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Spherical harmonic analysis and synthesis using FFT: Application to temporal gravity variation

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  • National Yang Ming Chiao Tung University

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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.
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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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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 ¼ffiffiffiffiffiffi
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
ffiffiffiffiffiffiffiffiffiffiffiffiffi
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
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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
¯
PnmðtÞ¼ð1Þnm¯
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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-30 -25 -20 -15 -10 -5 0 5 10 15 20 25
mbar
Fig. 2. Monthly averaged atmospheric pressure anomaly in December 1997.
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Fig. 4. Rates of geoid change due to oceanic mass change.
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4
J2 variations (1*e-10)
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
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(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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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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Supplementary resource (1)

... Figure 2 shows one period of function Pn(cosθ) in the direction of θ. In any 180°/n × 180°/n region of the sphere, the average gravity field is usually used for spherical harmonic synthesis instead of discrete values [17][18][19]. Therefore, we take the sense that some regular features would be explored from the gravity calculated with SHM in small size area. ...
... with the lumped coefficients: In any 180 • /n × 180 • /n region of the sphere, the average gravity field is usually used for spherical harmonic synthesis instead of discrete values [17][18][19]. Therefore, we take the sense that some regular features would be explored from the gravity calculated with SHM in small size area. ...
... a ρ n S nm P nm (cosθ) (11) In spherical harmonic analysis and synthesis, the surface of the Earth is gridded as numbers of subblock ∆θ × ∆λ [19]. We describe the disturbing potential values on a global grid as follows: ...
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For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS.
... The computation of definite integral of associated Legendre functions (ALF) is the basis of harmonic analysis on a sphere (Olver et al. 2010, Chapter 14 ). It is frequently used in geod- T. Fukushima (B) National Astronomical Observatory, Ohsawa, Mitaka, Tokyo 181-8588, Japan e-mail: Toshio.Fukushima@nao.ac.jp esy and geophysics as well as many other sciences (Hwang and Kao 2006). An arbitrary bivariate function defined on a unit sphere, f (θ, λ), can be expanded in terms of spherical harmonics (Heiskanen and Moritz 1967) as ...
... Then, except for the contribution from the polar caps, the harmonic coefficients are computed approximately (Hwang and Kao 2006) as ...
... Then, 10 will be a suitable size of such densified integration interval. At any rate, the summation along the parallel, i.e. that with respect to λ for constant θ , can be efficiently conducted by FFT (Hwang and Kao 2006 ). Therefore, the main computational problem of the spherical harmonic analysis reduces to the precise and fast evaluation of the definite integrals of ALF (Paul 1978). ...
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... The spherical harmonic analysis is a mathematical process of subdividing a function on a sphere into components of various wavelengths using surface spherical harmonics as base functions. The spherical harmonic synthesis combines components of various wavelengths to generate functional values on a sphere and it is the reverse process of a harmonic analysis (Colombo 1981;Hwang and Kao 2006). Among numerous authors addressing methods of spherical harmonic analysis and synthesis, we could mention studies by Colombo (1981), Dilts (1985), Healy et al. (2003), Hwang and Kao (2006), Kostelec et al. (2000), Mohlenkamp (1999), Potts et al. (1998), Sneeuw and Bun (1996), Suda and Takami (2002), Blais and Provins (2002), Wang et al. (2006), Gruber et al. (2011), Hirt (2012 and Claessens (2016). ...
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... [19] In the following development, we also need the timevarying gravity harmonic coefficients from the global surface mass variation and these are determined as [Wahr et al., 1998;Hwang and Kao, 2006] ...
... where Ds(q, l) is the change of the surface mass (expressed as thickness) at a specific location, P nm is the fully normalized associated Legendre functions, r ave is the average density of the Earth (5517 kg/m 3 ), and k n is the load Love number of degree n. Because surface mass anomalies are given on a 1 Â 1 grid, the FFT-based discrete formula of equation (28) [Hwang and Kao, 2006] is used to determine the coefficients. Because the hydrology data over the oceans are not provided by the GLDAS model, the values of Ds(q, l) are set to zero over the oceans. ...
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... where θ, λ, ρ ave , k n , ρ w and P nm are defined in Equation (3), h w is the GLDAS-derived hydrological EWH and C GLDAS nm and S GLDAS nm are the GLDAS-derived SHCs. Since the hydrological EWHs are given on a regular grid, the integral is implemented by the fast Fourier transform (FFT) algorithm of Hwang and Kao [32]. ...
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... Corrections due to the gravity effects of solid tide, ocean tide and atmosphere were applied to the raw GRACE gravity fields. At a given location of a spherical earth, the EWH can be derived from gravity change as (Wahr et al. 1998;Hwang and Kao 2006). ...
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High resolution transformations between regular geophysical data and harmonic model coefficients can be most efficiently computed by Fast Fourier Transform (FFT). However, a prerequisite is that the data grids are given in the appropriate geometrical domain. For example, if the data are situated on the ellipsoid at equi-angular reduced latitudes, spherical harmonic analysis can be employed and the coefficients subsequently converted by Jekeli’s transformation. This results in the spherical harmonic spectrum in the domain of geocentric latitudes. However, the data are most likely given at geodetic (ellipsoidal) latitudes which means that the FFT base needs to be shifted by latitude dependent phase lags in order to obtain the correct spherical harmonic spectrum. This requires appropriate sample rate conversion about the shifted latitudes by means of Fourier summation and cannot be treated efficiently by an FFT algorithm. In this article another solution is discussed instead. Since the variable heights between the spherical and ellipsoidal surfaces can be accurately approximated by a series of Tschebyshev polynomials, they can be convolved into the spherical basis. It will be shown how this new type of transformation to and from the ellipsoid in combination with Jekeli’s conversion of the spectra between the two surfaces allows eventually the sample rate conversion to shifted latitudes. This avoids the inexpedient Fourier summation mentioned previously. In this paper three applications for FFT in the domain of spherical and ellipsoidal surfaces, and using geocentric, reduced and geodetic latitudes are discussed. The Earth gravitational model EGM2008 of 5 arc minutes resolution has been used to demonstrate numerical results and computational advantages.
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1] Mass redistribution and motion in the oceans are major driving forces of geodetic variations, including polar motion, length of day, geocenter, and gravity field changes. We examine oceanic contribution to polar motion using estimates from a data-assimilating ocean general circulation model and satellite radar altimeter observations. The data include model estimates of variations in oceanic mass (OBP) and meridional and zonal velocities. Sea level anomalies from TOPEX/Poseidon (T/P) satellite altimeter measurements and steric sea surface height changes deduced from the model are also used to estimate OBP effects. Estimated oceanic contributions from both the model and T/P show considerably better agreement with polar motion observations compared with results from previous studies. The improvement is particularly significant at intraseasonal timescales. Both OBP and ocean current variations provide important contributions to polar motion. At intraseasonal timescales the oceans appear to be a dominant contributor to residual polar motion not accounted for by the atmosphere. The oceans also play an important role in seasonal excitation. Combined OBP and ocean current contributions explain much of the residual semiannual variability.
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We discuss the isostatic adjustment of the earth in response to Pleistocene deglaciation, and derive constraints on the earth's viscosity profile. We model the earth as viscoelastic, self-gravitating, realistically stratified and spherically symmetric. It is shown that the earth's relaxation eigenspectrum, besides possessing a small set of well-known discrete modes, also includes infinitely dense sets of modes (a continuous spectrum) which separate into classes that depend on the principal restoring force of the mode. Methods for estimating the continuous spectrum are discussed, and the effects on the predicted uplift are assessed. We find that the effects of the continuous spectrum are far less important than those of the discrete modes, though the presence of the continuous spectrum can make the discrete modes more difficult to identify. The continuous spectrum does become important when computing the rebound due to more recent loading, such as might be associated with present-day changes in polar ice. We use our model to refine predictions of the earth's response to Pleistocene glacial loading, with particular emphasis on implications for mantle viscosity. We have used time histories that are consistent with the recently recalibrated 14C time-scale. A special effort is made to match the geologic records of sea-level at sites in the vicinity of Hudson Bay. We find, consistent with the results of other authors, that the curvatures of the records are sensitive indicators of lower-mantle viscosity, but that there are inconsistencies between the records at different sites. However, we find we are also able to change the predicted curvatures by modifying the temporal and spatial distribution of the ice loads. If we assume a factor of 50 increase in viscosity between the upper and lower mantles, and if we assume that the ice on the eastern side of Hudson Bay melted a few thousand years later than the ice on the western side (a result consistent with recent geologic evidence), we are able to match the observed sea-level data about as well as we are able to, if we use, instead, a uniform mantle viscosity profile. A model with a factor of 50 jump in viscosity predicts a large free-air gravity anomaly over northern Canada that is reasonably consistent with what is observed, whereas a uniform viscosity model does not (though some authors have argued that the observed North American gravity anomaly may be mostly a consequence of mantle convection, and largely unrelated to postglacial rebound). Our objective here is not to present a preferred ice model or viscosity profile. Rather, we conclude only that the interpretation of the data is ambiguous, and that the mantle viscosity profile can not yet be uniquely inferred from them.
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The GRACE satellite mission, scheduled for launch in 2001, is designed to map out the Earth's gravity field to high accuracy every 2-4 weeks over a nominal lifetime of 5 years. Changes in the gravity field are caused by the redistribution of mass within the Earth and on or above its surface. GRACE will thus be able to constrain processes that involve mass redistribution. In this paper we use output from hydrological, oceanographic, and atmospheric models to estimate the variability in the gravity field (i.e., in the geoid) due to those sources. We develop a method for constructing surface mass estimates from the GRACE gravity coefficients. We show the results of simulations, where we use synthetic GRACE gravity data, constructed by combining estimated geophysical signals and simulated GRACE measurement errors, to attempt to recover hydrological and oceanographic signals. We show that GRACE may be able to recover changes in continental water storage and in seafloor pressure, at scales of a few hundred kilometers and larger and at timescales of a few weeks and longer, with accuracies approaching 2 mm in water thickness over land, and 0.1 mbar or better in seafloor pressure.
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An updated, new version (3.0) of the Generic Mapping Tools (GMT) has just been released. GMT is a public domain collection of UNIX tools that contains programs to manipulate (x,y) and (x,y,z) data and to generate PostScript illustrations, including simple x-y diagrams, contour maps, color images, and artificially illuminated, perspective, shaded-relief plots using a variety of map projections [Wessel and Smith, 1991]. GMT has been installed on super computers, workstations and personal computers, all running some flavor of UNIX. We estimate that approximately 5000 scientists worldwide are currently using GMT in their work.
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A method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast Fourier transform is described. Discussions of the operation count, storage requirements, accuracy, and an algebraic and a numerical example are included. A comparison with straightforward integration is made throughout. Also, a new method for evaluating the spherical harmonics is discussed.
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Earlier work by Driscoll and Healy has produced an efficient algorithm for computing the Fourier transform of band-limited functions on the 2-sphere. In this article we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions. All require at most O(N log2N) operations where N is the number of sample points. We also address implementation considerations and give heuristics for allowing reliable and computationally efficient floating point implementations of slightly modified algorithms. These claims are supported by extensive numerical experiments from our implementation in C on DEC, HP, SGI and Linux Pentium platforms. These results indicate that variations of the algorithm are both reliable and efficient for a large range of useful problem sizes. Performance appears to be architecture-dependent. The article concludes with a brief discussion of a few potential applications.