SIMULATION AND ANALYSIS OF THE GEM GRADIOMETRIC MISSION DATA SIMULAZIONE E ANALISI DEI DATI DELLA MISSIONE GRADIOMETRICA GEM

Transcription

1 A. Albertella, F. Migliaccio, F. Sansó DIIAR, Politecnico di Milano SIMULATION AND ANALYSIS OF THE GEM GRADIOMETRIC MISSION DATA ABSTRACT A new satellite gradiometric mission is under study in these days at thè European Sj Agency. This mission is considered as a follow on of thè abandoned Aristoteles mission, after parenthesis of thè study on thè possibility of exploiting a fundamental physics mission, STEP, for geodetic purposes; thè name of thè new mission is GEM, Gravity Explorer Missi In this paper we present a simulation of thè data of this mission and their analysis by m< of thè least squares method. SIMULAZIONE E ANALISI DEI DATI DELLA MISSIONE GRADIOMETRICA GEM RIASSUNTO Attualmente una nuova missione di gradiometria spaziale è allo studio presso l'age Spaziale Europea. Questa missione è considerata come il proseguimento dell'ormai abbandonata missione f toteles, dopo una parentesi di studio sulla possibilità di sfruttare una missione di fisica foi mentale, come STEP, per scopi geodetici; il nome della nuova missione è GEM, Gravity Expl Mission. In questo lavoro presentiamo una simulazione dei dati di tale missione e la loro analisi mezzo del metodo dei minimi quadrati. 1. INTRODUCTION Corning after a period of intensive researches on thè improvement of thè sensitivity of spaceborne gradiometer, thè new proposed accuracy for thè GEM mission is between IO"1 and 10-I35~2 (namely 10~2 EU and 10~4 EU) with thè value of l(t12s-2(10-3 EU) consid as realistically achievable. The design of thè instrument is not yet settled although thè solution of measuring th (radiai), xx (along track) and yy (out of piane) components seems to be prefèrred. We can observe that, if thè satellite rotations are perfectly reconstructed, thè harmon 181

2 condition AT = TIX + Tyy + Tzz = O shows that, at least as afirst step, TIZ + Tyy can be taker a further observation of Tzz, which by thè way is thè most important component of thè Man tensor; this is why for thè moment in our study we bave concentrated thè attention on reconstitution of a global geopotential model, T = ETim(*Y+*Ytm(<f>,\), from a simulatici thè GEM mission where only Tzz data have been considered. The reconstruction of thè anomalous potentini T can be done in severa! ways, one of wh is thè so-called spacewise approach basically consìsting in forming from true observation regular geographic grid of Tzz data (or alternatively of block averages) and then estimating coefficients Tim of our model, typically truncated so that.z: c _ i'mox i * _ /(t Previous simulation work has shown that in doing so an interpolation with polynomials of degree in 9, A can provide thè estimated value at thè center of ao.5xo.5 block with a practic negligible interpolation error (at thè level of 10~16s~2 or better); also, thè measurement n propagation follows very closely thè law rms = ffa (n= number of observations per block). This remark is very important because it has allowed to perforai thè heavy estimation w in a simpler way, namely directly estimating thè observables Tzz(Pjk) at thè centres Pjk regular grid with thè corresponding noise properly scaled at thè same points. 2. PREPARATION OF INPUT DATA The following parameters have been used to specify our simulation work satellite altitude inclination (7) mission duration measurement rate measurement noise for thè individuai observation (a0) 290 km months 0.33 Hz (every 3 s) 10~12s~2 This provides a total number of observations within thè mission of NT = evenly distributed among 0.5 X 0.5 blocks on thè sphere with a law NB = NT cos (f AV> AA 2*2 /dn21 - sin2? where B = number of observations per block = latitude of thè block centre 182

3 The total number of blocks considered is then , ranging over 334 values of latitude 720 values of longitude; thè observation simulated at thè centre of each block has thè analyt: expression gì i-r iso / Ta(Pik) = ^-j-(p>*)=e E Or t=\2m=-t <*imtlmylm(v3,\) <x(m = ^ +!)( + 2) (* r = R km, and thè noises associated with T z(pjk) is taken, in agreement with a previous remark, fro: normal population with zero average and The observations (data and noise separately) nave been generated by exploiting an I algorithm along parallels; yet thè computation has required ~ 180 hours of CPU on an HF machine. Let us observe here that in (2) thè Ytm functions are thè fully normalized spherical harmo and they refer to cos mx when m > O, and to sin \rn\\n m < 0. The model used to gene data has been OSU91A, from degree 13 up to 180, with ali orders according to formula (2) So finally thè observation equations can be written as t=l3m=-t where Ttm have to be recovered from T z(pjk), i.e unknowns from observatic 3. THE BLOCK-DIAGONAL L.S. ADJUSTMENT: FIRST RESULTS In thè described situation, no simple solution can be found like thè discretization of ort. onality relations. In fact, in spite of thè simplicity of thè observation operator considered. 32 ^-j, which is diagonal over thè solid spherical harmonics basis, thè fact that data do not e thè whole sphere, at satellite's altitude, prevents us from using such simple relations. Two options are then possible to solve (3) for Ttm, either by thè theory of bi-orthogonal si or by least squares. The two approaches have been proved to differ only for thè weight m; used, when thè number of unknowns is truncated, as in (3) (cf. [2]); here we shall use thè 1 squares approach. As it is known thè regular gridding óf data on thè sphere allows for a particular shape o: normal matrix. In fact by using well-known properties of discrete Fourier transforms one J that 183

4 720 E- fc=0 Now, noticing that, as (1) depends on <p only and is independent from k, thè normal system for a l.s. solution of (3) can be written as E- jk By exploiting (4) in (6) we find that it reduces to Tsm where we realize that for each fixed m we have a separate normal system capable of determi ali Tsm. At this point a comment is in order; in (7) thè index s has to satisfy contempor two lower bounds, namely it has to be ^ > 13, because thè coefficients Tsm(s < 13) are assu to be known and their effect removed from thè data; on thè same time it has to be s '. because there are no spherical harmonics Ym with m > s. The upper bound for 5 in ( indeed 180 in this work, unless differently stated, so (7) corresponds to a normal systen each m with 181 m unknowns (when m > 13), or 178 unknowns at most, thè solution of \ is particularly simple. Numerically, thè experiment has been conducted by a doublé adjustment; thè first one ta as known quantities thè true values Tzz(Pjk], thè second one taking as observations thè ol vational noises jk only. In this way we are able to test first of ali thè internai consistency o software, as well as to appreciate thè magnitude of thè purely numerical error, always pn in our computations. An index that represents thè precision of thè solution per degree is where Ttm = true OSU91A coefficients ST(m = T(m Tgm = true estimation errors. So (8) represents thè quadratic relative error per degree and in our case it is plotti Fig.l.A, while its root ((.) is plotted in Fig.l.B. 184

5 O.OOE+00 co o r- T- CM CM to T-comcMoiiocoor-.-* T-eomcMOtocoo «-T-CMtOCOTVOCO DEGREE Fig.l.A Quadratic relative error per degree 2( ) - cfr. (8) degree Fig.l.B Relative error per degree (t) As one can see thè maximum relative error is about 0.5% at degree 180; this result has 1 considered as fully satisfactory by thè authors. Moreover from thè solution with pure w noise as known terna we recover rim coefecients which represent thè propagation of thè ac observational noise to our solution. To evaluate thè performance of our approach against noise one could use thè signal to noise ratio, per degree, defined as \ t 7^2 m=-l Lt -2 = I tm tlùs quantity is displayed in Fig.2.A and 2.B with a solid line. 185

6 m=0-18c m=5-18c DEGREE Fig.2.A Signal to noise ratio per degree - cfr.(9) \-' V1,.. V'\ V \.,/,..,V.vv'W-'V" m=0-1; m=5-1i 1 4 o DEGREE Fig.2.B Same as Fig.2.A: centrai part amplified As one can see, roughly from degree 95 to 140 thè Rd( ) index is waving around 1 or below, indicating a strong degradation of thè information; this result has been considere very bad by thè authors. However, in thè effort of understanding this bad performanci became aware that ali thè problems were caused by only very few coefficients correspondir thè lowermost orders, below m = 4. Repeating thè same computations per degree, skippin thè coefficients with m < 4, we obtain thè dashed lines in Fig. 2.A and 2.B. As one can thè signal to noise ratio is now larger than 2 in thè worst cases, what has been considere satisfactory. To get a more precise picture of thè situation we have decided to compute thè signal to i ratio by order instead of by degrees. So we have computed thè index 186

7 R (m} = L T2 <=A ~tm where h is given by h = min[m, 13]. The result is shown in Fig.S.A and Fig.S.B, where thè first ordeders are more clearly re sented. As one can see as a matter of fact ali thè problems are concentrated in thè very low or and quickly tend to disappear over order 6. But what is happening with thè low orders, that degrades thè solution? In an attemp answer this question we bave looked into thè aliasing problem. Order 80 -r Fig.S.A Signal to noise ratio per order - cfr. (10) " m > O - m < O O Fig.S.B Same as Fig.S.A, amplified at low orders. 187

8 4. THE ALIASING PROBLEM Aliasing is a phenomenon of transfer of power form higher frequencies into lower frequt coefficients, related to thè spacing of data which does not allow to obtain information at quencies higher than thè Nyquist frequency. On thè sphere thè aliasing can be evidencec retrieving thè harmonic coefficients by a l.s. analysis in a model incorrectly representing signal, particularly in a model with a maximum degree which is lower than thè one conta: in thè signal; in this way one can see how much of thè non-modelled signal at high degret folded into thè lower degrees, due to thè particular design of thè network where data are gì In our case we have taken thè sphere generated with thè full model (3) up to degree 180 we have retrieved thè coefficients by solving a norma! System as in (7) with maximum degree to see thè effect of thè degrees between 161 and 180 incorrectly suppressed from thè observa equations. If we cali fem, m < i, i = 13, thè 1. s. solution so obtained, an aliasing ii can be computed per degree defined as A\ E t npì m.-l-ll representing thè mean power ratio of thè output over thè input per degree. The result is si in Fig.4 and it proves that thè aliasing effect is violent and spread over a large span of deg J$$ ii li! 'l'i"»' 1 f.j' I, i! i i!.» il 1 «i < 1 ' * i!" Ili i IIMi Jllllll fti ì!ì'k X-^" 'H%*^_ 0 linihininnhihinimtllmiiihiintltllhthiiiitlimimmihmhiitlhttmthhihmitthhnilhiilttnilthmithtth^mmlllllinilllllllttlhmmiiitlhh Nmax=180 Nmax=160 DEGREE Fig.4 Ratio output power/input power per degree - cfr (12) Again a closer inspection shows that ali thè problems come from very low orders, s decided to compute aliasing indexes by orders rather than by degrees. In Fig.S.A two ine are diplayed, computed by formulas E 160 rp2 l=h.-ll rp \ -Le.) t=h. Llm 188

9 where h is thè minimum between m and 13. Fig.S.B shows a zoom of Fig. 5.A at low ord As we can see by crossing thè plots 5.A, 5.B, ali thè damage practically comes from orde: and 2 and degrees above 50. This behaviour can be understood if we refer to thè pattern of Legendre functions of low orders and high degrees, particularly to PK, which are thè simp! As a matter of fact it is easy to see that PIO are very close to zero away from polar caps, as s as t increases; correspondingly, in a least squares analysis ali thè errors are so to say takei these harmonics and swept to thè polar caps, where there are no data to be interpolated ; through that, used to control these coefecients. 9 T O relative differences power out/power in ORDER Fig.S.A Ratio output power/input power and relative differences per order - cfr.(13),(14) relative differences power out/power in Fig.S.B Sanie as Fig.S.A., amplified at low orders This can be considered as thè origin of thè large errors in T(0 and Tt2- This result already that we can reconstruct most of thè coefficients of thè anomalous potential at a very satisfai level of accuracy. Yet thè question is stili open; can we do anything to obtain a better estim; thè coefficients Tto, T<2? In our opinion there are two possible answers to this question; thè fi 189

10 to abandon thè l.s. approach and try a specific forni of thè biorthogonality concept; thè seco; even simpler, is to take advantage of thè large amount of data that for geometrical reasons located in thè immediate neighbourhood of thè cap, to use a locai prediction inside thè cap i finally to use simple orthogonality relations to detennine ali thè coefficients. A rough theoreti computation shows that thè error committed in this case, or even taking Tzz O on thè caps in any way smaller than thè aliasing error coming from thè l.s. procedure. This point howe will be thè object of a specific analysis to be performed in future. 5. DISCUSSION The feasibility of thè GEM mission has been discussed and in particular it has been shc with an almost realistic simulation that most coefficients up to degree 180 can be retrieved f: a 6 months mission with a good accuracy. Some coefficients however cannot be safely retrie because of a strong aliasing effect. This subject has stili to be worked out at a theoret level. Yet thè potentiality of a mission like GEM has been clearly demonstrated and thè re: shows that, even using one observable only, interesting results can be derived. It has te expected that a slightly lower level of accuracy could be compensated by fully exploiting ali observables given by thè instrument; here it has not been taken into account thè contribuì that can be provided by adding thè gradiometer observations, that is thè non-gravitatk accelerations of thè satellite which, combined with GPS measurements, give a very effec information particularly on thè low frequency part of thè gravity spectrum. References [1] Brovelli M.A., Migliaccio F. (1993). The direct estimation of potential coefficiente by orthogonal sequences. In: Lecture Notes in Earth Sciences N. 50 "Satellite altimetr geodesy and oceanography", eds. R. Kummel, F. Sansó, Springer Verlag. [2] Migliaccio F., Sansó F. (1989). Data processing for thè Aristoteles Mission, Proceeding thè Italian Workshop on thè European Solid Earth Mission Aristoteles, Trevi (Italy), i 19S9. [3] Kummel R., Colombo O. (1985). Gravity field determination from satellite gradiomt Bulletin Geodesique N. 59. [4] Kummel R., van Gelderen M., Koop R., Schrama E., Sansó F., Brovelli M.A., Migliacci Sacerdote F. (1993). Spherical harmonic analysis of satellite gradiometry, In: Netherl; geodetic commission, Publications on geodesy, n.s. N. 39 [5] Sansó F., Usai S. (1995). The new GEM-ESA mission: data reduction theory and simulations, Report N. 1 to Alenia. [6] Sansó F., Usai S. (1995). The new GEM-ESA mission: interpolation errors, commi» errori and errors due to holes in thè data, Report N. 2 to Alenia. 190

11 The dependence of thè GEM mission on measurement noise F.Sanso', G.Sona In this short paper we discuss thè problem : how does thè performance of thè GEM mission depend on thè measurement noise? Whatever method is used to estimate thè gravity coefficients T!m, there are coefficients poorly estimable, on which a strong aliasing effect is present due substantially to thè configuration of thè measurement distribution (polar holes). In particular this has been proved for m=0, m=2; in any way ali thè low order coefficients display a very unfavourable signal to noise ratio, up to say m=4. In ali methods proposed (spacewise, timewise) thè inversion (estimation) can occur only on condition that these coefficients are dumped through a priori information. As a matter of fact this information can also be applied to ali thè coefficients, because thè ones that are well determined will not be constrained by it, as it is done in thè timewise approach. The advantage of thè spacewise method is that, by separating thè estimation order by order, it allows precisely to see where thè problem is. The type ofconstraint that can be applied to T\ is basically an information on thè order of magnitude per degree or a "degree variance" Of course a2 are not perfectly known, yet it is believed that they have a certain regular behaviour (decay) with /: thè simplest law known for this behaviour is Kaula's rute ^f UT5 - (2) This represents thè "mean" square amplitude of a single coefficient at ground level. The same quantity for Trr = ^-y, at satellite's altitude, is dr 0 (WM^±2)(^30 (70 = fl 0 (n. (3) R It could be remarked that our knowledge of T,m for low m, is as a matter of fact better ^han saying : 191

12 "if I substitute T[m=Q in thè representation of 7", I commit an error of thè order of a/(7"), given by (2)", as it is implied by thè statistical interpretation of thè degree varainces. In faci we could say that we know - from satellite observations T/m (m<4) up to degree 50~60 - from ground models up to degree 360 However thè GEM observations have thè possibility of defining thè low degrees T/m themselves, while ground models are doubtfùl due to thè uncertainty of thè height datum between different regions. ^ We believe that in principio it is preferable to avoid introducing into thè GEM solution drawbacks coming from ali thè other methods; therefore we believe that a constraint of thè type of Kaula's mie is thè only acceptable because it introduces only a vague a priori information. Indeed any other kind of "measurements" that could strengthen thè solution have to be used, if available; in this sense any observation from another true polar orbit mission is welcome, as well as any ground gravity measurement in thè polar regions. Let us describe how thè constraint (2) acts, when applied to thè spacewise approach. We summarise thè approach: Observation equations for Trr L I f t i /=/Q /W= / (4) /!y=(9y,xjfc) 6.5 <9y< = 1,2..N Fourier Transform along parallels (5) L /=/o + V; Vjm is stili a uniform white noise. 192

13 L.S. solution (order by order) : let us cali * = Vi." U, (n < 1 0), xn = (n >./(,) 4,= vi» (L.S. estimator) (6) Of course we could also take into account that thè observables Qtn can have different weights from one latitude band to another; this however would not change what follows The point is that for n < 4 thè normal matrix (A*An} is not well conditioned, so we can, for n < 4, and only in this case, introduce as pseudo observations T,n=0 (/»<4) and recali that these pseudo observations have thè known r.m.s., (2), If we cali 193

14 0 O a2 we can now treat an extended information schema, for instance for n=0 Qo = = Ix0+T 0 (7) note that in thè covariance of thè observations I has thè same dimension as Q), i.e. N. The L.S. estimator for this extended model is (8) and its covariance matrix is (9) The situation is perfectly analogous for n=2,3,4. In (8) it is quite evident that thè variances of 7/0 do not depend any more linearly on o2 '. It is because of this effect that thè error in N (geoid vo undulation) or Ag is not linear with 0. vo But this depends exactly from thè fact that we have introduced prior knowledge (pseudo-observations) in (7) ; in fact if we send -><», i.e. E'WO in (9) we go back to a linear dependence. If one would like to grasp thè reason why this happens, one can use a slightly simplified example with S=0?I (prior information) and assuming that AO AQ 'plcasc note that av are indecd proportional to thè originai square of thè noise of thè observations. 194

15 not only has a small eigenvalue (bad conditioning) but it really has a zero eigenvalue, i.e. there is a ^ such that ^ [ = 1 and Let us note that in this case so that if we computa thè variable we have for u «i.e. on thè component of x0 along we have indeed only thè prior Information which is not vanishing when o2 -> O Is then useless to increase thè accuracy of thè measurements? Of course not! In fact would thè observations have an extremely high accuracy (as it is with purely numerical noise) we could perfectly retrieve thè coefficients, our computation show that already at thè level of a0 ~ KHV2 we can expect that there is a significarli Information on ali thè {7},,,, m < 4} together, in any way it should not be forgotten that ali thè other coefficients (i.e. L (L-10) if we skip thè first four orders, which with L=180, 10=13, gives over coefficients) are in fact improving linearly with a<)2. The problem of decreasing thè aliasing should be in any way more thoroughly studied. 195 Jft

16 References Albertella A., Migliaccio F., Sanso'F. (1995) "Simulation and analysis of thè GEM gradiometric mission data", 13 GNGTS, Roma October Brovelli M.A., Migliaccio F. (1993) "The direct estimation of potential coefficients by bi-orthogonal sequences", Lecture Notes in Earth Sciences, N.50, 'Satellite altimetry in Geodesy and Oceanography', Springer Verlag. Migliaccio F., Sanso'F. (1989) "Data processing for thè Aristoteles Mission", Proc.Italian Workshop on thè European Solid Earth Mission Aristoteles - Trevi, Italy, May 30,31. Kummel R. et al. (1993) "Spherical harmonic analysis of satellite gradiometry", Publications on Geodesy - Netherland Geodetic Commission - N.39. pi, 196

17 AVERAGE SIGNAL TO NOISE RATIO (ORDER BY ORDER} O IO CO <- 05 Order AVERAGE SIGNAL TO NOISE RATIO (ORDER BY ORDER) 80 s so \j\j " * f\ m > u m < 0 / 0 I l~ 1 ^-f 1 1 C) X" / / t' 197 ia