Flechtner F.M., Gruber Th., G?ntner A., Mandea M., Rothacher M., Sch?ne T., Wickert J. (Eds.) System Earth via Geodetic-Geophysical Space Techniques

Подождите немного. Документ загружается.

Quality Evaluation of GOCE Gradients 271

tensor component V

is considered with respect to the disturbing potential T as T

Applying the remove-restore technique residual gravity anomalies g

= g

–

g

as differences between the terrestrial gravity anomalies g

and those from a

geopotential model g

are used in the computations. The resulting residual tensor

component

has to be restored using again the GPM (

Integral formulas with spectral weighting are applied to combine regional ter-

restrial gravity data with a geopotential model for the computation of the tensor

component based on the remove-restore procedure. For more theoretical details

see Wolf (2007). The spectral weights are based on the error degree variance

information of the GPM and of the terrestrial gravity data.

2.2 Synthetic Data

The input data (g

, g

) and the output data (T

[ij]

) are based on a syn-

thetic Earth model composed of a Digital Topography Model (DTM) and a

GPM. Simulated noise is added to the input data. The ground-truth GPM

(SYNGPM1300S) consists of the EIGEN-GRACE02S (n = 2...104, Reigber

et al., 2005), the EGM96 (n = 105...360, Lemoine et al., 1998) and the GPM

(n = 361...1,300, Wenzel, 1999). The characteristics of the synthetic scenarios

are chosen based on a statistical accuracy estimation using least-squares esti-

mation (Wolf, 2007). From the SYNGPM1300S, the test data sets are derived.

The gravity anomalies g

are based on the SYNGPM1300S up to a spheri-

cal harmonic degree n

max

= 360, the appropriate noise is generated according

to the coefﬁcients’ standard deviations of the SYNGPM1300S. For the terrestrial

data g

, the complete synthetic model is used and different noise scenarios are

investigated: 1 mGal uncorrelated (1UC), 1 mGal correlated (1C) and 5 mGal

correlated (5C).

Fig. 3 A number of 26 tracks

from a 29-days global data set

cover the test area

272 J. Müller et al.

The test area is located in Central Europe (Fig. 3), the data sets are computed in

a0.1

◦

geographical grid. The spectral signal content of the synthetic data is limited

to a resolution of about 8



max

= 1,300). The synthetic gravity data represent

gravity anomalies reduced from the gravitational effect of a residual terrain model

(RTM reduction). The gravity anomalies g

have a mean signal content of 10.07

± 33.36 mGal, the tensor values 2,747.107 ± 0.187 E.

2.3 Closed-Loop Differences in the Frequency Domain

The differences between the tensor component values directly computed from the

SYNGPM1300S and the values derived from the noisy synthetic gravity anomalies

(g

, g

) are analysed in an inner area (marked in Fig. 3) to avoid edge effects.

For the frequency domain analysis, the following steps are performed:

(a) computation of all tensor components in points of a three dimensional grid

(spherical surfaces with a 6



data grid; 5 km height difference between each

interpolation level),

(b) interpolation into points of a simulated GOCE orbit (SCVII, 2000) using a cubic

spline approach,

2006),

(d) derivation of closed-loop differences with error-free tensor values (directly

computed from SYNGPM1300S),

(e) computation of the spectral densities from the times series based on the closed-

loop differences.

Due to the limited regional data area, only parts of the GOCE tracks can be

evaluated. Within the test area, a number of 26 tracks from a global 29-days time

series cover each at least a time span of 225 s (Fig. 3).

In Fig. 4 the spectral density (for V

) of the simulated noise of the synthetic

GPM based on the global time series (grey) is compared with the spectral density

computed as mean curve from the 26 tracks (black). Within the MBW (dark grey

box) the track-wise computed spectral density can be used as approximation of the

global one. The solid lines in white mark the accuracy level of 7 and 11 mE/

√

Hz,

respectively. The white dashed lines mark the interesting frequency band for GOCE.

The line at 0.019 Hz corresponds to a spherical harmonic degree 105. The peak

formed by the spectral density based on the global times series is caused by the

much higher noise of the SYNGPM1300S in this frequency band resulting from

the EGM96. A similar peak can also be found in the spectral density curve when

the noise simulation is based on the accuracy of the EIGEN-GL04C, the current

combination model from GRACE, altimetry, terrestrial and other data (Förste et al.,

2006). The line at 0.036 Hz corresponds to a spherical harmonic degree of 200.

Here, the signal of the tensor components approaches zero at GOCE altitude and

the noise decreases, too.

Quality Evaluation of GOCE Gradients 273

Fig. 4 Spectral densities of

simulated noise in V

of the

synthetic GPM based on the

global time series (light grey)

and based on the 26 tracks

(black)

Fig. 5 Spectral densities of

the remaining noise in V

from the combined solution

adding terrestrial data to the

GPM information. Scenario

1C in black, scenario 5C in

light grey

In Fig. 5 the spectral density of the remaining noise in V

based on the combined

solution using the GPM and terrestrial data with a noise level of 5 mGal (5C, light

grey) and with a noise level of 1 mGal (1C, black) are shown. The curves are clearly

closer to zero than the spectral density based on the pure GPM solution. In scenario

1C the required accuracy (the Laplace threshold of 11 mE/

√

Hz is indicated as upper

white solid line) can be reached. This holds for the other tensor components as well,

for detailed ﬁgures see Wolf (2007).

3 Generation of Quality Reports

To document the quality of the investigated GOCE gradients’ measurements, the

results of both analysis procedures discussed above are represented in dedicated

quality reports. Those reports are implemented according to the ESA (2006) and

shall be included in the frame of the CMF activities as auxiliary validation tool.

274 J. Müller et al.

The performance of the input data (tensor elements, position, velocity, time and

orientation information) is monitored by calculating and representing XO differ-

ences, parameters of the error model according to Sect. 1 and some statistical

measures. The double-computation of XOs that have already been computed in the

past is avoided by strict distinction of “new data” and “reference data”. Within the

“new data” XOs have to be searched in crossings of new ascending and descending

arcs. In addition, XOs also result from the combination of “new arcs” and already

validated “reference arcs”. While the ﬁrst group assesses the internal quality of the

newly obtained data, the second group validates all data on a long-term basis. Thus,

potential “wrong measurements” can be identiﬁed. The quality report is divided into

four parts:

(a) All essential data ﬁles in the computation ﬂow, the used reduction model (see

Jarecki et al., 2006) as well as applied thresholds are represented.

(b) The operational procedures in the calculation (e.g., selected models, algorithms,

codes) are documented.

ences of the six components of V

. To trace their quality behaviour, general

statistics of all XOs is illustrated. The differences of the gravity gradients at

each XO are also reported with respect to the contributing orbit.

(d) In addition, the fourth part summarizes statistics, where XO differences that

belong to a certain portion of the orbit (like each single arc) are analyzed. Here,

ascending and descending arcs as well as XOs from “new data” and “reference

data” are distinguished. Also a trend and several Fourier coefﬁcients (e.g., up to

degree four) are estimated. Moreover, a plot of each data set under evaluation

and its XOs is generated, see Fig. 6.

−150 −100 −50 0 50 100 150

−80

−60

−40

−20

XO−differences global [E]

Longitude [°]

Latitude [°]

3.2e−007

Fig. 6 Ground tracks of nine arcs of “reference data” (solid line) and two arcs of “new data”

(dashed l ine). The cross-overs (XOs) are indicated with asterisk-in-circle symbols that are gray-

coded with respect to the XO difference in V

Quality Evaluation of GOCE Gradients 275

Similar reports are generated for the differences of observed GOCE gradients and

reference gradients based on upward-continued terrestrial gravity data as described

in Sect. 2. There, (a) and (b) correspond to the XO case, (c) represents the quality of

the upward-continued terrestrial data, and (d) provides the relevant information for

the gradients’ differences.

4 Conclusions

Cross-over analysis is a helpful tool for monitoring the quality of the GOCE

gradients at the required accuracy level during data acquisition. In addition, a selec-

tion of so-called calibration parameters can be determined, where, however, the

investigated data span and the kind of the parameter (bias, trend, periodic term)

plays a signiﬁcant role.

From upward (and transformed) terrestrial gravity data, highly accurate reference

gradients in GOCE altitude can be obtained. Our study shows that the remaining

noise of the computed reference values of the tensor component V

is well below the

required accuracy level of 11 mE/

√

Hz for the Laplace condition when combining

recent global potential models with RTM reduced terrestrial gravity data having a

noise level of better than 1 mGal.

The performance of the GOCE gradients as obtained by applying the two dis-

cussed procedures is monitored in special quality reports. There, the latter method

delivers information about the overall spectral quality of the gradients, whereas the

ﬁrst method indicates an anomalous behaviour of the gradiometer.

References

Bouman J, Koop R (2003) Error assessment of GOCE SGG data using along track interpolation.

Adv. Geosci. 1, 27–32.

Bouman J, Koop R, Tscherning CC, Visser P (2004) Calibration of GOCE SGG data using high–

low SST, terrestrial gravity data and global gravity ﬁeld models. J. Geod. 78(1–2), 124–137.

Denker H (2003) Computation of gravity gradients over Europe for calibration/validation of GOCE

data. In: Tziavos IN (ed.), Proceedings of IAG International Symposium “Gravity and Geoid”,

Thessaloniki, Greece, Aug 26–30, 2002, pp. 287–292.

Denker H, Jarecki F, Müller J, Wolf J (2003) Calibration and validation strategies for the

gravity ﬁeld mission GOCE. Status Seminar “Observation of the System Earth from Space”,

Geotechnologien Science Report, No. 3, Potsdam, pp. 32–35.

ESA (2006) GOCE Calibration & Validation Plan for L1b Data Products. EOP-SM/1363/MD-md,

ESTEC, Nordwijk.

Förste C, Flechtner F, Schmidt R, König R, Meyer U, Stubenvoll R, Rothacher M, Barthelmes

F, Neumayer KH, Biancale R, Bruinsma S, Lemoine J-M (2006) A mean global gravity ﬁeld

model from the combination of satellite mission and altimetry/gravimetry surface gravity data.

Geophy. Res. Abstr. 8, 03462.

Huber PJ (1981) Robust Statistics, John Wiley, New York.

Jarecki F, Wolf KI, Denker H, Müller J (2006) Quality assessment of GOCE gradients. In:

Flury J et al. (eds.), Observation of the Earth System from Space, Springer, New York,

pp. 271–285.

276 J. Müller et al.

Jarecki F, Müller J (2009) Robust trend estimates from GOCE SGG satellite track cross-over dif-

ferences. In: Sideris M (ed.), Observing Our Changing Earth. IAG Symposia 133, Springer,

New York, pp. 363–370.

Koop R, Bouman J, Schrama EJO, Visser P (2002) Calibration and error assessment of GOCE

data. In: Adam J et al. (eds.), Vistas for Geodesy in the New Millenium. IAG Symposia 125,

Springer, New York, pp. 167–174.

Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko

SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR

(1998) The Development of the Joint NASA GSFC and NIMA Geopotential Model EGM96,

NASA/TP-1998-206861. Technical Report, NASA, Greenbelt, MD.

Pail R (2003) Local gravity ﬁeld continuation for the purpose of i n-orbit calibration of GOCE SGG

observations. Adv. Geosci. 1, 11–18.

Reigber C, Schmidt R, Flechtner F, König R, Meyer U, Neumayer K-H, Schwintzer P, Zhu SY

(2005) An earth gravity ﬁeld model complete to degree and order 150 from GRACE: EIGEN-

GRACE02S. J. Geodynamics 39(1), 1–10.

SCVII (2000) Simulation Scenarios for Current Satellite Missions. IAG Special

Commission VII, http://www.geod.uni-bonn.de/apmg/lehrstuhl/simulationsszenarien/sc7/

satellitenmissionen.php (July 13, 2006).

Wenzel H-G (1999) Schwerefeldmodellierung durch ultra-hochauﬂösende Kugelfunktion-

smodelle. ZFV 124(5), 144–154.

Wolf KI (2007) Kombination globaler Potentialmodelle mit terrestrischen Schweredaten für

die Berechnung der zweiten Ableitungen des Gravitationspotentials in Satellitenbahnhöhe.

Deutsche Geodätische Kommission, Reihe C, Nr. 603.

Validation of Satellite Gravity Field

Models by Regional Terrestrial Data Sets

Johannes Ihde, Herbert Wilmes, Jan Müller, Heiner Denker,

Christian Voigt, and Michael Hosse

1 Introduction

The gravity ﬁeld mission GOCE aims at providing the global geoid and gravity

anomalies with accuracies of about 1–2 cm and 1 mGal (1 mGal = 10

–5

m/s

respectively, both at a spatial resolution of 100 km (ESA, 1999). For this purpose, a

three-axis gravity gradiometer and high-low satellite-to-satellite-tracking from GPS

satellites are combined to derive the gravity ﬁeld information. In this respect, many

error sources affect the ﬁnal GOCE products, and thus the calibration (procedure

to achieve “correct” observations) and validation (procedure to assess the quality

of end products) are essential tools for reaching the mission goals (Bouman and

Koop, 2003). Calibration and validation (cal/val) includes on-ground and in-orbit

techniques with internal and external (independent) data.

Within the GOCE-GRAND II project, internal and external cal/val procedures

were investigated. Internal cal/val studies included the use of cross-over differ-

ences for accuracy and integrity monitoring (chapter “Quality Evaluation of GOCE

Gradients” by Müller et al., 2009, this book; Jarecki and Müller, 2007) as well as the

utilization of the integrals of motion (Löcher and Ilk, 2007). External cal/val pro-

cedures were developed using terrestrial gravity ﬁeld data and ocean data (chapter

“Comparison of GRACE and Model-Based Estimates of Bottom Pressure Variations

Against In Situ Global Bottom Pressure Measurements” by Stammer et al., 2009,

this book). Further investigations involved upward continued regional gravity ﬁeld

data for the external calibration and validation of the GOCE gravity gradient obser-

vations (chapter “Quality Evaluation of GOCE Gradients” by Müller et al., 2009,

this book; Wolf, 2007).

In this contribution, the emphasis is put on the development of highly accurate

terrestrial data sets for the external validation of the GOCE products. This was also

one of the main issues of the regional validation and combination experiment car-

ried out in Germany within the framework of the GOCE-GRAND II project. In this

J. Ihde (B)

Bundesamt für Kartographie und Geodäsie (BKG), D-60598 Frankfurt am Main, Germany

e-mail: johannes.ihde@bkg.bund.de

277

F. Flechtner et al. (eds.), System Earth via Geodetic-Geophysical Space Techniques,

Advanced Technologies in Earth Sciences, DOI 10.1007/978-3-642-10228-8_22,



Springer-Verlag Berlin Heidelberg 2010

278 J. Ihde et al.

subproject, two extensive measurement campaigns were performed in order to inter-

nally validate the terrestrial data sets and to provide ground truth data for the GOCE

mission results.

The Earth’s gravity ﬁeld can be completely described by the gravity vec-

tor g and the (scalar) gravity potential W, which after linearization leads to

the disturbing potential T and the gravity anomaly and gravity disturbance vec-

tors g and δg, respectively. The disturbing potential T can be derived from

the solution of the geodetic boundary value problem or from GNSS/levelling in

discrete points. In this context, the spectral sensitivity, the data spacing and cov-

erage as well as the noise characteristics of different gravity ﬁeld parameters are

important.

Mostly terrestrial gravity observations and terrain data are available with high-

est resolution. In Germany, more than 260,000 gravity values exist in data bases.

The area coverage varies between one point per 1 km

to one point per 25 km

The random error is in general smaller than 0.1 mGal and reaches values up to 0.01

mGal for 1st order stations. The existing gravity data bases include measurements

from different epochs and instruments and also the data processing was not done

in a homogeneous way. Hence, the real accuracy, especially at long wavelengths,

is largely unknown. This was also the main reason for absolute gravity control

observations at about 100 points over the territory of Germany with an absolute

gravity meter A-10. In Sect. 2, the new absolute gravity observations are analyzed

and the differences to the existing gravity values from the national data base are

reported.

Section 3 describes the GPS/levelling control points available for Germany. The

data set includes about 900 GPS/levelling stations, which are more or less homo-

geneously distributed with spacing of 20–30 km. The GPS/levelling control points

are employed for the evaluation of gravimetric geoid computations as well as in

combination solutions. An evaluation of different global geopotential models by the

national GPS/levelling data set is given at the end of Sect. 3.

Section 4 deals with the development of gravimetric quasigeoid models for

Germany with a resolution of about 2 km and the cross-validation with the national

GPS/levelling data.

Section 5 includes the results of the second extensive measurement cam-

paign, consisting of about 300 astrogeodetic vertical deﬂections observed with the

zenith camera system TZK2-D. The astrogeodetic results are cross-validated with

GPS/levelling data and gravimetric quasigeoid models and the differences between

low-pass ﬁltered astrogeodetic vertical deﬂections and various global geopotential

models are presented. As the astrogeodetic vertical deﬂections are completely inde-

pendent of all other gravity ﬁeld data sets, they are quite valuable for manifold

validation purposes.

A general problem of the comparisons between point observations and global

geopotential models is the different spectral and spatial resolution and sensitivity.

In Sect. 6, a validation procedure is proposed for the comparison of point obser-

vations with global gravity ﬁeld models, employing a ﬁlter technique based on

wavelets.

Validation of Satellite Gravity Field Models 279

2 Gravity Data

High quality and high resolution regional gravity data sets are important inde-

pendent data sources for the validation of the GOCE products. Furthermore, the

combination of the terrestrial data with the GOCE global gravity ﬁeld models will

allow to determine the complete gravity ﬁeld spectrum (all wavelengths) with high

accuracy (about 0.01 m for geoid and 1 mGal for gravity). In this connection,

a thorough check of the existing gravity data sources with respect to systematic

errors (e.g., errors due to the used gravity reference stations, network structure or

gravimeter calibrations) and random errors is mandatory (Denker et al., 2007).

Within the project GOCE-GRAND II, Germany is used as a test area for vali-

dation and combination experiments. About 260,000 terrestrial gravity observations

with a spacing of 2–5 km are available for the territory of Germany at the Bundesamt

für Kartographie und Geodäsie (BKG), Frankfurt am Main, and the Institut für

Erdmessung (IfE), Leibniz Universität Hannover (see Fig. 1, left). In Germany, a

gravity base network (DSGN94) exists, which includes 30 station groups observed

with FG-5 absolute gravity meters in 1994/1995 (Torge et al., 1999). This network

was densiﬁed in the Eastern German states with relative gravimeters by the state sur-

vey agencies, yielding the ﬁrst order network DHSN96 with stations about 30 km

apart. In the Western German states older and already existing relative gravity mea-

surements of the DHSN were transformed to the level of the DSGN94 and both are

forming the DHSN96.

Fig. 1 Left: Terrestrial gravity data (thin dots) and GPS and levelling control points (thick dots)

for Germany; Right: Gravity control points in Germany observed with a bsolute gravity meter A-10

(triangles) and astrogeodetic proﬁles (thick lines)

280 J. Ihde et al.

Then the 2nd and 3rd order relative networks of the German federal states were

mounted to the DHSN96 network. Unfortunately, many of the other available ter-

restrial gravity data suffers from clear information about the gravity and position

reference systems.

In order to check the quality of the terrestrial gravity data base for Germany, spot-

checks at 100 evenly distributed stations (see Fig. 1, right) were performed by BKG

with two A-10 absolute gravity meters. Purposes of these measurements were detec-

tion of systematic errors and level differences within the existing terrestrial gravity

data ( mainly within 1st order gravity network), because all further measurements

are dependent on this hierarchy.

The selected stations are spaced about 60–80 km and cover the entire territory of

Germany. Stations are mostly part of the DHSN96 1st order gravity network and all

of them are ﬁeld stations. Hence, an absolute gravity meter of type A-10 had to be

employed, as it is the only instrument which can work under ﬁeld conditions (the

more accurate FG-5 instruments works only under laboratory conditions).

The A-10 accuracy was speciﬁed by the manufacturer as 10 μGal (1 μGal =

–8

–2

) for an observation time of about one hour, but it was found that it can be

increased by the following special measurement procedure and certain accuracy cri-

teria: On each site, two instrument set-ups in opposite directions (North and South)

were used to check for gross errors. The differences between both set-ups should

not exceed a value of 8 μGal. Furthermore, the entire observation procedure was

examined and optimized with respect to the number of measured drops and sets as

well as a proper instrument setup, wind shielding and power supply.

The frequencies of the HeNe-laser of the A-10 gravimeter were checked on

a regular basis at BKG using an iodine stabilized HeNe-laser and the Rubidium

frequencies were compared with the Cs-frequency standard at BKG to monitor

long-term stability of instrumental standards.

In order to check the accuracy of the A-10 instruments over the total project

time span, comparisons were done at three stations of the German GREF net-

work, that were measured with FG-5 periodically and, more importantly, with FG-5

results at the reference station in Bad Homburg before and after each particular ﬁeld

campaign. The gravity station Bad Homburg serves as a reference station with a

highly precise absolute gravity value due to periodic FG-5 observations with differ-

ent instruments and availability of long-term time series from two superconducting

gravity meters.

The synopsis of all above described approaches and more than 25 existing com-

parisons at the reference station Bad Homburg conﬁrm the accuracy speciﬁcation of

the manufacturer and, moreover, prove the accuracy increase to better than 10 μGal

for most of the occupied ﬁeld stations.

Between July 2006 and September 2008 a total of 13 ﬁeld campaigns were car-

ried out. Supplementary to the A-10 observations, the vertical gravity gradients

were measured using a Scintrex CG-5 relative gravity meter and a tripod with spe-

cial design. The gradient measurements became necessary in order to transfer the

observed absolute gravity values (sensor height about 0.7 m) to the corresponding

bench marks without loss of accuracy. On some locations the absolute gravity meter