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Computers & Geosciences 32 (2006) 442–451

Spherical harmonic analysis and synthesis using FFT:

Application to temporal gravity variation

$

Cheinway Hwang

, Yu-Chi Kao

Department of Civil Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan

Received 2 February 2005; received in revised form 2 July 2005; accepted 14 July 2005

Abstract

FFT and complex algebra-based methods of spherical harmonic analysis and synthesis are presented. Two computer

programs in FORTRAN are developed based on the methods. Both general and special cases are discussed. Special

cases involve the analyses of gravity changes of the hydrological origin and the atmospheric origin. Functionals of the

Earth’s gravity ﬁeld such as gravity anomaly and geoidal height can also be computed via synthesis. Thermal-corrected

sea level anomaly from TOPEX/Poseidon and atmospheric pressure from ECMWF are used to compute changes of

geopotential coefﬁcients due to oceanic and atmospheric mass redistributions. Interesting phenomena in the changes of

geopotential coefﬁcients have been identiﬁed. The two computer programs can facilitate analyses and syntheses of

gravity products from satellite missions such as GRACE.

r2005 Elsevier Ltd. All rights reserved.

Keywords: FFT; Geoid; J

2

; Spherical harmonics; Temporal gravity

1. Introduction

Spherical harmonic analysis is a process of decom-

posing a function on a sphere into components of

various wavelengths using surface spherical harmonics

as base functions. Spherical synthesis combines compo-

nents of various wavelengths to generate function values

on a sphere and is the reverse process of harmonic

analysis. Spherical harmonic analysis and synthesis have

been used in many occasions, e.g., ocean dynamic

topography (Engelis, 1985) and the Earth’s static gravity

ﬁeld (Lemoine et al., 1998) and temporal gravity ﬁelds

(Wahr et al., 1998). In particular, the temporal variation

of the Earth’s gravity ﬁeld is closely related to global

climate change. Satellite missions such as GRACE

(Tapley et al., 2004) have now routinely delivered

products that can be used to derive gravity variations.

Spherical harmonic analysis and synthesis are important

tools for investigating these gravity variations.

Existing works on spherical harmonic analysis and

synthesis can be found in, e.g., Colombo (1981),Dilts

(1985),Potts et al. (1998),Mohlenkamp (1999),Kostelec

et al. (2000),Suda and Takami (2002) and Healy et al.

(2003). There are two procedures in doing spherical

harmonic analysis and synthesis. One procedure is based

on numerical integration and the other based on least-

squares (e.g., Colombo, 1981). Both spherical harmonic

analysis (using numerical integration) and synthesis can

take advantage of fast Fourier transform (FFT). For

example, Colombo (1981) and Dilts (1985) have devel-

oped algorithms for spherical harmonic analysis using

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0098-3004/$ - see front matter r2005 Elsevier Ltd. All rights reserved.

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Corresponding author. Tel.: +88 63 5724739;

fax: +88 63 5716257

E-mail address: hwang@geodesy.cv.nctu.edu.tw

(C. Hwang).

FFT. In this paper, we will present two efﬁcient, FFT-

based computer programs for spherical harmonic

analysis and synthesis. Our methods will still employ

FFT but we will use different algorithms and the

complex algebra. We will emphasize special cases

involving the temporal variation of the Earth’s gravity

ﬁeld. Sea level data and atmospheric data will be used to

compute gravity variations and to demonstrate the

usages of the computer programs.

2. Spherical harmonic analysis

2.1. General case

Any spherical function, f(f,l), can be expanded into

series of surface spherical harmonics (Heiskanen and

Moritz, 1985)as

fðy;lÞ¼X

1

n¼0X

n

m¼0

¯

anm ¯

Rnmðy;lÞþ¯

bnm ¯

Snmðy;lÞ

, (1)

where ¯

anm and ¯

bnm are harmonic coefﬁcients, yand lare

co-latitude (polar distance angle from the north pole)

and geocentric longitude, respectively, ¯

Rnm ¼

¯

Pnmðcos yÞcos mland ¯

Snm ¼¯

Pnmðcos yÞsin mlare

fully normalized spherical harmonics, ¯

Pnmðcos yÞis the

fully normalized associated Legendre function, and n

and mare degree and order, respectively. In a

continuous case, ¯

anm and ¯

bnm can be obtained using

the orthogonal relationship of spherical harmonics as

(Heiskanen and Moritz, 1985, p. 29)

¯

anm

¯

bnm

()

¼1

4pZ2p

l¼0Zp

y¼0

fðy;lÞ

¯

Rnm

¯

Snm

()

sin ydydl. (2)

In practice, function values of fare always given at

discrete points, so (2) can only be implemented

numerically and approximately. Let t¼cos yand

Cnm ¼¯

anm þi¯

bnm

, where i ¼ﬃﬃﬃﬃﬃﬃﬃ

1

p. Given fon a

global, regular DyDlgrid (Dyis the sampling interval

in latitude and Dlis the sampling interval in longitude),

(2) can be approximated as (Hwang, 2001)

Cnm ¼1

4pqnX

M1

k¼1X

N1

l¼1

¯

fyk;ll

ðÞ

Ztkþ1

tk

¯

PnmðtÞdt

Zllþ1

ll

eimldl

¼gm

4pqnX

M1

k¼1

I¯

Pk

nm X

N1

l¼1

¯

fyk;ll

ðÞei2pmðl1Þ=ðN1Þ

¼gm

4pqnX

ðM1Þ=2

k¼1

I¯

Pk

nm ¯

fkðmÞþð1Þnm¯

fMkðmÞ

,ð3Þ

gm¼Dl;if m¼0;

sinðmDlÞþi1cosðmDlÞðÞ

½

=mif ma0;

((4)

where M¼ðp=Dyþ1Þis the number of grids in latitude

and N¼ð2p=Dlþ1Þis the number of grids in long-

itude, tk¼cosððk1ÞDyÞ,ll¼ðl1ÞDl,I¯

Pk

nm is the

integration of associated Legendre function (Paul,

1978), q

n

is a quantity dependent on nand the block

size ¯

fyk;lk

ðÞ(Rapp, 1989, p. 266) and is to make the

approximation in (3) to (2) as realistic as possible.

Colombo (1981, p. 76) suggested that q

n

be set to:

qn¼b2

n;0pnpL=3;

qn¼bn;N=3onoL;

qn¼1;n4L;

(5)

where b

n

is the Pellinen smoothing factor given by

bn¼1

1cos c0

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2nþ1

pPn1cos c0

Pnþ1ðcos c0Þ

(6)

with Lbeing the maximum degree of expansion and

c

0

being the radius of a spherical cap whose area is the

same as the block size of ¯

fyk;lk

ðÞ. The last equation in

(3) is due to the fact that I¯

PMk

nm ¼ð1ÞnmI¯

Pk

nm.As

such, the integration of associated Legendre functions

needs to be done only for the northern hemisphere. In

(3), ¯

fis the mean value in a block (or cell) and is

computed by the four-point average as

¯

ffk;ll

¼1

4fðkDy;lDlÞþfðkþ1ÞDy;lDlðÞ

½

þfðkþ1ÞDy;ðlþ1ÞDlðÞ

þfkDy;ðlþ1ÞDlðÞ

.ð7Þ

In (3), let K¼N1¼number of blocks in long-

itude, p¼l1. Then we have

¯

fkðmÞ¼X

K1

p¼0

¯

ffk;ll

ei2pmp=K;m¼0;...;K1, (8)

which can be computed efﬁciently by FFT for all m. The

maximum degree of expansion Lfollows the rule that

L¼p=Dy(Rapp, 1989). Since LoK1, for each ﬁxed

kwe will need the ¯

fkðmÞvalues in (8) only up to m¼L.

In our programming, the computations start from the

northernmost and southernmost latitude belts simulta-

neously and converge to the equator. The mean block

values (see (7)) from the northern and southern hemi-

sphere are stored in a complex array (one in the real part

and the other in the imaginary part), which is then

Fourier transformed to form the two needed Fourier

arrays. This process of simultaneously Fourier trans-

forming two real-valued arrays signiﬁcantly reduces the

computing time as compared to the process of trans-

forming real-valued array one by one.

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2.2. Case for variation of Earth’s gravity ﬁeld

Mass variation within the Earth system may be

induced by changes in ocean, atmosphere, precipitation

(snow and rainfall), water table, glacier, ice sheet, etc.

The origins of mass changes (Dm) may be classiﬁed into

two categories. One is of the hydrological origin and in

this case mass variation is related to the density of the

underlying quantity and the change of height (or

thickness). The other is of the pressure origin and in

this case mass variation is related to the change of

pressure. At any point (r, y,l)(r: geocentric distance)

exterior to the Earth, the perturbing potential relative to

a static Earth caused by surface mass variations on a

sphere of radius Rcan be expressed as

DVhðr;y;lÞ

DVpðr;y;lÞ

()

¼G

ZZ s

1

s

rDh

Dp=g

()

ds, (9)

where ds¼R2sin ydydl,sis the distance from a

surface area element to point (r,y,l), DV

h

is the

perturbing potential due to Dh( hydrological origin),

and DV

p

is the perturbing potential due to Dp(pressure

origin). The perturbing potential can be used to derive

variations in gravity, geoid, deﬂection of the vertical and

other functionals of the Earth’s gravity ﬁeld. It turns out

that conversion between surface mass variations and

perturbing potentials in the spherical harmonic domain

is much easier than conversion in the space domain as

expressed in (9). First, a perturbing potential in (9) can

be represented by a series of spherical harmonics as

DViðr;y;lÞ¼GM

rX

1

n¼0

a

r

nX

n

m¼0

D¯

Ji

nm ¯

Rnmðy;lÞþD¯

Ki

nm ¯

Snmðy;lÞ

hi

,ð10Þ

where D¯

Ji

nm and D¯

Ki

nm are changes of geopotential

coefﬁcients, i¼Dhor Dp, and ais a constant that is

roughly equal to the semi-major axis of the Earth’s

reference ellipsoid. The inverse of distance scan also be

expanded into a series of spherical harmonics as

1

s¼1

rX

1

n¼0

1

2nþ1

a

r

nX

n

m¼0

¯

Rnm y0;l0

¯

Rnmðy;lÞþ ¯

Snmðy0;l0Þ¯

Snmðy;lÞ

,ð11Þ

where y0;l0

are the spherical coordinates of a surface

area element. Since Dhand Dpare functions on a sphere,

they can also be expanded into series of spherical

harmonics (Section 2.1):

Dhðy;lÞ¼X

1

n¼0X

n

m¼0

¯

ah

nm ¯

Rnmðy;lÞþ¯

bh

nm ¯

Snmðy;lÞ

hi

, (12)

Dpðy;lÞ¼X

1

n¼0X

n

m¼0

¯

ap

nm ¯

Rnmðy;lÞþ¯

bp

nm ¯

Snmðy;lÞ

, (13)

Substituting (11), (12) and (11) into (9) and considering

surface loading effects leads to

D¯

Jh

nm

D¯

Kh

nm

8

<

:9

=

;¼4pra21þkn

ðÞ

Mð2nþ1Þ

¯

ah

nm

¯

bh

nm

()

, (14)

D¯

Jp

nm

D¯

Kp

nm

()

¼4pa21þkn

ðÞ

gMð2nþ1Þ

¯

ap

nm

¯

bp

nm

()

, (15)

where D¯

Jh

nm;D¯

Kh

nm

are changes of geopotential

coefﬁcients derived from Dh,D¯

Jp

nm;D¯

Kp

nm

are coefﬁ-

cients from Dp,k

n

is the loading Love number of degree

n(see, e.g., Han and Wahr, 1995), and Mis the mass of

the Earth (E5.973 10

24

kg). In deriving (14) and (15)

the orthogonal relationships of spherical harmonics are

used. Eqs. (14) and (15) express the relationships

between mass variations and perturbing potentials in

the spherical harmonic domain. In (14) and (15), the

coefﬁcients ¯

ah

nm;¯

bh

nm;¯

ap

nm, and ¯

bp

nm are derived from

global gridded data on a sphere with a radius of a.

Note that the geopotential coefﬁcients are unitless and

hence care must be exercised in using the units for

pressure and height change. Speciﬁcally, if g, M, a, and r

are in SI units, Dhshould be in meter and Dpshould be

in hpa (i.e., kg m

1

s

2

).

3. Spherical harmonic synthesis

3.1. General case

Given spherical harmonic coefﬁcients, harmonic

synthesis is to generate function values on a global

DyDlgrid as (see (1))

fðkDy;lDlÞ¼X

L

n¼0X

n

m¼0

¯

anm ¯

RnmðkDy;lDlÞþ¯

bnm ¯

SnmðkDy;lDlÞ

,

k¼0;...;M;l¼0;...;N1. ð16Þ

It is clear that fðkDy;NDlÞ¼fðkDy;0Þ, i.e., the values

at longitude ¼3601and 01are identical. To facilitate the

application of FFT, (16) is re-written as

fðkDy;lDlÞ¼X

L

0X

L

n¼m

¯

PnmðkDyÞ¯

anm

"#

cosðmlDlÞ

(

þX

L

n¼m

¯

PnmðkDyÞ¯

bnm

"#

sinðmlDlÞ)

¼X

L

0

CmcosðmlDlÞþSmsinðmlDlÞðÞ

k¼0;...;M;l¼0;...;N1. ð17Þ

ARTICLE IN PRESS

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451444

Note S0¼0. In (15), let Cm¼Sm¼0 for

m¼Lþ1;...;N1, and

Bm¼CmiSm

2. (18)

Then

fðkDy;lDlÞ¼2Re X

N1

m¼0

Bmei2pmlk=N

!

¼2ReðPÞ;l¼0;...;N1, ð19Þ

where Re stands for the real part of a complex number.

In (19), the sum Pcan be evaluated for all gridded values

along the same parallel (co latitude ¼kDy) by FFT.

Because the associated Legendre function satisﬁes

¯

PnmðtÞ¼ð1Þnm¯

PnmðtÞ, (20)

Dyshould be so chosen that 90

o

is an integer multiplier

of Dy. In this case, the associated Legendre functions

need to be computed only for the northern hemisphere,

and they can be used for the southern hemisphere with

only a selected change of sign based on the rule in (20);

see also Colombo (1981).

3.2. Case for functional of the Earth’s gravity ﬁeld

Given a set of geopotential coefﬁcients, we can

compute any functional of the gravity ﬁeld. For

example, geoidal height (or variation of geoidal height)

can be expanded into a series of spherical harmonics as

zðR;y;lÞ¼GM

RgX

1

n¼2

a

R

nX

n

m¼0

¯

Jnm ¯

Rnmðy;lÞþ ¯

Knm ¯

Snmðy;lÞ

¼X

L

m¼0X

L

n¼m

GM

Rg

a

R

n¯

PnmðyÞ¯

Jnm

"#

cosðmlÞ

(

þX

L

n¼m

GM

Rg

a

R

n¯

PnmðyÞ¯

Knm

"#

sinðmlÞ)

¼X

L

m¼0

Cz

mcosðmlÞþSz

msin ðmlÞ

,ð21Þ

where zis the geoidal height and Ris the radius of the

sphere where the expansion is made. Likewise, gravity

anomaly (or variation of gravity anomaly) can be

expanded into a series of spherical harmonics as

DgðR;y;lÞ¼GM

R2X

1

n¼2ðn1Þa

R

nX

n

m¼0

¯

Jnm ¯

Rnmðy;lÞþ ¯

Knm ¯

Snmðy;lÞ

¼X

L

m¼0X

L

n¼m

GM

R2ðn1Þa

R

n¯

PnmðyÞ¯

Jnm

"#

cosðmlÞ

(

þX

L

n¼m

GM

R2ðn1Þa

R

n¯

PnmðyÞ¯

Jnm

"#

sinðmlÞ)

¼X

L

m¼0

Cg

mcosðmlÞþSg

msinðmlÞ

.ð22Þ

One can see in (21) and (22) that, by properly scaling

the geopotential coefﬁcients, the algorithm of harmonic

synthesis presented in Section 3.1 can be used for

syntheses of gravity functionals. Also, the values of

functionals are not limited to the surface of the Earth;

rather they can be extended to an arbitrary height

deﬁned by R. The deﬁnitions of geopotential coefﬁcients

in (21) and (22) must be clariﬁed here. If the coefﬁcients

of gravity variation (see (14) and (15)) are given, then

one can use (21) and (22) directly to compute variations

of geoidal height and gravity anomaly. If the full

geopotential coefﬁcients are given, one must ﬁrst

subtract geopotential coefﬁcients of a reference ellipsoid

(see, e.g., Torge, 1989) from the full coefﬁcients, the

differences of coefﬁcients are then used in (21) and (22).

If the rates of geopotential coefﬁcients are available,

the rates of geoidal height and gravity anomaly can be

computed as

_

zðR;y;lÞ¼GM

RgX

1

n¼2

a

R

nX

n

m¼0

_

¯

Jnm ¯

Rnmðy;lÞþ _

¯

Knm ¯

Snmðy;lÞ

hi

,ð23Þ

D_

¯

gðR;y;lÞ¼GM

R2X

1

n¼2ðn1Þa

R

nX

n

m¼0

_

¯

Jnm ¯

Rnmðy;lÞþ _

¯

Knm ¯

Snmðy;lÞ

hi

,ð24Þ

where _

¯

Jnm and _

¯

Knm are the rates of geopotential

coefﬁcients. Again, in order to use FFT for an efﬁcient

computation, the expressions in (23) and (24) can be

changed to forms similar to those in (21) and (22).

4. Computer programs

Two computer programs, coded in FORTRAN90,

were developed for spherical harmonic analysis and

synthesis in various cases. Program sha is for analysis

and syn is for synthesis. Program sha accepts three types

of global grid: height anomaly, pressure anomaly, and

arbitrary surface function. Program syn reads fully

normalized harmonic coefﬁcients to generate a global

grid. The format of a global grid is called grd3. A

program, z2grd3, is developed to convert a netcdf grid of

GMT (Wessel and Smith, 1995) to a grd3 grid. The

ARTICLE IN PRESS

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451 445

computed harmonic coefﬁcients can be either in the fully

normalized form or the non-normalized form. The

default loading Love numbers are from Han and Wahr

(1995) and the maximum available degree is 696. Users

can supply their own Love numbers to program sha.

Appendix A shows the usages of sha and syn. Appendix

B and C show sample batch jobs of analysis and

synthesis.

5. Case studies

5.1. Altimeter and atmosphere data

Here we present two case studies using satellite

altimeter and atmospheric data. The altimeter data are

from the TOPEX/Poseidon (T/P) mission and the data

are supplied by AVISO (1996). T/P altimeter data

contain sea level anomalies (SLAs) at a 10-day interval

and cover a period from January 1993 to October 2001

(from T/P cycle 10 to cycle 344). The thermal-induced

sea level change, called steric height, represents the

volume change of the oceans and does not introduce

mass variation. Therefore, we subtracted the steric

heights from SLAs to create corrected sea level

anomalies (CSLAs), which reﬂect oceanic mass change.

The steric heights were supplied by Jianli Chen (2003,

private communication) at a 10-day interval and cover

the same time span as T/Ps; see also Chen et al. (2004)

for the modeling of the steric heights. For each T/P

cycle, we created a global 1111grid from the along-

track CSLAs. SLAs beyond 7661latitude were padded

with zeroes. The interpolation of SLAs onto a global

grid is done by ‘‘surface’’ of GMT. GMTs module

grd2xyz and program z2grd3 developed in this paper

were used to convert the netcdf grid to grd3 suitable for

use in sha. In order to reduce data noises, monthly SLAs

were created and were used in this paper. As an example,

Fig. 1 shows CSLAs in December 1997. As seen in Fig.

1, during the 1997–1998 El Nin

˜o the oceanic mass

increased in the equatorial, eastern Paciﬁc Ocean and

decreased in the western Paciﬁc Ocean. The maximum

changes of sea level in these two areas reach 40 cm.

Large changes of ocean mass also occurred in the Indian

Ocean. Such a large movement of sea water will

inevitably modify the Earth’s gravity ﬁeld.

The atmospheric pressure data are from the European

Centre for Medium-Range Weather Forecasts

(ECMWF). The data are monthly averaged atmospheric

pressures on a 2.512.51grid and cover the same period

as that of T/P SLAs. We ﬁrst computed a global mean

pressure ﬁeld by averaging all data. Monthly atmo-

spheric pressure anomalies (APAs) were then obtained

by subtracting the mean pressure from individual

monthly pressures. Fig. 2 shows APAs in December

1997. As seen in Fig. 2, it is typical that in December

atmospheric pressure highs occur over major continents

of the northern hemisphere and lows occur over major

continents of the southern hemisphere.

5.2. Analysis: time series of J

2

variation

The global grids of CSLA and APA over the period

from January 1993 to October 2001 were expanded into

ARTICLE IN PRESS

-80

-60

-40

-20

0

20

40

60

80

-8

0

-6

0

-4

0

-2

0

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

-40 -30 -20 -10 0 10 20 30 40

cm

Fig. 1. Monthly averaged corrected sea level anomaly (CSLA) from T/P in December 1997.

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451446

spherical harmonic series up to degree 50 using program

sha. The results are changes of geopotential coefﬁcients

due to oceanic and atmospheric mass redistributions.

The time series of a harmonic coefﬁcient can be used to

investigate long-term variations of certain geophysical

phenomena. For example, the degree-zero coefﬁcient

represents the total mass change of the underlying

geophysical ﬂuid (water or atmosphere). The degree-one

coefﬁcients are associated with the variation of geocen-

ter. The degree-two coefﬁcients have to do with the

variations of the Earth’s ﬂattening (the zonal term,

D¯

J20), polar motions (the tesseral terms, D¯

J21;D¯

K21) and

principal moments of inertia (the sectorial terms,

D¯

J22;D¯

K22); see Heiskanen and Moritz (1985).Asan

example, Fig. 3 shows the time series of D¯

J20 from

CSLAs and from APAs. The two time series show

strong annual variations and weak semi-annual varia-

tions. However, the phases of the annual variations from

the two sources are different. In general, the annual peak

of CLSA occurs in summer, while the annual peak of

APA occurs in winter. The phases of the semi-annual

variations from the two sources are also different.

Interestingly, the slopes of D¯

J20 from CSLAs before and

after December 1997 are 0.207 10

11

/year and

0.361 10

10

/year, respectively. This phenomenon

agrees with the result from satellite laser ranging

observations (Cox and Chao, 2002). Such a dramatic

change of the trend of the Earth’s ﬂattening is believed

to be caused by a recent surge in subpolar glacial melting

and by mass shifts in the Southern, Paciﬁc, and Indian

oceans (Dickey et al., 2002). The slopes of D¯

J20 from

APAs before and after December 1997 are

0.408 10

10

and, 0.608 10

10

, respectively, so

the rate of change is steady for 1993–2002. Further

investigations of the links between the J

2

variation and

geophysical phenomena are left to interested readers and

will not be elaborated here.

5.3. Synthesis: rate of geoid change

The time series of harmonic coefﬁcients obtained in

Section 5.2 were used to compute the rates of change for

individual coefﬁcients. The rates were then used in (23)

to compute the rates of geoid change up to degree 50.

Figs. 4 and 5 show the rates of geoid change from

CSLA’s and from APAs, respectively. The patterns of

geoid rate from the two sources are different. The

CSLA-implied geoid change contains a high near the

‘‘warm pool’’ northeast of Australia, where sea water

piles up before an El Nin

˜o occurs. Also, distinct lows

occur in the eastern Paciﬁc Ocean and the western

Indian Ocean. The atmospheric pressure-implied geoid

change contains distinct highs in the northeastern Paciﬁc

Ocean and the East Antarctica, and distinct lows in the

waters east of Australia and the Southeast Paciﬁc Basin.

Again, investigations of the phenomena seen in Figs. 4

and 5 are left to interested readers.

6. Conclusions

This paper presents FFT-based methods for spherical

harmonic analysis and synthesis. Two efﬁcient computer

programs coded in FORTRAN have been developed

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-80

-60

-40

-20

0

20

40

60

80

-80

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25

mbar

Fig. 2. Monthly averaged atmospheric pressure anomaly in December 1997.

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451 447

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-80

-60

-40

-20

0

20

40

60

80

-80

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

mm/yr

Fig. 4. Rates of geoid change due to oceanic mass change.

-4

-3

-2

-1

0

1

2

3

4

J2 variations (1*e-10)

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

J2 variations (1*e-10)

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

(a)

(

b

)

Fig. 3. Time series of J

2

variation due to mass changes in (a) ocean, and (b) in atmosphere.

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451448

based on the proposed methods. Both general cases and

special cases of analysis and synthesis were investigated.

The special cases involve the Earth’s gravity ﬁeld. We

demonstrate these computer programs using sea level

data from T/P and atmospheric data from ECMWF.

Interesting phenomena have been found in the J

2

time

series and the rates of geoid change. The programs

developed in this paper can facilitate the analysis and

synthesis of temporal gravity variations from satellite

missions such as GRACE.

Acknowledgments

This research is supported by the National Space

Organization (NSPO), Taiwan/ROC, under the project

‘‘Determination and analysis of Earth’s gravity ﬁeld

using COSMIC GPS data’’. We are grateful to JL Chen

of Center for Space Research, Austin, for supplying the

steric heights, and to two anonymous reviewers for their

constructive comments.

Appendix A. Usages of computer programs sha and syn

The usage of sha is

sha ﬁle.grd3 -Gcoef_ﬁle -TtypeId [-C –Ddensity -Lnmax

-Kloadnumbers.txt]

where

ﬁle.grd3: input ﬁle of grd3 grid containing data (must

cover the entire sphere)

-G: output ﬁle of fully normalized harmonic coefﬁcients

-T: type of gridded data

1¼height change of hydrological origin (e.g. sea

level, water table, snow, ice) (unit: m)

2¼pressure anomaly (unit: mbar)

3¼arbitrary surface function (eg., SST, height)

(unit: m)

Options:

-C: harmonic coefﬁcients will not be normalized

[default: fully-normalized]

-D: density (in kg/m

3

) associated with height change of

hydrological origin [default: 1000]

-K: ﬁle of loading Love numbers [default: Han and

Wahr (1995) up to degree 696]

-L: maximum degree of spherical harmonic expansion

[default: p/Dy]

The usage of syn is

syn coef_ﬁle -Idx/dy -Gﬁle.grd3 -Lnmax -Ttype [-Aa -B

-Dr –Mgm -R]

where

coef_ﬁle: input ﬁle of harmonic coefﬁcients

-I: sampling (grid) interval (in degrees) along longitude

and latitude

-G: output ﬁle of global grid in .grd3

-L: maximum degree of spherical harmonic expansion

-T: type of value to compute

0¼geoidal height

1¼gravity anomaly

2¼arbitrary function

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20

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60

80

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

mm/yr

Fig. 5. Rates of geoid change due to atmospheric mass change.

C. Hwang, Y.-C. Kao / Computers & Geosciences 32 (2006) 442–451 449

Options:

-A: scaling factor (a) associated with harmonic

coefﬁcients [default: a¼6378136:3m]

-B: ﬁle of harmonic coefﬁcient is a binary ﬁle [default:

ascii]

-D: radius (in meter) of sphere on which the expansion is

made [default: r¼a¼6378136:3m]

-M: product of Newtonian constant and the mass of the

earth [default: 3986004.415 10

8

m

3

/s

2

]

-R: subtract geopotential coefﬁcients of the GRS80

ellipsoid from the input coefﬁcients [default: do not

subtract]

Appendix B. Batch job of spherical harmonic analysis

using CSLAs

(1) Interpolate along-track CSLAs onto a global 1111

grid using GMTs ‘‘surface’’ surface tp_1997_12.xyh

-R0/360/-90/90 -I1/1 -T0 -Gtemp.grd

(2) Convert the GMT grd grid to a grd3 grid grd2xyz

temp.grd -Z | z2grd3 -R0/360/-90/90 -I1/1 -

Gtemp.grd3

(3) Perform spherical harmonic analysis to degree 50

using sha sha temp.grd3 -L50 -T1 –D1000

–Gtp_1997_12.coe

Appendix C. Batch job of spherical harmonic synthesis

using rates of geopotential coefﬁcients

(1) Compute rates of geoid change on a global 1111

grid using rates of geopotential coefﬁcients up to

degree 50

syn tp_coe_rate.dat -L50 -I1/1 -T0 -Gtemp.grd3

(2) Convert the grd3 grid to a GMT netcdf grid

grd3toz temp.grd3 | xyz2grd -Z -R0/360/-90/90 -I1/1

–Gtp_coe_rate.grd

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