Content uploaded by Yang Lu

Author content

All content in this area was uploaded by Yang Lu on Sep 29, 2015

Content may be subject to copyright.

An effective filtering for GRACE time-variable gravity: Fan filter

Zi-Zhan Zhang,

1,2

B. F. Chao,

2

Yang Lu,

1

and Hou-Tse Hsu

1

Received 19 June 2009; revised 28 July 2009; accepted 30 July 2009; published 5 September 2009.

[1] Spatial low-pass filtering is necessary for processing

the GRACE time-variable gravity (TVG) data which are

otherwise plagued with short-wavelength noises. Here we

devise a new non-isotropic filter, called the fan filter: In

terms of the spherical harmonic spectrum, the fan filter is

simply a 2-D double filter consisting of a low-pass along the

degree n (the same as the conventional isotropic filter)

simultaneously with a low-pass along the order m, whose

contour projection onto the (n, m) plane is fan-shaped. It is

deterministic and independent of a priori or external

information, its implementation is straightforward, and the

result is objective. Most importantly, we show that this

simple filter performs well among its counterparts under

similar conditions, in particular against the N-S striping

noises prevalent in the GRACE TVG solutions. We

demonstrate this with Gaussian weights at filter length

and hence spatial resolution as fine as 300 km. We also

deduce the fan filter’s nominal amplitude-reduction factor as

a function of the filter length for TVG signals that follow

the Kaula rule.

Citation: Zhang, Z.-Z., B. F. Chao, Y. Lu, and

H.-T. Hsu (2009), An effective filtering for GRACE time-variable

gravity: Fan filter, Geophys. Res. Lett., 36, L17311, doi:10.1029/

2009GL039459.

1. Introduction

[2] Launched i n 2002, the dual-satellite mission o f

GRACE (Gravity Recovery And Climate Experiment)

[Tapley et al., 2004] has enabled measurement of the Earth’s

(tiny) time-variable gravity (TVG), providing new and

precise information about large-scale mass transport on or

in the Earth. Given the nature of the measurement tech-

nique, the GRACE TVG solutions, in the form of monthly

spherical harmonic (SH) Stokes coefficients, are plagued

with short-wavelength noises. Spatial low-pass filtering, or

‘‘smoothing’’, becomes necessary.

[

3] The first such filter devised was, not surprisingly,

isotropic as exemplified by the Gaussian filter of Wahr et al.

[1998]. An isotropic filter has weight depending only on

the SH degree n, and hence wavelength, but not on the

orientation. Soon afterwards, it w as realized that the

GRACE noise structure itself is not isotropic – instead it

mainly manifests itself as near N-S ‘‘stripes’’ reflecting the

satellit e o rbit configuration and the resultant sensitivity

factors. This calls for non-isotropic filters depending also

on the SH order m in some designated manner.

[

4] There have since been many non-isotropic filters

proposed in the literature. Some of them are ‘‘adaptive’’,

with filter weights taken optimally according to either

numerical geophysical model output to which the GRACE

data are expected to ‘‘resemble’’ or correlate [e.g., Han et

al., 2005; Chen et al., 2006; Kusche, 2007; Klees et al.,

2008], or to the signal-to-noise ratio estimated for given n

and m in some a priori statistical criteria [e.g., Sasgen et al.,

2006; Davis et al., 2008]. Such filters have been demon-

strated to be useful in many specific examples. However,

they are often tedious to implement, and hard to duplicate or

be combined jointly with other data types with different

noise structure. Furthermore, as the filter design criteria

become subjective and time- and model-dependent, one

needs to take caution clarifying the geophysical interpreta-

tion of the results in quantitative applications.

[

5] Other types of filtering include those that are region-

ally effective [Swenson and Wahr, 2006; Velicogna and

Wahr, 2006], or employing localized basis functions such as

wavelet [e.g., Schmidt et al., 2006] or EOF [Schrama et al.,

2007; Wouters and Schrama, 2007]. In addition, one can

separately apply the so-called correlated-error filter, which

aims to reduce empirically the purported artifact noises

owing to certain even-odd degree correlation found in the

GRACE TVG solutions (Swenson and Wahr [2006] and see

later).

[

6] We here devise a non-isotropic filter, called the fan

filter, as a function of both SH degree n and order m.

Independent of a priori or external information, its imple-

mentation is straightforward and deterministic with fully

tractable performance. Equally important, we shall demon-

strate that this simple filter is indeed effective in smoothing

out the said noises even at relatively fine spatial resolutions.

2. Structure of GRACE Noise and the Fan Filter

[7] We use a total of 75 monthly GRACE TVG data

maps according to the Level-2 solution Release 4 by CSR

(CSR-RL04) spanning 2002/8–2008/12. Each map is com-

posed from the SH Stokes coefficients referenced to the

temporal-mean, or ‘‘static’’, values, for all 0 Q m Q n and n

from 2 to a maximum degree N takentobe60here

(altogether 1887 coefficients). As a usual practice we

disregard the degree-1 terms and replace the GRACE-

derived C

20

series with that better determined from satellite

laser ranging [Cheng and Ries, 2007].

[

8] Figure 1a shows the average (over the whole time span)

spectrum, or the per-degree, per-order root-mean-square

amplitude, of the TVG. It is evident that the nominal power

largely concentrates at the two corners of the 2-D (n, m)

space: Towards the low-n corner is where real geophysical

signals reside, whereas the high-n high-m region is plagued

by (short-wavelength) noises. In particular, the abnormally

GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L17311, doi:10.1029/2009GL039459, 2009

Click

Here

for

Full

A

rticl

e

1

Key Laboratory of Dynamic Geodesy, Institute of Geodesy and

Geophysics, Chinese Academy of Sciences, Wuhan, China.

2

College of Earth Sciences, National Central University, Chung-Li,

Taiwan.

Copyright 2009 by the American Geophysical Union.

0094-8276/09/2009GL039459$05.00

L17311 1of6

strong power along and near the m = n axis is what gives

rise to the spatial N-S stripe noises (see Figure 2a).

Incidentally, those occurring along n = 15 and its multiples

reflect the relatively poorer determination resulting fr om the

daily orbit resonance.

[

9] Figure 1b illustrates the Wahr et al. [1998] isotropic

(Gaussian) filter with filter weight depending only on n.

Here we select the filter length L, or the equivalent cutoff

wavelength, to be 300 km, which then represents the spatial

resolution of the data after filtering. The color contours (on

Plane A) shows the projection of the filter weight onto the

(n, m) plane. With no dependence on m, this isotropic

filter is not particularly effective against the stripe-raising

noises along and near the m = n axis, as stated above (see

Figure 2b).

[

10] Now in order to better treat the noise structure of

TVG shown in Figure 1a, we apply a low-pass along the

degree n simultaneously with a second low-pass along the

order m, i.e., a double filtering in the (n, m) space.

Specifically, for gravity anomaly, the filter weights W

n

and W

m

are implemented via constructing

Dg q; fðÞ¼

GM

a

2

X

N

n¼0

n 1ðÞW

n

X

n

m¼0

W

m

DC

nm

cos mfðÞþDS

nm

sin mfðÞðÞ½P

nm

cos qðÞ

ð1Þ

where GM is the gravitational constant and a the mean

equ atorial radius of the Earth, (q, 8) = (colati tudes,

longitude), DC

nm

and DS

nm

are the time-variable Stokes

coefficients referenced to the static or time-averaged value,

and P

nm

is the 4p-normalized associated Legendre function.

Note that equation (1) reduces to the isotropic filter if W

m

=1,

and to the original unfiltered TVG map if both W

n

and W

m

are identically 1. It can be easily modified to yield the

corresponding expressions for the geoid or the equivalent

water thickness (see below), if so desired. This way we

Figure 1. (a) Average per-degree and per-order root-mean-square amplitude of the Earth’s TVG; (b) isotropic Gaussian

filter weight and its projection onto the (n, m) plane (L = 300 km); (c) non-isotropic Gaussian fan filter weight and

projection (300 km).

L17311 ZHANG ET AL.: FAN FILTER FOR GRACE GRAVITY VARIATION L17311

2of6

construct a new, non-isotropic filter, as shown in Figure 1c

for the case of Gaussian weights with L = 300 km. On the

m = 0 plane (Plane B) or along the N-S direction, this

filter is identical to the isotropic filter (in fact, the latter

can be considered as a special case of the fan filter if the

m-dependence is ‘‘turned off’’.) The contour projection onto

Plane A becomes fan-shaped, hence the name fan filter. It

has a more severe reduction of the high-m noises along the

E-W direction hence reducing the N-S stripe noises more

efficiently. Note that strictly speaking the latter would mean

a loosening of the effective spatial resoluti on in the E-W

direction, which can be shown to be poorer by a factor of

sqrt(2) for the case of Gaussian filter. Thus, in our example

Figure 1c, the nominal resolution is 300 km in the N-S

direction, 420 km in the E-W direction on the Equator but

turning finer with higher latitudes by virtual of the nature of

the Earth’s sphericity and the orbit spacing.

[

11] The Gaussian weight which can be re adily and

recursively constructed on a sphere [Jekeli, 1981] is often

adopted in actual implementations as done here, although in

principle any low-pass filter of one’s choice can serve the

same purpose. In fact, the filters adopted along n and along

m need not be the same. For example, one can replace the

Gaussian with a more severe low-pass filtering along m

(hence a somewhat skewed fan-shape), if such is determined

to be desirable (e.g., to further de-stripe).

3. Demonstration of the Fan Filter

[12] There are two legitimate ways of displaying gravity

or TVG (relative to the ellipsoidal Earth) in terms of the SH

components: (i) the geoid height which is reconstructed

simply by summing up the SHs pre-multiplied by the given

Stokes coefficients, and (ii) the free-air gravity anomaly

which is of the same summation but with each SH term

multiplied by the extra factor n 1 (e.g., equation (1)).

Here we shall choose the latter because it renders higher

emphasis on higher-degree or shorter-wavelength compo-

nents. A third way, widely adopted for convenience, is to

give the equivalent surface water thickness obtained by the

same summation as (i) but each SH term multiplied by the

extra factor 2n + 1 (and further modified by the mass

loading effect). That practice however is inappropriate here

because the equivalent water thickness is meaningful only

for mass redistribution happening on the Earth surface and

hence, subject to the intrinsic non-uniqueness of the grav-

itational inverse problem [Chao, 2005], geophysically mis-

leading for processes internal to the Earth. Solid tides,

seismicity, post-glacial rebound, mantle and core convec-

tions are just some examples of the l atter.

[

13] Figure 2 shows a comparison conducted on GRACE

CSR-RL04 TVG solution for a typical month randomly

selected – April of 2008. Figure 2a is the raw data map up

to N = 60 without any filtering; Figure 2b is after applying

the isotropic Gaussian filtering at L = 300 km (cf. Figure 1b);

Figure 2c is the same but after applying instead the (non-

isotropic Gaussian) fan filtering at L = 300 km (cf. Figure 1c).

The improvement from Figure 2b to 2c is evident. Stripings

still, but barely, exist with much reduced amplitude and

hence considerably less contamination, particularly in land

areas where the signal-to-noise ratio is inherently higher.

Figure 2. For TVG gravity anomaly of April, 2008: (a) the raw data map up to N = 60 without any filtering; (b) applying

the isotropic Gaussian filtering (L = 300 km) on Figure 2a; (c) applying the non-isotropic Gaussian fan filtering (L = 300 km)

on Figure 2a; (d) the difference of Figures 2b and 2c; (e) after further applying the correlated-error filter on Figure 2b; (f) after

further applying the correlated-error filter on Figure 2c; (g) the difference of Figures 2e and 2f.

L17311 ZHANG ET AL.: FAN FILTER FOR GRACE GRAVITY VARIATION L17311

3of6

The difference of Figures 2b and 2c is plotted in Figure 2d,

showing the additional noises (mainly stripes) that are

removed by the fan filter. Presumably a small amount of

true signal is also removed, but recoverable in principle (see

Section 4).

[

14] Independent of the above, Swenson and Wahr [2006]

found that the GRACE data also suffer from certain even-

odd degree correlated artifacts, which can be decontami-

nated off by removing an empirical polynomial fit to the

variance of the even- or odd-n for a given m. Called the

correlated-error filter, it can be applied in combination with

other filters. Thus, we apply it on Figure 2b, with 5th-degree

polynomial fits for all m R 11 [cf. Chambers, 2006]; the

result is given in Figure 2e. Similarly we apply the same

to Figure 2c, resulting in Figure 2f. The difference of

Figures 2e–2f is plotted in Figure 2g, illustrating the

improvements by the fan filter, notably around the vast

equatorial region.

[

15] Here we shall consider the process leading to

Figure 2f our ‘‘best’’ result. An animation for the entire

time span is given in the auxiliary material, together with

the same but with L = 400 km.

1

[16] For a further test, we construct a synthetic TVG field

in the form of a rectangular box-car function (Figure 3a) and

run it respectively through three different filters, all at

effective filter length of 300 km under otherwise identical

processing: the isotropic Gaussian filter (resulting i n

Figure 3b), the Han et al. [2005] filter (resulting in

Figure 3c), and the fan filter (resulting in Figure 3d). The

Han et al. filt er is based on a simil ar ( deterministic)

philosophy as the fan filter but having an abrupt cutoff of

the filter weight, empirically chosen at m

1

=15[Han et al.,

2005, equation (11b)], although we have tried many other

choices but yielding qualitatively similar results). The main

differences are with respect to the N-S running Gibbs

leakage out of the box-car region, an undesirable feature

of the filter which proves to be negligible in the case of the

fan filter (Figure 3d).

4. Amplitude Recovery

[17] Filters inevitably result in signal amplitude reduc-

tion, the severity of which varies from case to case depend-

ing on the specific spectral power distribution in question

relative to the given filter parameters. It is sometimes

desirable to recover the amplitude in actual applications

such as comparison with respect to basin-wide hydrological

data, seismic model predictions, and various others involv-

ing geophysical interpretations. As an illustration we assess

the nominal amplitude reduction factor caused by the fan

filter in a global example (as opposed to regional variability

Figure 3. A synthetic test: (a) a ‘‘box-car’’ function (1 inside and 0 outside) in a rectangular test area; (b) applying the

isotropic Gaussian filter (L = 300 km)on Figure 3a; (c) applying instead the Han et al. [2005] non-isotropic Gaussian filter

for Lon = 400 km, Lat = 300 km, m

1

= 15; (d) applying instead the fan filter for L = 300 km. The contours show the Gibbs

leakage.

1

Auxiliary materials are available in the HTML. doi:10.1029/

2009GL039459.

L17311 ZHANG ET AL.: FAN FILTER FOR GRACE GRAVITY VARIATION L17311

4of6

that requires more specific treatment as Wahr et al. [1998]),

supposing the TVG in question follows the same type of

power-law as the Earth’s static gravity field, namely the

Kaula rule of thumb [Kaula, 1966] that the per-degree

root-mean-square of the TVG geoid is proportional to (2n +

1)

1/2

/n

2

. This has actually been justified to be true for the

hydrological fields from LaD and ECCO models [Kusche,

2007]. Sasgen et al. [2006] demonstrated that in some cases

the TVG deviates slightly from the Kaula rule in that the

power of n in the denominator can range between 1.5 to 2.

Whatever the case may be, one can accordingly deduce the

net effect of amplitu de reduction by the fan filter as a

function of the choice of L. Figure 4 gives the results of

the fan filter (with Gaussian weight) compared with the

corresponding isot ropic filter, for the geoid and gravity

anomaly TVG, for the cases of power 2. The inverse of

the reduction factor gives the nominal factor to be multi-

plied to the filtered signal to recover the true amplitude of

the TVG given the Kaula-type rule.

[

18] From Figure 4, one sees the following. (i) As

expected the fan filter is more severe in amplitude reduction

than the isotropic filter, but only mildly. (ii) Smaller L has

less reduction than larger L, because it tapers off more

sharply and leaves the largest signals towards lower degrees

(according to the Kaula-type rule) less affected by the filter.

(iii) The gravity anomaly suffers from considerably more

reduction in amplitude by the action of filtering than for the

geoid, as a result of its higher emphasis on the higher-

degree SH terms that are more affected by the filtering. For

example, the amplitude is reduced by as much as 25% for

gravity anoma ly by the fan filter at L = 300 km, compared to

only 2% for geoid.

5. Discussions

[19] The fan f ilter is based on a simple idea whose

implementation is straightforward and deterministic. We

have demonstrated its effectiveness in treating GRACE

TVG data, especially when combined with the correlated-

error filter of Swenson and Wahr [2006], producing our

‘‘best’’ results exemplified in Figure 2f. The latter is

practically free from the stripe nois es even at spatial

resolution as fine as 300 km, which is already the limit

resolution on the Earth surface corresponding to the SH

expansion up to degree 60. The application of the fan filter

is independent of a priori or external information or con-

ditions, in contrast to the adaptive filters whose implemen-

tation and hence quantitative results tend to be subjective

and time- or model-dependent.

[

20] While filter length L of 400 km or longer are often

used for experiment benchmark, we used L = 300 km for

demonstration. We have found that the fan filter performs

comfortably well even down to such resolution. For appli-

cations that require less stringent spatial resolution, filter

length L can be loosened, and the filter performance

becomes even more robust as expected.

[

21] In our demonstration we chose not to use the

equivalent water thickness (EWT) when presenting the

GRACE TVG maps because EWT only reflects the geo-

physical mass change when the latter is confined on the

Earth’s surface (see above). However, we should point out

that had we done so (physically meaningful or not) the

numerical improvements by the fan filter appear even more

striking. This is because EWT emphasizes the shorter-

wavelength SH components more than does the gravity

anomaly. On the other hand, determination of the

corresponding amplitude reduction (Section 4) by the fan

filter for EWT, call it R

e

, is less definite. In a nominal sense,

if the Kaula-type rule starts at n = 2, then R

e

is less severe

than that for gravity anomaly R

g

, at about 0.4R

g

asymptot-

ically as n increases (or somewhat less than that as modified

by the load Love number). This is because the multiplica-

tion factor for EWT, namely 2n + 1 (see Section 3), when

normaliz ed by 5 (the value at n = 2) approaches 0.4n

asymptotically while the corresponding factor n 1for

gravity anomaly approaches n asymptotically. Because this

argument in the case of R

e

is highly sensitive to the nominal

Kaula-type rule at the lowest degree, we recommend

alternative empirical approaches to assess R

e

(especially if

region-specific), such as using Monte Carlo simulation.

[

22] Acknowledgments. We thank the GRACE Project, and particu-

larly CSR of University of Texas for providing the data. The work is

supported by NSFC (40874037), ‘863’ pr oject (2006AA12Z128), the

Chinese Academy of Sciences (kzcx2-yw-143), and the NSC of Taiwan

(NSC97-2116-M-008-012).

References

Chambers, D. P. (2006), Evaluation of new GRACE time-variable gravity

data over the ocean, Geophys. Res. Le tt., 33, L17603, doi:10.1029/

2006GL027296.

Chao, B. F. (2005), On inversion for mass distribution from global (time-

variable) gravity field, J. Geodyn., 39, 223– 230, doi:10.1016/j.jog.2004.

11.001.

Chen, J. L., C. R. Wilson, and K.-W. Seo (2006), Optimized smoothing of

Gravity Recovery and Climate Experiment (GRACE) time-variable

gravity observations, J. Geophys. Res., 111, B06408, doi:10.1029/

2005JB004064.

Cheng, M., and J. Ries (2007), Monthly estimates of C20 from 5 SLR

satellites, GRACE Tech. Note 05, Jet Propul. Lab., Pas adena Calif.

(Available at http://podaac.jpl.nasa.gov/grace/documentation.html)

Davis, J. L., M. E. Tamisiea, P. Elo´segui, J. X. Mitrovica, and E. M. Hill

(2008), A statistical filtering approach for Gravity Recovery and Climate

Experiment (GRACE) gravity data, J. Geophys. Res., 113, B04410,

doi:10.1029/2007JB005043.

Han, S.-C., C. K. Shum, C. Jekeli, C.-Y. Kuo, C. Wilson, and K.-W. Seo

(2005), Non-isotropic filtering of GRACE temporal gravity for geophy-

sical signal enhancement, Geophys. J. Int., 163, 18 – 25, doi:10.1111/

j.1365-246X.2005.02756.x.

Figure 4. Amplitude reduction by the fan filter (solid

lines), compared to the isotropic filter (dashed lines), both

of Gaussian weight, as a function of the choice of filter

length, for the geoid (upper curves) and gravity anomaly

(lower curves) following the Kaula-type rule.

L17311 ZHANG ET AL.: FAN FILTER FOR GRACE GRAVITY VARIATION L17311

5of6

Jekeli, C. (1981), Alternative methods to smooth the Earth’s gravity field,

Tech. Rep. 327, Dep. of Geod. Sci. and Surv., Ohio State Univ., Columbus.

Kaula, W. M. (1966), Theory of Satellite Geodesy, Blaisdell, Waltham,

Mass.

Klees, R., E. A. Revtova, B. C. Gunter, P. Ditmar, E. Oudman, H. C.

Winsemius, and H. H. G. Savenije (2008), The design of an optimal filter

for monthly GRACE gravity models, Geophys. J. Int., 175, 417 – 432,

doi:10.1111/j.1365-246X.2008.03922.x.

Kusche, J. (2007), Approximate decorrelation and non-isotropic smoothing

of time-variable GRACE-type gravity field models, J. Geod., 81, 733 –749,

doi:10.1007/s00190-007-0143-3.

Sasgen, I., Z. Martinec, and K. Fleming (2006), Wiener optimal filtering of

GRACE data, Stud. Geophys. Geod., 50, 499–508, doi:10.1007/s11200-

006-0031-y.

Schmidt, M., M. Fengler, T. Mayer-Gu¨rr, A. Eicker, J. Kusche, L. Sa´nchez,

and S. C. Han (2006), Regional gravity modeling in terms of spherical

base functions, J. Geod., 81, 17 –38, doi:10.1007/s00190-006-0101-5.

Schrama, E. J. O., B. Wouters, and D. A. Lavalle´e (2007), Signal and noise

in GRAC E observed surface mass variations, J. Geophys. Res., 112,

B08407, doi:10.1029/2006JB004882.

Swenson, S., and J. Wahr (2006), Post-processing removal of correlated

errors in GRACE data, Geophys. Res. Lett., 33, L08402, doi:10.1029/

2005GL025285.

Tapley, B. D., S. Bettadpur, M. Watkins, and C. Reigber (2004), The

gravity recovery and climate experiment: Mission overview and early

results, Geophys. Res. Lett., 31, L09607, doi:10.1029/2004GL019920.

Velicogna, I., and J. Wahr (2006), Measurements of time-variable gravity

show mass loss in Antarctica, Science, 311, 1754 – 1756, doi:10.1126/

science.1123785.

Wahr, J., M. Molennar, and F. Bryan (1998), Time variability of the Earth’s

gravity field: Hydrological and oceanic effects and their possible detec-

tion using GRACE, J. Geophys. Res., 103, 30,205 – 30,229, doi:10.1029/

98JB02844.

Wouters, B., and E. J. O. Schrama (2007), Improved accuracy of GRACE

gravity solutions through e mpirical orthogonal function filtering of

spherical harmonics, Geophys. Res. Lett., 34, L23711, doi:10.1029/

2007GL032098.

B. F. Chao, College of Earth Sciences, National Central University,

Chung-Li 32001, Taiwan. (bfchao@ncu.edu.tw)

H.-T. Hsu, Y. Lu, and Z.-Z. Zhang, Key Laboratory of Dynamic Geodesy,

Institute of Geodesy and Geophysics, Chinese Academy of Sciences,

Xudong Street 340, Wuhan, Hubei 430077, China.

L17311 ZHANG ET AL.: FAN FILTER FOR GRACE GRAVITY VARIATION L17311

6of6