VXX , VYY and VZZ . GOCE gradients filtered to effectiveMB (blue); Model gradients (cyan); Model gradients filtered to effective MB (green); GOCE gradients minus model gradients (purple). The lower limit of theMB has been set to 4.0 mHz (solid black line).

C© 2011 The Authors, GJI, 187, 743–753



Figure 9. Geographical plot of relative model content in the rotated gradients in the LORF-v.

is therefore the largest for VXX and VYY . In addition, the relativemodel content in VYY is larger than in VXX because the rotationaround roll is larger than the rotation around pitch. Also the VYY

signal in the MB is smaller than that of VXX (see Fig. 3). Finally,VXX benefits from the VXZ measurements.

For the LNOF, the model contribution in VXX and VYY is thelargest at high latitudes because here the satellite tracks are almostorthogonal to the geographical meridian and must be rotated at leastby 90◦ in the yaw axis, which also affects VXZ . Note: the model con-tribution coming from VXY is largest at a yaw angle of 45◦. Thiscan be seen in the average latitude plot of VYY where the modelcontent decreases above ±80◦ latitude, see Figs 2 and 11. For VXX

the signal around ±80◦ comes (due to orthogonal projection) fromVYY . The model contribution in VZZ is small because it benefits fromthe double signal energy as compared with VXX and VYY , and fromthe small rotation angles around pitch and roll. The relative modelcontribution in the LORF-r and the LORF-v marginally differ, be-cause the GOCE orbit is nearly circular and therefore the velocityvector and the radial vector are almost orthogonal, that is, LORF-vand LORF-r hardly differ.

The less accurate tensor elements VXY and VYZ are purely deter-mined by model information in the GRF, but include measurementinformation after rotation. In case of the LORF-v this contribu-tion is low, as expected, but in case of the LNOF, the measurementinformation at the poles is significantly larger: on average the mea-surement contribution to VXY and VYZ in the LNOF is 39 per cent and36 per cent, respectively. This shows that the original measurementinformation of the accurate gradients is not lost but redistributed.

The average model contributions for the different gravity gradi-ents in the LNOF and LORF-r/v are summarized in Table 4. Thus,for the accurate gravity gradients the direct GOCE contribution inLNOF and LORF-r/v varies from 65 per cent to 98 per cent de-pending on the specific gravity gradient and reference frame. TheLORF-v has a slightly larger relative model content compared withLORF-r because the rotation around pitch is slightly larger. All inall, the pre- and post-multiplication of the rotational matrix for theframe rotation (1) may have a strong mixing effect.

6 C O N C LU S I O N S

The measured GOCE gravity gradient tensor is given in the in-strumental frame and may be rotated to local reference frames thatare directly related to the Earth. This rotation is restrained becausethe VXY and VYZ gradients are less accurate compared with VXX ,VYY , VZZ and VXZ that are highly accurate, and because the error ofthe accurate gravity gradients increases for low frequencies. Theseproblems can be circumvented by replacing the less accurate grav-ity gradients with model values, for example from a GOCE-onlygravity field model, and by replacing the gravity gradient signal ofthe accurate gravity gradients for low frequencies with model signalas well.

The combination of model gravity gradients and GOCE gravitygradients requires that both data sets are filtered and that a certaincut-on frequency is determined. We proposed and tested integrationmethods to determine the EMB of the accurate gravity gradients.

C© 2011 The Authors, GJI, 187, 743–753



Figure 10. Geographical plot of relative model content in the rotated gradients in the LNOF.

Figure 11. Average latitude plot of the relative model content in the rotated gradients in the LORF-v and LNOF.

C© 2011 The Authors, GJI, 187, 743–753



Table 4. Global average of the relative model con-tribution (per cent) in the effective MB (4 mHz tofrequency of SNR = 1 of Table 1).

Gravity gradient LORF-v LORF-r LNOF

VXX 3.8 3.8 25.8VYY 9.2 7.9 35.4VZZ 2.6 1.8 1.8VXY 91.7 92.6 61.2VXZ 3.1 2.9 21.7VYZ 92.2 93.1 64.4

The integration ends at a frequency at which the SNR of a gravitygradient has a specified value. This frequency determines the upperbound fu of the EMB and is kept fixed. The integration starts at alower frequency and the cut-on frequency for which the integratedSNR is maximized determines the lower bound fl of the EMB.

The lower bound of the EMB, the cut-on frequency, is relativelyinsensitive to the choice of the cut-off frequency (the upper bound ofthe effective MB). The evaluation of the integrated SNR in spectraland time domain, regionally or globally, yields more or less the sameoptimal cut-on frequency. In addition, the choice of the referenceglobal gravity field model seems to be of less importance as longas GRACE- or GOCE-based models are being used. The lowerbound of the effective MB varies between 3 and 4 mHz for the fouraccurate gravity gradients, which is well below the specified valueof 5 mHz.

The relative content of GOCE versus model gravity gradients indifferent local reference frames has been evaluated in the effectiveMB. The relative model content depends of course on the rotationalangles and on the signal size of the gravity gradients. The relativeamount of model information in the LNOF is much higher as com-pared with the LORF-r/v frames because the instrumental frame iskept aligned with the LORF. In the LNOF, the direct GOCE contri-bution to the rotated gravity gradients is 65 per cent, 74 per cent,78 per cent and 98 per cent for VYY , VXX , VXZ and VZZ , respectively.In the LORF-r/v these percentages are 81 percent, 96 per cent,97 per cent and 98 per cent. The LORF-r/v can therefore be used toanalyse GOCE-gradients locally with a small contribution (below5 per cent for VXX , VXZ and VZZ) of model information.

A C K N OW L E D G M E N T S

This study was performed in the framework of ESA’s project GOCEHigh-level Processing Facility (No. 18308/04/NL/MM). MartinFuchs was sponsored by the German Department for Education andResearch (Bundesministerium fur Bildung und Forschung) as partof the GEOTECHNOLOGIEN program. In addition, the first au-thor is supported by the TUM Graduate School’s Faculty GraduateCenter Bau-Geo-Umwelt at the Technische Universitat Munchen.The comments from the Editor (Duncan Agnew), Fernando Sansoand an anonymous reviewer helped to improve the paper.

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Rotation of GOCE gravity gradients to local frames

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