ISSN 10693513, Izvestiya, Physics of the Solid Earth, 2014, Vol. 50, No. 1, pp. 127–136. © Pleiades Publishing, Ltd., 2014. Original Russian Text © V.N. Koneshov, V.B. Nepoklonov, R.A. Sermyagin, E.A. Lidovskaya, 2014, published in Fizika Zemli, 2014, No. 1, pp. 129–138.
On the Estimation of Accuracy for Global Models of Gravitational Field of the Earth V. N. Koneshova, V. B. Nepoklonovb, R. A. Sermyaginc, and E. A. Lidovskayab a
Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 Russia b Moscow State University of Geodesy and Cartography, Gorokhovskii per. 4, Moscow, 105064 Russia c Central Scientific Research Institute for Geodesy, Aerial Photography, and Cartography, ul. Onezhskaya 26, Moscow, 125413 Russia Received February 25, 2013
Abstract—The methods and techniques for estimating the accuracy of global models of the Earth’s gravity field in the form of spherical harmonic expansion of the geopotential are analyzed. Various methods for obtaining the a priori and a posteriori estimates for the accuracy are considered and classified. The applica tion of different approaches is illustrated by numerical examples for nine models, including those recently developed using the modern methods of space geodesy. The basic requirements for the database and software for estimating the accuracy are formulated. Keywords: gravitational field of the Earth, model, spherical harmonics, quasigeoid height, gravity anomaly, estimate of accuracy DOI: 10.1134/S1069351313060074
In order to describe the external gravitational field of the Earth (EGF), the global EGF models in the form of spherical harmonic expansion of the geopo tential (hereinafter, EGF models) are widely used (Koneshov, Nepoklonov, and Stolyarov, 2012). The accuracy of EGF models is affected by the errors in the determination of the coefficients of expansion (har monic coefficients of the geopotential), which, in turn, are caused by the errors in the initial measure ments and methods of their processing. Limiting the order of the spherical harmonics included in the expansion of the geopotential by a certain threshold value nmax (truncation error) is another source of errors. The requirements for the EGF models are con stantly increasing. As a result, the questions associated with estimating the accuracy of these models remain topical. In the present paper, the methods and tech niques for solving this problem considering the recent advances in EGF modeling are discussed.
T ( ϕ, λ, r ) = n
×
∑ (C
nm
fM a
∑(
nmax
n =2
a r
)
n +1
×
(1)
cos mλ + S nm sin mλ )Pnm(sin ϕ),
m =0
where a is the major axis of ERE; ϕ, λ, and r are the spherical geocentric coordinates of the point (latitude, longitude, and radiusvector, respectively); fM is the product of the gravitational constant and the mass of the Earth; Pnm is the fully normalized Legendre func tions; and C nm, and S nm are the fully normalized coef ficients of serial expansion (Moritz, 1980). The accuracy of the models described by Eq. (1) is typically estimated in terms of the accuracy of the most common transforms of the disturbing potential, including the quasigeoid height (QGH) ζ, gravity anomaly (GA) Δg, and components of the vertical (plumbline) deflection (VD) in the meridian and in the first vertical (ξ and η, respectively):
The models of the considered class are based on the representation of the gravitational potential by the sum of the normal potential (the potential of the Earth’s reference ellipsoid, or briefly ERE) and dis turbing (anomalous) potential 127
ζ= n
×
∑P
nm (sin ϕ)
m =0
fM γr
∑ (ar )
nmax
n
×
n =2
(C nm cos mλ + S nm sin mλ);
(2)
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Table 1. The analytical estimates of the errors of global EGF models Parameter Error
ζ, N
Δg nmax
nmax
δ2
R
2
∑ δ 2n
∑
n
×
∑P
nm (sin ϕ)
nmax
∑ n =2
()
c n2
γ 20
n
a n −1 × ( ) r
nmax
∑ n =2
(3)
()
n
a × r
(4)
n
dPnm(sin ϕ) × (C nm cos mλ + S nm sin mλ); ϕ d m =0
∑
nmax
η=−
∑
∑ mP
nm (sin ϕ)
(5)
(S nm cos mλ − C nm sin mλ),
m =0
n
×
∑P
nm (sin ϕ)
fM γ e re
nmax
n
⎛a⎞ ⎜r ⎟ × n =2 ⎝ e ⎠
∑
(6)
(C nm cos mλ + S nm sin mλ),
m =0
where re is the radius vector of the point with the coor dinates ϕ and λ on the ERE surface; γe is the normal gravity acceleration at the point (ϕ, λ, re) (Moritz, 1980). The difference between the heights of the geoid and quasigeoid can be taken into account using the for mula (Sjoberg, 1995)
Δg B Δ g − δg B (7) h= h, γ0 γ0 where h is the height above sea level; γ0 is the mean normal gravity acceleration on ERE; Δ g B is the com mon Bouguer anomaly; δg B = 2π f ρ h is the Bouguer N −ζ ≈
n = nmax +1
reduction; and ρ is the density of the Bouguer layer (the standard value is ρ = 2.67 g/cm3). The accuracy estimates of EGF models can be classified into two major groups, namely, the a priori and a posteriori estimates. The a priori estimates rely on the statistical depen dences between the errors in the coefficients C nm, and S nm and errors in the output data. Assuming the men tioned sources of errors in the EGF models to be sta tistically independent, the a priori estimates can be calculated by the formula 2
where γ is the normal gravity acceleration at the point of calculations reduced to the telluroid surface. The set of the transforms also includes the height of the geoid: N =
∑ n ( n + 1) cn2
σ ( nmax ) = δ ( nmax ) + ε ( nmax ) ,
n
fM a × γ r 2 cos ϕ n=2 r
n
×
()
∞
∑ ( n − 1) 2 cn2
n= nmax +1
(C nm cos mλ + S nm sin mλ); fM γr 2
n =2
∞
m =0
ξ=−
∑ n ( n + 1) δ2n
2
n=2
n = n m ax + 1
fM Δg = 2 r
2
2
∞
R2
nmax
γ 0 ∑ ( n − 1) δ n
n =2
ε2
2 2 ϑ= ξ +η
2
(8)
where σ is the root mean square error (RMSE) of the model; δ and ε are the root mean square values of the corresponding total contributions of the errors in C nm, and S nm and the truncation error, respectively. The values of δ and ε are estimated by formal con version to RMSE in Eqs. (1)–(6) and in the corre sponding formulas for the remainder of the expansion. Typically, the global averaged estimates on the geo sphere of radius R, which is equal to the average radius of the Earth, are used. In analytical form, these esti mates are presented in Table 1, where cn2 and δ 2n are the powerlaw dispersions of coefficients C nm, and S nm and their MSE δC nm, and δS nm,, respectively (Moritz, 1980; Nepoklonov, 1998): n
cn2 =
∑ (C
2 nm
m =0
)
2 + S nm ,
(9)
)
(10)
n
δ 2n =
∑ ( δC
2 nm
2 + δS nm .
m =0
The truncation error is estimated using a suitable analytical model of the powerlaw dispersions at n ∈ [nmax + 1, ∞] . Typically, the powerlaw dispersions of Δg n2 are given, and the following formula is used: cn = Δg n γ 0 ( n − 1) . 2
2
2
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Table 2. The models of powerlaw dispersions of GA Nos.
Dependence of Δ g n2 on n
Author(s)
96 ( n − 1) ( 2n + 1) 2
1
(Kaula, 1966)
n
4
− 1.12
2
(Pellinen, 1970)
3
(Pellinen, 1992)
4
(Tscherning and Rapp 1974)
5
(Moritz, 1976)
⎛ 3.405s1n+2 140.03s2n+2 ⎞ + ⎟, n2 − 4 ⎠ ⎝ n +1 s1 = 0.998006, s2 = 0.914232, n > 2
6
(Jekeli, 1990)
161( n − 1) 2 n −2.898
166n
⎧⎪34 ( n − 1) 2 n −2.68 n ≤ 180 ⎨ 2 −3.409 n > 180 ⎪⎩1559 ( n − 1) n 425.28s n+2 ( n − 1) , s = 0.999617, n > 2 ( n − 2) ( n + 24)
( n − 1) ⎜
Table 3. The dependence of the truncation errors for QGH and GA on nmax εζ, m
Model
εΔg, mGal
Δg n2
360
720
1440
1800
2160
360
720
1440
1800
2160
1 2 3 4 5 6 Mean
0.18 0.11 0.14 0.22 0.30 0.22 0.20
0.09 0.05 0.06 0.10 0.12 0.11 0.09
0.04 0.03 0.03 0.04 0.04 0.06 0.04
0.04 0.02 0.02 0.03 0.02 0.05 0.03
0.03 0.02 0.02 0.02 0.01 0.04 0.02
25.2 26.1 18.5 25.2 28.7 34.0 26.3
22.5 25.1 16.1 20.1 20.1 30.8 22.4
19.3 24.0 13.9 14.5 9.8 26.9 18.1
18.1 23.7 13.3 12.7 6.8 25.4 16.7
17.1 23.5 12.8 11.2 4.8 24.2 15.6
Different models are suggested for GA powerlaw dis persions (Table 2) (Kaula, 1966; Pellinen, 1970, 1992; Moritz, 1976, 1980; Jekeli, Yang, and Kwon, 2009). The corresponding a priori estimates of the truncation errors for QGH and GA are presented in Table 3 as the functions of the characteristic values of nmax, consider ing the relevance of the models with nmax ≥ 360. Based on formulas (9)–(11) and Tables 1–3, we note the following points: (1) The errors in the coefficients C nm, and S nm and the truncation error change inversely: as nmax increases, δ increases while ε decreases, and vice versa. The main contribution in the total error is provided by (a) the truncation errors of the spherical harmonics in the case of GA, irrespective of nmax and in the case of QGH at relatively small nmax, and (b) the errors in the coefficients C nm, and S nm; in the case of QGH obtained from the ultrahighdegree models; (2) The dependences presented in Table 2 are close to each other in their asymptotics. Nevertheless, the IZVESTIYA, PHYSICS OF THE SOLID EARTH
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estimates ε ( nmax ) , with the same nmax can differ signif icantly, especially in the lowdegree models. There fore, the final (most reliable) estimate of the trunca tion error can be used in the form of the average value over a few models (the bottom line in Table 3); (3) The estimate of the truncation error depends on the shape of the GA powerlaw dispersions and is insensitive to the characteristics of the assessed EGF model. In contrast, the contribution of the errors in C nm, and S nm is individual for each particular case, even at fixed nmax. This is illustrated by the estimates obtained in the example of several modern combined EGF models, including the newest ultrahighdegree EIGEN6C and EIGEN6C2 models (Table 4). Judging by these estimates, the accuracy of determin ing the harmonic coefficients of the geopotential has been significantly improved since the mid2000s. Compared to the models developed in the second half of the 1990s and in the beginning of the 2000s, the accuracy of the presentday models is higher by a fac No. 1
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KONESHOV et al.
Table 4. The influence of the errors in the harmonic coefficients of the geopotential on the accuracy of the model values of QGH and GA EGF model EGM96 PZ2002 EIGENGLO4C GAO2008 EIGEN5C EGM2008 GIF48 EIGEN6C EIGEN6C2
Country
Year
USA Russia Germany, France Russia Germany, France USA USA Germany, France Germany, France
1996 2002 2006 2008 2008 2008 2011 2011 2012
tor of 4–5 and 2–3 for QGH and GA, respectively. The improvement has been achieved largely due to the application of new methods of satellite geodesy in the foreign projects, including the projects on satelliteto satellite tracking (CHAMP and GRACE) and low orbit satellite gravity gradiometry (GOCE mission). The main advantage of the a priori estimates is that they provide an idea of the accuracy characteristics of EGF models for the entire globe quite promptly and almost without involving any additional information. However, as is shown by the practice, these estimates are in some cases not quite adequate (excessively opti mistic), especially when it is required to estimate the accuracy of the model in some particular region. Therefore, the final conclusions on the accuracy of the EGF models should be based on the combination of the a priori and a posteriori estimates. The idea of the a posteriori estimates is to compare the output data of the model with the independent (conditionally) reference data. The accuracy of the model is assessed by its closeness to the reference data. The role of the measures of closeness is typically played by the statistical parameters of the discrepan cies, such as the extreme values (min, max), arith metic mean value μ, and standard deviation s. In the general case, the standard formulas for estimating μ and s can be adjusted by introducing the weighting fac tors, which allow the nonuniform distribution of the reference points in terms of their number per unit of area. The closeness can also be measured by the his tograms of the deviation between the model and ref erence values of the EGF parameters (Koneshov, Nepoklonov, and Stolyarov, 2012). Depending on the type of reference data, the a pos teriori estimates of the accuracy can be subdivided into three groups. The first group includes the estimates derived by the comparison of the studied model with a certain similar model that is conditionally assumed to be the reference. At present, EGM2008 expanded up to
nmax
δ QGH, m
360 360 360 360 360 2160 360 1420 1949
0.36 0.45 0.15 1.11 0.13 0.08 0.08 0.10 0.08
GA, mGal 8.5 10.8 4.4 1.9 3.8 4.2 2.4 3.6 3.3
degree 2160 is typically used as the reference model. In these group of estimates, the discrepancies between the model values of QGH, GA, and other transforms of the disturbing potential at the same points (typi cally, in the nodes of the uniform grid of the meridians and parallels of latitude on the entire Earth’s surface or within some region on the globe) are analyzed. For example, Koneshov et al. (2012) obtained the regional estimates for Arctic regions. In that work, the global scale comparison of the EGM2008 model with the models from Table 4 in terms of QGH and GA on the ERE surface is carried out. The obtained deviations (Table 5) can be treated as the errors of the model val ues of QGH and GA relative to the reference EGM2008 model of degree 2160. We note that comparison of the models can be con ducted both throughout the entire interval of n and fragmentarily, within any part of this range. Further more, the analysis of the discrepancies of the models in the spatial domain can be complemented by the analysis of their deviations in the frequency domain. In the latter case, the closeness of the models at differ ent frequencies can be assessed, for example, from the powerlaw dispersions δζ 2n and δg n2 for the differences in the model QGH (δζ) and GA (δg) expressed in terms of the differences in the coefficients C nm, and S nm between the first and second model. Application of this method is illustrated by the graphs in the figure, which present the comparison of the EGM96, GAO2008, and GIF48 models. Using this method, we can identify the interval of frequencies(the values of n), where δζ n ( δg n ) , characterizing the error spec trum of the studied model relative to the reference model (i.e., the error expanded in terms of the powers of spherical harmonics) do not exceed a certain value that determines the level of significance. The second group includes the a posteriori esti mates obtained by the orbital method using the satel lite ephemerides computed from the studied model.
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Table 5. The statistical characteristics of deviations of the EGF models from the reference EGM2008 model expanded to degree 2160 QGH, m nmax
Model
min
GA, mGal
max
µ
S
min
max
µ
s
EGM96
360
–10.43
11.98
0.00
0.72
–478.8
461.1
0.2
29.9
PZ2002
360
–10.66
10.60
0.01
0.74
–480.0
437.8
0.2
29.7
EIGENGLO4C
360
–6.01
7.22
0.01
0.36
–492.1
430.1
0.3
28.9
GAO2008
360
–8.56
8.45
0.01
0.58
–491.3
400.8
0.3
32.8
EIGEN5C
360
–6.31
7.04
0.00
0.33
–521.8
415.9
0.3
28.7
GIF48
360
–3.91
4.13
–0.01
0.19
–465.9
388.1
0.2
27.4
EIGEN6C
1420
–2.51
3.49
0.00
0.14
–334.1
315.0
0.2
24.0
EIGEN6C2
1949
–2.45
3.64
0.00
0.13
–220.5
231.4
0.2
23.0
The accuracy of the model is evaluated from the resid ual misties of the trajectory measurements and, for the altimeterequipped satellites, from the residual misties of the measured altitudes of the sea surface at the intersections of the altimeter paths. The trajectory measurement data are provided by precision laser and radio observations (DORIS, PRARE) from the Earth and by the precise determinations of satellite orbits using the space navigation systems. Application of the orbital method is illustrated by the accuracy estimates for the EGF models using the laser observations by the Envisat, ERS2, Jason1, Lageos1, Ajisai, Starlette, and Stella space vehicles (Cheng, Ries, and Cham
σζn, m
bers, 2009; Gruber et al., 2011; Forste et al., 2008) summarized in Table 6. Such estimates give an idea of the accuracy of the EGF models expressed in terms of the deviations of the predicted orbital parameters of a satellite from the measured parameters (in this case, measured by the precise laser observations). The glo bal coverage is the advantage offered by the orbital method; however, this method suffers from the low sensitivity of satellite orbits to the local anomalies in gravity and from the necessity to eliminate the non gravity perturbations of orbital motion (Gruber, 2004). δgn, mGal
(а)
(b)
0.8
0.06
GAO2008 EGM96 0.6
EGM96
0.04 0.4
GAO2008 0.02
0.2 GIF48 GIF48 0
100
200
300
400 n
0
100
200
300
400 n
The dependence of the error spectrum of the model (a) QGH and (b) GA on the studied model of the geopotential. IZVESTIYA, PHYSICS OF THE SOLID EARTH
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Table 6. The standard deviations of the predicted satellite range from the measured values, averaged over the orbit arcs Satellite Envisat Jason1 Lageos1 Ajisai Starlette Stella ERS2
Orbit height, km
EGM96
EIGENGLO4
EGM2008
EIGEN5
800 1340 5900 1480 810 810 800
0.087 0.042 0.015 0.056 0.050 0.091 n/a
0.058 0.041 0.046 0.014 0.031 0.026 n/a
n/a '' 0.015 0.053 0.046 0.029 n/a
n/a '' 0.014 0.047 0.031 0.026 0.043
Table 7. The statistical characteristics of the discrepancies between the model and gravimetric GA for different regions of the world Model
nmax
Region
Discrepancies, mGal
Number of points
min
max
µ
S
EGM96
360
Australia Arctic Antarctic Canada Scandinavia
1117054 56878 57140 14177 66904
–194.7 –193.3 –355.5 –124.9 –47.4
219.9 195.9 279.9 114.2 76.7
–0.3 –1.0 4.3 –0.1 –0.4
12.1 18.0 22.1 13.4 8.9
EIGENGLO4C
360
Australia Arctic Antarctic Canada Scandinavia
1117054 56878 57140 14177 66904
–192.1 –191.5 –356.4 –124.1 –50.9
218.3 193.4 282.2 105.9 83.9
0.3 –1.2 4.4 –0.1 –1.1
12.1 15.9 23.2 13.6 8.5
2160
Australia Arctic Antarctic Canada Scandinavia
1117054 56878 57140 14177 66904
–200.2 –193.6 –349.9 –100.7 –47.8
238.7 103.5 268.3 85.7 49.4
–0.3 –1.0 4.3 –0.9 –0.8
5.4 10.9 18.6 8.2 3.6
EGM2008
The third group includes the a posteriori estimates obtained by the comparison of the studied model with the reference values of QGH, GA, components of VD, and other transforms of the disturbing potential esti mated by different instrumental methods of EGF study, including gravimetry, satellite altimetry, satellite leveling, and astronomicalgeodetic method. The modern publications of Russian and foreign authors present numerous examples of such estimates for different EGF models. For example, the deviations of the GA model from the results of the groundbased gravity surveys in different regions of the world are pre sented in Table 7, which cites the data of Arabelos and Tscherning (2010). In this study, we obtained new esti mates for the accuracy of the EGF models by compar ing the model values of GA with the data of the air borne gravity survey in the Arctic carried out in 2011– 2012 by the geophysicists of the Institute of Physics of
the Earth, Russian Academy of Sciences with the use of the Russian gravimetric instruments. This survey included 40 flight paths with a total length of more than 2480 km, which provided more than 23000 gravi metric points for the resulting catalogue. The accuracy and the measurement step on the flight paths corre sponded to the requirements for the scale 1:200000 gravity survey. The statistical characteristics of the dis crepancies between the model and measured GA val ues are presented in Table 8. Judging by the estimates in Tables 7 and 8, the a posteriori accuracy estimates for the modern EGF models, which are derived from a comparison with the gravimetry data, quite closely agree with the a posteriori estimates presented above. This illustrates the positive effect on the accuracy of refining the harmonic coefficients of the geopotential and improving the resolution of the models (based on the geoid wavelengths). A similar conclusion can also
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ON THE ESTIMATION OF ACCURACY FOR GLOBAL MODELS Table 8. The statistical characteristics of the discrepancies between the model GA and the data of the airborne gravi metric survey carried out by the Schmidt Institute of Phys ics of the Earth, Russian Academy of Sciences in the Arctic Discrepancies, mGal Model EGM2008 EIGEN6C PZ2002
nmax 2160 1420 360
µ
S
–2.58 –3.04 –4.06
2.01 2.25 4.67
be deduced from the published results on the compar ison of global EGF models with the satellite altimetry data including both the sea surface (geoid) altitudes and gravity anomalies.
133
One of the main functions of the modern EGF models is to provide the altitude framework with the use of satellitebased technologies. Therefore, testing the EGF models in terms of the geoid (quasigeoid) heights yielded by the precision satellite leveling is par ticularly important. Typically, this testing only involves the accuracy estimates of the studied models in terms of the absolute altitudes at individual geodetic points. Examples of such estimates are presented in Table 9. In order to retrieve more complete information on the accuracy characteristics, it is practical to test the EGF models in both the absolute and relative altitudes, i.e., to analyze the differences in QGH (geoid altitudes) at two points binned in terms of the distance between the points (for example, 0 to 50 km, 50 to 100 km, etc.) (Nepoklonov, Zueva, and Pleshakov, 2007).
Table 9. The statistical characteristics of the discrepancies between the model and levelingbased QGH for several territories Model (nmax) EGM96(360)
Region Turkey Greece Australia Poland Algeria Republic of South Africa South America Greenland Belarus
EIGENGLO4C(360) Greece Algeria South America China EIGEN5C(360)
Turkey Poland
EGM2008(2160)
Turkey Greece South Korea Poland Czech Republic Italy Algeria Republic of South Africa South America Greenland Canada China Belarus
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Discrepancies, m
Number of points
min
max
µ
S
313 1542 1013 360 71 79 1190 78 196
–2.19 –1.06 –2.44 –0.54 –0.90 –0.95 –3.30 –0.52 –0.52
0.82 1.58 3.54 0.57 0.78 0.68 3.70 2.62 0.47
–0.81 –0.45 0.02 –0.04 –0.03 –0.24 0.24 0.71 0.01
0.46 0.42 0.50 0.19 0.34 0.35 0.80 0.52 0.22
1542 71 1190 652
–1.17 –0.63 –2.90 –2.26
1.77 0.64 3.10 1.80
–0.28 –0.02 0.22 –0.25
0.45 0.33 0.70 0.43
313 360
–3.33 –0.22
0.75 0.52
–0.87 0.10
0.66 0.11
313 1542 500 360 1024 977 71 79 1190 78 2579 652 196
–0.29 –0.44 –0.54 0.04 –0.52 –0.33 –0.67 –0.84 –3.30 –0.43 –0.92 1.89 –0.16
0.71 0.54 1.17 0.26 –0.33 0.34 0.61 0.02 3.40 1.60 0.09 1.64 0.11
0.29 –0.38 0.10 0.12 –0.42 0.00 –0.08 –0.42 0.22 –0.19 –0.38 –0.12 0.05
0.16 0.14 0.18 0.04 0.04 0.10 0.21 0.24 0.68 0.40 0.13 0.26 0.05
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Table 10. The statistical characteristics of the discrepancies between the model and levelingbased QGH for the territory of Russia Discrepancies, m Model EGM96 PZ2002 EIGENGLO4C GAO2008 EIGEN5C EGM2008 GIF48 EIGEN6C EIGEN6C2
nmax 360 360 360 360 360 2160 360 1420 1949
We note that the published comparisons of the modelbased and levelingbased QGH barely contain the data for Russia that are adequate for its territory. In order to fill this gap to some extent, in the present study with the participation of TsNIIGAiK, we obtained such estimates for 518 leveling points cover ing the entire territory of Russia. We analyzed all the EGF models listed in Table 4, including the newest GIF48, EIGEN6C, and EIGEN6C2 models, for which the data have not been published previously. The statistical characteristics of the discrepancies between the model and levelingbased QGH values at these points are presented in Table 10. We note that the GIF48 model of degree 360 is almost equally accu rate as the ultrahigh degree models. The data presented in Tables 7–10 illustrate the advances in refining the global EGF models since the mid2000s. The most striking progress has been achieved in improving the accuracy of QGH, largely due to ascertaining the lowfrequency part of the EGF spectrum using the modern methods of space geodesy. With the creation of the EGM2008 model, the accu racy of EGF determination in many large regions of the world came to within a decimeter. Moreover, the tests of the newest models relying on the satellite gra diometry data show that it will be possible to further improve the accuracy of QGH, in particular, for the territory of Russia. The practical application of the described methods for estimating the accuracy of global EGF models in the form of spherical harmonics of the geopotential are generally associated with the use of the modern geoinformational technology, which includes the rele vant data provision and software. The data provision should be implemented in the form of a computerized database containing the fol lowing main sections: —the global EGF models in the form of spherical harmonics of the geopotential, including the models used as reference;
min
max
µ
s
–2.77 –3.10 –2.66 –2.44 –2.10 –1.52 –1.73 –1.45 –1.38
2.33 2.28 2.31 3.01 2.33 1.76 2.58 1.59 1.58
0.15 0.19 0.18 0.16 0.18 0.14 0.15 0.15 0.15
0.56 0.58 0.49 0.65 0.46 0.43 0.44 0.42 0.42
—the global data for the surface topography of the Earth, including the coefficients of the spherical expansion of the heights (depths) (in particular, the DTM2006 model with a spectral resolution up to degree 2160 developed in United States), and the detailed array of the heights in the nodes of the uni form grid of meridians and parallels of latitude (for example, the GTOPO30 and ETOPO2 digital eleva tion models). These data are required in order to cor rectly take into account the altitudes of the points par ticipating in the calculations by formulas (2)–(5) and (7); —the initial data for calculating the a posteriori estimates of the accuracy by the orbital method, including the parameters of the models of motion, ini tial conditions of the orbital arcs, orbital parameters of geodetic satellites obtained from the highprecision trajectory measurements, and coordinates of the observation points on the ground; —the initial data for calculating the a posteriori estimates of accuracy based on the comparison of EGF model parameters (QGH, GA, VD components, etc.) and their counterparts obtained by the measure ments. Here, the following data can be used as refer ence: —the global and regional catalogues of the mean GA over the standard geographic trapezoids with sizes of 30' × 30' to 5' × 5' (5' x 7.5') and smaller; —the altimetrybased geoid heights and VD in the World Ocean (on the altimeter paths or on the uniform grid of meridians and parallels of latitude), as well as GA according to altimetry; —gravimetric GA on the land and in the oceanic regions determined by the route and areal surveys from the groundbased, marine, and air carriers (including the data obtained from the gravimetric maps and gravimetric altitudes of quasigeoid (geoid) and VD components); —the quasigeoid (geoid) heights obtained by sat ellite leveling (i.e., the differences between the geo
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detic and normal (orthometric) heights), and VD components estimated by the astronomicalgeodetic method. The listed reference data should be sufficiently accurate and as complete as possible in the sense of informational coverage of different regions of the globe. The set of computational programs should solve the basic computational problems associated with estimating the accuracy of global EGF models. Clearly, these problems should be solved using the database described above. The mentioned basic prob lems include the following problems: —computation of the model values of QGH, GA, VD components, and geoid heights by formulas (2)– (6) at the individual points and the nodes of the uni form grid of the meridians and parallels of latitude; —averaging the GA values determined by Eq. (3) over the geographic trapezoids of a given size; —calculation of the powerlaw dispersions for QGH, GA, and VD, and the powerlaw dispersions for their errors by formulas (9), (10) and relationships presented in Table 1; —calculation of the dispersions (RMSE) for the model values of QGH, GA and VD components, which are due to the errors in the initial harmonic coefficients of the geopotential, by the conversion to RMSE in formulas such as Eqs. (2)–(6); —calculation, according to Eq. (7), of the correc tion for the conversion from QGH to the geoid height or vice versa, and calculation of the corrections to QGH, VD components, and GA for the transition to another ERE; —calculation of the satellite orbital parameters for the given time moments, with the use of a given model of its orbital motion and the given initial conditions of the corresponding orbital arc; —the statistical analysis and visualization of the discrepancies between the EGF models in the coordi nate and frequency domains, the discrepancies between the model and reference values of QGH, GA, VD components, and other transforms of the disturb ing potential, and the discrepancies between the model and the reference values of the satellite orbit. Our analysis of the methods and techniques for estimating the accuracy of EGF models suggests the following conclusions: In order to meet the high presentday requirements to EGF models, the accuracy of the model character istics should be estimated with particular thorough ness, which implies the combined use of all methods for estimating the accuracy described in the present work. Implementation of these methods is provided by the corresponding geoinformation technology, whose main elements are the database and the software satis fying certain requirements; The a priori estimates are reasonable to use in the preliminary analysis of the accuracy characteristics of IZVESTIYA, PHYSICS OF THE SOLID EARTH
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the EGF models. The truncation error can be esti mated by combining different dependences for the powerlaw dispersions of GA presented in Table 2; The a posteriori estimates of the accuracy provide a more objective and detailed view of the accuracy char acteristics of the EGF models. Here, not only the EGM2008 model expanded to degree 2160 but also the new highdegree EIGEN6C and EIGEN6C models can be used as references for estimating the accuracy of the mediumresolution models; In the role of the reference data for a posteriori esti mation of the accuracy of EGF models, it is reason able to use the data that can be assumed to be indepen dent for the tested model (for example, the leveling based QGH, astronomicalgeodetic VD data, and detailed gravimetric GA, including the data of air borne gravimetric survey); The estimates presented in Tables 6–10 show that the accuracy of the EGF models has been constantly improving during the last 15 years. Depending on the region, the accuracy of the latest models in terms of QGH and GA is by a few percent to severalfold higher than in the similar models developed before the mid 2000s. REFERENCES Arabelos, D.N. and Tscherning, C.C., A comparison of recent Earth gravitational models with emphasis on their contribution in refining the gravity and geoid at continental or regional scale, J. Geod., 2010, vol. 84, pp. 643–660. Cheng, M., Ries, J.C., and Chambers, D.P., Evaluations of the EGM2008 gravity model, Newton’s Bull., 2009, no. 4, pp. 18–23. Foerste, C., Schmidt, R., Stubenvoll, R., Flechtner, F., Meyer, U., Koenig, R., Neumayer, H., Biancale, R., Lem oine, J.M., Bruinsma, S., Loyer, S., Barthelmes, F., and Esselborn, S., The GeoForschungsZentrum Pots dam/Groupe de Recherche de Geodesie Spatiale satellite only and combined gravity field models: EIGENGL04S1 and EIGENGL04C, J. Geod., 2008, vol. 82, no. 6, pp. 331–346. doi: 10.1007/s0019000701838 Gruber, T., Validation concepts for gravity field models from satellite missions, in Proc. 2nd Int. GOCE user workshop “GOCE, The Geoid and Oceanography”, 2004. Gruber, T., Viesser, P.N.A.M., Ackermann, C., and Hosse, M., Validation of GOCE gravity field models by means of orbit residuals and geoid comparions, J. Geod., 2011, vol. 85, pp. 845–860. Jekeli, C., Yang, H.J., and Kwon, J.H., Evaluation of EGM08—globally, and locally in South Korea, Newton’s Bull., 2009, no. 4, pp. 38–49. Kaula, W.M., Theory of satellite geodesy: Applications of Sat ellites to Geodesy (Waltham, Mass.: Blaisdell, 1966). Koneshov, V.N., Nepoklonov, V.B., and Stolyarov, I.A., Study of the anomalous gravity field in the Arctic based on modern geopotential models, Izv., Phys. Solid Earth, 2012, vol. 48, nos. 7–8, pp. 587–593. No. 1
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Translated by N. Astafiev
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