Analysis of Gravity Signals and Gravity Potential Determination Using Order 8 Multiresolution Analysis Discrete Wavelets

Transcription

1 Aalysis of Gravity Sigals ad Gravity Potetial Deteriatio Usig Order 8 Multiresolutio Aalysis Discrete Wavelets KENNETH J. FRIESEN, NICK PANAGIOTACOPULOS Departet of Electrical Egieerig, Califoria State Uiversity at Log Beach; Log Beach,CA 984, USA ad WILLIAM L. SJOGREN Jet Propulsio Laboratory, Califoria Istitute of Techology; Pasadea,CA 9119, USA Abstract: Our study focused upo coparig the effectiveess of the discrete wavelets i extractig gravitatioal data fro orbital velocity sigals of satellites orbitig the oo versus extractig this data by the ethod of least squares ad polyoial approxiatio. The results of our study show that the ultiresolutioal aalysis(mra) of order 8 discrete wavelets shows proise as a alterate filterig process i covertig orbital velocity sigals to gravitatioal data for ay plaet uder cosideratio. As a filter the discrete wavelets have several properties that ake the a desirable choice: the property of perfect recostructio of sigals ad high coputatioal speed. Key Words: wavelet, ultiresolutioal aalysis (MRA), oo gravity field, potetial. 1 Itroductio Oe of the iportat products of NASA s Luar Prospector issio to the oo was easureet of the lie-of-sight(los) velocity obtaied fro Doppler radio sigals set by orbitig satellites. These velocity sigals were recorded by earth based trackig statios at Goldstoe (Califoria), Madrid(Spai), ad Caberra(Australia) This velocity cotais iforatio ecessary to ake a deteriatio of the gravitatioal field of the oo. The curret procedure of processig these sigals is first to reove all velocity copoets fro kow sources such as the velocity due to earth's rotatio the velocity of the space craft due to orbital aeuvers, the cetrifugal velocity of the space craft due to the oo's cetral gravity, orbital otio of the oo ad all the velocity cotributios that do ot have sigificace. This is followed by a least squares fit estiatig the space craft positio ad velocity as well as ay paraeters of the gravity field leavig residual velocity that still has systeatic sigatures. The residual velocity is the approxiated by cubic splies which reoves the rado oise preservig oly the systeatic sigatures. The splie fit ca the be differetiated producig acceleratio profiles which represet the gravitatioal effects fro luar oblateess ad fro local surface coditios producig aoalies i the gravitatioal field. The goal of our research was to ivestigate a ew approach i processig the residual velocity usig order 8 MRA discrete wavelets to see if the results would be of higher resolutio tha those of the curret ethod. The approach is producig a higher resolutio acceleratio profile of the oo s gravitatioal field. We begi with a descriptio of the fudaetals of geopotetial ad wavelet theory followed by descriptios of both the curret ethod ad the use of wavelets i covertig residual velocity to acceleratio. 2 Fudaetals Fudaetal to our presetatio are the gravitatio odel ad the essetials of wavelet theory. 2.1 Gravity Potetial The gravitatioal potetial fuctio for all plaetary bodies is P( r, θ, φ) = GM / r[ = 1 ( C cos( θ ) + S where r is the radius fro the plaet ceter, a p is the surface radius, θ is the logitude, ϕ is the latitude, G is the Uiversal Gravitatioal Costat, M is the plaet ass, C ad S are the spherical haroic coefficiets, ad P is the associated Legedre polyoial of degree order. By restrictig the liit of suatio of the degree idex to 7, this =. ( ap / r)... si( θ )) P (siϕ)]

2 fuctio is said to be the potetial fuctio of degree 7 ad order 7. Sice a gravity surface of the oo of degree 7, order 7 had bee produced prior to the Luar Prospector Missio the acceleratios produced fro processig the residual velocity are oly refieets of the existig 7 th degree, 7 th order gravity odel. I the cetral regio of the oo: 3 W to 3 E logitude by 3 S to 3 N latitude the predoiat ter i the odel for gravitatioal effects due to oblateess ad surface aoalies is the haroic acceleratio which is the radial copoet of gravitatioal acceleratio ius the ter defiig the plaet's cetrifugal gravity. The haroic acceleratio is the partial derivative of the potetial fuctio with respect to r ius the cetrifugal ter GM/r 2 A r(7,7) = 1 ( C 2 = GM / r [ 7 = cos( θ ) + S ( + 1)[( a The first suad o the right had side is the copoet of zoal haroics kow as the "J" ter, ad the secod suad o the right had side is the copoet of tesseral haroics ad is kow as the " haroics" ter. I the cetral regio refieets are produced by additio of the acceleratios derived fro the residual velocity to the haroic acceleratio A r (refied) = A r(7,7) + a where a is either the CS acceleratio or the wavelet acceleratio. 2.2 Wavelet Theory Discrete wavelets are fuctios that for a basis for the fuctioal space L 2 (R), the space of all fuctios for which the itegral of the square of its absolute value take over the tie iterval (-,). The wavelet basis for L 2 (R) actually is a dual basis cosistig of a pair of basis widow fuctios kow as the scalig fuctio ad the wavelet fuctio respectively. As widow fuctios these fuctios are defied oly o copact itervals; each basis ca oly represet the portio of a fuctio that is defied i the widow. To cover the iterval (-,), the widow ust be traslated by a itegral idex k This fuctioal pair partitios the space L 2 (R) ito a decopositio that has a special algebraic structure, which is kow as a ultiresolutial aalysis decopositio (MRA) of L 2 (R). Both the scalig fuctio ϕ(t), ad the wavelet fuctio ψ(t), geerate a sequece of subspaces, {V } by the scalig fuctio ad {W }, by the wavelet p / r) C si( θ )) P P (siφ))] (si( φ))... fuctio whose uio geerates L 2 (R) as. Both fuctios have the special atheatical property that the bases fuctio for each idexed subspace is the scaled versio of that fuctio scaled by the factor 2, V is geerated by ϕ(2 t) ad W is geerated by ψ(2 t). The two sequeces of subspaces differ however i their iteral structures: The sequece {V } is utually iclusive whereas the sequece {W } is utually orthogoal. Ay sigal f(t) L 2 (R) has two series expasios relative to the pair of wavelet bases. The first is a series expasio relative to the scalig fuctio bases j f ( t) hj, kφ(2 t k) = j= k= The secod series expasio is relative to the wavelet bases j f ( t) dj, kψ (2 t k) = j= k= where the coefficiets i the wavelet series {d j,k } are the called the discrete wavelet trasfor of f(t) relative to the wavelet basis ψ(2 j t-k). The wavelet series expasio acts as a filter o the sigal; segetig the sigal ito its copoets i disjoit frequecy bads as well ito local values cotaied withi the widow iterval for ψ(2 j t-k), [t-k/2 j, t+ k/2 j ]. Geeratio of a wavelet series expasio for the sigal is the decopositio phase of the wavelet filterig operatio. Each scaled wavelet fuctio subdivides the spectru [,π] ito frequecy bads decreasig by iverse powers of 2, ( (π/2, π], (π/4, π/2], ). ]. This property of the wavelet expasio coes fro the orthogoality of the subspaces {W }. For fiite sigals, the rage of this series expasio i the scalig idex is N, where N is the expoet of 2 that equals or exceeds the uber of data poits ad the rage of each traslatio idex is 2 j, for j <= N. The segetatio of the frequecy spectru is fro (,π/2 N ] to (π/2,π]. Geeratio of a scalig fuctio series expasio is the recostructio phase of the wavelet filterig operatio sice this operatio cobies the disjoit segets of the sigal i the wavelet series expasio ito a series expasio where the bases fuctios are ot utually orthogoal.

3 2.2.2 Orbits The orbits coprisig the coplete data base that we used traversed the ear side of the oo fro the luar orth pole to the luar south pole i a ear perpedicular plae to the luar equator; extedig fro -16 W to 16 E ( a logitudial drift of oly few degrees characterized each orbit as it traversed the face of the oo). With few exceptios the logitudial offset of each of these orbital tracks was approxiately 1 degree at the luar equator. Each poit i the track was defied by its altitude above the surface ad by its logitude ad latitude. Thus as each track passed over its respective strip of the oo s surface, it registered velocity iduced by the gravity at that particular poit. 3 Proble Stateet The residual velocity cotais high frequecy oise, origiatig fro easureet ad sigal trasissio that corrupts deteriatio of the uderlyig velocity profile. The oise is reoved by cubic splie approxiatio. The oe proble i this ethod is that o polyoial approxiatio procedure ca extract a velocity fuctio fro discrete velocity values without approxiatio errors; however, as the order of approxiatio icreases, the errors becoe extreely sall. The precise deteriatio of local topographical aoalies requires that all short-tie perturbatios i the velocity profile be retaied i the approxiatio process regardless of how sall these perturbatios ay be. These trasiet features appear as short duratio sharp peaks or dips of relatively sall agitude. We preset a coparative descriptio of the curret procedure ad of wavelet filterig i the productio of the acceleratio profile fro the iitial LOS velocity data. Velocity (/sec) Fig. 1 Residual Velocity 3.1 Curret Method of Covertig Residual Velocity to Acceleratio The LOS velocity is coposed of the followig copoets, each fro a distict source of acceleratio: (1) Velocity fro Newtoia poit-ass acceleratio. relative to the oo's ceter. (2) Velocity due to oblate acceleratios fro the earth, su ad earby plaets (3) Velocity due to acceleratio fro radiatio pressure, spacecraft cotrol operatios, ad gas leaks. (4) Velocity due to acceleratio fro geeral relativity. () Velocity due to acceleratio fro the oblateess of the oo. The first four of these copoets is reoved fro the LOS velocity by coputatio producig the residual velocity. The residual velocity cotais systeatic sigatures ad high frequecy rado oise. To produce the required acceleratio iduced by the gravitatioal field of the oo it is ecessary to reove all rado oise. This is accoplished by by a cubic splie fit. The resultig cubic splie (CS) velocity is a cotiuous oise free velocity profile fro which the requisite acceleratio ca be coputed by first order differetatio. (See Fig 2 for CS velocity ad Fig 4. for CS acceleratio). The output of this procedure is a table for each orbital track cotaiig the followig data ites: (1)tie that sigal was trasitted fro the spacecraft, (2)the residual velocity, (3)spacecraft positio (altitude, logitude,latitude), (4)CS velocity, ()CS acceleratio ad, (6) haroic acceleratio. These tables for the orbital tracks of the Luar Prospector provided us with the database for our research. 3.2 Covertig Residual Velocity to Acceleratio Usig Wavelets Wavelet filterig replaces least squares fit ad cubic splie approxiatio i reovig rado oise fro the residual velocity. Noise reoval is accoplished by wavelet decopositio ad partial recostructio producig a oise free velocity profile fro which a acceleratio profile ca be produced by differetatio Sigal Decopositio ad Recostructio We perfored a coplete order 8 MRA decopositio of the residual velocity fro each orbital track. Sice each track cotaied 7-8 data poits a order 8 wavelet filter produces 8 decopositio levels ( level 2 was the lowest decopositio level ad level 1 was

4 the highest decopositio level). We the recostructed the sigals first to level, level 6, ad level 7 respectively. Exaiatio of radoly selected saples of levels 8 through 1 recostructed sigals revealed i each case that these sigals cotaied too uch of the high frequecy oise to be acceptable cadidates for coputig acceleratio. We ade careful coparisos of the level ad level 6 recostructed sigals to a selected saple of CS velocities. If a particular level of recostructed velocity was to be accepted for a particular orbit it ust be i good correspodece to the CS velocity over the relatively sooth regios, deviatig oly fro the CS velocity by sall perturbatios of sharp peaks or dips. I this effort we foud that level recostructios were overly sooth ad did ot show good correspodece. Level 6 recostructios showed very good correspodece to the selected saple of CS velocities over all sooth areas, deviatig oly by sall perturbatios. Level 7 recostructios copared as favorably to soe of the selected saples but ot i every case. Therefore, we chose to copute our wavelet based acceleratios fro level 6 recostructios i all the cases. 3.3 Acceleratio Sice our level 6 recostructed residual velocities existed as discrete data sets rather tha as a cotiuous velocity, the oly way i which we could covert these sigals to acceleratio was to use first order differeces divided by the iterveig tie iterval. We faced the iitial proble of excessive uerical error if we siply divided the sequetial velocity poits by the iterveig tie iterval i secods. The agitude of the velocity sigals were i the rage of.1 ad 1 /sec while the average tie iterval is. secods. Sice discrete wavelets coverge to a cotiuous wavelet fuctio we ca justify usig the discrete first order approxiatio to the actual derivative as a eas of coputig the derivative if the tie iterval is sufficietly sall. To overcoe the proble of covergece we chose to scale the tie iterval i iutes rather tha i secods (. secods.8i). Thus we coputed the acceleratio i /(i 2 ) iitially ad the coverted this to /(sec 2 ) by dividig by 6. To validate this ethod, we applied it to the CS velocity ad copared the results to the CS acceleratio; obtaiig perfect agreeet i each case. 4 Results We ade a extesive aalysis of both the resultat velocity ad the coputed acceleratio to deterie the validity of our iitial hypothesis. 4.1 Velocity Aalysis I coparig the processed residual velocities ad the correspodig acceleratios we selected a saple orbit: The orbit traced a vertical path crossig the luar equator at 2 degrees west logitude. Figure 2 cotais a plot of the CS velocity, ad Figure 3 cotais a plot of the MRA level 6 (L6) recostructed velocity. Coparig the two figures oe sees oticeable differeces i the rage of 6 deg to 2 deg latitude where the L6 velocity is ore defiitive. The lobes at deg ad deg latitude i both figures is early idetical; however, the lobes i the level 6 velocity have ore slightly jagged edges due to sall perturbatios. As we said i the precedig sectios the sectios the recostructed velocity ust correspod to the CS velocity i all sooth regios; differig oly i sall itervals where the recostructed velocity shows deviatios ore clearly. Velocity (/sec) Velocity (/sec) Fig. 2 Cubic Splie Velocity Latitude(degrees) Fig. 3 L6 Velocity

5 4.2 Acceleratio Aalysis Figures 4 ad cotai plots of the CS acceleratio ad the wavelet L6 acceleratio. The uit of acceleratio used by NASA is the illigal(gal), 1 illigal = 1-2 /sec 2. The differeces betwee the two velocities shows up as ore proouced differeces i the two acceleratios. I the rage of 6 deg to 2 deg latitude where the L6 velocity showed grater defiitiveess, the L 6 acceleratio has oticeably grater agitude ad sharper oscillatios tha does the CS acceleratio. I the reaiig regios the two acceleratios are of siilar agitude but the L6 acceleratio shows sharper oscillatios which is due to the ay ore sall perturbatios of the L6 velocity. Accleratio (gals) refied radial acceleratio produced fro the L6 acceleratio ad the CS acceleratio respectively. Our illustrative orbit track (2 W) passes through two proiet craters: Ptoleaeus (2 W,8 S) ad Alphosus(3 W,14 S); ad the two refied acceleratios should show local differeces i ajor craters. Figs 6 ad 7 are gravity surfaces i the regio 1 W to logitude ad S to S latitude for the CS refied acceleratio ad the wavelet L6 refied acceleratio. Copariso of the two acceleratio profiles shows that the L6 acceleratio has a steeper gradiet reachig a greater local axiu ad a lesser local iiu i the regios of these two craters tha does the CS acceleratio. I the regio of the crater Ptoleaeus, the L6 refied acceleratio has two low cotour levels: gal ad 4gal; whereas the CS refied acceleratio shows o cotour lies at all i this regio. I the regio of Alphosus, the L6 refied acceleratio shows a elogated cotour of 4gal cotaiig a depressio at gal. The CS refied acceleratio shows a siilar elogated cotour oly at 4 gal that is actually west of the locale of Alpohosus, ad the eclosed area is saller. This is evidece of the lower level refied acceleratios produced by the wavelet L6 acceleratio Fig.4 Cubic Splie Acceleratio Fig. 6 Gravity Map: CS Refied Fig. 6 Gravity Map CS Acceleratio Accleratio (gals) Logitude (degrees) Fig. L6 Acceleratio Most iportat i acceleratio aalysis is the copariso betwee the refieets to the radial acceleratio of the 7 th degree, 7 th order produced fro the CS acceleratio ad the L6 acceleratio. Figures 6 ad 7 cotai acceleratio surfaces of the Fig 6. Gravity Map CS Acceleratio

6 evidece of this. We had to cocetrate our velocity ad acceleratio aalysis o a sigle orbit track, but the differeces betwee the L6 velocity ad the CS velocity ad the resultig two acceleratios are typical for all reaiig orbits; i soe cases they are slightly less proouced i other cases they are slightly ore proouced. Exaiatio of this ad other cases shows that the use of MRA order 8 discrete wavelets does produce higher resolutio residual velocity ad as a cosequece higher resolutio acceleratio Fig. 7 Gravity Map L6 Acceleratio Logitude (degrees) Fig. 7 Gravity Map L6 Acceleratio Coclusio Careful exaiatio of Fig's 1, 2, ad 3 will show that the L6 velocity better preserves ay of the sigatures of the residual velocity tha does the CS velocity. Due to the higher resolutio of the L6 velocity, the L6 acceleratio has greater agitude ad frequecy. Whe added to the haroic acceleratio the L6 acceleratio produces a local gravity ap that reveals a steeper gradiet i regios with sigificat surface aoalies: the two sigificat acceleratio troughs at the sites of the craters Ptoleaeus ad Alphosus give Refereces [1] Matheatical Forulatio of the Double-Precisio Orbit Deteriatio Progra, Techical Report ; Theodore J. Meyer; Jet Propulsio Laboratory, Pasadea, CA; [2] Iproved Gravity Field of the Moo fro Luar Prospector; A.S. Koopliv, et all: Sciece, vol 281, Sept [3] P.M. Mueller ad W.L. Sjogre; Sciece, 161, 68, Ackowledgeets We would like to thak Dr. Alexader Koopliv, Jet Prolusio Laboratory, for providig the data ad for techical support ad Prof Courtey Colea Harvey Mudd College, for suggestios i atheatical odel developet