3 Measurement systems, models and data
Level 2 products
3.3 Data
• CHAMP, Swarm A+B+C, and the Earth observation satellites TerraSAR-X18and TanDEM-X19. GRACE contributes to low- and medium-resolution spectral content approximately up to degree 150, whereas the significant GOCE information predominates from degree 120 on.
3.2.3 Regional model: GCG2011
The German Combined QuasiGeoid 2011 (GCG2011) is the official German height reference of the ”Arbeits-gemeinschaft der Vermessungsverwaltungen der Länder der Bundesrepublik Deutschland (AdV)“ describing quasigeoid heights, according to Eq. (2.63) w.r.t. the reference ellipsoid GRS80.
It is provided in terms of a 1′ longitude times 1.5′latitude geographic grid byBKG (2011). The corresponding spatial reso-lution yields about 1.8 km × 1.7 km; the position coordinates refer to European Ter-restrial Reference System 1989 (ETRS89) (Boucher and Altamimi, 1992). The quasi-geoid heights in Fig. 3.8 are given in the ”Deutsches Haupthöhennetz (DHHN) 1992“ (Weber, 1994). The DHHN is the pri-mary leveling network of Germany. The res-olution counts 1 mm, the accuracy reaches 1 cm to 2 cm and varies depending on (1) geographical structures and (2) data avail-ability. Over the Alps, the values are less accurate with 3 cm to 4 cm, and over the North and Baltic Sea even worse with 4 cm to 10 cm. The GCG2011 is obtained by aver-aging two independent solutions from BKG and IfE (Institut für Erdmessung der Leib-nitz Universität Hannover). Whereas BKG uses an adjustment approach based on point masses, IfE applies integration and colloca-tion. Based on a new release DHHN2016, a new release GCG2016 is planned, as well
(AdV, 2014). Figure 3.8: German combined quasigeoid 2011; source:BKG(2011).
Figure 3.9: Distribution of the observations in the test areaNorthern Germany(green bordered): satellite altimetry (dark green), shipborne (red), airborne (orange), and terrestrial data (yellow). Thin black lines mark rivers and canals.
in Fig. 3.9, further encompass gravity measurements from the German gravity archive and from Deutsches Geodätisches ForschungsInstitut (DGFI), department 1+2 (nowadays DGFI-TUM and Institut für angewandte Geodäsie (IfAG), Frankfurt), taken in the 1950s.
The BKG homogenized the terrestrial data for the computation of the GCG2011 and provided a consistent data set, in the following denoted as ”BKGterr“. It contains in total 31 703 gravity valuesgreferenced to IGSN71, at observation sites with ellipsoidal coordinatesλ, βin the ETRS89 w.r.t. GRS80, and normal heightsHnorm, as defined in Eq. (2.81), w.r.t. DHHN92.
The spatial resolution of the observation sites yields<1 km inMecklenburg-West Pomeraniaand Schleswig-Holstein. InLower Saxonydata gaps of several kilometers occur and lead to a worse spatial resolution. An average point distance of 5 km is reached for the three federal states (Lieb et al., 2016).
Pre-processing
In order to make the data set usable for the regional gravity modeling approach, the absolute gravity values ggiven at observation sites P(ellipsoidal coordinatesλ, β) with potentialWP, are transformed into gravity anomalies∆g(cf. Fig. 2.13) at the referring locations described by spherical coordinatesλ, ϕ. The following steps are performed for each observation point P:
(1) computing normal gravityγQ(Q)withUQ =WP at heightHnorm, cf. Eq. (2.51), (2) subtractingγQfromgPin order to obtain gravity anomalies∆g, cf. Eq. (2.67), (3) transforming ellipsoidal to spherical coordinates(λ, β,Hnorm)→(λ, ϕ,r ), cf. Eq. (2.20).
Note: the normal height Hnorm of an observation sitesP (geopotential valueWP) refers to the quasigeoid.
The distance along the ellipsoidal normaln′can be counted from the reference ellipsoid up to the according telluroid pointQwithUQ =WP, as visualized in Fig. 2.16. Consequently, after coordinate transformation, the spherical coordinates relate to telluroid points Q(λ, ϕ,r ) and the values∆g refer to reference ellipsoid GRS80.
3.3.2 Shipborne data set
The terrestrial data set provided by BKG originally also contained off-shore data in the Baltic Sea. They stem from a prototype of gravimetric shipborne measurements after A. Graf and were collected in the 1950s (Neunhöfer et al., 1997). The federal stateMecklenburg-West Pomeraniareprocessed the data at sea level.
BKG integrated the data within the GCG2011 model and provided the valuesgat observation sitesP(λ, β,0.0).
As the measurement technique of those off-shore data, their pre-processing, their corresponding spectral and
spatial resolution, and the time epoch entirely differ from those of the terrestrial observations, they are separated from the latter according to their heightHnorm, which was set to zero by BKG. Consequently, the values are given on the quasigeoid, cf. Fig. 2.16. The spatial resolution of the generated data set ”BKGship“ averages 4 km. The measurement accuracy is assumed to be low due to the early period of shipborne gravimetry (Lieb et al., 2016). Detailed information is not available.
Pre-processing
The pre-processing of the shipborne data at locationsP(λ, β,Hnorm)with potentialWP comprises:
(1) computing normal gravityγQ(Q)(withUQ=WP) at a height of 0.0 m, i. e. at the reference ellipsoid,
cf. Eq. (2.51), (2) subtractingγQfromgPresulting in gravity anomalies∆g, cf. Eq. (2.67), (3) transforming ellipsoidal to spherical coordinates(λ, β,0.0)→(λ, ϕ,r ), cf. Eq. (2.20).
The gravity anomalies∆gat positions(λ, ϕ,r )refer to the reference ellipsoid GRS80.
3.3.3 Airborne data sets
The airborne gravity data stem from two flight campaigns, operated by Danish National Space Center (DNSC):
”BALGRACE06“ (BG06) over theBaltic Seain 2006 and ”NORTHGRACE07_08“ (NG0708) over theNorth Sea in 2007/2008. The gravity data, provided in terms of gravity disturbances δg, refer to the ISGN71.
Reference system of the according ellipsoidal coordinatesλ, βof the observation sites along the flight tracks is ETRS89 (reference ellipsoid GRS80); reference system of the according normal heights Hnorm at flight altitude (around 30 m) is DHHN92.
Both data sets BG06 (6834 observations) and NG0708 (6063 observations) are pre-processed by DNSC and compared with terrestrial and shipborne gravity data by BKG. Due to flight turbulence during the first cam-paign in 2006, some observations show larger differences. Removing those outliers results in a reduced data set BG06red (6677 observations).
Within the framework of homogenizing the measurements for their contribution to GCG2011, BKG did further embracing evaluations. As a result, both data sets were reduced by a constant offset of−2.0 mGal (BG06), and−0.6 mGal (NG0708), respectively.
The east-west oriented flight tracks in Fig. 3.9 are related to BG06, the north-south oriented flight tracks to NG0708. The along- and cross-track spatial resolution averages 10 km (Lieb et al., 2016).
Pre-processing
Due to the comprehensive pre-processing of the airborne data by BKG and DNSC, only transforming the ellipsoidal to spherical coordinates (λ, β,Hnorm) → (λ, ϕ,r ), cf. Eq. (2.20), is necessary. The gravity disturbance valuesδggiven at normal heightsHnormrefer to telluroid pointsQ(λ, ϕ,r )w.r.t. GRS80.
3.3.4 Altimetry data
The altimetry data originate from satellite missions of different agencies as listed in Tab. 3.3. DGFI-TUM provides a broad variety of data products at OpenADB ”openadb.dgfi.tum.de“, derived from the official level 2 GDRs (Geophysical Data Record). The instantaneousDOTvaluesDOTi, according to Eq. (3.15) are obtained from cross-calibrated range measurements, referring to 1 HzSSHdata from MMXO14. The geoid undulations N are obtained from the SH model GOCO03s. All data are referenced to the ellipsoid ”TOPEX/Poseidon“, cf. Tab. 3.6.
The sampling rate defines the along-track resolution of each satellite mission. 1 Hz data ensure a balanced signal to noise ratio. It corresponds to about 7 km along-track spatial resolution at the Earth’s surface. The cross-track resolution depends on the orbit configuration of each mission. Table 3.3 lists the corresponding mean values. Due to the meridian convergence with increasing latitude, the spatial along- and cross-track resolution of the altimeter missions further depends on the geographic region.
Pre-processing
For each point P(λ, β,SSHi) of the different altimetry data sets, the following pre-processing procedure is applied:
(1) subtracting instantaneousDOTi values fromSSHidata (resulting in geoid undulationsNi),
cf. Eq. (3.13), (2) transformingh′from the TOPEX to the GRS80 reference ellipsoid:
h′(GRS80) =h′(TOPEX)−dh′,
cf. Eq. (2.22), (3) computing normal gravityγon the ellipsoid ath′=0 (γaandkw.r.t. GRS80), cf. Eq. (2.51), (4) transforming ellipsoidal to spherical coordinates(λ, β,h′)→(λ, ϕ,r ), cf. Eq. (2.20), (5) computing disturbing potentialT withNi from (1) andγfrom (3), cf. Eq. (3.14).
Consequently, a consistent data set of valuesT is obtained for each pointP(λ, ϕ,r )w.r.t GRS80.
3.3.5 GOCE SGG data
The GOCE SGG data refer to the final release of Level 2 products ”EGG_NOM_2“, cf. Tab. 3.5, from March 06, 2014. They can be accessed from ESA through the GOCE Virtual Online Archive (VOA) http://eo-virtual-archive1.esa.intand are provided in a time series (one day temporal coverage) from Nov 1, 2009 until Nov 11, 2013 along the GOCE orbit. The spatial resolution averages 8 km along track, derived from 1 Hz spectral resolution.
The nominal data given in GRF are chosen, as a general aim of this work is, to keep all data in their most original, untouched, non-preprocessed mode. Within the product ”EGG_TRF_2“, the tensor rotation would transfer model information to the originally observed tensor elements (Fuchs and Bouman, 2011).
Consequently, the tensor rotation has to be applied vice versa in this study: the GOCE observation equations (see next chapter, Tab. 4.7) of the estimation model, which is set up in Sec. 5.2, comprise a rotation of the GGs from the Earth-bound LNCS into the GRF in order to keep most valid GOCE information.
Pre-processing
Due to errors in the long wavelengths and increasing errors in the high frequencies, GOCE measured gravity gradients are band-pass filtered from 7.5 mHz to 100 mHz, followingFuchs and Bouman(2011). In order to reduce the anomalous signal in theVy y component close to the magnetic poles, the according values are band-pass filtered from 15 mHz to 120 mHz. The signal below the MBW is replaced by model GGs derived from the global SH model GOCO03s. The model is low-pass filtered with the complement of the band-pass filter.
Outliers in the four accurate GGs are eliminated using a 3.5-sigma threshold of the along track standard deviation w.r.t. GOCO03s. The less accurate components remain with the approximately 200 times higher noise level.
The total observation period is split into three parts due to different accuracies and resolutions of the data:
11/2009 - 02/2010 less accurate data due to the erroneous on-board Central Processing Unit (CPU)-A side (a switch to CPU-B side caused impacts on the error characteristics; the residuals ofVz z, for instance, become around 1.5 times smaller (Bouman et al., 2014)),
03/2010 - 07/2012 nominal phase at a mean altitude of 255 km,
08/2012 - 11/2013 lower orbit phase (higher sensitivity of the gradiometer due to a step-wise orbit lowering, down to around 225 km).
Consequently, each of the three data sets contains observation sitesP(λ, ϕ,r )in the GRF with the following information:
Vx x,Vy y,Vz z,Vx y,Vx z,Vy z GGs in the GRF,
λ, ϕ,r spherical coordinates,
r11,r12,r13,r21,r22,r23,r31,r32,r33 components of the rotation matrixRLNCSGRF .
3.3.6 GRACE level 2 data
GRACE measurements are not directly integrated in the regional gravity field modeling approach presented in this work. Observation equations would have to be set up for GRACE measurements processed by one of the strategies (a) - (e), presented in Sec. 3.1.5. However, those processed data are not publicly available. Thus, instead, gravitational potential differences according to Eq. (2.61) are computed from level 2 SH products.
The GRACE release 05 data are available from April 2002 up to present in terms of monthly solutions from GFZ (Dahle et al., 2012). The SH coefficientsCl,m,Sl,mare provided up to degree and order 90 athttp://
icgem.gfz-potsdam.de/ICGEM/shms/monthly/gfz-rl05/and refer to the normal potential parameters a=6 378 136.460 m,GM =3 986 004.415×108km3/s2.
Pre-processing
From those coefficients, gravitational potential values V (xi) and V (xii) are computed at positions xi = (λi, ϕi,ri)T andxii = (λii, ϕii,rii)T of the two GRACE satellites (i),(ii) along their orbits with a sampling rate of 5 s. Gravitational potential differences∆V then are obtained between each two neighboring positions:
(1) computing gravitational potential valuesV (xi)andV (xii)from SH coefficients with corresponding normal potential valuesa,GM,
cf. Eq. (2.40), (2) computing gravitational potential differences∆V (xi,xii), cf. Eq. (2.61).
The time span from Nov 2009 until Nov 2012 is pre-processed, related to the availability of GOCE data, in order to generate an overlapping time period. The GRACE data set contains the values∆V (λi, ϕi,ri, λii, ϕii,rii).
Overview of pre-processed data sets
The pre-processed and homogenized data sets, witch will be used in the sequel of this work, are summarized in Tab. 3.7.
Table 3.7: Overview of pre-processed data sets.
number type name functionalY[ ˜V] coordinates height normal potential
[1] terr BKGterr ∆g λ, ϕ,r Hnorm GRS80
[2] ship BKGship ∆g λ, ϕ,r Hnorm GRS80
[3] air BG06 δg λ, ϕ,r Hnorm GRS80
[4] air NG0708 δg λ, ϕ,r Hnorm GRS80
[5] alti ERS-1e T λ, ϕ,r h′ GRS80
[6] alti ERS-1f T λ, ϕ,r h′ GRS80
[7] alti TOPEX T λ, ϕ,r h′ GRS80
[8] alti Jason-1 T λ, ϕ,r h′ GRS80
[9] alti Envisat T λ, ϕ,r h′ GRS80
[10] alti Jason-2 T λ, ϕ,r h′ GRS80
[11] alti Cryosat T λ, ϕ,r h′ GRS80
[12] goce GOCE0911-1002 (Vab) λ, ϕ,r
[13] goce GOCE1003-1207 (Vab) λ, ϕ,r
[14] goce GOCE1208-1309 (Vab) λ, ϕ,r
[15] grace GRACE0911-1211 ∆V λi, ϕi,ri, λii, ϕii,rii)