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Validation of Satellite Gravity Field Models 281

Fig. 2 Left: Histogram of differences between A-10 observations and DHSN96 1st order net-

work gravity values; Right: Spatial distribution of differences between A-10 absolute gravity

observations and corresponding values from DHSN96 1st order network in Germany

could not be placed directly above the intended measurement point. In these cases

eccentric observations were connected to the corresponding bench marks using rel-

ative measurements as well. The necessary height transfer from the absolute sites to

the DHSN96 bench marks was done by spirit levelling.

In order to evaluate the accuracy of the DHSN96 1st order gravity network, the

gravity values were compared with the observed A-10 absolute gravity values at 93

ﬁeld stations, measured during the GOCE-GRAND II project (7 points were left

out due to instrumental problems or environmental issues). The differences between

the A-10 and DHSN96 gravity values (which are inﬂuenced by observation errors

of the A-10 absolute and CG-5 relative gravimeter as well as by the errors of the

DHSN96 values themselves) show a mean of ±0.2 μGal and a standard deviation

of ±18.7 μGal (see Fig. 2). About 90% of the measured points differ by not more

than 20 μGal from the DHSN96 values. With the results presented above, the accu-

racy estimates for the DHSN96 network were conﬁrmed, and thus no systematic

impacts on geoid modelling have to be worried about. The quality of subordinate

networks or independent data sets is still questionable and requires further inves-

tigations. Regarding the source, measurement method, accuracy and homogeneity,

the existing data base includes a variety of different data sets. A proper weighting

strategy of these different data sets in the course of the quasigeoid modelling should

be derived from this evaluation in the future.

Furthermore, the existing gravity data base, e.g., used for the computation of the

national quasigeoid model GCG05 (German Combined QuasiGeoid 2005; Liebsch

et al., 2006), was validated by the new A-10 absolute gravity observations. For this

purpose, Bouguer anomalies were predicted from the GCG05 data base, which

were then compared with corresponding values for the A-10 stations. However,

since the accuracy of the predicted values is two orders of magnitude worse than

the accuracy of the absolute measurements, meaningful conclusion appear to be

impossible.

282 J. Ihde et al.

3 GPS and Levelling Data

GPS and levelling data are essential in two respects. First, they are important for the

direct validation of global gravity ﬁeld models including the ones from the GOCE

mission, and secondly they can be used for cross-validating astrogeodetic and gravi-

metric geoid and quasigeoid models. Moreover, the GPS and levelling data can be

combined with the aforementioned data sets. In this context, the ﬁtting of gravimet-

ric geoid and quasigeoid models to GPS and levelling control points has become

a standard procedure for GPS height determination with a targeted accuracy of

about 1 cm.

In Germany, a national GPS and levelling data was collected by BKG, consist-

ing of 907 stations with a spacing of 25–30 km. The GPS coordinates are based on

the re-adjusted SAPOS reference station coordinates, which were introduced at the

beginning of 2004 (Beckers et al., 2005). However, the Federal States of Germany

used individual procedures to implement the new SAPOS reference frame, rang-

ing from re-observation and re-adjustment to different transformation procedures,

which may have introduced some inhomogeneities in the GPS coordinates. The

gravity ﬁeld related heights are normal heights referring to the ofﬁcial DHHN92

system (AdV, 1995). For further details cf. Liebsch et al. (2006). In this context,

future improvements will result from the renewal of the German 1st order level-

ling network until 2011 in connection with a parallel GNSS observation campaign

in 2008.

The German GPS/levelling data set was compared with the global geopotential

models EGM1996 (Lemoine et al., 1998), EGM2008 (Pavlis et al., 2008) and sev-

eral models developed at the Helmholtz Centre Potsdam, GFZ German Research

Centre for Geosciences (ICGEM, 2009). All geopotential models include terres-

trial gravity and satellite altimeter data in combination with satellite observations.

Apart from EGM1996, all other models are based on data from the GRACE mission.

Table 1 provides the statistics of the differences between the German GPS/levelling

data set and the above mentioned geopotential models. In all cases, a constant bias

was subtracted from the differences in order to account for different zero levels and

very long wavelength errors in all data sets involved (GPS, levelling, quasigeoid).

Table 1 Comparison of 907 GPS and levelling stations in Germany with different global

geopotential models. A constant bias is subtracted. Units are m

Geopotential model Maximum degree used RMS Min Max

EGM1996 360 0.251 –0.588 +1.187

EGM2008 360 0.135 –0.552 +0.663

EGM2008 2,190 0.030 –0.107 +0.144

EIGEN-CG01C 360 0.211 –0.734 +0.693

EIGEN-CG03C 360 0.189 –0.723 +0.704

EIGEN-GL04C 360 0.180 –0.736 +0.724

EIGEN-5C 360 0.153 –0.541 +0.756

Validation of Satellite Gravity Field Models 283

From the models to spherical harmonic degree 360, EGM2008 performs best with

a RMS difference of 0.135 m, while the latest EIGEN-5C models yields a RMS

difference of 0.153 m, compared to 0.251 m for EGM1996. This shows the signif-

icant improvements due to the inclusion of the GRACE data but also due to the

improved terrestrial data sets. Regarding the EIGEN models f rom GFZ, it can be

observed that the results improve with time and the inclusion of GRACE data over a

longer time period. Finally, with the EGM2008 model to degree 2,190, a RMS dif-

ference of only 0.030 m is obtained, which is close to the best regional quasigeoid

models.

4 Gravimetric Quasigeoid Models

The computation of regional high-resolution geoid and quasigeoid models is com-

monly based on the remove-restore technique, combining terrestrial gravity data

(with a spacing of a few km), a detailed digital terrain model (with a grid size

down to 30 m) and a global geopotential model. In this context, the global geopo-

tential model and the regional data sets complement each other in a favorable

manner, i.e., the global model provides the long wavelength gravity ﬁeld structures

with high accuracy, while the regional data sets contribute the medium and short

wavelength gravity ﬁeld components, which are not included in the global model.

Up-to-date global geopotential models are based on the GRACE satellite mission

(e.g., EGM2008; Pavlis et al., 2008) and provide the global geoid with an unprece-

dented accuracy of about 1 mm to spherical harmonic degree 40 and 1 cm to degree

80. On the other hand, high quality regional terrestrial gravity ﬁeld data sets exist in

several parts of the world and can be combined with a global model. In the combi-

nation approach, e.g., least-squares collocation or spectral combination techniques,

the signal and error characteristics of all involved data sets have to be taken into

account. As a result, the long wavelength components of the regional geoid and

quasigeoid models and of the global geopotential model match to a great extent.

Therefore, comparisons of regional geoid and quasigeoid models with GPS/levelling

data can be regarded also as a validation tool for the underlying global geopotential

models.

In the following, the high resolution European quasigeoid models derived within

the framework of the European Gravity and Geoid Project (EGGP) are considered

as representative examples (cf. Denker et al., 2009). The latest model EGG2008

was computed on the basis of the 2008 European gravity and terrain data sets

and the EGM2008 global model, utilizing the spectral combination approach. In

order to show the improvement with time, the 2008 terrestrial data was also

combined with other recently published global models as well as the EGM1996

model (Lemoine et al., 1998). In addition, the previous European quasigeoid model

EGG1997 is considered as a reference. All results were evaluated by comparisons

with the German GPS/levelling data set. Table 2 shows the statistics of differ-

ences between the GPS/levelling data and different European quasigeoid models.

A constant bias was subtracted from the differences. First, the results based on

284 J. Ihde et al.

Table 2 Comparison of 907 GPS and levelling stations in Germany with different quasigeoid

models. A constant bias is subtracted. Units are m

Terrestrial data Underlying geopotential model RMS Min Max

1997 (EGG1997) EGM1996 0.094 –0.185 +0.327

2008 EGM1996 0.072 –0.132 +0.296

2008 EIGEN-CHAMP03S 0.049 –0.118 +0.258

2008 EIGEN-GRACE01S 0.038 –0.113 +0.148

2008 EIGEN-GRACE02S 0.037 –0.081 +0.119

2008 EIGEN-GL04S1 0.028 –0.096 +0.082

2008 EIGEN-5S 0.027 –0.097 +0.070

2008 EIGEN-5C 0.027 –0.098 +0.073

2008 (EGG2008) EGM2008 0.026 –0.090 +0.074

EGM1996 show that the updated terrestrial data sets improve the results; the RMS

difference reduces from 0.094 m for the 1997 terrestrial data to 0.072 m for the

2008 data. Then further signiﬁcant improvements can be found when using a

CHAMP based model (EIGEN-CHAMP03S, 0.049 m RMS) and the GRACE based

EIGEN and EGM2008 models. It can be observed that the GRACE based models

improve with time with the recent EIGEN-5C/S and EGM2008 models resulting in

a RMS difference of 2.7 and 2.6 cm, respectively. In total, this is a very signiﬁcant

improvement of more than 70%, as compared to the previously published EGG1997

quasigeoid model. Hence, the new satellite mission data have a signiﬁcant impact on

Fig. 3 Comparison of 907 GPS and levelling stations in Germany with the quasigeoid solutions

EGG1997 (left part) and EGG2008 (right part). A constant bias is subtracted. Stations (dots)and

differences (positive: up; negative: down) are depicted

Validation of Satellite Gravity Field Models 285

the performance of the r egional quasigeoid models and further improvements down

to the level of about 1 cm can be expected from the GOCE mission. However, on

the other hand, also the quality of the terrestrial data sets is of vital importance. In

the end, Fig. 3 depicts the differences between the German GPS/levelling data set

and the EGG1997 and EGG2008 gravimetric quasigeoid models.

Furthermore, the GPS/levelling data can be included as input data in the quasi-

geoid modelling. This was done in Germany and lead to the combined GCG2005

quasigeoid model (Liebsch et al., 2006). This model was developed as a joint effort

of BKG and IfE and serves as a standard in Germany for the conversion between

GPS ellipsoidal heights and normal heights. The evaluation of the model with inde-

pendent GPS and levelling points suggested an accuracy of about 1–2 cm (Liebsch

et al., 2006).

5 Astrogeodetic Vertical Deﬂections

In recent years, a new generation of zenith camera systems has become available

for the determination of astrogeodetic vertical deﬂections. The increased accuracy

of 0.1



or better (Hirt, 2004) and the automation of the vertical deﬂection processing

are the main improvements. Two very similar zenith camera systems exist in Europe,

namely the TZK2-D at IfE (Hirt, 2004) and DIADEM at ETH Zurich (Bürki et al.,

2004). After an extensive test phase, both systems were applied in high-resolution

gravity ﬁeld modelling investigations. The TZK2-D was employed extensively in

the local validation of gravimetric geoid models along proﬁles in the German Alps

(Hirt and Flury, 2007) and in the Harz Mountains (Hirt et al., 2008). The results

showed the great accuracy potential of the astrogeodetic vertical deﬂections within

local test areas.

Within the framework of the GOCE-GRAND II project, the IfE zenith cam-

era system TZK2-D was employed in extensive campaigns to collect vertical

deﬂections on longer proﬁles of several hundred kilometres length (regional

scales), aiming at an independent regional validation of other gravity ﬁeld data

sets such as GPS/levelling data, regional quasigeoid models or global geopo-

tential models. Between 2006 and 2008, astrogeodetic vertical deﬂections were

determined on 148 stations along a North-South proﬁle and on 134 stations

along an East-West proﬁle. The observations were collected during 54 nights in

total.

Both proﬁles have a spacing of about 2.5–5 km between adjacent stations, result-

ing in a total length of 540 km (North–South) and 533 km (East–West), respectively.

In addition, astrogeodetic vertical deﬂections observed within previous projects

(Hirt and Flury, 2007; Hirt et al., 2008) were embedded in the North-South proﬁle,

resulting in a unique data set of in total 498 highly accurate astrogeodetic vertical

deﬂections (see Fig. 1, right). Due to the high degree of automation of the TZK2-

D system, up to 12 vertical deﬂection stations could be observed in a single night,

but it should also be pointed out that unfavourable nightly weather conditions may

signiﬁcantly limit the efﬁciency of the astronomical observation technique.

286 J. Ihde et al.

In order to assess the quality of the astrogeodetic vertical deﬂections, repeated

observations were carried out at 48 identical stations in different nights. The stan-

dard deviation of the differences was found to be 0.16



for both ξ and η. However,

these values were affected by observations with a reduced accuracy due to worse

weather conditions, e.g. fog; after excluding corresponding observations from the

analysis, the repeatability agrees well with the stated accuracy of 0.1



for the

TZK-2D system (Hirt, 2004). Furthermore, vertical deﬂection determinations and

calibrations were carried out regularly at the IfE astronomic reference station in

Hannover (Hirt and Seeber, 2007) in order to verify the instrumental stability.

5.1 Astrogeodetic Validation of GPS/Levelling Data

and Gravimetric Quasigeoid Models

In order to allow a cross-validation with the national GPS/levelling data set, the

astrogeodetic proﬁles traverse through 46 nearby GPS/levelling control points in

total (see Fig. 1). The method of astronomic levelling was used to compute quasi-

geoid height differences (Torge, 2001). In this context, a digital terrain model with

agridsizeof1



× 1



was utilized in order to smooth the observed vertical deﬂec-

tion data (Voigt and Denker, 2007). The accuracy estimate for the quasigeoid height

difference over an entire proﬁle length of more than 500 km is 2.2 cm, implying

random noise with a magnitude of 0.1



for the vertical deﬂection data and neglecting

omission errors. The results of the comparisons between the astrogeodetic quasi-

geoid proﬁles and the GPS/levelling data as well as the gravimetric quasigeoid

models German Combined QuasiGeoid 2005 (GCG05; Liebsch et al., 2006) and

the European Gravimetric Quasigeoid 2008 (EGG2008; Denker et al., 2009) are

shown in Table 3 and Fig. 4. Since the astrogeodetic method only provides differ-

ences of height anomalies, the absolute l evel is taken from the GCG05 model at the

ﬁrst station.

The cross-validation of the astrogeodetic results, based on the method of astro-

nomic levelling, with GPS/levelling and gravimetric quasigeoid data revealed partly

systematic differences in the order of a few cm over some 100 km length (Voigt

et al., 2009), which could not be fully explained so far, although repeat and densiﬁ-

cation observations were carried out with the TZK2-D instrument. Possible reasons

for the discrepancies in the cross-validation may be inhomogeneities in the GPS

Table 3 Statistics (cm) of the differences ζ between the astrogeodetic quasigeoid heights and

GPS/levelling as well as gravimetric models along the North–South and the East–West proﬁle

North–South proﬁle East–West proﬁle

Data set No Std Min Max No Std Min Max

GPS/lev 23 3.5 −1.3 10.7 23 2.7 –6.8 2.1

GCG05 364 3.4 −1.1 9.9 134 2.6 –6.3 2.0

EGG08 364 4.3 0.0 12.3 134 1.4 –0.6 4.0

Validation of Satellite Gravity Field Models 287

Fig. 4 Comparison of the astrogeodetic quasigeoid solution with GPS/levelling and quasigeoid

models GCG05 and EGG2008 along the North–South proﬁle (top) and the East–West proﬁle

(bottom)

results due to different processing strategies used in the individual Federal States of

Germany (ranging from re-observation and re-adjustment to different transforma-

tion procedures), systematic effects or insufﬁcient spacing of the vertical deﬂection

observations, and to some extent also large-scale levelling errors. It is manda-

tory to further study and enlighten the reasons for the discrepancies by additional

observations, data re-processing and development of realistic error models.

5.2 Astrogeodetic Validation of Global Geopotential Models

The main difference between the astrogeodetic vertical deﬂections and correspond-

ing values derived from a global geopotential model (GGM) is the different spectral

content. Whereas the GGM deﬂections have only a limited spectral content due to

the limited degree of the spherical harmonic series, the astrogeodetic deﬂection data

contain the complete spectrum (all frequencies).

In order to ﬁlter out all high-frequency signals from the terrestrial data, which

are not included in the GGM (i.e. outside the measurement bandwidth with regard to

GOCE), the Gaussian ﬁlter technique was applied with a radius of 50 km. Due to the

delay of the GOCE mission, the analyses were carried out using the currently avail-

able models EGM96 (Lemoine et al., 1998), EIGEN-GL04C (Förste et al., 2008a)

and the ultra-high degree model EGM2008 (Pavlis et al., 2008). Figure 5 shows the

unﬁltered and Gaussian ﬁltered differences of the vertical deﬂection components ξ

288 J. Ihde et al.

Fig. 5 Comparison between the unﬁltered (top) and Gaussian ﬁltered (bottom) astrogeodetic and

GGM vertical deﬂection components ξ and η along the North–South proﬁle

and η along the North–South proﬁle. The comparisons with the models EGM96 and

EIGEN-GL04C show RMS values from 1.7



to 2.8



for the unﬁltered differences

and 0.8



–2.1



for the Gaussian ﬁltered values, with mean values of 1.0



or below.

The validation of the ultra-high degree model EGM2008 shows RMS values from

0.6



to 1.2



for the unﬁltered differences – slightly better than the RMS differences

of 1.1



to 1.3



in CONUS (Contiguous United States) and Australia (Pavlis et al.,

2008) – and 0.1



to 0.3



for the Gaussian ﬁltered values, with mean differences of

0.2



or below. These results conﬁrm the high accuracy level of the EGM2008 model

and show that the astrogeodetic vertical deﬂections are suitable for an external and

completely independent validation of the GGM data.

6 Global Model Validation by Wavelet Techniques

In general there are two kinds of gravity ﬁeld validation. The ﬁrst one is a global

approach: the gravity ﬁeld covering the whole Earth is compared to another global

gravity ﬁeld. Both ﬁelds are spectrally band limited. There are some well-known

techniques to ﬁlter and to estimate the errors introduced by the chosen ﬁlter.

The second approach is to compare a global gravity ﬁeld with regional point

observations, covering only a small part of the Earth. These two data sets are not

compatible. The global gravity ﬁeld is spectrally band limited, while the local obser-

vations are not. Thus, validation has to be performed on the observation points and

it has to be band-limited by low pass ﬁltering the terrestrial validation data set.

Apart from the ﬁltering, additional uncertainties in such a validation process might

result from the data combination technique and regularization method applied in the

global (GOCE) gravity ﬁeld analysis and from possible datum off-sets, biases and

trends in the terrestrial validation data sets. Because available data sets cover only a

small part of the Earth, validation results might not be representative for the globe

and long wavelengths of the global model might not be well represented in the test

data sets.

Validation of Satellite Gravity Field Models 289

Table 4 Available GPS/levelling data sets

Country/continent Number of points References

Australia 197 Johnston and Manning (2003, personal communication)

Canada 430 Verroneau (2007, personal communication)

Europe 1,233 Kenyeres et al. (2006)

Germany 675 Ihde and Sacher (2002)

Japan 837 Nakagawa (2003, personal communication)

USA 5,167 NGS (1999)

For all approaches it is necessary to use data sets of sufﬁcient consistency

and precision for the global model validation. Astrogeodetic vertical deﬂections

(see Sect. 5) and GPS/levelling observations (see Sect. 3) are natural candidates

for this. In the following study, GPS/levelling heights in different areas of the

world are applied for validating global models. In particular it is referred to the

German GPS/levelling observations described in Sect. 3. As already pointed out,

it is expected that GOCE gravity ﬁeld models will have a precision of a few cen-

timeters in geoid height with a resolution of 100 km or better. Hence, the regional

data must have the same or even better precision. As a further requirement for a

satisfactory validation there should exist different data sets evenly spread over the

whole Earth. Table 4 speciﬁes the GPS/levelling geoid height data sets applied for

GOCE gravity ﬁeld validation. As one can see in Table 4, only very few data sets are

available for validation. Especially there are no test data in South America, Africa

and most parts of Asia.

The input data and the results of this work will be stored in a newly designed

validation database. This database is able to store all validation data as well as the

GOCE gravity ﬁeld results. It is hosted on a server at IAPG. A second server is

used as a web interface to the database. With the web interface one can access all

data in the validation database. The interface assists the user by showing only those

data sets that match the selections already made. As a last step, the data can be

downloaded in various graphical and textual formats.

6.1 Filtering Terrestrial Data by Second Generation Wavelets

The calculation of global gravity ﬁelds is commonly done with spherical harmon-

ics. They have a lot of well-known advantages for computing global ﬁelds, but for

locally bounded problems this globality is a disadvantage, because spherical har-

monics cannot be restricted in frequency and space simultaneously without loss of

accuracy. As mentioned before, in order to make both data sets compatible, it is nec-

essary to ﬁlter regional data with respect to the frequency content, because the global

ﬁeld is band limited in the frequency domain. There are several ﬁlter techniques to

achieve this: Least-Square-Collocation (LSC), Spherical-harmonic synthesis (SHS)

with ultra-high degree spherical harmonic models and second generation wavelets

(SGW). The latter approach is described in the following paragraphs.

290 J. Ihde et al.

Wavelets can ﬁlter a data set in the frequency and space domains simultane-

ously. This makes wavelets suitable for ﬁltering regional data in order to compare

them with global models. Usually wavelets rely on regularly spaced data because

of an underlying Fourier transform, requiring a regular grid of data. To apply a

wavelet transform to GPS/levelling data, the data would have to be interpolated to

this regular grid. However, interpolating data can add aliasing to data and removes

(mainly high-frequency) information (Soltanpour et al., 2006). To avoid this inter-

polation, Sweldens (1998) developed wavelets usable f or irregular spaced data, the

so called second generation wavelets (SGW). Besides the ability to work on irreg-

ular data, there are some other differences to “normal” wavelets: First generation

wavelet functions are dyadic translations and dilations of one mother wavelet

j,m

(x) = ψ (2

x −m).

So each wavelet ψ

j,m

at a level j and location m can be retrieved from the mother

wavelet ψ. This is not possible with second generation wavelets. In terms of signal

processing this means that the new wavelets have different ﬁlter characteristics for

each level depending on the underlying data.

In a ﬁrst step, the wavelet coefﬁcients have to be determined. In general one needs

to know the neighbours of each calculation point. With one-dimensional calculation,

only the direct neighbours have to be taken into account. With two dimensional data,

the determination of the neighbourhood of each vertex is not a trivial task anymore.

In this work, the Delaunay triangulation (Fig. 6 left) is used for this purpose as

proposed by Delouille et al. (2006).

To determine the area of support of the scaling functions, the Voronoi polygons

(Fig. 6 right) have to be built for each vertex. At the ﬁnest level, the integral of

the scaling functions is initialized with the area of the Voronoi polygon around the

vertex it was calculated for, i.e. v

. After that, the vertex with the smallest area is

removed, the surrounding vertices are updated to keep the support constant, and

the triangulation has to be partly rebuilt. On each level, only one new wavelet

coefﬁcient is calculated for a vertex. These steps are repeated until only bound-

ary vertices remain. Figure 7 shows the calculated coefﬁcients. In this case, the ﬁrst

Fig. 6 Delaunay triangulation