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Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients
Accepted Manuscript Title: Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients Authors: Wan Xiaoyun, Yu Jinhai PII: DOI: Reference:
S0264-3707(13)00031-8 http://dx.doi.org/doi:10.1016/j.jog.2013.02.005 GEOD 1215
To appear in:
Journal of Geodynamics
Received date: Revised date: Accepted date:
20-10-2012 15-1-2013 3-2-2013
Please cite this article as: Xiaoyun, W., Jinhai, Y., Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients, Journal of Geodynamics (2013), http://dx.doi.org/10.1016/j.jog.2013.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients Wan Xiaoyun[1][2], Yu Jinhai[1][2] 1, Key Laboratory of Computational Geodynamics, Chinese Academy of Sciences, Beijing 100049,China 2, College of Earth Science, University of Chinese Academy of Sciences, Beijing 100049,China
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ABSTRACT: The formula for computing the radial gradient of the gravity is derived from the
gravity gradients. Since GOCE satellite attitude has such property that one axis in the gradiometer
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almost has the same direction as the radial one of the earth, the formula can be used to compute
the radial derivative of second order of the disturbing potential. This means that the boundary
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value problem (BVP) with the radial gradient of the gravity can be established on the satellite orbit. Hence, the recovery of the gravity field from GOCE data can be realized by solving the BVP. In
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order to examine the accuracy of the BVP, an arithmetic example is given, the computational results of which illustrate that the efficient degrees/orders of the gravity field model recovered by
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the BVP can satisfy the requirements of the satellite gravity gradients. In addition, a gravity field model is solved out from the BVP by using actual GOCE data.
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1. Introduction
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Keywords: GOCE, Non-full tensor, Harmonic analysis
GOCE (Gravity field and steady-state Ocean Circulation Explorer) (Balmino et al., 1999;
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Albertella et al., 2002) is an explorer belonging to ESA (European Space Agency), and one of its
main aims is to recover an accurate static gravity field model with high degrees and orders. GOCE has provided a huge amount of data about its position，attitude and the gravity gradients on the
orbit since its launch on 17 March 2009. Obviously, it is widely concerned how to recover the gravity field models by using these data. The research focuses mainly on time-wise and space-wise approach.
Rummel (1993) introduced the principles of the time-wise and space-wise approach. Keller (1997) investigated space-wise approach by constructing boundary value problems. Klees et al. (2000) proposed an efficient algorithm for recovering gravity field in time domain. Reguzzoni (2003) discussed the observation process from time domain to space domain. Albertella et al. (2004) analyzed the filtering techniques for gradients data processing in time and space domain. Milani et al. (2005) simulated the GOCE mission by time-wise approach whereas Migliaccio et al. 1
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(2007) conducted a simulation by space-wise approach. Reguzzoni et al. (2008) studied the application of step-wise solution in space domain. Pertusini et al. (2009) evaluated the structure of the covariance in space-wise solution. Pail et al. (2010) recovered the gravity field model GOCO01S using time-wise method by combining GOCE and GRACE data. The GOCE gravity
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field models from different methods were compared by Pail et al (2011). This paper is aimed at doing some work in space domain, by constructing BVP.
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Some notations are introduced here to make treatments clear. Let X 1 , X 2 and X 3 be
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three axes of GRF (Gradometer Reference Frame), where the direction of X 3 is very close to radial one of the earth. Again letting v denote the gravitational potential of the earth, and
! 2v , vij ( i, j ! 1, 2,3 ) can be measured along the satellite orbit by the gradiometer and ! X i! X j
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vij !
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are called SGG (Satellite Gravity Gradients). Because of poor accuracies of v12 and v23 , only diagonal elements and v13 in SGG can be used to recover the gravitational field. Generally,
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recovery of the gravitational field from SGG can be attributed to solving the normal system of
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equations. The advantage of establishing the normal system of equations is that SST data (Satellite-to-Satellite Tracking) can be easily used to construct equations together with SGG data,
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but the issue is that the computation is very complicated since the unknowns exceed 40 000 if the degree of the model is higher than 200, although some researchers, such as Kusche (2002), Ditmar (2003), and Preimesberger (2003) et al., investigated the methods of rapid resolution of GOCE gravity field models.
The approach by BVPs (Boundary Value Problems) with boundary condition of the radial
gradients has its special advantage because of computing efficiently. In the early researches about GOCE, many researchers made deep discussions on BVPs. For example, Rummel et al. (1992) analyzed the spectral characteristics of
! ! zz ! ，! !
xz
,!
yz
! ，! !
xx
! !
yy
,2!
xy
!
and gave their two
dimension Fourier expressions. van Gelderen et al. (2001) summarized the solution of the general geodetic BVPs by least-square methods and gave the solution under the boundary conditions of gravity gradients. Martinec (2003) discussed Green function to gradiometric BVP. However, SGG is not a full-tensor measurement because of the absence of
v12 and v23 , thus it seemingly lacks
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some terms to establish BVP with the boundary conditions of the radial gradient. Recently, Yu et al. (2010) constructed the boundary conditions based on the invariants of the gravitational gradients, which can be represented as the radial component of the gravitational gradient tensor. This makes that BVPs can be used to deal with SGG data, which means that the model of the
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gravitational field can be recovered by the method of spherical harmonic analysis. The method of the invariants is not only simple, but affected with few attitude errors also. Consequently, Yu and
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Wan (2012) processed the actual SGG data and obtained a GOCE gravitational field model with higher than 200 degrees.
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In fact, it is a key problem to derive the boundary conditions about the radial gradient in BVPs.
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Apart from the methods of the invariants (Sacerdote and Sansò,1989), the boundary condition of the radial gradient can also be derived from the coordinate transformation. However, in the actual GOCE data processing, the latter has not been paid enough attention. One reason may be that SST
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data can not be used as BVPs and a whole model of the gravitational model can not be recovered using only SGG data due to the limitations of the measure bandwidth. The other is that the radial
v23 can not be used due to their poor accuracies.
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while v12 and
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gradient is different from v33 in the SGG so that it has to be composed of vij ( i, j ! 1, 2,3 )
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In the paper, it is discussed how to establish the boundary condition about the radial gradient from the coordinate transformation. Then the accuracy of the boundary condition is analyzed using the satellite attitude data. Finally, some arithmetic about recovery of the gravitational field from the boundary condition is given.
2. Establishing the boundary condition 2.1 Computation of the radial gradient A reference gravitational potential V is chosen firstly, and then let T ! v ! V be the disturbing potential. Generally, V should be a very good approximation to the actual gravitational potential v . For example, V can be chosen as EGM08 (Pavlis et al., 2012) or EIGEN_5C (Foerste et al., 2008) models. Letting ! 1 , !
2
and !
3
denote the angles between
X 1 , X 2 and X 3 in GRF and the radial direction of the earth, they can be computed from GOCE satellite attitude. By derivation, we have
3
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! 2T ! T33 cos 2 ! 3 ! T22 cos2 ! 2 ! T11 cos2 ! 1 ! 2T12 cos ! 1 cos ! 2 !r ! 2T13 cos ! 1 cos ! 3 ! 2T23 cos ! 2 cos ! 3
2
(1)
where
! 2T ! vij ! Vij , ! X i! X j
( i, j ! 1, 2,3 )
(2)
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Tij !
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Since vij are measured from the gradiometer and Vij can be computed from GOCE satellite’s
position and attitude, T11 , T22 , T33 and T13 are all known while T12 and T23 are unknown
from Eq. (1).
! 2T cannot be computed directly ! r2
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due to the poor accuracies of v12 and v23 . This means that
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On the other hand, although the actual X 3 -axis is different with the radial direction of the
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earth, their difference is small. This means that ! 1 ! 900 , !
2
! 900 and !
3
! 00 . Thus, it can
be concluded that the term related to T33 is the main one in Eq. (1) and the terms related to T12 3
is, the attitude data of GOCE satellite
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and T23 are small ones. To understand how small !
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during the period from 1 November to 10 November 2009 are chosen and the values of !
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during the period are computed. Table 1 summarizes the statistics of !
3
3
and Fig. 1 gives its
distribution during this period.
Table 1
Statistics of ! Number 690613
3
( unit: degree)
Min 0.0008
Max 1.3577
Mean 0.3859
Std 0.2106
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Fig. 1. Angle between X 3 -axis of GRF and radial direction
cos!
3
! cos1.50 ! 0.9997
is small and its maximum is less than 1.50 . ,
cos 2 ! 1 ! cos 2 !
2
! cos 2 !
3
cos ! 1 , cos !
and
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Hence,
3
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According to Fig. 1, angle !
! 1 , that is, compared with cos ! 3 , cos !
2
1
! 0.0262 and cos !
since
2
can be
considered as small quantities. So, Eq. (1) can be written as
! 2T ! T33 cos 2 ! 3 ! T22 cos 2 ! 2 ! T11 cos 2 ! 1 ! 2T13 cos ! 1 cos ! ! r2 where the neglected magnitude is
! ! 2T12 cos ! 1 cos ! 2 ! 2T23 cos ! 2 cos !
In fact, !
3
(3)
(4)
3
is the error of the boundary condition (3). Assuming that all the components in Tij
have the same magnitudes, the relative magnitude of the error !
! !
2cos ! 1 cos ! 2 ! 2cos ! 2 cos ! cos2 ! 3
3
is approximately equal to (5)
By computation, we know that ! ! 5.5% . Also from Fig. 1, it can be seen that the average of the angle !
3
is less than 0.50 . Hence, the relative error !
of the boundary condition (3) should be
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smaller. The value of !
given by report of ESA is 2% (Rummel and Gruber, 2012).
How to estimate the absolute error of ! ? This depends on the choice of the reference gravitational potential V . The more accurate the reference gravitational potential V is, the smaller the error !
is. This is why EGM08 or ENGEN_5C are recommended as the reference
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gravitational potential. In addition, the iteration computations for T12 and T23 are also required,
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that is, T12 and T23 can be computed and then inserted into Eq. (1) after the disturbing potential
T is solved firstly from the boundary condition (3). In fact, the effect for Eq. (1), caused by the T12 and T23 , can be completely eliminated by such iteration process (Migliaccio et
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absence of al, 2004a).
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In sum, from Eq. (1) or Eq. (3), we can obtain the following BVP
(6)
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!! T ! 0 ! 2 !! T ! 2 ! f ! !r S ! T ! O (r ! 1 ), !
at infinity
2.2 The solution of the BVP
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where S is the orbit surface of GOCE satellite and f is a known function on S .
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As the GOCE satellite’s orbit is almost circular, S can be seen as a sphere, that is,
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S ! {r ! a} , where a is the average radius of the orbit. Moreover considering that f is known on S , f can be expressed as the following spherical harmonic series
f !
GM a3
!
n
! !
Pnm (cos ! )(Cnm cos m! ! S nm sin m! )
where GM is the gravitational constant of the earth,
Pnm ! cos!
!
(7)
n ! 0 m! 0
! r ,! , ! !
are the spherical coordinates,
is the fully normalized Associated Legendre Functions with degree n and order m,
and Cnm and Snm are the fully normalized spherical harmonics coefficients. Now we also expand the disturbing potential T as
T!
GM r
Rn !n! 0 m!! 0 r n Pnm (cos! )(anm cos m! ! bnm sin m! ) !
n
(8)
where R is the mean radius of the earth. By computation, we have (Rummel et al, 1993)
! 2T GM ! 3 ! r2 r
!
n
n! 2 m! 0
n
R! ! Pnm (cos! )(anm cos m! ! bnm sin m! ) ! r!
! ! ! ! n ! 1!! n ! 2 ! !
(9)
6
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Putting Eq. (7) and (9) into BVP (6), we have(Rummel et al, 1993)
(10)
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n! 3 ! R3 ! a! f ! Pnm (cos ! ) cos m! sin ! d! d! ! anm ! ! ! 4! GM (n ! 1)( n ! 2) ! R ! 0! !!!! ! ! 0 ! ! ! 2! ! ! n! 3 3 R ! ! a! !!! ! f ! Pnm (cos! )sin m! sin ! d! d! ! bnm ! 4! GM (n ! 1)(n ! 2) !! R !! 0 ! ! !! 0 ! ! ! 2!
So far, the main process for dealing with SGG is given, that is, f is computed according
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to Eq. (3) firstly; and then from Eq. (10) and (8), the disturbing potential T can be obtained preliminarily; and if the accuracy of T cannot satisfy the requirements, f can be computed
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again from Eq. (1) by inserting T12 and T23 into Eq. (1) which are computed from the obtained
T . By such iteration, the gravitational field of the earth can be given finally.
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2.3 Discussion on data reduction, polar gaps and low frequency error
BVP (6) should be given on a sphere in order to obtain its solution. The prerequisite condition
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that the spherical harmonic coefficients of the disturbing potential T can be expressed as Eq. (10) is that the orbit surface S is replaced by its average sphere. In fact, the reduction of BVP (6)
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from S onto its average sphere S ! {r ! a} is required to solve BVP (6). In order to show the
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statistics of actual S , Table 2 shows the orbits information on 1 November 2009.
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Table 2 Statistics of orbit data Variable
Min
Max
Mean
Std
r ( unit : m)
6620526.72
6651805
6637703
10893.02
r! a a
2.89283E-08
0.002588 0.001474
0.000721
In Table 2, a is the mean value of r , which equals 6637.703km. Definitely, the actual
position of GOCE satellite is not strictly located on the average sphere surface and the reduction is necessary. Correspondingly, the reduction formula can be obtained by using Taylor expansion, that is,
! 2T ! r2
! r! a
! 2T ! r2
! S
!
! 3T ! r3
where the magnitudes less than O r ! a
!r ! a!
(11)
r! a 2
!
are neglected. Based on Eq. (11), the concrete
iterative method for the reduction can be written as 7
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! ! T1 ! 0 ! 2 ! ! T1 ! 2 ! f ! !r S ! !1 at infinity ! T1 ! O (r ), where f is given by Eq. (3); as well as
(12)
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! ! Tn ! 1 ! 0 ! 2 ! 3Tn ! ! Tn ! 1 f (r ! a) ! ! ! 2 ! r3 r! a ! !r S ! !1 at infinity ! Tn ! 1 ! O (r ),
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(13)
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Yu and Wan (2011) illustrated that the reduction formula is not required under the accuracy of SGG if the reference gravitational potential V is chosen as EGM08 model. Even if the reference
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model is not accurate enough, the iteration can be done. By that, the disturbing potential T becomes increasingly smaller which also makes the term neglected in Eq. (11) smaller. As for the polar gap caused by the satellite’s inclination, though the influence can not be
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neglected (Koop 1993, van Gelderen 1997, Sneeuw 1997, Reguzzoni and Tselfes 2009, et al), two main strategies can be adopted. One is technique of regularization (Rudolph et al, 2002, Kusche
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and Klees 2002, Ditmar et al, 2003, Metzler and Pail, 2005) which is appropriate to least-square
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method, and the other is using discrete values from other observations or a prior gravity field model (Migliaccio et al.,2006, Yi, 2012). In this paper the latter strategy is adopted which is that
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the data in polar district are given by EGM08 as grids of 25! ! 25! . Since it’s not a new problem, we do not discuss it here.
It is a very important issue in dealing with GOCE data how to filter SGG data since the
recommended measurement bandwidth (MBW) for SGG is 0.005~0.1Hz. It can be seen from the released SGG data that the error in low frequency range of SGG is very large. Many filter methods for SGG data have been proposed according to different approaches to recover the gravitational field. For example, Schuh (2003) and Klees et al. (2003) studied ARMA filter method, Migliaccio et al. (2004b) and Reguzzoni et al. (2009) studied Wiener filter to deal with the error of SGG in low frequency. Particularly, Migliaccio et al. (2004b) pointed out that the high and low frequency tails in SGG could have some effects on the recovery of the gravitational field. The filter method, i.e., band-pass FIR, proposed by Yu and Wan (2011) is adopted in this paper. To overcome the issue caused by the high and low frequency tails of SGG, these tails are
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replaced by the corresponding ones from some known model which is achieved by remove-restore method (Yu and Wan,2011). In this paper, the reference model is EIGEN-5C. The algorithm for the band-pass FIR filter is applied along the forward direction and then along the backward direction in order to eliminate the effect of phase-shift.
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3. Arithmetic examples 3.1 Simulated example
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EGM08 model with former 300 degrees/orders is used as the actual gravitational potential v ,
and then EIGEN_5C model with former 300 degrees/orders is chosen as the reference
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gravitational potential V . Considering the poor accuracy，the C20 of EIGEN_5C is replaced by that of EGM08 . Then GOCE data during the period of 1 November 2009 to 1 June 2011 are
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selected. At first, vij and Vij ( i, j ! 1, 2,3 ) can be computed in GRF using the satellite position and attitude data provided by SST_PSO_2 and EGG_NOM_2 respectively. Then computing
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Tij ! vij ! Vij and replacing Tij into Eq. (3), the value of
! 2T on the orbit surface S can be ! r2
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obtained from Eq. (3) where v12 and v23 (or T12 and T23 ) are not used. Divide the orbit surface S into 25! ! 25! grids along the longitude and latitude and compute the average value of
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! 2T on each patch of the grids. Finally, the spherical harmonic coefficients of T can be ! r2 ! 2T computed from Eq. (10) where the real value of on the polar gap is adopted, and T (0) is ! r2
used to denote the obtained disturbing potential. Furthermore, the iterative process is executed from T (0) , and the disturbing potential after the iteration is denoted by T (1) . Table 3 gives the statistics of !
defined by Eq. (4) during the period of 1 November 2009 to
1 January 2010 to show the influence of non-full tensor. !
! 1 is the value after one iteration. ! Table 3 Statistics of ! Variable
0
is the value of the initial data and
is the mean relative ratio of !
to
! 2T . ! r2
( unit: s-2 )
!
Number
Min
Max
Mean
0
5183997
0.0
5.542741E-13
1.126832E-14
3.09%
!1
5183997
0.0
1.167528E-13
1.958139E-15
3.17%
!
9
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Definitely, the mean value of !
are far smaller than 1mE and after one iteration, the value
becomes smaller by nearly one magnitude. The relative ratio of ! which supports the conclusions of section 2.1. !
is about 3.09% or 3.17%
can be seen as error from the non-full tensor.
Obviously, compared to the observation errors of gradient data from the GOCE satellite, it is not a
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dominate one.
To exhibit the accuracies of recovering the gravitational field, the degree variances of the
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obtained solutions to EGM08 are shown in Fig. 2. Correspondingly, the cumulated geoid height
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differences to EGM08 are shown in Fig.3
Fig. 2. Degree variances of the gravity field models recovered from simulated data
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Fig. 3. Cumulated geoid height differences of EIGEN_5C, the two models recovered from the simulated data, and EGM08
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In Fig. 2, EGM GRACE denotes the degree variance difference of EIGEN_5C to EGM08,
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while Degree variance0 and Degree variance1 mean the degree variances of T (0) and
T (1) respectively. According to Fig. 2, T (0) , as the solution in the absence of v12 and v23 , can
recover the model with over 265 degrees/orders and its accuracy of determining the spherical harmonic coefficients is obviously higher than the measured SGG (see Fig. 4). Compared with
T (0) , T (1) has higher accuracy to the simulated model. According to Rummel (2012), we know
the degree variances of models from GOCE are a little larger than 10-10 and the highest degree is
about 250, so it can be concluded that the approach proposed by this paper is effective and accurate. According to Fig.3, through iteration, the geoid difference to EGM08 is only about 1 cm at 200 degree and 4 cm at 250 degree. The accuracy meets the requirements of GOCE data processing, one of whose main objectives is to determine the geoid with an accuracy of 1~2 cm up to 200 degree (Balmino et al., 1999; Albertella et al., 2002). 11
Page 11 of 18
It should be emphasized that this study used the actual position and attitude data of GOCE and only four components of the gradient tensor like actual observation, so it is also proved that the spherical harmonic analysis of the radial gradient of the gravity can be used in dealing with SGG data although the observations are non-full tensor.
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3.2 Experiment in actual data processing
Actual gradient observation data from EGG_NOM_2 are used in this section. Our emphasis is
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that the only difference, compared to above section, is the gradient data which contain observation noise here but not in the above section. FIR filter (Yu and Wan, 2011) was adopted to remove the
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observation errors of this experiment in order to weaken the effect of low frequency error. Finally, a gravity field model is recovered. The degree variance differences and cumulated geoid height
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d
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differences to EGM08 are shown in Fig. 4 and Fig.5 respectively.
Fig. 4. Degree variance differences of the gravity field models recovered from actual observation data and EGM08
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Page 12 of 18
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Fig. 5. Cumulated geoid height differences of the gravity field models recovered from actual observation data and EGM08 In Fig.4, Variance _ EGM _ D3 and Variance _ EGM _ T 3 are degree variance
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differences of models from ESA which are derived using direct-wise and time-wise methods respectively; Variance _ EGM _ Trr is degree variance difference of the gravity field model
recovered
from
actual
observation
data
in
this
study.
In
Fig.
5,
EGM _ D3 ! EGM 08 , EGM _ T 3 ! EGM 08 and EGM _ Trr ! EGM 08 denote the
cumulated geoid differences respectively. Easily to know, the approach of this paper can successfully recover a gravity field model with higher than 200 degrees, although the accuracy is a little poorer after degree 200. It is important to note satellite tracking data is also used in the recovery of EGM_D3 and EGM_T3, but present study only uses gradient data which are sensitive only to high degrees and orders of the gravity field model due to low frequency error. Hence, the low degrees of the models are much different, especially the degrees between 27 and 100, which leads to differences in several centimeters in cumulated geoid heights. Nevertheless, at 200 degree, their differences are smaller than 4 cm. Filtering methods are also different with other groups’ 13
Page 13 of 18
inversion (Schuh 2003, Reguzzoni 2009 et al.), because this paper uses spherical harmonic analysis but other studies mainly use least square or least square collocation methods. Generally speaking, the accuracy is consistent in higher degrees. The FIR filter is mainly used to remove low frequency errors in this experiment. However,
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random and gross errors also exist. Thus, if some strategies can be used to process these errors, the accuracy may well be improved further. This is next work of this paper where Innovation
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Algorithm (Peter and Richard, 2006) and grid processing are being experimentally used. However, according to the comparison of Figs. 2 and 4 and of Figs. 3 and 5, it’s not difficult to conclude the
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error from the non-full tensor given by Eq. (4) is smaller than the observation errors. Hence, the boundary condition in Eq. (3) is accurate enough for GOCE data processing.
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On the other hand, since solving a large number of normal equations is a complicated task because the unknown parameters would exceed 40 000 if a gravity field model with 200 degrees is recovered, the approach of this paper is more optimal in computing efficiency, such as if getting
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Trr on orbit surface, the calculation time is shorter than 1 hour in an ordinary computer. It is meaningful for rapid solution of GOCE gravity field model.
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4. Conclusions
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The attitude of GOCE satellite has the following property: one axis of the gradiometer loaded in GOCE almost points at the same direction as the radial direction of the Earth. The angle is
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denoted as ! 3 . One main objective of the paper is to analyze this factor in detail and calculate the
mean value and standard deviation (Table 1). Starting from the formula of radial gravity gradients by the full tensor, the boundary
condition of radial gradients on satellite orbits is established in the absence of gradient components
v 12
and
v
23
, based on the attitude property of GOCE.
The accuracy of the
conditions is also analyzed according to the statistics of ! 3 . The boundary condition is examined by data processing. The results show that accuracy of the gravity field model recovered from simulated gradient data by using this condition is definitely higher than the accuracy of models from actual gradient data. It means that the approach of computing boundary conditions on satellite orbits is effective, both theoretically and practically. Finally, the approach is used in actual GOCE data processing, and a gravity field model is recovered, the accuracy of which is nearly consistent with the models from ESA. 14
Page 14 of 18
Acknowledgments The authors would like to thank European Space Agency (ESA) for providing GOCE data. We also thank the two reviewers for their helpful suggestions. This work is supported by the National Natural Science Fund of China (No. 41074015, 41104047) and the open fund of Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University(11-01-07).
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Klees, R., Koop, R., Visser, P., et al. 2000. Efficient gravity field recovery from GOCE gravity gradient observations. Journal of Geodesy, 74(7-8), 561-571. Klees, R., Ditmar, P., Broersen P., 2003. How to handle colored observation noise in large-square problems. Journal of Geodesy, 76, 629---640. Koch, K.R., Kusche, J., 2002. Regularization of geopotential determination from satellite data by variance components. Journal of Geodesy. 76(5), 259-268. Koop, R., 1993. Global gravity field modelling using satellite gravity gradiometry . Publications on Geodesy 38 , Delft . Kusche, J., Klees, R., 2002. Regularization of gravity field estimation from satellite gravity gradients .
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Journal of Geodesy, 76(6-7), 359-368. Kusche, J., 2002. On fast multigrid iteration techniques for the solution of normal equations in satellite gravity recovery. Journal of Geodynamics, 33(1-2), 173-186. Kusche, J., Klees, R., 2002. Regularization of gravity field estimation from satellite gravity gradients . Journal of Geodesy . 76(6-7), 359-368. Martinec, Z., 2003. Green’s function solution to spherical gradiometric boundary-value problems.
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Journal of Geodesy, 77, 41-49.
Metzler, B., Pail, R., 2005. GOCE data processing: the spherical cap regularization approach. Studia Geophysica Et Geodaetica , 49(4), 441-462.
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Migliaccio, F., Reguzzoni, M., Sansò, F., Zatelli, P., 2004a. GOCE: dealing with large attitude variations in the conceptual structure of the space-wise approach, in Proceedings of the 2nd GOCE User Workshop, March 8–10, Frascati, Italy, ESA SP-569.
us
Migliaccio, F., Reguzzoni, M., Sansò, F., 2004b. Space-wise approach to satellite gravity field determination in the presence of coloured noise. Journal of Geodesy. 78(4-5), 304-313. Migliaccio, F., Reguzzoni, M., Sansò, F., et al., 2006. The latest test of the space-wise approach for SP-627).
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GOCE data analysis. In: Proceedings of the third international GOCE user workshop (ESA Migliaccio, F., Reguzzoni, M., Tselfes, N., 2007. GOCE: a full-gradient simulated solution in the space-wise approach. Dynamic Planet: Monitoring and Understanding a Dynamic Planet with
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Geodetic and Oceanographic Tools. 130, 383-390.
Milani, A., Rossi, A., Villani, D., 2005. A timewise kinematic method for satellite gradiometry: GOCE simulations. Earth Moon and Planets. 97(1-2), 37-68.
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Pail, R., Goiginger, H., Schuh, W. D. et al., 2010. Combined satellite gravity field model GOCO01S derived from GOCE and GRACE. Geophysical Research Letters. 37.
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Pail, R., Bruinsma, S., Migliaccio, F., et al., 2011. First GOCE gravity field models derived by three different approaches. Journal of Geodesy. 85(11), 819-843.
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Pavlis, N. K., Holmes, S. A., Kenyon, S. C., Factor, J. K., 2012. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), Journal of Geophysical Research-Solid Earth, 117. B04406, doi:10.1029/2011JB008916.
Pertusini, L., Reguzzoni, M., Sansò, F., 2010. Analysis of the Covariance Structure of the GOCE Space-Wise Solution with Possible Applications. Gravity, Geoid and Earth Observation. 135, 195-202.
Peter, J., Richard, A., 2003. Time Series: Theory and Methods ( Second Edition ). Springer . New York. 132~136.
Preimesberger, T. Pail, R., 2003. GOCE quick-look gravity solution: Application of the semianalytic approach in the case of data gaps and non-repeat orbits. Studia Geophysica Et Geodaetica. 47(3). 435-453. Reguzzoni, M., 2003. From the time-wise to space-wise GOCE observables. Advances in Geosciences. 2003(1), 137~142. Reguzzoni, M., Tselfes, N., 2009. Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis. Journal of Geodesy, 83(1), 13-29. Reguzzoni, M., Tselfes, N., Venuti, G., 2008. Stepwise Solutions to Random Field Prediction Problems. Vi Hotine-Marussi Symposium on Theoretical and Computational Geodesy. 132, 263-268. Reguzzoni, M., Tselfes, N., 2009. Optimal multi-step collocation: application to the space-wise
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approach for GOCE data analysis. Journal of Geodesy. 83(1), 13-29. Rummel, R., van Gelderen, M., 1992. Spectral-Analysis of the Full Gravity Tensor. Geophysical Journal International, 111(1), 159-169. Rummel, R., van Gelderen M., Koop, R., Schrama, E., Sansò, F., Brovelli, M., Migliaccio, F., Sacerdote, F., 1993. Spheical Harmonic Analysis of Satellite Gradiometry, Netherlands Geodetic
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Commission Publications on Geodesy, New Series, Number 39, Delft, The Netherlands.
Rummel, R., Gruber. Thomas., 2012. Product Acceptance Review (PAR)- Level 2 Product ReportRep., ESA.
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Sacerdote, F., Sansò, F.,1989. Some problems related to satellite gradiometry. Bull Geod, 63,405-415.
Schuh, W.D., 2003. The processing of band-limited measurements: Filtering techniques in the least
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squares context and in the presence of data gaps. Space Science Review. 108(1-2), 67-78.
Sneeuw , N. , van Gelderen , M., 1997. The polar gap .Lecture Notes in Earth Sciences, 65, 559-568. Yi, W., 2012. An alternative computation of a gravity field model from GOCE. J. Adv. Space Res.
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http://dx.doi.org/10.1016/j.asr.2012.04.018.
Yu, J., Wan, X., 2011. Reduction for gradiometry and corresponding imitation. Chinese J. Geophys. (in Chinese). 54(5), 1182~1186.
Yu, J., Wan, X., 2011. The Frequency Analysis of Gravity Gradients and the Methods of Filtering
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Processing. Proc. of ‘4th International GOCE User Workshop’, Munich, Germany, 2011 (ESA SP-696, July 2011).
Yu, J., Wan X., 2012. Recovery of the Gravity Field from GOCE Data by Using the Invariants of
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Gradient Tensor. Sci China Ser D-Earth, doi: 10.1007/s11430-012-4427-y. Yu, J., Zhao D., 2010. The gravitational gradient tensor’s invariants and the related boundary
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conditions. Sci China Ser D-Earth, 53(5), 781-790.
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Highlights: 1. The boundary condition of radial gradients is established for gravity field recovery using GOCE data whose gradient observations are non-full tensors. 2. The influence of non-full tensors is analyzed in detail by using actual orbit and attitude data.
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3. Compared to the value of radial gradients, relative error from non-full tensors is around 3%.
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4, The experiment on gravity field recovery using simulated data proves the effectiveness of the method based on the BVP.
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an
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5, A gravity field model is derived from actual GOCE data, whose accuracy is nearly consistent with models from ESA in high degrees.
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S0264-3707(13)00031-8 http://dx.doi.org/doi:10.1016/j.jog.2013.02.005 GEOD 1215
To appear in:
Journal of Geodynamics
Received date: Revised date: Accepted date:
20-10-2012 15-1-2013 3-2-2013
Please cite this article as: Xiaoyun, W., Jinhai, Y., Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients, Journal of Geodynamics (2013), http://dx.doi.org/10.1016/j.jog.2013.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Derivation of the radial gradient of the gravity based on non-full tensor satellite gravity gradients Wan Xiaoyun[1][2], Yu Jinhai[1][2] 1, Key Laboratory of Computational Geodynamics, Chinese Academy of Sciences, Beijing 100049,China 2, College of Earth Science, University of Chinese Academy of Sciences, Beijing 100049,China
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ABSTRACT: The formula for computing the radial gradient of the gravity is derived from the
gravity gradients. Since GOCE satellite attitude has such property that one axis in the gradiometer
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almost has the same direction as the radial one of the earth, the formula can be used to compute
the radial derivative of second order of the disturbing potential. This means that the boundary
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value problem (BVP) with the radial gradient of the gravity can be established on the satellite orbit. Hence, the recovery of the gravity field from GOCE data can be realized by solving the BVP. In
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order to examine the accuracy of the BVP, an arithmetic example is given, the computational results of which illustrate that the efficient degrees/orders of the gravity field model recovered by
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the BVP can satisfy the requirements of the satellite gravity gradients. In addition, a gravity field model is solved out from the BVP by using actual GOCE data.
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1. Introduction
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Keywords: GOCE, Non-full tensor, Harmonic analysis
GOCE (Gravity field and steady-state Ocean Circulation Explorer) (Balmino et al., 1999;
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Albertella et al., 2002) is an explorer belonging to ESA (European Space Agency), and one of its
main aims is to recover an accurate static gravity field model with high degrees and orders. GOCE has provided a huge amount of data about its position，attitude and the gravity gradients on the
orbit since its launch on 17 March 2009. Obviously, it is widely concerned how to recover the gravity field models by using these data. The research focuses mainly on time-wise and space-wise approach.
Rummel (1993) introduced the principles of the time-wise and space-wise approach. Keller (1997) investigated space-wise approach by constructing boundary value problems. Klees et al. (2000) proposed an efficient algorithm for recovering gravity field in time domain. Reguzzoni (2003) discussed the observation process from time domain to space domain. Albertella et al. (2004) analyzed the filtering techniques for gradients data processing in time and space domain. Milani et al. (2005) simulated the GOCE mission by time-wise approach whereas Migliaccio et al. 1
Page 1 of 18
(2007) conducted a simulation by space-wise approach. Reguzzoni et al. (2008) studied the application of step-wise solution in space domain. Pertusini et al. (2009) evaluated the structure of the covariance in space-wise solution. Pail et al. (2010) recovered the gravity field model GOCO01S using time-wise method by combining GOCE and GRACE data. The GOCE gravity
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field models from different methods were compared by Pail et al (2011). This paper is aimed at doing some work in space domain, by constructing BVP.
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Some notations are introduced here to make treatments clear. Let X 1 , X 2 and X 3 be
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three axes of GRF (Gradometer Reference Frame), where the direction of X 3 is very close to radial one of the earth. Again letting v denote the gravitational potential of the earth, and
! 2v , vij ( i, j ! 1, 2,3 ) can be measured along the satellite orbit by the gradiometer and ! X i! X j
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vij !
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are called SGG (Satellite Gravity Gradients). Because of poor accuracies of v12 and v23 , only diagonal elements and v13 in SGG can be used to recover the gravitational field. Generally,
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recovery of the gravitational field from SGG can be attributed to solving the normal system of
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equations. The advantage of establishing the normal system of equations is that SST data (Satellite-to-Satellite Tracking) can be easily used to construct equations together with SGG data,
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but the issue is that the computation is very complicated since the unknowns exceed 40 000 if the degree of the model is higher than 200, although some researchers, such as Kusche (2002), Ditmar (2003), and Preimesberger (2003) et al., investigated the methods of rapid resolution of GOCE gravity field models.
The approach by BVPs (Boundary Value Problems) with boundary condition of the radial
gradients has its special advantage because of computing efficiently. In the early researches about GOCE, many researchers made deep discussions on BVPs. For example, Rummel et al. (1992) analyzed the spectral characteristics of
! ! zz ! ，! !
xz
,!
yz
! ，! !
xx
! !
yy
,2!
xy
!
and gave their two
dimension Fourier expressions. van Gelderen et al. (2001) summarized the solution of the general geodetic BVPs by least-square methods and gave the solution under the boundary conditions of gravity gradients. Martinec (2003) discussed Green function to gradiometric BVP. However, SGG is not a full-tensor measurement because of the absence of
v12 and v23 , thus it seemingly lacks
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some terms to establish BVP with the boundary conditions of the radial gradient. Recently, Yu et al. (2010) constructed the boundary conditions based on the invariants of the gravitational gradients, which can be represented as the radial component of the gravitational gradient tensor. This makes that BVPs can be used to deal with SGG data, which means that the model of the
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gravitational field can be recovered by the method of spherical harmonic analysis. The method of the invariants is not only simple, but affected with few attitude errors also. Consequently, Yu and
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Wan (2012) processed the actual SGG data and obtained a GOCE gravitational field model with higher than 200 degrees.
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In fact, it is a key problem to derive the boundary conditions about the radial gradient in BVPs.
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Apart from the methods of the invariants (Sacerdote and Sansò,1989), the boundary condition of the radial gradient can also be derived from the coordinate transformation. However, in the actual GOCE data processing, the latter has not been paid enough attention. One reason may be that SST
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data can not be used as BVPs and a whole model of the gravitational model can not be recovered using only SGG data due to the limitations of the measure bandwidth. The other is that the radial
v23 can not be used due to their poor accuracies.
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while v12 and
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gradient is different from v33 in the SGG so that it has to be composed of vij ( i, j ! 1, 2,3 )
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In the paper, it is discussed how to establish the boundary condition about the radial gradient from the coordinate transformation. Then the accuracy of the boundary condition is analyzed using the satellite attitude data. Finally, some arithmetic about recovery of the gravitational field from the boundary condition is given.
2. Establishing the boundary condition 2.1 Computation of the radial gradient A reference gravitational potential V is chosen firstly, and then let T ! v ! V be the disturbing potential. Generally, V should be a very good approximation to the actual gravitational potential v . For example, V can be chosen as EGM08 (Pavlis et al., 2012) or EIGEN_5C (Foerste et al., 2008) models. Letting ! 1 , !
2
and !
3
denote the angles between
X 1 , X 2 and X 3 in GRF and the radial direction of the earth, they can be computed from GOCE satellite attitude. By derivation, we have
3
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! 2T ! T33 cos 2 ! 3 ! T22 cos2 ! 2 ! T11 cos2 ! 1 ! 2T12 cos ! 1 cos ! 2 !r ! 2T13 cos ! 1 cos ! 3 ! 2T23 cos ! 2 cos ! 3
2
(1)
where
! 2T ! vij ! Vij , ! X i! X j
( i, j ! 1, 2,3 )
(2)
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Tij !
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Since vij are measured from the gradiometer and Vij can be computed from GOCE satellite’s
position and attitude, T11 , T22 , T33 and T13 are all known while T12 and T23 are unknown
from Eq. (1).
! 2T cannot be computed directly ! r2
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due to the poor accuracies of v12 and v23 . This means that
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On the other hand, although the actual X 3 -axis is different with the radial direction of the
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earth, their difference is small. This means that ! 1 ! 900 , !
2
! 900 and !
3
! 00 . Thus, it can
be concluded that the term related to T33 is the main one in Eq. (1) and the terms related to T12 3
is, the attitude data of GOCE satellite
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and T23 are small ones. To understand how small !
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during the period from 1 November to 10 November 2009 are chosen and the values of !
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during the period are computed. Table 1 summarizes the statistics of !
3
3
and Fig. 1 gives its
distribution during this period.
Table 1
Statistics of ! Number 690613
3
( unit: degree)
Min 0.0008
Max 1.3577
Mean 0.3859
Std 0.2106
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Fig. 1. Angle between X 3 -axis of GRF and radial direction
cos!
3
! cos1.50 ! 0.9997
is small and its maximum is less than 1.50 . ,
cos 2 ! 1 ! cos 2 !
2
! cos 2 !
3
cos ! 1 , cos !
and
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Hence,
3
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According to Fig. 1, angle !
! 1 , that is, compared with cos ! 3 , cos !
2
1
! 0.0262 and cos !
since
2
can be
considered as small quantities. So, Eq. (1) can be written as
! 2T ! T33 cos 2 ! 3 ! T22 cos 2 ! 2 ! T11 cos 2 ! 1 ! 2T13 cos ! 1 cos ! ! r2 where the neglected magnitude is
! ! 2T12 cos ! 1 cos ! 2 ! 2T23 cos ! 2 cos !
In fact, !
3
(3)
(4)
3
is the error of the boundary condition (3). Assuming that all the components in Tij
have the same magnitudes, the relative magnitude of the error !
! !
2cos ! 1 cos ! 2 ! 2cos ! 2 cos ! cos2 ! 3
3
is approximately equal to (5)
By computation, we know that ! ! 5.5% . Also from Fig. 1, it can be seen that the average of the angle !
3
is less than 0.50 . Hence, the relative error !
of the boundary condition (3) should be
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smaller. The value of !
given by report of ESA is 2% (Rummel and Gruber, 2012).
How to estimate the absolute error of ! ? This depends on the choice of the reference gravitational potential V . The more accurate the reference gravitational potential V is, the smaller the error !
is. This is why EGM08 or ENGEN_5C are recommended as the reference
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gravitational potential. In addition, the iteration computations for T12 and T23 are also required,
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that is, T12 and T23 can be computed and then inserted into Eq. (1) after the disturbing potential
T is solved firstly from the boundary condition (3). In fact, the effect for Eq. (1), caused by the T12 and T23 , can be completely eliminated by such iteration process (Migliaccio et
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absence of al, 2004a).
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In sum, from Eq. (1) or Eq. (3), we can obtain the following BVP
(6)
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!! T ! 0 ! 2 !! T ! 2 ! f ! !r S ! T ! O (r ! 1 ), !
at infinity
2.2 The solution of the BVP
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where S is the orbit surface of GOCE satellite and f is a known function on S .
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As the GOCE satellite’s orbit is almost circular, S can be seen as a sphere, that is,
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S ! {r ! a} , where a is the average radius of the orbit. Moreover considering that f is known on S , f can be expressed as the following spherical harmonic series
f !
GM a3
!
n
! !
Pnm (cos ! )(Cnm cos m! ! S nm sin m! )
where GM is the gravitational constant of the earth,
Pnm ! cos!
!
(7)
n ! 0 m! 0
! r ,! , ! !
are the spherical coordinates,
is the fully normalized Associated Legendre Functions with degree n and order m,
and Cnm and Snm are the fully normalized spherical harmonics coefficients. Now we also expand the disturbing potential T as
T!
GM r
Rn !n! 0 m!! 0 r n Pnm (cos! )(anm cos m! ! bnm sin m! ) !
n
(8)
where R is the mean radius of the earth. By computation, we have (Rummel et al, 1993)
! 2T GM ! 3 ! r2 r
!
n
n! 2 m! 0
n
R! ! Pnm (cos! )(anm cos m! ! bnm sin m! ) ! r!
! ! ! ! n ! 1!! n ! 2 ! !
(9)
6
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Putting Eq. (7) and (9) into BVP (6), we have(Rummel et al, 1993)
(10)
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n! 3 ! R3 ! a! f ! Pnm (cos ! ) cos m! sin ! d! d! ! anm ! ! ! 4! GM (n ! 1)( n ! 2) ! R ! 0! !!!! ! ! 0 ! ! ! 2! ! ! n! 3 3 R ! ! a! !!! ! f ! Pnm (cos! )sin m! sin ! d! d! ! bnm ! 4! GM (n ! 1)(n ! 2) !! R !! 0 ! ! !! 0 ! ! ! 2!
So far, the main process for dealing with SGG is given, that is, f is computed according
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to Eq. (3) firstly; and then from Eq. (10) and (8), the disturbing potential T can be obtained preliminarily; and if the accuracy of T cannot satisfy the requirements, f can be computed
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again from Eq. (1) by inserting T12 and T23 into Eq. (1) which are computed from the obtained
T . By such iteration, the gravitational field of the earth can be given finally.
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2.3 Discussion on data reduction, polar gaps and low frequency error
BVP (6) should be given on a sphere in order to obtain its solution. The prerequisite condition
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that the spherical harmonic coefficients of the disturbing potential T can be expressed as Eq. (10) is that the orbit surface S is replaced by its average sphere. In fact, the reduction of BVP (6)
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from S onto its average sphere S ! {r ! a} is required to solve BVP (6). In order to show the
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statistics of actual S , Table 2 shows the orbits information on 1 November 2009.
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Table 2 Statistics of orbit data Variable
Min
Max
Mean
Std
r ( unit : m)
6620526.72
6651805
6637703
10893.02
r! a a
2.89283E-08
0.002588 0.001474
0.000721
In Table 2, a is the mean value of r , which equals 6637.703km. Definitely, the actual
position of GOCE satellite is not strictly located on the average sphere surface and the reduction is necessary. Correspondingly, the reduction formula can be obtained by using Taylor expansion, that is,
! 2T ! r2
! r! a
! 2T ! r2
! S
!
! 3T ! r3
where the magnitudes less than O r ! a
!r ! a!
(11)
r! a 2
!
are neglected. Based on Eq. (11), the concrete
iterative method for the reduction can be written as 7
Page 7 of 18
! ! T1 ! 0 ! 2 ! ! T1 ! 2 ! f ! !r S ! !1 at infinity ! T1 ! O (r ), where f is given by Eq. (3); as well as
(12)
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! ! Tn ! 1 ! 0 ! 2 ! 3Tn ! ! Tn ! 1 f (r ! a) ! ! ! 2 ! r3 r! a ! !r S ! !1 at infinity ! Tn ! 1 ! O (r ),
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(13)
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Yu and Wan (2011) illustrated that the reduction formula is not required under the accuracy of SGG if the reference gravitational potential V is chosen as EGM08 model. Even if the reference
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model is not accurate enough, the iteration can be done. By that, the disturbing potential T becomes increasingly smaller which also makes the term neglected in Eq. (11) smaller. As for the polar gap caused by the satellite’s inclination, though the influence can not be
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neglected (Koop 1993, van Gelderen 1997, Sneeuw 1997, Reguzzoni and Tselfes 2009, et al), two main strategies can be adopted. One is technique of regularization (Rudolph et al, 2002, Kusche
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and Klees 2002, Ditmar et al, 2003, Metzler and Pail, 2005) which is appropriate to least-square
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method, and the other is using discrete values from other observations or a prior gravity field model (Migliaccio et al.,2006, Yi, 2012). In this paper the latter strategy is adopted which is that
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the data in polar district are given by EGM08 as grids of 25! ! 25! . Since it’s not a new problem, we do not discuss it here.
It is a very important issue in dealing with GOCE data how to filter SGG data since the
recommended measurement bandwidth (MBW) for SGG is 0.005~0.1Hz. It can be seen from the released SGG data that the error in low frequency range of SGG is very large. Many filter methods for SGG data have been proposed according to different approaches to recover the gravitational field. For example, Schuh (2003) and Klees et al. (2003) studied ARMA filter method, Migliaccio et al. (2004b) and Reguzzoni et al. (2009) studied Wiener filter to deal with the error of SGG in low frequency. Particularly, Migliaccio et al. (2004b) pointed out that the high and low frequency tails in SGG could have some effects on the recovery of the gravitational field. The filter method, i.e., band-pass FIR, proposed by Yu and Wan (2011) is adopted in this paper. To overcome the issue caused by the high and low frequency tails of SGG, these tails are
8
Page 8 of 18
replaced by the corresponding ones from some known model which is achieved by remove-restore method (Yu and Wan,2011). In this paper, the reference model is EIGEN-5C. The algorithm for the band-pass FIR filter is applied along the forward direction and then along the backward direction in order to eliminate the effect of phase-shift.
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3. Arithmetic examples 3.1 Simulated example
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EGM08 model with former 300 degrees/orders is used as the actual gravitational potential v ,
and then EIGEN_5C model with former 300 degrees/orders is chosen as the reference
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gravitational potential V . Considering the poor accuracy，the C20 of EIGEN_5C is replaced by that of EGM08 . Then GOCE data during the period of 1 November 2009 to 1 June 2011 are
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selected. At first, vij and Vij ( i, j ! 1, 2,3 ) can be computed in GRF using the satellite position and attitude data provided by SST_PSO_2 and EGG_NOM_2 respectively. Then computing
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Tij ! vij ! Vij and replacing Tij into Eq. (3), the value of
! 2T on the orbit surface S can be ! r2
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obtained from Eq. (3) where v12 and v23 (or T12 and T23 ) are not used. Divide the orbit surface S into 25! ! 25! grids along the longitude and latitude and compute the average value of
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! 2T on each patch of the grids. Finally, the spherical harmonic coefficients of T can be ! r2 ! 2T computed from Eq. (10) where the real value of on the polar gap is adopted, and T (0) is ! r2
used to denote the obtained disturbing potential. Furthermore, the iterative process is executed from T (0) , and the disturbing potential after the iteration is denoted by T (1) . Table 3 gives the statistics of !
defined by Eq. (4) during the period of 1 November 2009 to
1 January 2010 to show the influence of non-full tensor. !
! 1 is the value after one iteration. ! Table 3 Statistics of ! Variable
0
is the value of the initial data and
is the mean relative ratio of !
to
! 2T . ! r2
( unit: s-2 )
!
Number
Min
Max
Mean
0
5183997
0.0
5.542741E-13
1.126832E-14
3.09%
!1
5183997
0.0
1.167528E-13
1.958139E-15
3.17%
!
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Page 9 of 18
Definitely, the mean value of !
are far smaller than 1mE and after one iteration, the value
becomes smaller by nearly one magnitude. The relative ratio of ! which supports the conclusions of section 2.1. !
is about 3.09% or 3.17%
can be seen as error from the non-full tensor.
Obviously, compared to the observation errors of gradient data from the GOCE satellite, it is not a
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dominate one.
To exhibit the accuracies of recovering the gravitational field, the degree variances of the
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obtained solutions to EGM08 are shown in Fig. 2. Correspondingly, the cumulated geoid height
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d
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an
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differences to EGM08 are shown in Fig.3
Fig. 2. Degree variances of the gravity field models recovered from simulated data
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Page 10 of 18
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Fig. 3. Cumulated geoid height differences of EIGEN_5C, the two models recovered from the simulated data, and EGM08
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In Fig. 2, EGM GRACE denotes the degree variance difference of EIGEN_5C to EGM08,
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while Degree variance0 and Degree variance1 mean the degree variances of T (0) and
T (1) respectively. According to Fig. 2, T (0) , as the solution in the absence of v12 and v23 , can
recover the model with over 265 degrees/orders and its accuracy of determining the spherical harmonic coefficients is obviously higher than the measured SGG (see Fig. 4). Compared with
T (0) , T (1) has higher accuracy to the simulated model. According to Rummel (2012), we know
the degree variances of models from GOCE are a little larger than 10-10 and the highest degree is
about 250, so it can be concluded that the approach proposed by this paper is effective and accurate. According to Fig.3, through iteration, the geoid difference to EGM08 is only about 1 cm at 200 degree and 4 cm at 250 degree. The accuracy meets the requirements of GOCE data processing, one of whose main objectives is to determine the geoid with an accuracy of 1~2 cm up to 200 degree (Balmino et al., 1999; Albertella et al., 2002). 11
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It should be emphasized that this study used the actual position and attitude data of GOCE and only four components of the gradient tensor like actual observation, so it is also proved that the spherical harmonic analysis of the radial gradient of the gravity can be used in dealing with SGG data although the observations are non-full tensor.
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3.2 Experiment in actual data processing
Actual gradient observation data from EGG_NOM_2 are used in this section. Our emphasis is
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that the only difference, compared to above section, is the gradient data which contain observation noise here but not in the above section. FIR filter (Yu and Wan, 2011) was adopted to remove the
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observation errors of this experiment in order to weaken the effect of low frequency error. Finally, a gravity field model is recovered. The degree variance differences and cumulated geoid height
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d
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an
differences to EGM08 are shown in Fig. 4 and Fig.5 respectively.
Fig. 4. Degree variance differences of the gravity field models recovered from actual observation data and EGM08
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Fig. 5. Cumulated geoid height differences of the gravity field models recovered from actual observation data and EGM08 In Fig.4, Variance _ EGM _ D3 and Variance _ EGM _ T 3 are degree variance
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differences of models from ESA which are derived using direct-wise and time-wise methods respectively; Variance _ EGM _ Trr is degree variance difference of the gravity field model
recovered
from
actual
observation
data
in
this
study.
In
Fig.
5,
EGM _ D3 ! EGM 08 , EGM _ T 3 ! EGM 08 and EGM _ Trr ! EGM 08 denote the
cumulated geoid differences respectively. Easily to know, the approach of this paper can successfully recover a gravity field model with higher than 200 degrees, although the accuracy is a little poorer after degree 200. It is important to note satellite tracking data is also used in the recovery of EGM_D3 and EGM_T3, but present study only uses gradient data which are sensitive only to high degrees and orders of the gravity field model due to low frequency error. Hence, the low degrees of the models are much different, especially the degrees between 27 and 100, which leads to differences in several centimeters in cumulated geoid heights. Nevertheless, at 200 degree, their differences are smaller than 4 cm. Filtering methods are also different with other groups’ 13
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inversion (Schuh 2003, Reguzzoni 2009 et al.), because this paper uses spherical harmonic analysis but other studies mainly use least square or least square collocation methods. Generally speaking, the accuracy is consistent in higher degrees. The FIR filter is mainly used to remove low frequency errors in this experiment. However,
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random and gross errors also exist. Thus, if some strategies can be used to process these errors, the accuracy may well be improved further. This is next work of this paper where Innovation
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Algorithm (Peter and Richard, 2006) and grid processing are being experimentally used. However, according to the comparison of Figs. 2 and 4 and of Figs. 3 and 5, it’s not difficult to conclude the
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error from the non-full tensor given by Eq. (4) is smaller than the observation errors. Hence, the boundary condition in Eq. (3) is accurate enough for GOCE data processing.
an
On the other hand, since solving a large number of normal equations is a complicated task because the unknown parameters would exceed 40 000 if a gravity field model with 200 degrees is recovered, the approach of this paper is more optimal in computing efficiency, such as if getting
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Trr on orbit surface, the calculation time is shorter than 1 hour in an ordinary computer. It is meaningful for rapid solution of GOCE gravity field model.
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4. Conclusions
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The attitude of GOCE satellite has the following property: one axis of the gradiometer loaded in GOCE almost points at the same direction as the radial direction of the Earth. The angle is
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denoted as ! 3 . One main objective of the paper is to analyze this factor in detail and calculate the
mean value and standard deviation (Table 1). Starting from the formula of radial gravity gradients by the full tensor, the boundary
condition of radial gradients on satellite orbits is established in the absence of gradient components
v 12
and
v
23
, based on the attitude property of GOCE.
The accuracy of the
conditions is also analyzed according to the statistics of ! 3 . The boundary condition is examined by data processing. The results show that accuracy of the gravity field model recovered from simulated gradient data by using this condition is definitely higher than the accuracy of models from actual gradient data. It means that the approach of computing boundary conditions on satellite orbits is effective, both theoretically and practically. Finally, the approach is used in actual GOCE data processing, and a gravity field model is recovered, the accuracy of which is nearly consistent with the models from ESA. 14
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Acknowledgments The authors would like to thank European Space Agency (ESA) for providing GOCE data. We also thank the two reviewers for their helpful suggestions. This work is supported by the National Natural Science Fund of China (No. 41074015, 41104047) and the open fund of Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University(11-01-07).
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Highlights: 1. The boundary condition of radial gradients is established for gravity field recovery using GOCE data whose gradient observations are non-full tensors. 2. The influence of non-full tensors is analyzed in detail by using actual orbit and attitude data.
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3. Compared to the value of radial gradients, relative error from non-full tensors is around 3%.
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4, The experiment on gravity field recovery using simulated data proves the effectiveness of the method based on the BVP.
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5, A gravity field model is derived from actual GOCE data, whose accuracy is nearly consistent with models from ESA in high degrees.
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