Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B9, 193, doi:10.109/001jb000576, 00 Method for inferring al urfae-ma anomalie from Gravity Reovery and Climate Experiment (GRACE) meaurement of time-variable gravity Sean Swenon and John Wahr Department of Phyi and Cooperative Intitute for Reearh in Environmental Siene, Univerity of Colorado, Boulder, USA Reeived 8 May 001; revied 9 Marh 00; aepted 14 Marh 00; publihed 19 September 00. [1] The Gravity Reovery and Climate Experiment, GRACE, will deliver monthly average of the pherial harmoni oeffiient deribing the Earth gravity field, from whih we expet to infer time-variable hange in ma, averaged over arbitrary having length ale of a few hundred kilometer and larger, to auraie of better than 1 m of equivalent water thikne. Thee data will be ueful for examining hange in the ditribution of water in the oean, in now and ie on polar ie heet, and in ontinental water and now torage. We deribe method of extrating al ma anomalie from GRACE gravity oeffiient. Spatial averaging kernel were reated to iolate the gravity ignal of individual while imultaneouly minimizing the effet of GRACE obervational error and ontamination from urrounding glaial, hydrologial, and oeani gravity ignal. We then etimated the probable auray of averaging kernel for of arbitrary hape and ize. INDEX TERMS: 1836 Hydrology: Hydrologi budget (1655); 483 Oeanography: General: Water mae; 1655 Global Change: Water yle (1836); 143 Geodey and Gravity: Spae geodeti urvey; 1640 Global Change: Remote ening; KEYWORDS: GRACE, atellite gravity, hydrology, time-variable gravity, al water torage Citation: Swenon, S., and J. Wahr, Method for inferring al urfae-ma anomalie from Gravity Reovery and Climate Experiment (GRACE) meaurement of time-variable gravity, J. Geophy. Re., 107(B9), 193, doi:10.109/001jb000576, Introdution [] Time-variable gravity hange are aued by a ombination of potglaial rebound, flutuation in atmopheri ma, and the reditribution of water, now, and ie on land and in the oean. The patial reolution of gravity data obtained from atellite meaurement ha not yet been uffiient to eparate the effet of thee proee from one another. Data from GRACE, launhed in 00 hould provide dramatially improved time-variable gravity meaurement. GRACE will deliver monthly average of the pherial harmoni oeffiient deribing the Earth gravity field at ale of a few hundred kilometer and larger. [3] From the gravity field etimate, we expet to infer time-variable hange in ma, averaged over arbitrary having length ale of a few hundred kilometer and larger, to auraie of better than 1 m of equivalent water thikne. Thee data will be ueful for examining hange in the ditribution of water in the oean, in now and ie on polar ie heet, and in ontinental water and now torage. Thee quantitie an then be ued to ae and improve limate model, to better undertand large-ale hydrologial proee, and to monitor the ditribution of land-baed water for agriultural and water reoure appliation. Combined with radar altimetry over the oean, thee data an improve etimate of the time-varying oean Copyright 00 by the Amerian Geophyial Union /0/001JB000576$09.00 heat torage, a well a deep oean urrent. In polar, GRACE data an be ued to tudy potglaial rebound and, in onjuntion with laer altimetry, to ontrain the ma balane of ie heet. [4] Beaue the patial reolution of GRACE i on the order of a few hundred kilometer, an etimate of a urfaema anomaly will not be a point meaurement, but rather a patial average. Wahr et al. [1998] introdue an averaging method baed on a imple Gauian filter. However, thi method doe not iolate a peifi. In order to ue GRACE to etimate al hange in urfae ma, tehnique mut be developed whih extrat al ma anomalie from GRACE gravity oeffiient. In thi paper, we deribe method of reating patial averaging kernel whih iolate the gravity ignal of individual while imultaneouly minimizing the effet of GRACE obervational error and the ontamination from urrounding glaial, hydrologial, and oeani gravity ignal. We then etimate the probable auray of averaging kernel for of arbitrary hape and ize.. Inferring Surfae Ma Change From the Time-Variable Gravity Field [5] It i uual to repreent the Earth gravity field in term of the hape of the geoid, the equipotential urfae that mot loely oinide with mean ea level over the oean. The geoid, N, an be expanded a a um of ETG 3-1

2 ETG 3 - SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY normalized aoiated Legendre funtion, ~P [ee, e.g., Chao and Gro, 1987]: Nðq; fþ ¼ a X1 l¼0 X l ~P ðo qþfc o mf þ S in mfg ð1þ where q i olatitude, f i longitude, a i the mean radiu of the Earth, and C and S are dimenionle Stoke oeffiient. GRACE Projet peronnel will ue the GRACE meaurement to olve for the Stoke oeffiient up to degree l 100 every 30 day, and thee oeffiient will be made available to uer. The patial ale, l, aoiated with a partiular Stoke oeffiient i inverely proportional to it angular degree, l, and an be found approximately by the relation l = 0,000 km/l, o that l 100 orrepond to length ale of 00 km and larger. [6] Uing thee oeffiient, it will be poible to infer hange in the gravity field from one 30-day period to the next, and o to tudy proee involving the reditribution of ma within the Earth and on or above it urfae. GRACE will be aurate enough to be enitive to hange in the Earth gravity field aued by flutuation in ontinental water torage and the polar ie heet, a well a by hange in atmopheri and oeani ma ditribution. The ontribution from the atmophere an be etimated from independent atmopheri data and largely removed [Veliogna et al., 001; Swenon and Wahr, 00]. Beaue mot of the remaining oure of time-variable ma hange are onfined to a thin layer at the Earth urfae, one an approximate the vertially integrated water and ie ma a a urfae ma denity. With thi approximation, Wahr et al. [1998] how that a loal hange in urfae ma denity, (q, f), an be related to hange in the Stoke oeffiient, C and S,by ðq; fþ ¼ ar X 1 E 3 l¼0 X l þ S in mfg ðl þ 1Þ P ð1 þ k l Þ ~ ðo qþfc o mf where r E i the average denity of the olid Earth and k l are the load love number repreenting the effet of the Earth repone to urfae load. The love number an be obtained from Wahr et al. [1998]. [7] The Stoke oeffiient produed by GRACE will ontain meaurement error, d and d, uh that C GRACE S GRACE ¼ C true þ d ðþ ¼ S true þ d : ð3þ Satellite meaurement error inlude ytem noie error in the interatellite range rate, aelerometer error, error in the ultratable oillator, and error in the orbit. [8] If the oeffiient of the atellite error and the urfae ma anomaly at n time t 1, t,...t n are unorrelated for every (l, m) and (l 0, m 0 ), 1 n 1 n X n C true t i X n S true t i ð Þd l 0 m ð t 0 iþ ¼ 1 n ð Þd l 0 m ð t 0 iþ ¼ 1 n X n C true t i X n S true t i ð Þd l 0 m ð Þ ¼ 0 0 t i ð Þd l 0 m ð Þ ¼ 0; 0 t i ð4þ then two term omprie the expreion for the variane of the inferred urfae ma anomaly at the point (q, f): varðþ GRACE ¼ varðþ true þ varðþ at ; where the ontribution to the variane due to atellite meaurement error i varðþ at ¼ X l;m X l 0 ;m 0 K l K l 0 ð5þ ~P ðo qþ ~P l 0 m 0ðo qþ ð6þ ll 0 mm o mf o 0 m0 f þ ll 0 mm o mf in 0 m0 f þ ll 0 mm 0 in mf o m0 f þ ll 0 mm 0 in mf in m0 f : ð7þ K l ¼ ar E 3 ðl þ 1Þ ð1 þ k l Þ onvert the geoid oeffiient to urfae ma oeffiient, and ll 0mm are the ovariane matrie of the GRACE 0 meaurement error. If, for example, the tatitial propertie of the meaurement error were the ame for eah of the n month t i, i =1,n, then ð8þ ll 0 mm ¼ 1 X n d 0 ð n t iþd l 0 m t 0 i ð Þ; ll 0 mm ¼ 1 X n d 0 ð n t iþd l 0 m ð t 0 iþ; ll 0 mm ¼ 1 X n d 0 ð n t iþd l 0 m ð t 0 iþ; ll 0 mm ¼ 1 X n d 0 ð n t iþd l 0 m ð t 0 iþ: ð9þ ð10þ ð11þ ð1þ While alulation utilizing the full ovariane matrie are outlined in Appendix A, the alulation in thi paper are made with ertain aumption whih greatly implify the ovariane matrie. Firt, we aume d and d l 0m are 0 unorrelated for all value of l, l 0, m, m 0 ; equation (10) and (11) therefore vanih. Seond, we aume d and d l 0 m 0 are unorrelated, and d and dl 0 m0 are unorrelated, unle l = l 0 and m = m 0, o that equation (9) and (1) vanih in thoe ae. Lat, error are aumed to depend on patial ale but not orientation; that i, d = d and thee oeffiient depend on l but not m. With thee aumption, equation (7) beome where B l ¼ 1 n X n varðþ at ¼ X l;m X l K l B l ; ð13þ h d ð t þ i iþ d ð Þ : ð14þ t i

3 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG Spatial Averaging to Improve Auray [11] The auray of etimate of urfae ma anomalie an be improved by patial averaging, at the expene of patial reolution. An exat averaging kernel, J(q, f), i a funtion whih deribe the hape of the bain (e.g., a river bain, a of the oean floor, an ie heet, a politial boundary) aording to 0 outide the bain Jq; ð fþ ¼ 1 inide the bain ð15þ The hange in vertially integrated water torage averaged over an arbitrary i ¼ 1 Z ðq; fþjq; ð fþd; ð16þ Figure 1. Etimate of the quare root of the ontribution to the variane of the inferred urfae ma anomaly due to GRACE atellite meaurement error, a a funtion of pherial harmoni degree, uing equation (13) and value of B l from B. Thoma and M. Watkin (JPL) onitent with the GRACE SMRD. Beaue of our aumption onerning d and d,the ontribution of the atellite meaurement error to the variane of the urfae ma anomaly etimate (equation (13)) i independent of loation. B l are the degree amplitude of the variane of the atellite error and deribe the ontribution of the error at a partiular wavelength to the variane of the geoid. In thi paper we ue preliminary etimate of B l a a funtion of l provided by B. Thoma and M. Watkin at the Jet Propulion Laboratory (peronal ommuniation, 1996) that are onitent with thoe deribed in Jet Propulion Laboratory [001]. [9] Figure 1 how the quare root of var() at a a funtion of l. For value of l greater than about 5, the GRACE atellite error etimate inreae rapidly with inreaing l. In priniple, the ummation in equation () and (13) inlude ontribution from all wavelength, up to l = 1, and the atellite error will lead to extremely inaurate reult. Beaue GRACE will deliver C and S only to l 100, the um will in pratie be trunated at l trn = 100. Although negleting the ontribution of oeffiient with l > 100 will redue the amount of atellite error preent in the etimate of (q, f), the error from the larger remaining value of l will till eriouly degrade the olution. In addition, uing a trunated um in equation () i not equivalent to a point meaurement, (q, f) beaue it lak omponent having length ale le than 00 km. [10] Thee iue an be avoided by averaging (q, f) over a. Spatial averaging redue the ontribution from large l to the ummation in equation (), reduing the effet of both atellite error and mirepreentation of the gravity field due to the abene of hort-wavelength omponent. where d = in q dq df i an element of olid angle. Integrating J(q, f) over the phere give, the angular area of the of interet. Uing equation (), equation (16) an be reexpreed by a um of Stoke oeffiient a ¼ a r X 1 E 3 l¼0 X l ðl þ 1Þ ð1 þ k l Þ J C þ J S ; ð17þ where J and J are the pherial harmoni oeffiient deribing J(q, f): Jq; ð fþ ¼ 1 X 1 X l 4p l¼0 8 < : J J 9 = Z ; ¼ ~P ðo q Jq; ð f Þ J o mf þ J in mf ð18þ 8 9 < o mf = Þ~P ðo qþ d: ð19þ : ; in mf One effet of the bain oeffiient in equation (17) i to redue the ontribution to from the Stoke oeffiient for large l. For example, Figure how the degree amplitude petrum, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux l J l ¼ t J ; ð0þ þ J for di-haped bain having radii of 1000, 500, and 100 km. The petrum ha been normalized by dividing by J 0 for eah ae. A the bain ize inreae, it degree amplitude i onentrated at relatively maller l, orreponding to longer wavelength. Average over larger therefore are influened le by poorly known hortwavelength ignal than are maller. [1] Bain average alulated uing Stoke oeffiient provided by GRACE will differ from the true bain average due to the preene of atellite meaurement

4 ETG 3-4 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY error and the abene of oeffiient for l > 100. Failure to inlude all J and J reult in an inaurate repreentation of the bain hape. Figure 3 how ro etion of J reontruted from J and J for different value of l trn, the value of l at whih the ummation in equation (18) i trunated. The ringing preent in thee reontrution near the boundarie of the bain mak, alled the Gibb phenomenon, i due to the abene of bain oeffiient with l > l trn. Ringing inreae a l trn dereae, reulting in a bain average whih ample more of the outide the bain. [13] With thee error in mind, we write the bain average omputed uing Stoke oeffiient provided by GRACE a the um of the true bain average, atellite error, and trunation error: GRACE where ¼ true error þ atellite þ trunation error ; ð1þ GRACE ¼ Xltrn l¼0 X l K l J CGRACE þ J SGRACE ; ðþ atellite error Xltrn X l ¼ l¼0 K l J d þ J d ; ð3þ Figure. Degree amplitude of the exat averaging kernel for di-haped bain of (a) 1000 km radiu, (b) 500 km radiu, and () 100 km radiu. The maximum amplitude ha been normalized by 1/J 0. trunation error ¼ X1 X l l¼l trnþ1 K l : J Ctrue þ J Strue ð4þ Figure 3. Cro etion of reontruted bain mak for variou value of l trn.

5 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG Reduing Satellite Meaurement Error [14] Beaue GRACE will provide Stoke oeffiient only to degree and order 100, omponent of urfaema variability orreponding to l > 100 will be abent from the alulation of the bain average. The ontribution of high l Stoke oeffiient to large bain average may be unimportant, beaue, a hown in Figure, the bain oeffiient beome quite mall for high l, but a bain ize dereae their ontribution beome more ignifiant. In addition to error due to trunation, GRACE data will ontain atellite meaurement error. The implet way to redue thee error would be to trunate the um in equation (3) at ome l trn < 100. However, dereaing the value of l trn inreae equation (4), the amount of trunation error. We would like to find a method of dereaing the atellite error in the etimate of bainaveraged urfae ma hange without inreaing the error due to trunation. [15] The exat averaging kernel equation (15) hange diontinuouly from a value of 1 to 0 at the bain boundarie, reulting in the preene of hort-wavelength J and J in the expanion of J(q, f). Beaue the atellite error are greatet at large l, the ontribution from thee hort-wavelength oeffiient dominate equation (3), the bain-averaged atellite meaurement error. The expanion of an averaging kernel whih varie moothly aro the bain boundary ha le power in it hortwavelength oeffiient than doe the exat averaging kernel. However, the bain average omputed uing a moothed averaging kernel will le aurately repreent the true bain average. Thi approximate average will be influened by ma ignal outide the bain, referred to a leakage, a well a over- or under-etimating the ontribution of the ignal inide the bain. While the approximation error annot be eliminated, we will how that one an reate moothed averaging kernel whih produe a redution in atellite meaurement error while keeping the amount of leakage error in the bain-averaged etimate of urfae-ma hange to an aeptable level for bain having length ale of a few hundred kilometer and larger. We deribe two type of averaging kernel. The firt kind, deribed in etion 4.1, inorporate a Gauian filter, and i relatively imple to viualize and ompute. The eond kind, deribed in etion 4., i more ompliated but provide a mean of minimizing either the leakage error or the atellite meaurement error, or the um of the two. [16] An approximate bain average an be obtained by replaing the exat averaging kernel, J(q, f), by an approximate averaging kernel, W(q, f), in equation (16): f ¼ 1 Z ðq; fþw ðq; fþ d; ð5þ where f denote the approximate bain average. Expanding W a W ðq; fþ ¼ 1 X ltrn X l 4p l¼0 ~P ðo q Þ W o mf þ W in mf ; ð6þ the approximate bain average an be expreed in term of Stoke oeffiient a f ¼ X l;m K l W C þ W S : ð7þ When omputed uing thi approximate averaging kernel, the ontribution of atellite meaurement error to the variane of the average urfae ma anomaly beome varðe at Þ ¼ 1 X l;m Kl B l l þ 1 W þ W : ð8þ 4.1. Gauian Smoothing [17] A mooth averaging kernel, W (q, f), may be reated in a traightforward way by onvolving the bain funtion, J(q, f), with a Gauian filter: Z W ðq; fþ ¼ W ðq; f; q 0 ; f 0 ÞJq ð 0 ; f 0 Þ d 0 ; ð9þ where equation (9) i integrated over olid angle and, following Jekeli [1981], the Gauian filter, W(q, f, q 0, f 0 ), depend only on the angle g between two point (q, f) and (q 0, f 0 ), i.e., o g = o q o q 0 + in q in q 0 o(f f 0 ), Wðq; f; q 0 ; f 0 Þ ¼ W ðgþ ¼ b p lnðþ b ¼ 1 o r1 =a : exp½bð1 ogþš 1 e b ; ð30þ ð31þ r1 =a i the half width of the Gauian moothing funtion; when g = r1 =a, WðgÞ ¼1Wð0Þ. The new averaging kernel, W, hange moothly from a value of 1 inide the boundary to a value of 0 outide the boundary over a horizontal ditane of approximately r1. For thi type of averaging kernel, the weighting oeffiient in (6) are defined aording to W J W ¼ p W l J ; ð3þ where 1 W l ¼ pffiffiffiffiffiffiffiffiffiffiffiffi l þ 1 Z p 0 W ðgþ ~P l0 ðo gþin g dg: ð33þ W l may be omputed reurively by the following relation: W 0 ¼ 1 p ; W 1 ¼ 1 1 þ e b p 1 e b 1 ; ð34þ b W lþ1 ¼ l þ 1 W l þ W l1 b [18] Figure 4 how the reult of uing a Gauian filter to mooth the exat averaging kernel, in thi ae that of the Miouri river bain. Changing the half width of the Gauian filter allow one to ontrol the relative amount

6 ETG 3-6 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY Figure 4. Example of hanging the half width of the Gauian filter ued to mooth the bain mak. of atellite and leakage error. Small value of r1 =a lead to an approximate averaging kernel whih hange rapidly at it boundarie; the weighting oeffiient therefore ontain power at hort-wavelength and the error budget i dominated by atellite error. Large value of r1 =a redue the amplitude of the hort-wavelength weighting oeffiient, but inreae the amount of error due to leakage from urrounding the bain of interet. [19] Figure 4a how the exat averaging kernel. It ha a value of 1 inide the bain and 0 outide the bain. A previouly noted, the ringing at the boundarie i due to the finite upper limit (l trn = 100) in the ummation in equation (6) ued to alulate W(q, f). Figure 4b and 4 how the effet of moothing uing 00 and 400 km Gauian filter, repetively. At 00 km, the averaging kernel hange ontinuouly aro the bain edge, while at the ame time nearly reproduing the bain hape. When a 400 km filter i applied, the averaging kernel beome muh moother than that in Figure 4b and ample a larger portion of the urrounding. 4.. Optimizing the Averaging Kernel [0] While moothing the exat averaging kernel with a Gauian filter provide a imple and intuitive way of reating an averaging kernel with dereaed hort-wavelength omponent, it may not provide the mot aurate etimate of the bain average. We outline two minimization tehnique whih inorporate meaure of atellite and leakage error to reate an optimal averaging kernel. In the firt ae we minimize the um of the atellite and leakage error, while in the eond ae we fix one type of error to a peifi value and minimize the other type. [1] If J i the exat averaging kernel, then the total approximation error, or leakage, in our etimate of the ma anomaly of the bain at time t i i the differene between equation (16) and (5): Z eðt i Þ ¼ Wðq; fþ Jq; ð fþ q; ð f; ti Þ d: ð35þ The variane of the leakage i then varðe lkg Þ¼ 1 n Z ¼ X n ½eðt i ÞŠ Wðq; fþjq; ð fþ W ð q 0 ; f 0 ÞJq ð 0 ; f 0 Þ " # 1 X n q; ð f; t i Þq ð 0 ; f 0 ; t i Þ n d d 0 : ð36þ While the ontrution of an optimal averaging kernel an inorporate a ignal ovariane funtion having any angular dependene (deribed in Appendix A), the expreion for var(e lkg ) i greatly implified if we aume that there exit a orrelation whih depend only on the angular ditane, a, between (q, f) and (q 0, f 0 ), o that 1 X n n q; ð f; t i Þ q ð 0 ; f 0 ; t i Þ ¼ 0 GðaÞ; ð37þ where 0, the variane of the urfae ma ignal at any point, i aumed to be uniform over the bain and it urrounding, and G(a) i an appropriate funtion for deribing the orrelation (in etion 5, G(a) will be hoen to be a Gauian funtion of a). Replaing W and J with

7 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG 3-7 their repetive pherial harmoni expanion, the variane of the bain average leakage beome X 1 var e lkg ¼ 0 l¼0 where Z G l ¼ X l G l h W þ J W J i ; ð38þ GðaÞP l ðo aþ in a da: ð39þ Minimization of total error [] To find averaging kernel oeffiient whih minimize the total error, we et the partial derivative with repet to and W of the um of the error variane to ; varðe at Þþvar e lkg ¼ 0; ð40þ whih produe the following et of equation: W W ¼ 1 þ K l B 1 l J 0 G l ðl þ 1Þ J : ð41þ Equation (41) make it poible to find optimal averaging oeffiient for a peifi bain, from knowledge of the ignal variane ( 0 ), the ignal orrelation funtion (G l ) and the degree variane of the GRACE meaurement error (B l ) Lagrange multiplier method [3] While the previou method fulfill the goal of minimizing the total error, it require a priori etimate of both the amplitude and angular dependene of the ovariane funtion to do o. If thee etimate are unavailable or unreliable, it may deirable to reate an averaging kernel whih doe not depend on a foreknowledge of the ignal harateriti. An alternative definition of leakage, independent of the ignal, i the ratio of the patial variane of the differene between the exat and approximate averaging kernel to that of the exat averaging kernel, R W ðq; fþjq; ð fþ d varðe lkg Þ¼ R ½Jq; ð fþš d 1 X h ¼ W þ J W J i 4p l;m One may ue the method of Lagrange multiplier to reate an averaging kernel whih minimize thi leakage error ubjet to a ontraint on the value of atellite meaurement error. Let d be the deired variane of the average atellite meaurement error equation (8) and let = d. Denoting the Lagrange multiplier by l, we determine the value of W, W, and l that minimize the quantity x ¼ X n W þ J W J o l;m ( ) Kl B l þ l l þ 1 W þ W ; X l;m ð4þ ð43þ where we have aborbed the 4p in equation (4) into l. Setting the partial derivative of x with repet to W and W equal to zero give the et of equation W W ¼ 1 þ l K l B 1 l J l þ 1 J : ð44þ Setting the partial derivative of x with repet to l equal to zero return the requirement that the effet of the atellite meaurement error be equal to : X Kl B l l þ 1 W þ W ¼ ; ð45þ l;m whih an be ombined with (44) to give an equation for l X Kl B l J þ J h i l þ 1 l;m 1 þ l K B l ¼ : ð46þ l lþ1 One l i determined from equation (46), it an be ued in equation (44) to olve for W and W. In general, there are approximately l trn value of l whih are olution to equation (46). However, there exit only one olution whih i poitive, and it i thi root whih provide the true leakage minimum; the negative root are only loal extrema. If one inadvertently peifie a value for whih i greater than that obtained by uing the exat averaging kernel oeffiient, J and J in equation (8), then no poitive root of equation (46) exit. [4] To reate an averaging kernel whih fixe the leakage error (4) to a peifi value while minimizing the atellite error, one till ue equation (44). l, however, i determined from 3 X J þ J ¼ R 4p l;m 1 þ l K B l ; l lþ1 ð47þ where R i the deired leakage ratio defined in equation (4). 5. Gauian Smoothing Veru Minimization [5] To ae the probable auraie of the approximate averaging kernel, we firt examine the one-dimenional ae of a di-haped bain. To ompute the leakage error, one require an etimate of the length ale, d, of the patial orrelation of the urfae-ma hange ignal. One oure of urfae-ma variability i terretrial water torage. While urfae water, now water, and groundwater are all oure of water torage variability, Rodell and Famiglietti [001] determined that the larget omponent of variability in the Amerian Midwet i oil moiture. We aume that thi relationhip haraterize all of the bain examined in thi tudy. Studie uh a Entin et al. [000], Vinnikov et al. [1996], and Cayan and Georgakako [1995] have hown the patial oherene of oil moiture to have a length ale ranging from 00 to 800 km. While Entin et al. [000] employ a ovariane funtion whih deay exponentially

8 ETG 3-8 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY Figure 5. Leakage a a funtion of the peified RMS of the atellite meaurement error,, for a dihaped, 500 km radiu bain for different urfae-ma anomaly orrelation length, d. Dot denote the Lagrange multplier minimization method, and aterik denote the Gauian moothing method. (a) d = 10 km, (b) 100 km, () 00 km, and (d) 500 km. Dahed line repreent trunation error. with ditane, we have hoen to ue a Gauian ditribution (equation (30)) to deribe the orrelation between urfaema variability at different loation. Thi aumption i baed on the eae of manipulation of the Gauian funtion in the pherial harmoni domain. In thi ae, the oeffiient G l, of the patial ovariane funtion (equation (37)) are omputed in the ame manner a the moothing oeffiient W l, uing the reurion relation (34) where d, the patial orrelation length ale, replae r1. [6] To ompare the Gauian moothing proe to the minimization proe, we firt reate Gauian kernel for di-haped bain with filter of variou half width. For eah kernel, we ompute the atellite error uing equation (8). Next, we ue the Lagrange multiplier method to reate an optimal averaging kernel for that value of atellite error, and ompare the leakage ratio for the two type of averaging kernel. We define the leakage ratio, R lkg,athe quare root of the ratio of the variane of the error aued by Figure 6. Same a Figure 5, but for a di-haped bain having a radiu of 00 km.

9 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG 3-9 the approximate averaging kernel to the variane of the true bain-averaged ignal. R lkg i thu independent of the amplitude of the ovariane funtion. Beaue the leakage ontain the effet of both the approximate averaging kernel and the finite upper limit in the ummation in the expreion for the bain average, we plit the um in equation (4) at l trn. Dividing by the expreion for the orret urfae-ma variane give P ltrn P h l R lkg ¼ l¼0 G l W þ J W J i P 1 P l l¼0 G l J þ J ð48þ P 1 P l l¼l þ trnþ1 G l J þ J P 1 P l l¼0 G l J þ J : Figure 7. Cro etion of averaging kernel for (a) a 500 km radiu bain and (b) a 00 km radiu bain. The olid line repreent the true bain hape, and the dahed line repreent the bain hape omputed uing only bain oeffiient of l l trn. The urfae-ma anomaly orrelation length i d = 00 km. The averaging funtion are omputed o that the RMS ma error due to the atellite meaurement error i 1 m in eah ae. The firt term on the right-hand ide of equation (48) deribe the approximation error made by replaing the true bain funtion, J(q, f), by the approximate averaging kernel, W(q, f). In thi paper we aume GRACE will only upply Stoke oeffiient to degree l trn = 100. The eond term deribe the trunation error due to negleting term l > 100 when alulating the bain average. While the approximation error depend on the form of a partiular averaging kernel, the trunation error ha a ontant value fixed by the value of l trn and, through the G l, the value of d. The maximum value of atellite error for a given bain hape an be found by etting w and w to J and J in (8) when olving for varðe at Þ. The exat averaging kernel, J, ha an additional property: beaue it approximation error i identially zero, it leakage ratio i entirely due to trunation error. Figure 5 how the leakage ratio a a funtion of atellite error and patial orrelation ale of the ignal, d, for a 500 km radiu, di-haped bain. The level of trunation error an be een in Figure 5 a the value taken by R lkg a the atellite error approahe it maximum value (dahed line). From Figure 5a it an be een that when urfae-ma anomalie are orrelated to about 10 km, trunation error i about 30% of the ignal variane, and dominate R lkg for atellite error greater than about 3 mm. A the ignal orrelation length, d, inreae, the amplitude of the hort-wavelength omponent of the urfae-ma ignal dereae, and therefore the trunation error dereae. A the trunation error diminihe, the differene between the minimization method and the Gauian moothing method beome more apparent. For example, the leakage due to the Gauian method at 1 m atellite error and 00 km orrelation length i aot double that due to the minimization method. The effetivene of the minimization method in reduing leakage error i even more pronouned in the ae of a 00 km radiu di (Figure 6). [7] Figure 7 how the reontruted ro etion of the Gauian and optimal averaging kernel deigned to produe a atellite error of 1 m for two di-haped bain. Eah plot aume a orrelation length, d, of 00 km. Figure 7a how a 500 km di. In order to ahieve a 1 m atellite error, the Gauian averaging kernel ha a rolloff, r1, of 15 km. A viual omparion of the two averaging kernel onfirm Figure 5; the optimal kernel more loely reemble the trunated bain hape (dahed line) and therefore produe le leakage than the Gauian averaging kernel. The trunated bain hape lak the hort-wavelength omponent needed to reprodue the exat bain hape (olid line), but beaue the orrelation length i uffiiently large, the omponent of R lkg due to trunation error i 1. [8] Figure 7b how a 00 km di. The Gauian averaging kernel in thi ae employ a roll-off of 15 km to produe the required 1 m atellite error. Again, the trunated bain hape doe not well repreent the exat bain hape, but Figure 6 how the trunation error omponent to be <5%. However, the leakage i greater than for the 500 km di beaue neither averaging kernel loely reemble the trunated bain hape. At 1 m atellite error, the leakage ratio in Figure 6 i nearly 0.5. Thi large approximation error i due to the moothne in the averaging kernel required by the ondition that the atellite error mut be 1 m. 6. Comparion With Real Drainage Bain [9] While the alulation in the previou etion employ di-haped bain in order to better undertand

10 ETG 3-10 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY Figure 8. North Amerian river bain. the effetivene of eah of the averaging kernel, a real bain, whether it deribe an oeani, a polar ie tream, a river bain, or a politial boundary, i rarely haped o imply. For ix North Amerian river bain (Figure 8) of varying hape and ize, we alulate the average atellite and leakage error for eah bain uing the Gauian moothing method and the Lagrange multiplier method. The river bain mak were extrated from the HYDRO 1K Elevation Derivative Databae maintained by the U.S. Geologial Survey EROS Data Center. [30] For a bain average with 1 m atellite error, leakage ratio obtained uing a Gauian averaging kernel range from approximately 0.4 for the mallet bain (0,000 km in area) to 0.1 for the larget bain (1,400,000 km in area) for a orrelation length of 00 km (Figure 9). Leakage ratio for the minimization method are about half thi amount, ranging from 0. to Taking 4 m a a typial value for the RMS variation in midlatitude oil moiture [Entin et al., 000, Table 1], one would then expet leakage due to the ue of the Gauian averaging kernel to aount for about Figure 9. Leakage a a funtion of RMS atellite meaurement error for ix North Amerian river bain. d, the urfae-ma anomaly orrelation length, i 00 km. Dot denote the Lagrange multiplier minimization method and aterik denote the Gauian moothing method. (a) Interbain, (b) Red River, () Upper Miiippi River, (d) Arkana River, (e) Ohio River, and (f) Miouri River.

11 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG 3-11 Figure 10. Same a Figure 9, but for a urfae-ma anomaly orrelation length, d, of 500 km. 1.6 m of error for bain (Figure 9a) and 0.4 m of error for bain (Figure 9f ). Ue of the minimization method would redue the leakage to approximately 0.8 m for bain (Figure 9a) and 0. m for bain (Figure 9f ). If the orrelation length of oil moiture variability i 500 km (Figure 10), the leakage ratio for a bain with 1 m atellite error range from about 0. for the mallet bain to le than 0.05 for the larget bain when the Gauian moothing method i ued. Applying the optimal kernel reult in leakage error whih are negligible for even the mallet of thee bain. In thi ae, one ould et the atellite error to a low a 0. m without produing a ignifiant amount of leakage in the bain averaged etimate of urfae ma variability. 7. Summary [31] GRACE, heduled for launh in 00, will deliver monthly average of the Earth geoid, in the form of Stoke oeffiient. The finite number of Stoke oeffiient provided by GRACE prohibit the etimation of urfae ma anomalie at a point, but al average an be made. Beaue the expeted atellite meaurement error inreae rapidly for hort wavelength, an average alulated uing the exat repreentation of the bain hape, the petrum of whih ha power at all wavelength, i influened by all the error aoiated with thee gravity data. A mooth averaging kernel ontain le power at hort wavelength, and it ue in omputing bain average will lead to a maller atellite error. However, any approximate averaging kernel will no longer exatly ample the of interet, and ignal from urrounding area will leak into the etimate of the bain average. [3] We have outlined different method for reating approximate averaging kernel whih extrat the average urfae-ma anomaly of a deired from the GRACE geoid oeffiient. Eah of thee method redue the amount of atellite error preent in the bain average. One method ue a Gauian filter (30) to mooth the exat averaging kernel (equation (15)), providing a oneptually and omputationally imple mean of reating a new averaging kernel. In thi ae, W and W ued in equation (7), are given by equation (3) (34). However, tuning the width of the Gauian filter to hange the amount of atellite error in the average require a ertain

12 ETG 3-1 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY amount of trial-and-error. In addition, the leakage introdued by thi moothed averaging kernel may not be minimized. [33] Two minimization tehnique for finding W and W were outlined, one of whih reated an averaging kernel whih minimized the total error (equation (41)). The other tehnique ued a Lagrange multiplier (equation (44)). The Lagrange multiplier, l, either minimized the leakage for a given value of atellite error (equation (46)) or minimized the atellite error for a given leakage level (equation (47)). Both method, deribed in Appendix A, an inorporate the full ovariane matrie of atellite error and/or expeted ignal. Full atellite error ovariane matrie will be provided by GRACE Projet peronnel when the Stoke oeffiient are made available to uer. A priori ignal ovariane matrie may be obtainable from hydrologial model or other data et. However, both minimization tehnique implify oniderably under the aumption of azimuthal ymmetry. [34] The method for developing weighting oeffiient derived by minimizing the equation deribing the total error require an a priori deription of the expeted ignal ovariane. Beaue of the additional omplexitie of analyzing thee data, we will ae thi method in further tudie. Intead, we ued a Lagrange multiplier method to reate averaging kernel whih were optimized without the ue of an expeted ignal variability. The Lagrange multiplier minimization method i omewhat more ompliated than applying a Gauian filter, but it dereae the leakage error oniderably. Calulation for a few drainage bain in North Ameria having area ranging from to km, indiate that urfae-ma hange ignal, patially orrelated at length ale of about 00 km, an be retrieved uing an averaging kernel whih ha been moothed with a Gauian filter to an auray of 10 40% of the ignal amplitude plu <1 m atellite meaurement error. Uing the Lagrange multiplier minimization method redue the amount of leakage error by about half, o that ma anomalie an be omputed to an auray of 5 0% of the ignal amplitude plu <1 m atellite meaurement error. Surfaema hange ignal patially orrelated at longer length ale an be retrieved with aot no leakage error and atellite error of le than a few millimeter. Appendix A: Inverion With Arbitrary Covariane Funtion [35] While more ompliated than the ae of an azimuthally ymmetri ovariane funtion, it i poible to ue the full ignal and atellite error ovariane matrie to ontrut an optimized averaging kernel. Beginning with (36), W, J, and are replaed with their repetive pherial harmoni expanion. The variane of the leakage beome X X var e lkg ¼ l;m l 0 ;m 0 W J þ W J W þ W J W þ W J W W l 0 m 0 J l 0 m 0 l 0 m 0 J l 0 m 0 l 0 m 0 J l 0 m 0 l 0 m 0 J l 0 m 0 ll 0 mm 0 ll 0 mm 0 ll 0 mm 0 ll 0 mm0 ða1þ; where ll 0mm0 are the ovariane matrie of the ignal: ll 0 mm ¼ 1 X n 0 n ðt iþ l 0 m 0ðt iþ; et: ðaþ The variane of the bain-averaged atellite meaurement error i varðe at Þ ¼ X X K l K l 0 W W l 0 m 0 ll 0 mm þ W 0 W l 0 m 0 ll 0 mm 0 l;m l 0 ;m 0 þ W W l 0 m 0 ll 0 mm þ W 0 W l 0 m 0 ll 0 mm 0Š; ða3þ where ll0 mm are the ovariane matrie of the GRACE 0 meaurement error (ee equation (9) (1)). A1. Minimization of Total Error [36] Let ¼ Then J ¼ ll 0 mm 0 ll 0 mm 0 ll 0 mm 0 ll 0 mm 0 J J ; W ¼ W W ; ða4þ ; ¼ K lk l 0 ll 0 mm K lk 0 l 0 K l K l 0 ll 0 mm K lk 0 l 0 T varðe lkg Þ¼ J W J W varðe at Þ¼W T W: ll 0 mm 0 ll 0 mm 0 ða5þ ða6þ ða7þ To find the averaging kernel oeffiient whih minimize the total error, we et the gradient with repet to W of the um of the error variane to zero, r W whih produe h T T J W J W þ W W h i þ W J ¼ 0: i ¼ 0; ða8þ ða9þ Equation (A9) i a linear ytem, whih an be olved numerially for W. A. Lagrange Multiplier Method [37] A an alternative to equation (A6) we may define the leakage a the ratio of the patial variane of the differene between the exat and approximate averaging kernel to that of the exat averaging kernel, var e lkg R W ðq; fþjq; ð fþ d ¼ R ½Jq; ð fþš d ¼ 1 T J W J W : 4 p ða10þ Thi definition of leakage i free from the aumption regarding the form of the expeted ignal required by equation (A6). However, thoe aumption provided a mean of diretly omparing leakage error to atellite error. In the abene of uh a onnetion, one annot minimize

13 SWENSON AND WAHR: METHODS FOR GRACE SURFACE MASS RECOVERY ETG 3-13 the total error. Intead, one may ue a Lagrange multiplier tehnique to minimize either leakage error or atellite error, ubjet to a ontraint on the other type of error. Denoting the Lagrange multiplier by l, we determine the value of W and l that minimize the quantity T n o T x ¼ J W J W þ l W W ; ða11þ where i the deired variane of the atellite meaurement error, averaged over the bain. The 4p in equation (A10) ha been aborbed into l. Setting the gradient of x with repet to W equal to zero give h i I þ l W ¼ J; ða1þ where I i the identity matrix. Setting the partial derivative of x with repet to l equal to zero return the requirement that var(e at )=, whih an be ombined with to give an equation for l varðe at Þ¼J T h i 1 T h i 1 I þ l I þ l J; ða13þ One l i known, it an be ued with equation (A1) to olve for W. [38] Thi method an alo be applied to the onvere of thi problem: minimizing atellite error for a given value of leakage. In thi ae, equation (A1) till deribe the averaging kernel oeffiient. However, l i determined now by olving h i J T 1 T h i 1 I I þ l I I þ l J ¼ R 4p ; ða14þ where R i the deired ratio of the patial variane (equation (A10)). [39] Aknowledgment. We wih to thank Steven Jayne and Iabella Veliogna for their helpful diuion. With the aitane of Lynda Latowka, the river bain boundarie were extrated from the HYDRO 1K Elevation Derivative Databae maintained by the U.S. Geologial Survey EROS Data Center from their web ite at gtopo30/hydro/. Thi work wa partially upported by NASA grant NAG and NAG to the Univerity of Colorado, by a CIRES Innovative Reearh Grant, and a NASA Earth Sytem Siene Fellowhip awarded to Sean Swenon. Referene Cayan, D., and K. Georgakako, Hydrolimatology of ontinental waterhed, Water Reour. Re., 31(3), , Chao, B. F., and R. S. Gro, Change in the Earth rotation and lowdegree gravitational field indued by earthquake, Geophy. J. R. Atron. So., 91, , Entin, J., A. Robok, K. Vinnikov, S. Hollinger, S. Liu, and A. Namkhai, Temporal and patial ale of oberved oil moiture variation in the extratropi, J. Geophy. Re., 105(D9), 11,865 11,877, 000. Jekeli, C., Alternative method to mooth the Earth gravity field, Rep. 37, Dep. of Geod. Si. and Surv., Ohio State Univ., Columbu, Jet Propulion Laboratory, GRACE Siene and Miion Requirement Doument, GRACE 37-00, Rev. D, JPL Publ. D-1598, 001. Rodell, M., and J. Famiglietti, An analyi of terretrial water torage variation in Illinoi with impliation for GRACE, Water Reour. Re., 37, , 001. Swenon, S., and J. Wahr, Etimated effet of the vertial truture of atmopheri ma on the time-variable geoid, J. Geophy. Re., 107, /000JB00004, in pre, 00. Veliogna, I., J. Wahr, and H. van den Dool, Can urfae preure be ued to remove atmopheri ontribution from GRACE data with uffiient auray to reover hydrologial ignal?, J. Geophy. Re., 106(B8), 16,415 16,434, 001. Vinnikov, K., A. Robok, N. Sperankaya, and C. Shloer, Sale of temporal and patial variability of midlatitude oil moiture, J. Geophy. Re., 101(D3), , Wahr, J., M. Molenaar, and F. Bryan, Time variability of the Earth gravity field: Hydrologial and oeani effet and their poible detetion uing GRACE, J. Geophy. Re., 103(B1), 30,05 30,9, S. Swenon and J. Wahr, Department of Phyi and Cooperative Intitute for Reearh in Environmental Siene, Univerity of Colorado, Campu Box 390, Boulder, CO , USA. (weno@olorado.edu; wahr@lemond.colorado.edu)