Journal of Geodynamics 47 (2009) 9–19

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Journal of Geodynamics

journa l homepage: ht tp : / /www.e lsev ier .com/ locate / jog

Atmospheric effects on satellite gravity gradiometry data

Mehdi Eshagh ∗, Lars E. SjöbergRoyal Institute of Technology, Tekninkringen 72, SE-10044 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 25 January 2008Received in revised form 30 May 2008Accepted 5 June 2008

Keywords:

a b s t r a c t

Atmospheric masses play an important role in precise downward continuation and validation of satellitegravity gradiometry data. In this paper we present two alternative ways to formulate the atmosphericpotential. Two density models for the atmosphere are proposed and used to formulate the external andinternal atmospheric potentials in spherical harmonics. Based on the derived harmonic coefficients, thedirect atmospheric effects on the satellite gravity gradiometry data are investigated and presented in theorbital frame over Fennoscandia. The formulas of the indirect atmospheric effects on gravity anomaly and

Satellite gradiometryDirect and indirect atmospheric effectsAtmospheric density

geoid (downward continued quantities) are also derived using the proposed density models. The numer-ical results show that the atmospheric effect can only be significant for precise validation or inversion of

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. Introduction

Satellite gravity gradiometry (SGG) is a technique by which theecond order derivatives of the gravitational potential are mea-ured based on differential accelerometry. The atmospheric masseselow satellites affect SGG data. The Gravity field and Ocean Cir-ulation Explorer (GOCE) [see, e.g., Balmino et al., 1998, 2001; ESA,999; Albertella et al., 2002] is an upcoming satellite mission basedn this technique, and it is expected to produce an Earth’s gravityeld model with an improved resolution from space. The SGG dataan directly be continued downward to the sea level for the localravity field determination. Therefore, it is necessary to considerhe gravitational effects of the topography and static atmosphereo smooth the gravity field and simplify the downward continua-ion process. Temporal variations of the atmospheric density areeyond the scope of this paper, but such variations will be an oblig-tory part of data processing of GOCE as it was for GRACE as wellFlechtner et al., 2006). These effects should also be considered ifalidation of the SGG data is of interest. The topographic effect hasargely been investigated by many scientists and in different appli-ations (see, e.g., Martinec et al., 1993; Martinec and Vanícek, 1994;jöberg, 1998, 2000, 2007; Sjöberg and Nahavandchi, 1999; Tsoulis,001; Heck, 2003; Seitz and Heck, 2003; Wild and Heck, 2004a,b;

akhloof and Ilk, 2005, 2006; Makhloof, 2007; Eshagh and Sjöberg,

008a,b).Some primary assumptions are needed for considering the

tmospheric effects. The atmosphere is usually assumed to be

∗ Corresponding author. Tel.: +46 8 7907369; fax: +46 8 7907343.E-mail addresses: eshagh@kth.se (M. Eshagh), sjoberg@kth.se (L.E. Sjöberg).

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264-3707/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.oi:10.1016/j.jog.2008.06.001

the mE level.© 2008 Elsevier Ltd. All rights reserved.

ayered spherically above a sphere approximating the Earth sur-ace. Also Moritz (1980), Sjöberg (1993, 1998, 1999, 2001), Sjöbergnd Nahavandchi (1999) and Nahavandchi (2004) used sphericalpproximation of sea level and considered the topographic heightsith respect to this sphere. Similar assumptions are considered for

he atmospheric density through this study although we know thathe semi-major axis of the ellipsoidal Earth is 21 km longer thanhe semi-minor one, this approximation results an overestimationnd an underestimation in the atmospheric effect at the poles andquator, respectively. In order to solve this problem the ellipsoidalarmonics should be used and more complicated formulation iseeded which is not in scope of this paper neither. This matter wasell investigated by Sjöberg (2006a,b).

Ecker and Mittermayer (1969) used an ellipsoidal approach totudy the atmospheric gravitational potential and acceleration;hey proposed a mathematical model for the direct atmosphericffect (DAE), which is well-known as the IAG (International Associa-ion of Geodesy) approach. Anderson and Mather (1975) consideredhe effect of the atmospheric masses in physical geodesy problemsnd computed the global values of the atmospheric effect on grav-ty and the geoid. Sjöberg (1993) investigated the effect of terrainn the atmospheric gravity and geoid corrections, Sjöberg (1998)resented the atmospheric correction on the gravity anomaly,eoid and on the satellite derived geopotential coefficients. Sjöberg1999) found some shortcomings in the IAG approach in atmo-pheric geoid correction and proposed a new strategy to solve them.

jöberg and Nahavandchi (2000) investigated the direct and indi-ect effect of the atmosphere in modified Stokes’ formula and theyhowed that the DAE on the geoid can reach 40 cm. Novák (2000)resented a density model for the atmosphere based on a simpleolynomial fitting and used that model to compute the atmospheric

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ffect on the geoid. The polynomial function was used only fortmospheric mass density modeling within the topography, i.e.,elow the elevation of highest spot on the Earth (approximately0 km above the mean sea level). The atmospheric masses above0 km elevation can be considered by using United States Standardtmospheric model (USSA76) (United State Standard Atmosphere,976). For more details about his method (see, e.g., Novák, 2000).jöberg (2001) investigated the atmospheric correction and foundhat the atmosphere contributes with the zero-degree harmonicf magnitude of 1 cm on the geoid. Nahavandchi (2004) presentednother strategy for the direct atmospheric gravity effect in geoidetermination. He numerically compared his new strategy with oldormulas in Iran and found 17 cm difference on the geoid betweenoth formulas in that region, but as Sjöberg (1998) argued theirect effect is considerably reduced after restoring the atmosphericffect. Sjöberg (2006a,b) showed that the atmospheric effect ineoid determination needs a correction for the geometry whenpplying the spherical approximation of Stokes formula. Accord-ng to his conclusions the correction needed to the atmosphericffect in spherical Stokes formula varies between 0.3 and 4.0 cmn the geoid at the equator and pole, respectively. Tenzer et al.2006) considered the effect of atmospheric masses for Stokesroblem with concentration on the direct and secondary indi-ect atmospheric effects. They found the complete effect of thetmosphere on the ground gravity anomaly varies between 1.75nd 1.81 mGal in Canada, and the effects are mainly dependentn the accuracy of the atmospheric density model (ADM). Nováknd Grafarend (2006) proposed a method to compute the effectf topographic and atmospheric masses on spaceborne data basedn spherical harmonic expansion with a numerical study in Northmerica.

In this paper, another approach to the atmospheric effect onhe SGG data is proposed than that used by Novák and Grafarend2006). The main difference is related to the used ADM. We expandhe atmospheric potential in spherical harmonics based on theunglia Tekniska Högskolan Atmospheric density model (KTHA)nd we modify that model based on the USSA76 and compare itith the original KTHA. Finally, we propose a new KTHA (NKTHA)

ased on Novák’s atmospheric density model (NADM) and theTHA. The DAE on the SGG are derived in orbital frames overennoscandia; and their statistics, based on different methods, areompared and discussed. Since the SGG can be continued down-ard using inversion of second order derivatives of the extended

tokes or Abel-Poisson integrals, we will formulate the internal typef the gravitational effect of the atmosphere on gravity anomaly andeoid (restoration of the atmospheric effects) as the results of thenversion. It should be mentioned that such gravitational effect, cane considered on the downward continued SGG data too. The paper

s arranged as follows.In the next section different ADMs including the NKTHA are

ntroduced. In Sections 3 and 4 we formulate the external andnternal atmospheric gravitational potentials in spherical harmon-cs based on the KTHA and NKTHA, respectively. In Section 5 wexplain how to compute the DAE on the SGG data in the orbitalrame. Section 6 presents the formulas for the indirect atmosphericffects (IAE) on the gravity anomaly and geoid. Section 7 deals withumerical studies on the ADMs and DAE on the SGG data overennoscandia, and the article is ended by the conclusions in Section.

. Atmospheric density models (ADMs)

There are different models for the density of the atmosphere.ne of the most well-known models was issued by NOAA (Nationalceanic and Atmospheric Administration), NASA (National Aero-

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eodynamics 47 (2009) 9–19

autics and Space Administration) and USAF (United State Airorce) as United States Standard Atmospheric model in 1976USSA76) (United State Standard Atmosphere, 1976). This models a complicated model depending on atmospheric pressure and

olecular-scale temperature. For considering these parameters,he atmospheric masses (up to the height of 86 km), are dividednto seven layers, and in each layer a molecular-scale temperaturend pressure are defined according to some other mathematicalodels. However, this model is not preferred in geodesy, and sim-

ler approximating models are sought. In the following we reviewwo ADMs and present a new ADM which is formulated based onhe two previous ADMs.

.1. KTH atmospheric density model (KTHA)

Sjöberg (1998) assumed that the atmospheric density is laterallyayered and changes only with elevation. This assumption is not soar from reality as the atmospheric density reduces by increasingeight. Based on the results of Ecker and Mittermayer (1969)hich were derived from the US standard atmospheric densityodel presented in 1961 (USSA61) (Reference Atmosphere

ommittee, 1961), Sjöberg (1998) proposed the followingDM:

a(r) = �0

(R

r

)�, (1)

here, �a(r) is the atmospheric density, R (set to 6,378,137 m) ishe Earth’s mean radius, R ≤ r ≤ R + Z is the geocentric radius of anyoint inside the atmosphere, �0 = 1.2227 kg/m3 is the atmosphericensity at the sea level, and � = 850 is an estimated constant. Weame this model KTHA.

.2. Novák’s atmospheric density model (NADM)

The maximum value of the atmospheric density is at theea level and decreases fast with increasing elevation. Novák2000) proposed the following model to approximate the verticalehaviour of the atmospheric density (NADM):

a(r) = �0[1 + ˛H + ˇH2], (2)

here, ˛= −7.6492 × 10−5 m−1, ˇ = 2.2781 × 10−9 m−2 and H = r − Rith 0 ≤ H < 10 km. The other parameters of this model are the same

s defined in previous section. As Novák (2000) mentioned hisodel fits the USSA76 with the accuracy of about 10−3 to up to

0 km above sea level. For higher elevations the USSA76 should besed.

.3. New KTH atmospheric density model (NKTHA)

We now propose the following model (NKTHA) for the atmo-pheric density, which in fact is a direct combination of the NADMnd KTHA:

a(r) ={�0[1 + ˛H + ˇH2], 0 ≤ H ≤ H0

�a(H0)(R+H0

r

)v′′, H0 ≤ H ≤ Z , (3)

here H0 = 10 km and �a(H0) = 0.4127 kg/m3 is based on the NADM.

y using this ADM we approximate the atmospheric density atigher levels by a simple mathematical model and it can be consid-red up to the satellite level. The parameter v′′ = 890 was derivedased on a simple least-squares fit to the USSA76 elevations ofbove 10 km.

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. Atmospheric potential based on KTHA

In this section we derive the atmospheric potentials consideringhe KTHA as the ADM. The atmospheric potential can be expressedccording to the well-known Newtonian volume integral as

a(P) = G∫ ∫

�

∫ rZ

rS

�a(rQ)r2Q drQlPQ

d�, (4)

here, G is the Newtonian gravitational constant, Va(P) stands forhe atmospheric potential at point P,�a(rQ) is the atmospheric den-ity function at point Q (integration point), � is the full solid anglef integration, rS and rZ are the topographic surface, and the upperimit of the atmosphere, lPQ is the distance between computationoint P and integration point Q. drQ and d� are the radial and hor-

zontal integration elements, respectively. In order to obtain thexternal atmospheric potential 1/lPQ is expanded into a Legendreeries of external type as:

1lPQ

= 1rP

∞∑n=0

(rQrP

)nPn(cos PQ). (5)

here rP ≥ rQ is the geocentric radius of the computation point P,nd PQ is the geocentric angle between P and Q. By substitutingq. (5) into Eq. (4) and considering Eq. (1) as the ADM we have:

aext(P) = G

∞∑n=0

�0R�

rn+1P

∫ ∫�

∫ rZ

rS

rn+2−�Q drQPn(cos PQ) d�, (6)

nd after integrating radially and regarding rS = R + H and rZ = R + Zwhere H is a function of position whereas Z is a constant) we obtain

aext(P) = G�0

∞∑n=0

Rn+3

rn+1P (n+ 3 − �)

∫ ∫�

[(1 + Z

R

)n+3−�

−(

1 + H

R

)n+3−�]Pn(cos PQ) d�. (7)

he two terms in square bracket can be expanded into a binomialeries and eventually truncated, since the series is converging fast.n our investigation we consider the expansion to fourth order. Cf.un and Sjöberg (2001). Further simplifications yield

aext(P) = G�0

∞∑n=0

Rn+3

rn+1P

∫ ∫�

F(Q)Pn(cos PQ) d�, (8)

here

(Q) = Z −HR

+ (n+ 2 − �)Z2 −H2

2R2

+(n+ 2 − �)(n+ 1 − �)Z3 −H3

6R3. (9)

ccording to the addition theorem of the fully normalized sphericalarmonics

n(cos PQ) = 12n+ 1

n∑m=−n

Ynm(Q)Ynm(P), (10)

here Ynm(P) and Ynm(Q) are the spherical harmonics at any pointand Q and∫

�

Ynm(Q)Yn′m′ (P) d� = 4�ınn′ımm′ , (11)

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eodynamics 47 (2009) 9–19 11

here ı is Kronecker’s delta, we have

aext(P) = 4�G�0R

2∞∑n=0

(R

rP

)n+1 12n+ 1

n∑m=−n

(Faext)nmYnm(P), (12)

here

Faext)nm = Zın0 −Hnm

R+ (n+ 2 − �)

Z2ın0 −H2nm

2R2

+(n+ 2 − �)(n+ 1 − �)Z3ın0 −H3

nm

6R3, (13)

here Hnm, H2nm and H3

nm are the spherical harmonic coefficients of, H2 and H3, respectively, derived in a global spherical harmonicnalysis of topographic heights. Considering in Eq. (12) that

M = 4��eGR3

3, (14)

here �e = 5500 kg/m3 (Novák and Grafarend, 2006) is the meanensity of the earth, and M is the earth’s mass, we finally obtain

aext(P) = GM

R

∞∑n=0

(R

rP

)n+1 n∑m=−n

(vaext)nmYnm(P), (15)

here

vaext)nm = 3�0

(2n+ 1)�e(Fa

ext)nm. (16)

q. (16) represents the spherical harmonic coefficients of the exter-al atmospheric potential. The above formula is analogous to thatbtained by Novák and Grafarend (2006). The only difference iselated to the ADM, which leads to the following harmonic coeffi-ients in Novák and Grafarend (2006):

Faext)nm = Zın0 −Hnm

R+ (n+ 2 − ˛R)

Z2ın0 −H2nm

2R2

+[(n+ 2)(n+ 1 − 2˛R) + 2ˇR2]Z3ın0 −H3

nm

6R3. (17)

It is now worth to compare Eq. (17) with Eq. (13). By comparinghe second order terms in those equations, we observe that theres ˛R = −596.55 in Novák and Grafarend’s model, while we have= 850 in KTHA. Also, the coefficient of the third term in Eq. (13)

an be written as [(n+ 2)(n+ 1 − �) − (n+ 1)�+ �2], and as one canee, an extra term appears vs. Novák and Grafarend’s model. Theonstant terms of both formulas are not comparable as they areˇR2 = 226623.09 and �2 = 722,500. However, when the terms areivided by 6R3 the effect of the third term is considerably reduced.

n comparison with Sjöberg (1998), we can say that, since Sjöberg’smphasis was on the DAE of the gravity anomaly and geoid, thepper limit of the radial integral in Eq. (4) was set to infinity, andhe internal type of Legendre expansion was used instead of Eq. (5),ut in our case that we want to obtain the DAE on the SGG data weave to limit this upper bound to the specific value rZ = 6,628,137 m.

t means that we assume the massive part of the atmosphere iselow rZ level from the earth’s mean sphere. It is also possible toonsider the satellite elevation as this specific value.

Now, consider the computation point to be below the atmo-pheric masses, in such a case the internal atmospheric potentialhould be formulated. Similar to the external atmospheric potentialormulation in spherical harmonics, we start with the Newtonianolume integral, Eq. (4) and consider Eq. (1) as the ADM (KTHA). If

e expand the 1/lPQ into Legendre series of internal type

1lPQ

= 1rP

∞∑n=0

(rPrQ

)n+1

Pn(cos PQ), (18)

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nd put it back into Eq. (4), after some further simplifications weave

aint(P) = G

∞∑n=0

�0R�rnP

∫ ∫�

∫ rZ

rS

r1−n−�Q drQPn(cos PQ) d�. (19)

n a similar way as for the external atmospheric potential we obtain

aint(P) = G�0

∞∑n=0

−R−n+2rnPn+ �− 2

∫ ∫�

[(1 + Z

R

)−(n+�−2)

−(

1 + H

R

)−(n+�−2)]Pn(cos PQ) d�. (20)

f the two terms in the square bracket are expanded into a binomialeries up to fourth order, after some simplifications we obtain

aint(P) = G�0

∞∑n=0

R−n+2rnP2n+ 1

n∑m=−n

∫ ∫�

Faint(Q)Ynm(Q) d�Ynm(P), (21)

here

aint(Q) = Z −H

R− (n+ �− 1)

Z2 −H2

2R2+ (n+ �− 1)(n+ �)

Z3 −H3

6R3.

(22)

gain, spherical harmonic expansion of H and Z yields

aint(P) = 4�G�0R

2∞∑n=0

12n+ 1

(rPR

)n n∑m=−n

(Faint)nmYnm(P), (23)

here

Faint)nm =

{Zın0 −Hnm

R− n+ �− 1

2R2(Z2ın0 −H2

nm)

+ (n+ �− 1)(n+ �)6R3

(Z3ın0 −H3nm)

}, (24)

nd according to Eq. (14), we finally arrive at

aint(P) = GM

R

∞∑n=0

(rPR

)n n∑m=−n

(vaint)nmYnm(P), (25)

here

vaint)nm = 3�0

(2n+ 1)�e(Fa

int)nm. (26)

his derivation is comparable with that obtained by using theADM for the internal atmospheric potential in spherical harmon-

cs yielding (Eshagh and Sjöberg, 2008b)

Faint)nm = Zın0 −Hnm

R− (n− 1 − ˛R)

Z2ın0 −H2nm

2R2

−[(1 − n)(n+ 2˛R) − 2ˇR2]Z3ın0 −H3

nm

6R3. (27)

Novák (2000) proposed to use atmospheric shells with differ-nt densities (based on the USSA76) for generating the atmosphericotential between 10 and 86 km levels. The potential of such atmo-pheric shells can be considered as an additional value to theero-degree harmonic coefficient; see Novák (2000). On the con-rary, there is no restriction in elevation for the KTHA, and it can

heoretically be considered up to infinity. However, the approxi-

ations used in generating the atmospheric potential may not beccurate enough for higher elevations based on this model. As Zn Eq. (9) is a constant it only contributes to the zero-degree har-

onic. It is obvious that when increasing the elevation of the upper

eodynamics 47 (2009) 9–19

oundary of the atmosphere, the magnitude of this harmonic canncrease unboundedly. In order to solve this problem we propose tose the following relation for the zero-degree harmonic coefficientf the atmospheric potential (which follows from Eq. (6) for n = 0).

vaext)0 = 3�0

�e(3 − �)

{(1 + Z

R

)3−�− H

}, (28)

here

¯ = 14�

∫ ∫�

(1 + H

R

)3−�d�. (29)

he integral of Eq. (29) can be solved numerically.

. Atmospheric potential based on the NKTHA

Now we consider the NKTHA that we proposed in Section 2.3.his model is a combination of the NADM and the KTHA. In theollowing we express how to use this NKTHA for formulating thexternal and internal atmospheric potentials in spherical harmon-cs.

Inserting the NKTHA into Eq. (4) and considering Eq. (3) and thexternal type of expansion of 1/lPQ we obtain

aext(P) = G

∞∑n=0

1

rn+1P

∫ ∫�

[∫ H0

rS

�a(rQ)rn+2Q drQ

+∫ rZ

H0

�a(rQ)rn+2Q drQ

]Pn(cos PQ) d�. (30)

he ADM in the first integral in the square bracket is related thepper function of Eq. (3) and the second integral relates to the lowerunction. Therefore, the solution of the first term is the same as inq. (17). Considering the second part of Eq. (3) as the ADM above0, the solution of the second integral becomes

�(H0)(R+H0)�∫ R+Z

R+H0

rQn+2−� drQ

= �(H0)Rn+3

n+ 3−�

[(1 + H0

R

)�(1 + Z

R

)n+3−�−(

1+H0

R

)n+3], (31)

ince this term is a constant w.r.t. integration point Q in Eq. (31), itssociates just with the zero-degree harmonic. The unitless spheri-al harmonic coefficients of the external atmospheric potential canhus be written in the following form

vaext)nm = 3

(2n+ 1)�e(�0(Fa

ext)nm + �(H0)Gn), (32)

here (Faext)nm was defined in Eq. (17), and

n= 1n+ 3−�

[(1 + H0

R

)�(1 + Z

R

)n+3−�−(

1+H0

R

)n+3]ın0. (33)

nserting Eqs. (33) and (17) into Eq. (32) and setting Zın0 = H0ın0 weave

Vaext)nm = 3

(2n+ 1)�e

{�0

[H0ın0 −Hnm

R

+(n+ 2 − ˛R)H2

0ın0 −H2nm

2R2

+[(n+ 2)(n+ 1−2˛R) + 2ˇR2]H3

0ın0−H3nm

3+ �(H0)

6R n+ 3 − �

×[(

1+H0

R

)�(1+ZR

)n+3−�−(

1+H0

R

)n+3]ın0

}(34)

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M. Eshagh, L.E. Sjöberg / Journ

or generating the external atmospheric potential based on theKTHA it is sufficient just to insert the harmonic coefficients gen-rated according to Eq. (34) into Eq. (15).

We can also consider our new atmospheric model NKTHA toenerate the internal type of the atmospheric potential. By consid-ring Eq. (18) and Eq. (3) (the NKTHA) and reinserting into Eq. (4)e obtain

aint(P) = G

∞∑n=0

rnP

[∫ H0

rS

�a(rQ)r−n+1Q drQ

+∫ rZ

H0

�a(rQ)r−n+1Q drQ

]Pn(cos PQ) d�. (35)

he first integral in the square bracket is the same as in the internalype of the NADM and is also the same as Eq. (27). The second partan be written

�(H0)(R+H0)�∫ R+Z

R+H0

r−n+1−�Q drQ

= �(H0)R−n+2

−n− �+ 2

[(1 + H0

R

)�(1 + Z

R

)−n−�+2−

(1 + H0

R

)−n+2],

(36)

nd the harmonic coefficients of the internal atmospheric potentialan thus be written

vaint)nm = 3

(2n+ 1)�e(�0(Fa

int)nm + �(H0)Kn), (37)

here (Faint)nm is the same as in Eq. (27) and

n = 1−n− �+ 2

[(1 + H0

R

)�(1 + Z

R

)−n−�+2

−(

1 + H0

R

)−n+2]ın0. (38)

inally we obtain

Vaint)nm = 3

(2n+ 1)�e

{�0

[H0ın0 −Hnm

R

−(n− 1 − ˛R)H2

0ın0 −H2nm

2R2−[(1 − n)(n+ 2˛R)−2ˇR2]

×H30ın0 −H3

nm

6R3

]+ �(H0)

−n− �+ 2[(1 + H0

R

)�(1 + Z

R

)−n−�+2−

(1 + H0

R

)−n+2]ın0

}.

(39)

t should be noted that these harmonic coefficients should benserted into Eq. (25) for generating the internal atmosphericotential.

. DAE on the SGG data

According to the spherical harmonic coefficients of the externaltmospheric potential, the DAE can easily be computed by puttinghe harmonic coefficients into the spherical harmonic expansion of

he SGG data either in a geocentric spherical, local, or orbital frame.n our study we use the orbital frame to present the effects. Theon-singular expression of the gradients in this frame is preferrable

or us, as we do not have to compute the associated Legendreunction derivatives. These formulas were originally presented by

oˇA

e

eodynamics 47 (2009) 9–19 13

etrovkaya and Vershkov (2006). The orbital frame is defined by u,and w axes so that w axis coincides with z and upward, v points

owards the instantaneous angular momentum vector and u com-lements the right-handed triad. The mathematical models of theGG data in such a frame are presented as follows (Petrovkaya andershkov, 2006):

uu(P) = GM

R3

N∑n=2

n∑m=−n

(R

rP

)n+3(va

ext)nm{Qm(�P)[cos 2˛fn,m,1

− cos2 ˛(n+ 1)(n+ 2)Pn,|m|] + Q−m(�P) sin 2˛fnm,2} (40)

vv(P) = −GMR3

N∑n=2

n∑m=−n

(R

rP

)n+3(va

ext)nm{Qm(�P)[cos 2˛fnm,1

+ sin2 ˛(n+ 1)(n+ 2)Pn,|m|] + Q−m(�P) sin 2˛fnm,2} (41)

uv(P) = −GMR3

N∑n=2

n∑m=−n

(R

rP

)n+3(va

ext)nm{Qm(�P)[sin 2˛fnm,1

− cos˛ sin˛(n+ 1)(n+ 2)Pn,|m|] + −Q−m(�P) cos 2˛fnm,2}(42)

uw(P) = GM

R3

N∑n=2

n∑m=−n

(R

rP

)n+3(va

ext)nm[Qm(�P) cos˛fnm,3

+Q−m(�P) sin˛fnm,4] (43)

vw(P) = −GMR3

N∑n=2

n∑m=−n

(R

rP

)n+3(va

ext)nm[Qm(�P) sin˛fnm,3

−Q−m(�P) cos˛fnm,4] (44)

ww(P) = GM

R3

N∑n=2

n∑m=−n

(n+ 1)(n+ 2)(R

rP

)n+3(va

ext)nmQm(�P)Pn,|m|

(45)

here, Pn,|m| = Pn,|m|(cos �P) and fnm,1 = fnm,1(�P)

nm,1(�P) = anmPn,|m|−2 + bnmPn,|m| + cnmPn,|m|+2, (46)

nm,2(�P) = dnmPn−1,|m|−2 + gnmPn−1,|m| + hnmPn−1,|m|+2, (47)

nm,3(�P) = ˇnmPn,|m|−1 + �nmPn,|m|+1, (48)

nm,4(�P) = nmPn−1,|m|−1 + �nmPn−1,|m|+1, (49)

m(�P) ={

cosm�P, m ≥ 0sin |m|�P, m < 0

(50)

here, (vaext)nm is the spherical harmonic coefficients of the exter-

al atmospheric potential, �P and �P and rP are the co-latitude,ongitude and geocentric radius of the point P or the satellite posi-ion. N is the maximum degree of harmonic expansion, and Pn,|m| ishe fully normalized associated Legendre function of degree n and

rder m.˛ is the satellite track azimuth. anm, bnm, cnm, dnm, gnm, hnm,nm, �nm, nm and �nm are the constant coefficients presented inppendix A.

These formulas were generally designed for gravitational gradi-nts of the disturbing potential. We can include the effect of zero-

14 M. Eshagh, L.E. Sjöberg / Journal of Geodynamics 47 (2009) 9–19

F al axia nces t

ai

6

grerrmdwitlsth

h

ı

ıwp

ı

a

ı

T

ı

7

mtiae

sm2tanors

TD

VVVVVV

ig. 1. (a) NADM in green, KTHA in red and USSA76 in blue versus elevation (verticnd USSA76 in red, respectively, to 10-km elevation. (For interpretation of the refere

nd first-harmonics separately. The contribution of these harmon-cs is presented in Appendix B.

. Remote–compute–restore scheme

In the remove–compute–restore approach, the external topo-raphic and atmospheric potentials are removed and theesult will be no-topography and no-atmosphere potentials. Theffect of the topographic and atmospheric masses must beestored after computations (this is why the method is calledemove–compute–restore). The gravity field can also be deter-ined locally from SGG data using inversion of the second order

erivatives of extended Stokes or Abel-Poisson integrals. The down-ard continued gravity anomaly or disturbing potential at sea level

s the results of this inversion process. One can also downward con-inue the SGG data directly to geoid height or gravity anomaly at seaevel. In any case, the effect of the removed atmospheric potentialhould be restored on these quantities. In the following we presenthese indirect atmospheric effects on the gravity anomaly and geoideight.

According to the fundamental equation of physical geodesy weave (Heiskanen and Moritz, 1967, p. 86):

gaind(P) = −∂V

aint(P)

∂rP− 2rPVa

int(P). (51)

gaind(P) is the indirect atmospheric effect on gravity anomaly

hich should be restored. By putting the internal atmospheric

otential given by Eq. (25) we obtain

gaind(P) = −GM

R2

N∑n=0

(n+ 2)(rPR

)n−1 n∑m=−n

(vaint)nmYnm(P), (52)

tl

eu

able 1AE on the SGG data at 250 km level based on the NADM and the KTHA up to 10 km (unit

NADM

Max Mean Min Std

uu −2.0069 −2.2153 −2.391 ±0.0762vv −1.9238 −2.2247 −2.4123 ±0.1062ww 4.6257 4.4400 3.9364 ±0.1520uv 0.0729 −0.0276 −0.1119 ±0.0403uw 0.2976 0.0110 −0.2984 ±0.1125vw 0.3544 0.0125 −0.2043 ±0.0879

s is in logarithmic scale), (b) deference between NADM and USSA76 in blue, KTHAo color in this figure legend, the reader is referred to the web version of the article.)

nd at the very approximate geoid rP = R we have

gaind(P) = −GM

R2

N∑n=0

(n+ 2)n∑

m=−n(va

int)nmYnm(P). (53)

he indirect atmospheric effect on the geoid will be

Naind(P) = −GM

R�

N∑n=0

n∑m=−n

(vaint)nmYnm(P). (54)

. Numerical studies

Let us start with a simple investigation of the mathematicalodels approximating the atmospheric density. First we compare

he NADM and KTHA with the USSA76 up to 86 km elevation (whichs the maximum level of the USSA76). In the following figures, thesepproximating density models are visualized with respect to thelevation.

Fig. 1 shows that the NADM is valid up to 10 km elevation aboveea level (see Novák, 2000). The KTHA agrees with the USSA76ore or less but underestimates the atmospheric density below

0 km and overestimates it in higher altitudes. It should be men-ioned that the main aim of Novák (2000) was to formulate thetmospheric topography. This is why he considered a simple poly-omial to model the atmospheric density up to 10 km. Since 80%f atmospheric masses are below 12 km (Lambeck, 1988), it iseasonable to use a simple polynomial to express the effect of atmo-pheric roughness although Wallace and Hobbs (1977) believes

hat 99% of the masses lie within the lowest 30 km above seaevel.

For investigating the DAE on the SGG data, we generate thexternal atmospheric potential coefficients considering Z = 10 kmsing both the NADM and the KTHA. The orbital frame is used

: 1 mE)

KTHA

Max Mean Min Std

−0.6957 −0.8831 −1.0405 ±0.0693−0.6159 −0.8831 −1.0491 ±0.0966

1.9352 1.7662 1.3171 ±0.13920.0679 −0.0229 −0.1004 ±0.03700.2632 0.0063 −0.2706 ±0.10380.3213 0.0140 −0.1822 ±0.0809

M. Eshagh, L.E. Sjöberg / Journal of Geodynamics 47 (2009) 9–19 15

spheric potential versus various Z. (a) Z = 0–50 km and (b) Z = 0–250 km.

ttcetas

tmthatVoih

h

NhilmbHmEduatF

Table 2Values of the atmospheric mass density based on the USSA61 and USSA76 (unit:kg/m3)

Elevation (km) USSA61 USSA76 Difference

0 1.2225 1.2227 0.00022 1.0067 1.0047 −0.00204 8.2362 × 10−1 8.1780 × 10−1 −0.0582 × 10−1

6 6.6511 × 10−1 6.5888 × 10−1 −0.0623 × 10−1

8 5.2702 × 10−1 5.2479 × 10−1 −0.0223 × 10−1

10 4.0937 × 10−1 4.1273 × 10−1 0.0336 × 10−1

12 3.1131 × 10−1 3.1135 × 10−1 0.0004 × 10−1

14 2.3211 × 10−1 2.2742 × 10−1 −0.0469 × 10−1

16 1.7007 × 10−1 1.6615 × 10−1 −0.0392 × 10−1

18 1.2289 × 10−1 1.2142 × 10−1 −0.0147 × 10−1

20 8.8025 × 10−2 8.8741 × 10−2 0.0716 × 10−2

3 −2 −2 −2

45

dKtp(

iwabd

TS

VVVVVV

Fig. 2. Behaviour of the unitless zero-degree harmonic of the atmo

o generate the DAE at 250 km level with 2-months revolution ofhe GOCE satellite over Fennoscandia. The position of the satellitean be generated using a numerical integration technique and anxisting geopotential model. For more details the reader is referredo, e.g., Hwang and Lin (1998), Eshagh (2003, 2005) and Eshaghnd Najafi-Alamdari (2006). The statistics of the computations areummarized in Table 1.

It illustrates that the DAE on the SGG data is dependent onhe ADM. The KTHA shows small DAE since it underestimates the

assive part of the atmosphere in the lower altitudes. This underes-imation of the atmospheric density directly affects the zero-degreearmonic coefficient of the atmospheric potential. Since Vuu, Vvvnd Vww include this harmonic in their formulation, we can expecto see larger DAE based on the NADM than the KTHA. Vuv, Vuw andvw are more or less in the same order in both approaches becausef their independency from the zero-degree harmonic. However,t should be noted that Vuw and Vvw include also the first-degreearmonics.

Fig. 2 shows more details about the behaviour of the zero-degreearmonic with respect to elevation.

Fig. 2 illustrates values of the zero-degree harmonic based on theADM and KTHA. As we know the NADM is valid just up to 10 kmeight, and it is not surprising to see larger values when apply-

ng it for the external potential to higher elevations. The horizontaline presented in the figure is the true value of the zero-degree har-

onic computed by Eq. (28). As it is expected, this harmonic shoulde treated as a bounded function when increasing the elevation.owever, Fig. 2a shows that the approximations of the true mathe-atical expressions with binomial expansion which was used in

q. (13) is good just for elevations lower than 20 km. The zero-

egree harmonic has smaller value up to 50 km when KTHA issed vs. the NADM, while it is larger at 250 km. It means that thepproximation used Eq. (13) is not good enough for higher eleva-ions and can destroy the solution even worse than the NADM (seeig. 2a). The simplest way to get a non-diverging value for the zero-

Aca

T

able 3tatistics of DAE on the SGG data at 250 km level based on the KTHA and the modified KT

KTHA

Max Mean Min Std

uu −0.8718 −1.0590 −1.2166 ±0.0693vv −0.7920 −1.0590 −1.2249 ±0.0966ww 2.2875 2.1180 1.6694 ±0.1392uv 0.0680 −0.0229 −0.1004 ±0.0370uw 0.2632 0.0063 −0.2706 ±0.1038vw 0.3213 0.0140 −0.1822 ±0.0809

0 1.8410 × 10 1.8375 × 10 −0.0035 × 100 3.9957 × 10−3 3.9878 × 10−3 −0.0079 × 10−3

0 1.0269 × 10−3 1.0248 × 10−3 −0.0021 × 10−3

egree harmonic is to avoid approximations in this harmonic in theTHA using Eq. (53). As we mentioned previously we can consider

he atmospheric shells above 10 km and compute the atmosphericotential corresponding to each shell and add it in this harmonicsee Novák, 2000).

The next point of our discussion is to consider a possiblemprovement of KTHA model. As we explained before the KTHAas derived using the results of Ecker and Mittermayer (1969)

nd USSA61 but the NADM is based on the USSA76. The differenceetween these two models is due to the differences in the originalata used (see Table 2).

Fig. 1 shows that the KTHA are not well-fitted to the USSA76.ssuming the densities generated by the USSA76 as true values, we

an modify the KTHA to obtain better fit to the USSA76. In Fig. 3and b results of this modification to the KTHA is presented.

By fitting the KTHA to the USSA76 we obtain the value �=930.his modified KTHA is visualized in Fig. 3a, which figure shows

HA (unit: mE)

Modified KTHA

Max Mean Min Std

−0.7800 −0.9661 −1.1228 ±0.0690−0.7005 −0.9657 −1.1306 ±0.0961

2.1005 1.9319 1.4859 ±0.13850.0677 −0.0227 −0.0998 ±0.03690.2615 0.0061 −0.2691 ±0.10330.3196 0.0141 −0.1811 ±0.0805

16 M. Eshagh, L.E. Sjöberg / Journal of Geodynamics 47 (2009) 9–19

Fig. 3. (a) Modified KTHA in red (vertical axis is in logarithmic scale), (b) NKTHA based on the USSA76 in red (vertical axis is in logarithmic scale), (c) differences betweena retativ

tistDa

mobKsoouimtip

febKatws

vlTf

F

1i

d4a360. Two-month revolution of the satellite with 30 s integrationstep size was considered in our investigation over Fennoscandia.The lower boundary of the atmosphere was considered the Earthsurface and the upper bound at satellite level. The JGP95e global

Table 4Statistics of the DAE on the SGG data at 250 level based on the NKTHA (unit: mE)

NKTHA

Max Mean Min Std

V −2.3564 −2.5643 −2.7406 ±0.0762

pproximating models of KTHA, modified KTHA New KTHA and USSA76. (For interpersion of the article.)

hat, although the modified KTHA has very good fit to the USSA76n higher elevations, it underestimates the density of the most mas-ive part of the atmosphere in low levels. Therefore, it is expectedo see small DAE on the SGG data. In Table 3 the statistics of theseAE based on the KTHA and the modified KTHA over Fennoscandiare presented.

The differences are again mainly related to the diagonal ele-ents Vuu,Vvv andVww of the gradiometric tensor as they, unlike the

ther elements, include the zero-degree harmonic. The differenceetween the DAEs generated based on the original and modifiedTHA is related to the underestimation of the atmospheric den-ity in the modified KTHA and also to significant overestimationf KTHA in higher elevation. However, since the most massive partf the atmosphere lies below 10 km level we can expect that thenderestimation of the atmospheric density in the modified KTHA

s the main reason for these differences. The only advantage of theodified KTHA relative to the original KTHA is thus to have a bet-

er fit in the higher elevations. Consequently, the modified KTHAs inferior to the original KTHA in context of the gravimetric datarocessing.

We also consider another approach in which the NADM is usedor the heights below 10 km and another modified KTHA for consid-ring higher levels. In this case the atmospheric density generatedy the NADM at 10 km is considered as a reference value and the

THA model is modified to get best fit to the densities of the USSA76fter 10 km. Fig. 3b shows the result of this fitting. The figure showshat the atmospheric densities are overestimated by the NKTHAith respect to the USSA76 between 20 and 60 km levels. However,

ince the atmospheric density decreases fast by increasing the ele-

VVVVV

on of the references to color in this figure legend, the reader is referred to the web

ation we can expect that such misfitness is insignificant in higherevels. In this case we estimate �′′ = 890.1727=890 (see Eq. (3)).he RMSEs of the model fittings are 0.038, 0.032 and 0.0027 kg/m3

or KTHA, modified KTHA and new KTHA, respectively.In Table 4 the statistics of the DAE on the SGG data over

ennoscandia are presented based on the NKTHA.Since in the NKTHA we use the NADM for the elevations below

0 km, we expect to see a good fit to the DAE for these ADMs, whichs confirmed by the statistics in Tables 1 and 4.

In the following we present the maps of the DAE on the SGGata over Fennoscandia. In order to simulate the satellite orbit theth order Runge–Kutta integrator was used to integrate the satelliteccelerations generated from EGM96 geopotential model to degree

uu

vv −2.2733 −2.5738 −2.7617 ±0.1062ww 5.3250 5.1381 4.6355 ±0.1520uv 0.0729 −0.0276 −0.1119 ±0.0403uw 0.2976 0.0110 −0.2984 ±0.1125vw 0.3544 0.0125 −0.2043 ±0.0879

M. Eshagh, L.E. Sjöberg / Journal of Geodynamics 47 (2009) 9–19 17

(d), (e

tm

NVVi

ovadsfidhi

8

tmoUmmttabwwcn

ti(oiria

A

ePewu

A

(

a

b

Fig. 4. DAE on the SGG at 250 km based on the NKTHA, (a), (b), (c),

opographic model was also used to generate the topographic har-onics to degree and order 360.Fig. 4 illustrates the DAE on the SGG data at 250 km based on the

KTHA. (a), (b), (c), (d), (e) and (f) are Vuu, Vvv, Vww , Vuv, Vuw andvw , respectively. The largest DAE is about 5.1381 mE and related toww as would we expect, and the smallest DAE (about −0.0276 mE)

s related to Vuv.The numerical results show that the DAE on the SGG data based

n all ADMs is at mE level. Such effects are significant in precisealidation of the SGG data. On the other hand, Xu (1992, 1998)nd Xu and Rummel (1994) concluded that in inverting the SGGata for the determination of the local gravity anomaly at the meanphere of the Earth with 5 mGal level of accuracy, 0.01E accuracyor the SGG data is enough. Hence if such an accuracy for grav-ty anomaly be required one can safely ignore the DAE on the SGGata. It should be mentioned that the atmospheric effect is alsoighly variable in time, which would have to be taken into account

n future gradiometric missions.

. Conclusions

The KTH atmospheric density model (KTHA) was modified sohat it delivers better fit to the United States atmospheric density

odel (USSA76) than the original one (which was derived basedn the United States atmospheric density model presented in 1961,SSA61), and the constant � = 930 was obtained for the modifiedodel. Numerical studies show that this model underestimates theost massive part of the atmosphere which is below 10 km more

han the original one, but it has better fit for higher elevations, andhis effect on the satellite gravity gradiometry is less than 1 mEnd negligible. A combination of both density models presented

y Novák (2000) and Sjöberg (1998) with a simple modificationas proposed as a new model (NKTHA). This model has good fitith the USSA76 for low and high levels. The spherical harmonics

oefficients of the atmospheric potential generated based on thisew model were used to compute the DAEs on the SGG data in

c

) and (f) are Vuu , Vvv , Vww , Vuv , Vuw and Vvw , respectively. Unit: mE.

he orbital frame. Numerical results show that the maximum DAEs related to Vww and about 5.13 mE. In comparison with Novák2000) model, which generate the corresponding value 4.44 mE,ne can state that the effect of the atmospheric masses above 10 kms less than 1 mE but significant for precise validation of the SGG. Weecommend the use of NKTHA in considering atmospheric effectsn satellite gradiometry data processing and any other gravimetricpplications.

cknowledgments

The Swedish National Space Board (SNSB) is cordially acknowl-dged for its financial support. The first author would like to thankrof. P. Novák for the stimulating discussions about atmosphericffect. Also we appreciate the thorough work of the two reviewers,ith many detailed remarks, that helped in making the paper morenderstandable.

ppendix A

The constant coefficients related to Eqs. (40)–(45) arePetrovkaya and Vershkov, 2006):

nm={

0, |m| = 0,1√1+ı|m|,2

4

√n2−(|m|−1)2

√n+|m|

√n− |m|+2, 2 ≤ |m| ≤ n

(A.1)

nm =

⎧⎨⎩

(n+ |m| + 1)(n+ |m| + 2)2(|m| + 1)

, |m| = 0,1

n2 +m2 + 3n+ 2, 2 ≤ |m| ≤ n

(A.2)

2

nm=

⎧⎨⎩

√1+ı|m|,0

4

√n2−(|m|+1)2

√n−|m|

√n+ |m| + 2, |m| = 0,1

14

√n2 − (|m| + 1)2

√n− |m|

√n+ |m| + 2, 2 ≤ |m| ≤ n

(A.3)

1 al of G

d

g

h

ˇ

�

v

w

A

Ega

V

V

V

V

V

V

wdn

R

A

A

B

B

E

E

E

E

E

E

E

F

H

HH

L

M

M

M

8 M. Eshagh, L.E. Sjöberg / Journ

nm =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, |m| = 1

− m

4|m|

√2n+ 12n− 1

√1 + ı|m|,2

×√n2 − (|m| − 1)2√

n+ |m|×

√n+ |m| − 2, 2 ≤ |m| ≤ n

(A.4)

nm =

⎧⎪⎪⎨⎪⎪⎩

m

4|m|

√2n+ 12n− 1

√n+ 1

√n− 1(n+ 2), |m| = 1

m

2

√2n+ 12n− 1

√n+ |m|

√n− |m|, 2 ≤ |m| ≤ n

(A.5)

nm =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

m

4|m|

√2n+ 12n− 1

√n− 3

√n− 2

× √n− 1

√n+ 2, |m| = 1

m

4|m|

√2n+ 12n− 1

√n2 − (|m| + 1)2

× √n− |m|

√n− |m| − 2, 2 ≤ |m| ≤ n

(A.6)

nm ={

0, |m| = 0n+ 2

2

√1 + ı|m|,1

√n+ |m|

√n− |m| + 1, 1 ≤ |m| ≤ n (A.7)

nm =

⎧⎪⎨⎪⎩

−(n+ 2)

√n(n+ 1)

2, |m| = 0

−n+ 22

√n− |m|

√n+ |m| + 1, 1 ≤ |m| ≤ n

(A.8)

nm = − m

|m|(n+ 2

2

)√2n+ 12n− 1

√1 + ı|m|,1

√n+ |m|

√n+ |m| − 1

(A.9)

nm = − m

|m|(n+ 2

2

)√2n+ 12n− 1

√n− |m|

√n− |m| − 1 (A.10)

here ı is Kronecker’s delta.

ppendix B

The contribution of the zero- and the first-degree harmonics toqs. (40)–(45) can be derived based on the original formulas of theravitational gradients in the orbital frame as (see also Petrovkayand Vershkov, 2006, Eq. (45)):

0,1uu (P) = V0,1

xx (P) = −GMRr4P

{rPRC00 + 3

√3[C10 cos �P + (C11 cos�P

+S11 sin�P) sin �P]}

(B.1)

0,1vv (P) = V0,1

yy (P) = −GMRr4P

{rPRC00 + 3

√3[C10 cos �P

+(C11 cos�P + S11 sin�P) sin �P]}, (B.2)

0,1ww (P) = V0,1

zz (P) = 2GMR

r4P

{rPRC00 + 3

√3[C10 cos �P

+(C11 cos�P + S11 sin�P) sin �P]}, (B.3)

0,1uv (P) = V0,1

xy (P) = 0, (B.4)

M

M

MN

N

eodynamics 47 (2009) 9–19

0,1uw (P) = 3

√3GMR

r4P{cos˛[C10 sin �P + (C11 cos�P

+S11 sin�P) cos �P] + sin˛(−C11 sin�P + S11 cos�P),

(B.5)

0,1vw (P) = 3

√3GMR

r4P{− sin˛[−C10 sin �P

+(C11 cos�P + S11 sin�P) cos �]

+ cos˛(−C11 sin�P + S11 cos�P) (B.6)

here superscripts of 0 and 1 stand for the zero- and the first-egree harmonics, respectively. C00, C10, C11, S10 and S11 are the fullyormalized zero- and the first-degree geopotential coefficients.

eferences

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