Journal of Geodetic
Science
• 2(2) • 2012 • 144-155 DOI: 10.2478/v10156-011-0033-8 •
An evaluation of recent GOCE geopotential models in Brazil Research article G. N. Guimarães∗ , A. C. O. C. Matos, D. Blitzkow
Laboratory of Surveying and Geodesy, Department of Transportation - University of São Paulo, Postal Code 61548, São Paulo, São Paulo, Brazil
Abstract: Several global geopotential models based on Gravity eld and steady-state Ocean Circulation Explorer (GOCE) data have been published in the last two years. Some of these models use combinations of different satellite missions, while others use only GOCE data. This paper presents the evaluation and analysis of each approach using GOCE data in the Southeast of Brazil. Two assessments have been made. We compared the geoid heights derived from GOCE-based models with the geoidal heights from 176 GPS stations on leveling benchmarks. The ndings show an improvement in GOCE-based models TIM_R3 (0.40 m) and DIR_R3 (0.39 m) for degree and order 210 in relation to EGM2008 (0.44 m) in terms of RMS. For the other models the results did not exceed 0.44 m. The second evaluation reports the comparison in terms of gravity disturbances between terrestrial gravity data and the models. The results, in terms of RMS and up to degree and order 210, indicate slightly low GOCO 02S values (10.34 mGal), TIM_R2 (10.37 mGal) and TIM_R3 (10.47 mGal) compared to EGM2008 (10.66 mGal). We also applied the residual terrain model and, as a result, the RMS errors were reduced by ∼35% (∼6.0 mGal) in the entire area and by ∼45% in the mountain region. Keywords: GOCE • validation • EGM2008 • spherical harmonic coefficients • terrestrial data © Versita sp. z o.o. Received 10-01-2012; accepted 17-06-2012
1. Introduction
Earth’s static gravity eld (e.g. Visser et al. 2002; Drinkwater et al. 2003). It was launched on 17 March 2009, and, according to Rum-
In the past few years satellites dedicated to gravity eld observa-
mel (2005), the main objective of the GOCE mission is to obtain a gravity eld model at the ∼ 1–2 cm accuracy level for geoid undu-
tions, such as CHAMP (Challenging Minisatellite Payload), GRACE (Gravity Recovery and Climate Experiment), and GOCE (Gravity eld and steady-state Ocean Circulation Explorer), have allowed detailed study of the Earth’s gravitational eld. As a result, several Global Gravitational Models (GGMs) have been published. These models are expected to contribute to the improvement of the accuracy of geoid modeling. Some of the latest available models share a common feature: they combine data from different satellite
lations and at the 1–2 mGal level for gravity anomalies. At the time of the GOCE launch, a geodetic project was started in São Paulo, Brazil, supported by the State of São Paulo Research Foundation (FAPESP). One of the aims of this project is to increase the density of gravity measurements in the state of São Paulo and its surroundings. To aid this effort, much eld work has been carried out to ll the gaps in coverage across the state.
missions.
Using this geodetic project and on the recent GOCE-based GGMs, this paper aims to assess all of the models using GOCE data cur-
The last satellite launched, GOCE, was developed by the European
rently published, in the southeast of South America—speci cally, in the State of São Paulo, Brazil, and its surroundings.
Space Agency (ESA) and deploys gravity gradiometry in space to produce homogeneous, highly accurate, near-global maps of the
∗
E-mail:
[email protected]
In this study, two analyses were performed. The rst analysis compared GOCE-based GGMs and 176 GPS points on leveling benchmarks. In the second analysis, we compared the “observed” gravity
Journal of Geodetic disturbances with the model values from GOCE-based GGMs and EGM2008.
Table 1.
This study evaluates a total of 12 GOCE-based models, available at the International Centre for Global Earth Models (ICGEM). We also included EGM2008 in our analysis to identify any improvement with respect to it. In the following we provide a brief description
and a space-wise solution (SPW) (Migliaccio et al., 2010). The differences among these solutions are the processing strategies applied and the level of a priori knowledge introduced. Table 1 shows their data periods. The DIR approach is based on the least-squares solution of the inverse problem (Pail et al., 2011), and all associated releases go to degree and order (d/o) 240. The data reduction procedure is a combination of the normal equations for going from GPS satellite to satellite tracking (SST) observations and normal equations for satellite gravity gradiometry (SGG) observations. In the third approach, the GOCE-SSG normal equation was fully combined with a GRACE normal equation. In order to improve the gravity eld solution, very-low degree harmonics (degree 2 and 3) were estimated using LAGEOS-1 and LAGEOS-2 normal equations over the same period as GRACE. The TIM is the only solution based solely on GOCE data. It also has three releases so far, the rst of which goes to d/o 224 while the other two go to d/o 250. This solution considers the gravity gradient and orbit observations (orbital kinematics) as time series measured along the satellite orbit. According to Pail et al. (2011), this is especially bene cial considering the highly correlated gravity gradient observations. This approach allows independent evaluation of GOCE’s abilities. The solution may be compared directly with complementary gravity eld information and potential insufficiencies can be detected (Pail et al., 2011).
Data periods of solutions and releases
DIR_R3 DIR_R2 DIR_R1 TIM_R3 TIM_R2 TIM_R1 SPW_R2 SPW_R1
2.1. GGMs
on GOCE data have been available to users: a direct solution (DIR) (Bruinsma et al., 2010), a time-wise solution (TIM) (Pail et al., 2010a)
145
Solution From d/m/y To d/m/y Number of days
2. Data Sets
of each solution, though for the sake of legibility some product names will be abbreviated. Since July 2010, three solutions based
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01/11/2009 01/11/2009 01/11/2009 01/11/2009 01/11/2009 01/11/2009 30/10/2009 30/10/2009
19/04/2011 30/06/2010 11/01/2010 17/04/2011 05/07/2010 11/01/2010 05/07/2010 11/01/2010
536 242 72 534 247 72 248 74
number represents the release (e.g. “DIR_R3” is the third release from the direct solution by Bruinsma et al. (2010)). The newest EIGEN solution is divided into EIGEN-6S and EIGEN-6C (Förste et al., 2011). The rst one is a satellite-only gravity eld of a maximum d/o 240. It consists of 6.5 years of LAGEOS (SLR) and GRACE (GPS-SST and K-band range rate) data and 6.7 months of GOCE (satellite gradiometry) data. The second solution is a combined global gravity eld of maximum d/o 1420. It also used the same data from EIGEN-6S, as well as the DTU2010 (Andersen, 2010) global gravity anomaly data set obtained from altimetry and gravimetry. The solution was obtained from one full normal equation to d/o 365 and a block diagonal solution (for the terrestrial data only) to d/o 1420. The GOCO (Gravity Observation COmbination) is a project initiative following the framework of ESA’s GOCE Data AO, aiming to compute high-accuracy and high-resolution static global gravity eld models. The rst solution, GOCO01S (Pail et al., 2010b), was published to d/o 224 and it used GOCE and GRACE satellite data. The GOCO02S (Goiginger et al., 2011) consists of 8 months of GOCE, 7 years of GRACE, 8 years of CHAMP and 5 years of SLR and it was developed up to d/o 250. The last model used in this paper is the EGM2008 (Pavlis et al., 2008). It is based on ITG-GRACE03S (57 months of GRACE data) altimetric and terrestrial data, making it the only model in this paper not to use GOCE data. The model was fully developed to degree 2159 in ellipsoidal harmonics. Conversion from ellipsoidal to spherical harmonics preserves the order but not the degree. This is why the model extends to degree 2190 but order 2159 (Arabelos and Tscherning, 2010).
The SPW solution has two available releases, one up to d/o 210 and the other up to d/o 240. It is based on a collocation solu-
2.2.
tion (Tscherning, 2001), the aim of which is to estimate the spherical harmonic coefficients of the geopotential model by exploiting
We used the national gravity data set of Brazil (Blitzkow et al., 2010).
spatial correlations in the terrestrial gravity eld (Pail et al., 2011). The solution makes use of both satellite tracking data derived from
The study area consists of 46,290 stations (Figure 1) kindly provided by the National Observatory, the Brazilian Oil Company, the Brazil-
the onboard GPS receiver and gravity gradients observed by the
ian Institute of Geography and Statistics, the Institute of Astron-
onboard electrostatic gradiometer, uses kinematic orbits for SST gravity eld recovery, and uses reduced dynamic orbits for geo-
omy, Geophysics and Atmospheric Science, and the School of Engineering of the University of São Paulo. It is important to mention
locating gravity gradients. In order to simplify the notation of results, the following names will be used such that the rst three let-
that the FAPESP thematic project contributed signi cantly towards the availability of the terrestrial gravity data used, especially in the
ters represent the solution’s name and the letter R followed by a
State of São Paulo. Subsets of the data also cover the neighboring
Terrestrial data
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Figure 1.
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Location of 46,290 gravity stations over the Southeast of Brazil.
Figure 3.
Standard deviation of heights.
Out of this total, 139 stations belong to the Laboratory of Geodesy and Topography of the USP (USP-LTG). The ellipsoidal height accuracy is about 0.06m (Sa, 2006) and it is not possible to de ne the orthometric height, since the network was not adjusted. The other 37 stations were provided by the IBGE, and 26 of them are included in the latest Brazilian altimetric adjustment (IBGE, 2011). Figure 3 illustrates the standard deviations of both quantities. In terms of orthometric height, the standard deviations vary between 0.06 and 0.07 m, while the accuracy of ellipsoidal heights ranges from a few millimeters to 0.10 m. The accuracy estimated for the geoid undulations from GPS/leveling is 0.09 m, and the maximum value is 0.17 m. 3. Comparison in Southeast Brazil
Figure 2.
Location of 176 GPS/leveling over the State of São Paulo.
countries of Paraguay and Argentina. With an area of more than one million km2 , this data set is large enough to provide feedback on the GOCE models. The accuracy of the Brazilian terrestrial gravity data is ∼ 0.1 mGal or better (Blitzkow et al., 2010). In the São Paulo, Paraná, and Santa Catarina states, the spatial resolution of the data is ∼5–8 km; in the northwest and northeast, the resolu-
As described in the introduction, two assessments were carried out in this study to validate GOCE-based GGMs. The study area extends from 56–42 ◦ W longitude and 28–17 ◦ S latitude. The study region is shown in the context of South America in Figure 4. This area includes all of the State of São Paulo (shown inside the rectangle area), as well as some of its and surroundings. The models were computed using the International Centre for Global Earth Model (ICGEM) website’s “Calculation Service”, avail-
tion is about 10 km, with some gaps.
able at http://icgem.gfz-potsdam.de/ICGEM/ICGEM. html. A regular 5’ x 5’ grid was employed, with WGS84 as the ref-
As second data set, we selected 176 GPS leveling stations over the State of São Paulo (Figure 2). The spirit leveling was carried out
erence system and “tide free” as the tide system. The computed functions were the geoid and the gravity anomaly. Table 2 shows
by the Brazilian Institute of Geography and Statistics (IBGE) and
a summary of the spherical harmonic degrees used for each model computation.
the Institute of Geography and Cartography (IGC) (formerly known as the Institute of Geography and Geology (IGG)). The orthometric
(nmax)
The omission error (σomission ) ensues from the discreteness of or
heights are referred to the Imbituba tide gauge local height datum, and the ellipsoidal heights are in reference to the WGS84 (World
lack of resolution in the gravimetric data. This error of omission is composed of high-frequency gravity eld signals. No gravity eld
Geodetic System 1984) ellipsoid.
features occurring at scales ner than the spatial resolution of the
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rate modeling of the gravity eld’s ne structure than the RTM approach alone. This is because the RTM technique is usually based on a simpli cation of the distribution of mass-densities inside the topography. Often, a standard rock density is used uniformly for the complete RTM, thus neglecting the impact of any local density variations (Hirt, 2010). In regions with insufficient distribution or scarce availability of gravity data, the local gravimetric re nement of the quasigeoid through the rst (remove-restore) option is of limited use or sometimes even impossible. Particularly in mountainous terrain, the RTM option represents a simple and promising alternative. Sjöberg (2011) estimated for EGM2008 that this error is -0.7 ∆g (in millimeters), where ∆g is the regional mean gravity anomaly in mGal (1 mGal = 10−5 m/s2 ). For a ∆g of 10 and 100 mGal, this truncation error becomes -7 and -69 (mm), respectively. The gravity anomaly RMS computed from EGM2008 is 10.32 mGal, thus truncation error is -7 mm. When terrestrial data are used in comparison with GGMs, some conditions should be met in their evaluation (Hirt et al., 2011): 1. Large amounts of terrestrial data (making the results in terms of RMS a feasible analysis)
Figure 4.
2. Large areal coverage of the data set (making the analysis representative, once the spatial resolution of the GOCE-
The study area (delimited by a square).
based GGMs is ∼100 km), and
Table 2.
3. Comparison data that is independent of the data set
Degree and order used in the comparisons
Models DIR_R3 TIM_R3 EIGEN-6C EIGEN-6S GOCO02S DIR_R2 TIM_R2 SPW_R2 DIR_R1 TIM_R1 SPW_R1 GOCO01S EGM2008
sourced by the GGMs
n max 210 √ √ √ √ √ √ √ √ √ √ √ √ √
224
240 250 √ √ √ √ √ √ √ √ √
√ √ √
In our comparisons, GPS/leveling data are independent of all GGMs used here. The EGM08 is not completely independent of the terrestrial gravity data because ∼80% of it was used in the model development. 3.1.
Comparison in terms of geoid heights
In the past, spirit leveling was a very time-consuming operation. The advent of GPS was revolutionary; from GPS leveling, it is possible to obtain the geoid height by the basic expression (Heiskanen; Moritz, 1967)
√
N∼ =h−H
√
(1)
This equation relates the geoid height (N ) to the ellipsoidal height (h) and the orthometric height (H ). If h is measured by GPS and H GCM can be represented by a truncated spherical harmonic series expansion (Torge, 2001).
is provided by leveling, then N may be easily computed. The comparison between geoid height from GPS/leveling and the geopo-
Hirt et al. (2010) describe two ways to model the high-frequency signals not provided by a truncated GGM series expansion, thus
tential models is a traditional way to evaluate these models and gives a reasonable indication of the accuracy of both the geopo-
reducing the signal omission error. The rst way is the well-known
tential and geoid model.
remove-restore approach from regional geoid/quasigeoid modeling via Stokes’ or Molodensky’s integrals. The second uses RTM
According to Featherstone et al. (2001), in absolute veri cations, the GPS network must have been previously tied to a geocentric,
data for source-modeling of high-frequency gravity eld signals. Hirt et al. (2010) explain that in regions with enough terrestrial
international terrestrial reference frame. Furthermore, these data can be used to apply constraints on the zero-degree term in or-
gravity data coverage, the rst case generally allows more accu-
der to account for inexact knowledge of the mass of the Earth and
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Table 3.
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Statistics of the geoid height (GPS/leveling) minus GGMs, in meters.
GGMs (degree/order) DIR_R3 (210) TIM_R3 (210) EIGEN-6C (210) EIGEN-6S (210) GOCO02S (210) DIR_R2 (210) TIM_R2 (210) SPW_R2 (210) DIR_R1 (210) TIM_R1 (210) SPW_R1 (210) GOCO01S (210) EGM2008 (210) TIM_R1 (224) GOCO01S (224) EGM2008 (224) DIR_R3 (240) EIGEN-6S (240) DIR_R2 (240) SPW_R2 (240) DIR_R1 (240) EGM2008 (240) TIM_R3 (250) EIGEN-6C (250) GOCO02S (250) TIM_R2 (250) EGM2008 (250)
(nmax)
(nmax)
Mean Std.Dev. RMS diff. Max. Min. σomission σcommission 0.18 0.19 0.20 0.21 0.21 0.21 0.21 0.20 0.19 0.17 0.21 0.19 0.21 0.16 0.18 0.15 0.11 0.14 0.13 0.19 0.09 0.12 0.10 0.09 0.14 0.15 0.10
0.35 0.36 0.38 0.37 0.36 0.39 0.36 0.38 0.39 0.38 0.42 0.39 0.38 0.37 0.38 0.34 0.32 0.34 0.35 0.37 0.33 0.32 0.33 0.29 0.32 0.32 0.29
0.39 0.40 0.42 0.42 0.42 0.44 0.42 0.42 0.43 0.42 0.46 0.43 0.44 0.41 0.42 0.37 0.34 0.37 0.38 0.41 0.34 0.34 0.32 0.31 0.35 0.36 0.31
1.01 1.01 1.07 1.05 1.03 1.12 1.04 1.03 1.06 1.04 1.31 1.02 1.08 1.05 1.04 1.05 1.00 1.05 0.98 1.01 0.83 0.93 1.03 0.87 0.99 1.03 0.92
-0.68 -0.70 -0.60 -0.71 -0.59 -0.70 -0.63 -0.70 -0.69 -0.68 -0.77 -0.65 -0.62 -0.65 -0.66 -0.64 -0.70 -0.64 -0.66 -0.67 -0.76 -0.52 -0.71 -0.64 -0.56 -0.59 -0.59
0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.29 0.29 0.29 0.27 0.27 0.27 0.27 0.27 0.27 0.26 0.26 0.26 0.26 0.26
0.18 0.20 0.23 0.22 0.20 0.25 0.20 0.23 0.25 0.23 0.29 0.25 0.23 0.23 0.25 0.18 0.17 0.21 0.22 0.25 0.19 0.17 0.20 0.13 0.19 0.19 0.13
the potential of the geoid (Hofmann-Wellenhof; Moritz, 2006). Al-
results (0.31 m). The DIR model, release 3, reduces the RMS errors
ternatively, GPS/leveling may be used in a relative sense to evaluate the precision of the gravimetric geoid or global geopotential
by 0.04 m (9.30% reduction) compared to release 1 and by 0.05 m (11.30% reduction) compared to release 2, for d/o 210. With re-
model (Featherstone et al., 2001). The zero-degree term is due to the difference between the geopo-
spect to d/o 250, releases 1 and 3 present similar results. Concerning the TIM model, the third release also presents the smallest re-
tential constant GM of the geopotential model and the ellipsoid was adopted in all models. In this study, we used a value of -0.41 m,
sults for d/o 210 and 250. The decrease between TIM R3 and the previous one is ∼0.05 m. Regarding the SPW model, release 1 pre-
the same as assumed in the EGM2008. Equation 2 shows the zero degree term (Heiskanen; Moritz, 1967):
sented the highest result (0.46 m) for d/o 210 and the difference between it and the second release is 0.04 m.
GδM δW N0 = − Rγ0 γ0
(2)
The constant GδM is given by the geopotential models and reference ellipsoid. Thus, the rst term in equation 2 can be simply determined. The second term must be further reduced. The statistics of the differences between geoidal heights from GPS/levelling and GGMs are reported in Table 3. Looking at the statistics in Table 3, one can see that release 3 from the model based on the direct approach (DIR) presented the smallest RMS (0.39 m) taking into account only the degree and order 210. For spherical harmonic degrees 224 and 240, EGM2008 presents the smallest RMS (0.37 m and 0. 34 m), respectively, while for degree and order (d/o) 250 this model and EIGEN-6C have the same
Figure 5 shows the results in graphical form. The solid black line represents EMG2008 results and it is present in all plots to serve as a reference. For degree and order 210 EGM2008 shows higher results than the other models (except DIR R2 and SPW R1). This shows that for this spectral frequency, GOCE-based models improved as compared to EGM2008. For d/o 250, EGM2008 and EIGEN6C provided the smallest results (0.31 m). Equation 3 also shows the global estimate of the standard devi(nmax)
ation of the omission error (σomission ) developed by W.M. Kaula (Kuala, 1966, p. 98) as
√ (n
)
max σomission =
∑ n=nmax +1
(
σN2
) n
≈
64 nmax
(3)
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where σN2 is the degree variance, and also shows the commission (nmax)
error (σcommission ) for this region. This last error is independent of
Table 4.
the omission error, such that
(
(n
)
max σcommission
)2
(
(n
)
max 2 = σtotal − σcmission
)2 (4)
For a local computation of the geoid undulation from a particular GGM, the standard deviation of the omission error may be signi cantly lower or higher because, as previously described, Kaula’s rule is a global model for the standard deviation of the omission error. This rule signi cantly overestimates the power spectral density of the geoid undulation at the lower frequencies and underestimates it at the high frequencies (Jekeli et al., 2009). Thus, it is very difficult to estimate the omission error and the commission error. We disregarded errors in the GPS/leveling data for this result, but the accuracy estimated for the geoid undulations from GPS/leveling is 0.09 m (Section 2.2). 3.2. Comparison in terms of gravity disturbances In Geodesy, the gravity disturbance (δg) is de ned as the scalar dif( ) ference between the Earth’s gravity on the geoid gp and normal
( )
gravity at the same point γp (Hofmann-Wellenhof; Moritz, 2006). In this second assessment, the gravity disturbance from terrestrial gravity data was compared with the GOCE-based models gravity disturbance. The ‘observed’ gravity disturbances have been computed by subtracting the normal gravity at the ellipsoidal height of the station from the observed gravity.
Statistics of gravity disturbance differences (observed minus models) with RTC, in mGal.
GGMs (degree/order)
Mean
RMS Max.
Min.
DIR_R3 (210) TIM_R3 (210) EIGEN-6C (210) EIGEN-6S (210) GOCO02S (210) DIR_R2 (210) TIM_R2 (210) SPW_R2 (210) DIR_R1 (210) TIM_R1 (210) SPW_R1 (210) GOCO01S (210) EGM2008 (210) TIM_R1 (224) GOCO01S (224) EGM2008 (224) DIR_R3 (240) EIGEN-6S (240) DIR_R2 (240) SPW_R2 (240) DIR_R1 (240) EGM2008 (240) TIM_R3 (250) EIGEN-6C (250) GOCO02S (250) TIM_R2 (250) EGM2008 (250)
1.01 0.72 0.42 0.92 0.84 0.82 0.79 0.69 0.51 1.36 -0.98 1.37 0.35 1.05 1.47 0.77 1.17 1.32 1.22 0.80 -0.69 0.43 1.09 0.72 1.14 1.10 0.62
10.52 10.47 10.60 10.63 10.34 10.65 10.37 10.73 10.66 11.06 11.75 11.03 10.66 26.56 11.22 10.12 10.31 11.23 11.19 10.74 10.21 10.16 10.28 9.69 10.29 10.38 9.72
-143.15 -142.78 -140.94 -140.31 -137.51 -141.71 -136.89 -138.57 -144.77 -142.46 -141.20 -143.48 -138.84 -126.94 -144.05 -137.49 -147.09 -144.31 -145.76 -139.98 -143.68 -136.53 -146.37 -139.74 -138.77 -138.32 -136.95
116.02 117.28 116.82 115.09 116.47 115.71 116.78 112.44 118.54 116.42 113.66 115.78 117.49 135.63 113.97 113.74 111.40 110.58 110.44 111.37 116.89 114.71 111.50 116.34 111.75 111.56 115.66
This topic shows two comparisons. In the rst, the RTC (Residual Terrain Correction) was added to GGMs gravity disturbances. This correction was computed for each station using the TC program
Table 5.
Statistics of gravity disturbance differences (observed minus models) without RTC, in mGal.
(Forsberg, 1984) and the digital terrain model SAM3s_V2 (Blitzkow et al., 2009).This model consists of SRTM3 (Farr et al., 2007), but
GGMs (degree/order)
Mean
RMS Max.
Min.
EGM96 (Lemoine et al., 1998a and 1998b) geoidal heights used in the SRTM3 were replaced by EIGEN-GL04C (Förste et al., 2006),
DIR_R3 (210) TIM_R3 (210) EIGEN-6C (210) EIGEN-6S (210) GOCO02S (210) DIR_R2 (210) TIM_R2 (210) SPW_R2 (210) DIR_R1 (210) TIM_R1 (210) SPW_R1 (210) GOCO01S (210) EGM2008 (210)
0.59 0.30 0.00 0.50 0.42 0.40 0.37 0.27 0.09 0.93 0.55 0.42 -0.08
16.57 16.47 16.48 16.61 16.43 16.56 16.43 16.72 16.58 17.06 17.50 16.43 16.47
-119.84 -119.47 -117.63 -117.00 -114.20 -118.40 -113.58 -115.26 -121.46 -119.15 -117.89 -114.20 -115.53
in order to derive the orthometric height. The gaps were substituted by digitized maps and DTM2002 topographic model (Saleh and Pavlis, 2002). The RMS taking into account the RTC is reported in Table 4. Table 5 shows the results without the RTC up to degree and order 210. The omission error comprises high-frequency gravity eld signals that cannot be represented by a truncated spherical harmonic series expansion of the GGMs (Torge 2001). The RTC estimates these signals, so omission errors are modelled. The use of RTC omission
120.90 122.16 121.70 119.97 121.35 120.59 121.66 117.32 123.42 121.30 118.54 121.35 122.37
error corrections are comparable to the RTM omission error corrections and it should be stated which data set was used as longwavelength reference in the RTC computations.
smaller than EGM2008. Furthermore, GOCO02S also presented val-
In general, the results in terms of RMS (Table 4) are very similar except for the model TIM_R1 (26.56 mGal). The GOCE approaches
ues lower than EGM2008. This shows that the results of the latest releases have improved a little bit as compared to EGM2008.
(DIR, TIM and SPW) present a slight decrease in terms of RMS at d/o 210, but model TIM_R3 presented an increase as compared
Comparing Tables 4 and 5, the RTC augmentation reduces the RMS errors to ∼6.00 mGal (∼35%). The most signi cant improvement
to release 2. The results obtained for DIR_R3 and TIM_R3 are
was near the coast line, where there are mountains up to 2100 m
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Figure 5.
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RMS of the geoid height (GPS/leveling) minus GGMs.
tall. Some comparisons involving EGM2008 were carried out by Hirt et al., (2010a; 2011) that demonstrate the improvement after applying RTC. Figure 6 depicts the differences for ve models up to d/o 210, with results without the RTC on the left and results after applying the correction on the right.
4. Conclusion The aim of this paper was to evaluate and analyze GOCE-based models from terrestrial data in the Southeast of Brazil. As a rst evaluation, we used geoid height data from GPS/leveling and from GOCE-based models. This is a powerful evaluation technique to check GGM performance. The results showed a gain in accuracy of
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Figure 6.
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Observed gravity disturbance minus the modeled value, shown without RTC (left) and with RTC (right). See text for details.
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Figure 6.
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(Continuation) Observed gravity disturbance minus the modeled value, shown without RTC (left) and with RTC (right). See text for details.
the GOCE-based models with respect to EGM2008 (spherical harmonic degree and order 210). In the GPS/leveling comparisons, for
results (0.31 m). Another important point is that the models lost accuracy for resolutions above degree and order 210. This
degree and order 210, the GOCE models improved over EGM2008. Out of 12 models evaluated, 10 of them presented results smaller
suggests that GOCE-based models do not present the same performance for the spectral band of spherical harmonics above
than EGM2008 in terms of RMS. Comparisons at higher degrees (e.g. degree and order 240) include the effect of gravity attenua-
degree 210. This agrees with demonstrations of similar results in other regions (Gruber et al., 2011).
tion in GOCE models and the use of terrestrial or predicted gravity in EGM2008. In our second evaluation, we compared the gravity disturbances The DIR_R3 and TIM_R3 models provided the best t to GPS/leveling, with errors of 0.39 m and 0.40 m, respectively,
derived from terrestrial gravity data and from GOCE-based models. These results indicate a slight decrease in the latest releases,
for degree and order 210 and presented results slightly closer
with DIR_R3 at 10.52 mGal, TIM_R3 at 10.47 mGal, and GOCO02S
to EGM2008 at degree and order 240 (at 0.34 m). The release 3 solutions DIR_R3 and TIM_R3 (degree and order 210) improved
at 10.34 mGal, as compared with EGM2008, which is 10.66 mGal at degree and order 210. Also, the RTC re ected its contribution to
signi cantly against those of release 1 and 2. Similarly, release 2 of the SPW solution presented better results than release 1. Degree
reduce the RMS, especially in mountainous areas, where the highest discrepancies were found. In this area, the mean improvement
and order 250 EGM2008 and EIGEN 6C presented the smallest
among all models was ∼45%.
Journal of Geodetic
Science
153
The models DIR_R3 and TIM_R3 presented similar performance, which is indicated in both the GPS/leveling and gravity compar-
Earth gravity eld from space—from sensors to earth sciences. In the Space Sciences Series of ISSI, v. 18, Kluwer Academic
isons. While DIR_R3 tted slightly better with GPS/leveling (0.39 m
Publishers, Dordrecht, pp 419–432. ISBN: 1-4020-1408-2.
vs. 0.40 m), TIM_R3 performed somewhat better than DIR_R3 in terms of terrestrial gravity data (10.47 mGal vs. 10.52 mGal).
Farr t., Rosen P., Caro E., Crippen R., Duren R., Hensley S.,
In summary, this comparison showed signi cant results improvements in GOCE-based models, indicating that the GOCE mission is proceeding nicely towards its goals. Deeper investigations will likely be made in Brazil and elsewhere in South America that demonstrate the worth of the GOCE mission, especially in the Ama-
Kobrick, M., Paller, M., Rodriguez, E., Roth, L., Seal, D., Shaffer, S., Shimada, J., Umland, J., Werner, M., Oskin, M., Burbank, D., Alsdorf, D., 2008, The Shuttle Radar Topography Mission, Rev. Geophys., 45, RG2004. DOI:10.1029/2005RG000183.
zon region where gravity data is currently lacking.
Featherstone W.E., Kirby J.F., Kearsley A.H.W, Gilliand J.R.,
Acknowledgements
Jonhston G.M, Steed J., Forsberg R. and Sideris M.G., 2001, The AUSGeoid98 geoid model of Australia: data treatment,
The authors acknowledge the State of São Paulo Research Foundation (FAPESP) for supporting the Thematic Project (Process number: 2006/04008-2). We would also like to thank the reviewers and the editor for their comments and suggestions on this manuscript.
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