UNIVERSIDAD COMPLUTENSE FACULTAD DE CIENCIAS MATEMATICAS CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS INSTITUTO DE ASTRONOMIA y GEODESIA (Centro Mixto C.S.I.C. - U.C.M.). MADRID Publicación núm. 184 MARE NOSTRUM GEOMED Report - 2 Edited by M. J. SEVILLA MADRID 1992 FOREWORD GEOMED is a project for the determination of the geoid and of the Sea Surface Topography in the Mediterranean Sea. The existing set of data available on the Mediterranean in already quite significant and it is going to be improved in these years because of the new altimetric data coming from ERS1 and the TOPEX-POSEIDON missions. In the execution of the project we use, mainly, radar altimetric data (f rom SEASAT, GEOSAT and ESR1 missions) gravimetrie data, bathymetric models, as well as land gravi ty data set and DTMs of surrounding regions: these data sets are generally poor on the southern side of the Mediterranean. More geophysical data, like tidal and Moho depth models, will be user too. The partieularity of this projeet is in that the dynamics of the Mediterranean, as a elosed sea, eannot be easily modelled, so that the time dependent part of the altimetric signal has to be modelled empirically together wi th the radial orbi terror; on the other hand, sinee the satellite ares are very short, this choice is well justified. The separation between the stationary Sea Surface Topography and the geoid can be done in such an area using both data sets, the altimetric and the gravimetric one, whieh is available eontrary to the situation in open oceanic areas. The groups that are working on this projeet are: Dept. of Environmental Engineering, Milan, Italy. (Coordinator); Dept. of Geodesy and Surveying. Thessaloniki, Greeee; Inst. of Astronomy and Geodesy, Madrid, Spain; Inst. of Mathematical Geodesy, Graz, Austria; Dept. of Geophysies, Copenhagen, Denmark; National Survey and Cadastre, Copenhagen, Denmark; Finnish Geodetic Institute, Helsinki, Finland. This projeet in supported by a eontract wi thin the Program Seienee of the European Eeonomic Community. This volume, the second of the MARE NOSTRUM Series, eolleet the works done by many groups involved in the GEOMED organization, in the last year, as informed in the GEOMED Meeting held in Madrid at october 1992, and will be followed by others with new eontributions. INDEX ~ GEOMED Groups: page M.A. Brovelli and F. Sanso The Geomed Project: the state of the art 1 D. Arabelos, S.D. Spatalas and I.N. Tziavos Altimeter Data from ERS-l in the Mediterranean Sea 35 M.J. Sevilla, G.Rodríguez-Caderot and A.J. Gil Progress in the gravimetric geoid computations 43 R. Vieira and C. de Toro Ocean tide charts in the Mediterranean Sea 55 W. Fürst, W.Hausleitner, E.Hock, W.D. Schuh and H. Sünkel Crossover adjustment of Satellite Altimeter Data 75 C.C. Tscherning and M.S. Reierup First analysis of gross-errors in ERS-l altimeter data in the Mediterranean Sea 91 P. Knudsen Evaluation of the Ocean Tides in the Mediterranean Sea in a collinear Analysis of Satellite Altimetry 99 M. Vermeer Geoid Determination with mass point frequency domain inversion in the Mediterranean 109 ~ Invited Group M.A. Andreu, C.Simó 121 Catalan geoid 91: summary of results The Geomed Project: the state oí the art M.A. Brovelli*, F. Sansó** * I.Ge.S. - Istituto Nazionale di Geofisica (Milano) ** D.I.I.A.R. - Politecnico di Milano 1. Introduction It was more than one year ago that 7 scientific groups from 6 nations (Spain, Italy, Austria, Greece, Denmark and Finland) met with the purpose of setting up an international cooperative effort to determine the geoid and the Sea Sur- face Topography (SST) on the Mediterranean Sea. The Geoid, or better some equipotential surface of the gravity field suitably fitting the physical surface of the Mediterranean, is determined as the basic surface in Geodesy to which ort home- tric heights are referred; a good knowledge of the geoid could for instance allow for a much better reattachment of the different national height systems usually conventionally referred to some tide gauges as zero points. The SST, i.e. the stationary height of the sea above the ellipsoid, is a fundamental parameter of physical oceanography strongly related to the steady circulation pat- tern involving surface as well as deep water streams (geostrophic flow) (cfr. e.g. C. Wunsch, [1992]). These two surface can nowadays be separated because the geoid can be determined by measurements related to the gravity field only, while the physical surface of the ocean can be achieved by the radaraltimetric measurements performed by dedi- cated satellite missions (like the now flying ERS1 and Topex Poseidon satellites) after several corrections (firstly the radial orbital correction) and time averaging are applied (cfr. G. Balmino, [1992]and V. Zlotnicki, [1992]). In the most advanced approaches the global analysis of the available data sets (gravimetric, altimetric, satellite tracking, etc.) proceeds as follows (cfr. R. H. Rapp, [1989b]): 1) let us call D.g the field of mean block values of gravi ty anomalies usually known on land areas (e.g. over 10 x 10 blocks), T the anomalous gravity potential, t the stationary sea surface topography, h the measured heights of the sea above the ellipsoid (already corrected by the time varying components), Tp a reference (prior) model of the anomalous potential developped in spherical harmonics up to some degree N max (e.g. N max = 50) and derived from the adjustment of pure satellite tracking data, ~r the radial orbital error of the orbit computed for the flying altimetric satellite from Tp (this error is due to both imperfect knowledge of the initial state and errors contained in Tp propagated to the orbit); then we can write observation equations of the form: IIARE NOSTRUII 2, (1992), pp 1-33 2 MARE NOSTRUK 2 8T 2 6.g = - - - - T + 1/ar r 9 (1.1) T h = - + t + ~T + V« I (1.2) Tp = T + I/T, (1.3) where I is the normal gravity and I/g, I/n , I/T are independent noises; (1.1) holds on continental areas, while (1.2) holds on oceanic areas. 2) in the above equations usually T is modelled as a sum of spherical harmonics up to the same degree Nmax as Tv; t is also represented by some truncated development, e.g. again by spherical harmonics up to some degree which in principle can be as high as Nmax but usually is much lower (e.g. degree 12 or 20), ~T is also parametrized by a small number of parameters (e.g. 5 ~ 6) over some time span (e.g. 1day) in consideration of the fact that most of the power of this perturbation is known to be concentrated at the frequencies of once per rey and twice per rey (cfr. C. Wagner, [1989]); all these unknowns are then estimated by applying a big least squares process to (1.1), (1.2) and (1.3) where, as one can easily recognize, ~T is separated from T and t as it is the only time varying unknown, while T is separated from t mainly by virtue ofthe equations (1.1) and (1.3); all this happens at the degree of resolution given by Nmax which is supposed to be enough to get a good estimate of t and ~T. 3) Once t and ~T are known, track by track, they are subtracted from h at block averaged to obtain estimates of mean values of T with a much higher resolution; from the mean values of 6.g on land and of T on sea one can derive high degree global models (e.g. up to Nmax = 360) of T by applying one of several known techniques (see for instance R.H. Rapp, [1992J or M. A.Brovelli and F.Migliaccio, [1992J and F. Sansó , [1992]). This approach, a1though criticizable in some point, has certainly contributed an enormous improvement in the knowledge of global gravity field models. Unfortunately however it cannot contribute as much in an area like the Mediterranean for two reasons: a) the time dependent pattern of the sea surface is generally more complicated in closed seas than in open ocean where simpler tidal corrections hold; b) the orbital corrections ~T are more difficult to be estimated because of the shortness of the ares which cannot last longer than few minutes before hitting continental areas; moreover the bad performance of global models on Eastern Eu- ropean countries (due to the non availability of gravity material there) makes the radial orbital error to display systematic effects and larger values (cfr. P. Knud- sen and M.A. Brovelli, [1991]). On such a sea however there are available many gravity measurements derived from marine gravimetry; this allows for an indepen- dent computation of the gravimetric geoid which compared with the stationary surface rising from altimetric measurements can supply the sought SST. Very sim- ilar reasons and reasonings apply to the case of Baltic sea and this, beyond the Brove II i and Sanso 3 scientific and personal closeness of the groups, is another justification for their cooperation. We conclude this paragraph by mentioning that also other groups have decleared their interest for the project and cooperate directly or indirectly in it like: CERGA- France (Dr. F. Barlier), University of Barcellona (Dr. M.A. Andreu), Institut of Cataluña (Dr. 1. Colomina), General Command of Mapping, Geodetic Computing - Ankara (Dr. A. Ayhan) 2. Data We try to summarize in this paragraph the type of data we have been able to collect till now focussing on their validation and on the need of new data to solve some ambiguous case. The main data files used for the purpose of Geomed concern: - Marine Gravity Data: these are F.A. anomalies given on the sea surface (Fig. 1), mainly derived by digitizing the famous maps by Morelli (cfr. C. Morelli, [1970]; T.D. Allan and C. Morelli, [1971];C. Morelli et al., [1975a], C. Morelli et al., [1975b]; C. Morelli et al. [1975c]; D. Arabelos, [1980]; D. Arabelos, [1987]; D. Arabelos and C.C. Tscherning, [1988];D. Arabelos and I.N. Tziavos, [1989]), although other gravity files are now available and in future they will be compared with the above for the scope of validation. By the way the actual data have all been scrutinized and essentially submitted to internal validation, so that doubious or possibly spourious data are now properly flagged in our files. - Land Gravity Data: we have (available) the national archives of Spain, Portu- gal and Italy; the Greek gravity data are not open, however they can be used by the Thessaloniki University in computations of the geoid in the Eastern Mediter- ranean. To these a few more data must be added, mainly provided by the Bureau Gravimetrique International which is considering to deliver a set of low resolution gravity for France. Moreover there is a possibility to get similar data for northern Africa from the University of Leeds. (see Fig. 2) - Digit al Terrain Models: these include both topographic heights on land and the bathymetry of Mediterranean (Fig. 3). As for land data only a small part of what would be needed with the proper resolution, is available. On the other hand on the whole region, including bathymetry, we have two global models, namely TUG87 and ETOP05U both with a resolution of 5' x 5'. Moreover the bathymetric maps of Morelli (resolution 5' x 7.5', equidistance 200 m) are also available in a digitized form thanks to the work of the Thessaloniki group. Some work has been already done by using the TUG87 Model, however there are several doubts about its effectiveness due to a recent experience in the computation of the geoid in Italy where it was shown that there are large discrepancies with the national DTM particularly in Southern Italy. Furthermore looking at any 4 MARE NOSTRUM 2 contour map, it seems quite obvious that it is unrealistically too flat in the whole central Mediterranean; the good point on the other hand is that there seems to be a fair agreement with the shore line and with the islands locations, proving that in applying a remove-restore technique the highest frequency contribution to the geoid should be possibly guessed. Some work will be done in the next future in order to obtain an improved bathymetry by merging the existing data. - Altimetric Data: we have collected the available altimetric data for Mediter- ranean concerning the Seasat mission (Fig. 4) as well as the Geosat mission (Fig. 5), the last restricted to the (ERM) Exact Repeat Mission (of periof 17 days) for the first 22 repetitions. These data have already been cleaned and processed in global adjustments by the OSU University and in particular for Geosat the radial orbital error has been corrected for (Y.M.Wang and R.H. Rapp, [1990]). Natu- rally the correction for the radial orbital error in a global treatment suffers of the drawbacks we have already discussed in §1, as a local postprocessing has made clear. A new altimetric data set is now in the process of being collected and validated, namely that produced by the ERS1 mission; for the moment our files include the ERM(1-4) with a 35 days period (Fig. 6) and the ERM(1-4) with a 3 days period (Fig. 7). - Global Geopotential Models: many global models are availbale in the Geomed Files, including lfE SS, OSU7S, GPM2, OSUS1, OSUS6E, OSUS6F, OSUS9A, OSUS9B, OSU91A, DGFI92A, GEM10C . Al! these models have been tested sta- tistically in the area of interest against gravity or altimetric data to decide which one could conveniently represent the data locally. At the end the choice has been for OSU91A (cfr. Fig. S) as, although its performance was comparable to that of lfESS, it is credited to have superior global representativity. Beyond these data which are essential either in computing the gravimetric geoid or the stationary sea surface, other two data sets are currently collected in the Geomed Project as, so to say, subsidiary data, namely: - Tide Gauge Data: these are currently corrected by the Madrid and the Thes- saloniki groups and at tempts are now made to set up an empirical tidal model for the Mediterranean, split into 3 basins (Western, Central, Eastern); - Geophysical Data: in particular we have collected information on the Moho depth in order to be able to smooth as much as possible the gravity field and to be able to predict it as accurately as possible. The Graz group has already performed some experiments in this direction. 3. Methods and first results In this paragraph we try to summarize the different methods proposed to solve our problem as well as the first results obtained, trying to make it clear which are Brove 111 and Sanso 5 the problems still open. A) For altimetry only This treatment is essentially an adjustment of cross-over values based on the ob- servation equations h = (N + t) + (~r + T) + V, (3.1) where N = TI" t, ~r have the samemeaning as in (1.2), while T is a time varying component. Let's assume that ~r and T are so smooth that on a time span of a few minutes (so long can last at the maximum a track on the Mediterranean before hitting a land) they can be well approximated by a linear function of time ~r + T = aT + b; (3.2) this is certainly true for ~r (cfr. E.J.O. Schrama, [1989]) and probably true, at least roughly, for T at least when the subsatellite point is not too close to a coast. Due to the very regular shape of the satellite orbit which is close to a circle, in (3.2) the variable time T can be substituted by A. Now assume also that the observation (3.1) refers exactly to a point where two tracks cross each other (if this is not the case one can always perform an interpo- lation along the track), then since (N + t) is the same in both tracks i and j we can write h ; - hj = (aiA + b¡) - (ajA + bj) + Vij (3.3) A system of equations of the type (3.3) can be adjusted by a least squares approach once the relevant rank deficiency prablem is solved; in practice one can show that a bilinear surface (z = Axy + Ex + Cy + D in planar coordinates) cannot be determined by this system of equations (cfr. R. Barzaghi et al., [1990]) so that some constraint has to be imposed. The most convenient of such constraints is to minimize the sum of the squares of the differences: li¡ - NMod - (aiA + bi) = lli (3.4) on condition that realistic weights be chosen for (3.4) (on this subject cfr. R. Barzaghi et al., [1992]). It is interesting to observe that to strengthen the solution also different data sets can be adjusted, while "almost" separated tracks can be stacked together (collinear analysis) to obtain stronger profiles. In this way for instance an altimetric geoid for the Mediterranean has been com- puted by the Copenhagen and the Milan graups jointly from the available Seasat and Geosat data. To perceive the effectiveness of the adjustment we can say that (cfr. P.Knudsen and M.A. Bravelli ,[1991]) from row data to adjusted we have discrepancies IIwith the model (see (3.4)) with s.d. going from 60 cm down to 36 cm and, even more important, crossover residuals v (see (3.3)) with s.d. decreasing fram 30 cm down to 5 cm. 6 HARE NOSTRUM 2 B) The gravimetric geoid This can be computed in several different ways following the classical approach of the collocation method; in either cases it is convenient first of all to modify the gravity data set by a process, called remove-restore, which has the effect to smooth and regionalize the gravity field. Essentially first of all the free air anomalies f:::...gF are filtered at the long wave- lengths by subtracting the anomalies computed by a global model f:::...gMj with this manipulation the data set is regionalized in the sense that in principie it becomes devoid of signals at wavelengths larger or even comparable with the sides of the window where the data are given. The remaining signal is therefore well estimable with the available data and we can neglet the data outside the window, which are not available. Second we reduce considerably the power of the signal by further subtracting the effect of the Residual Terrain Modelling, f:::...g¡, (cfr. R. Forsberg, [1985]) i.e. the high frequency part of the Terrain correction; therefore we are left with a residual field (3.5) which is both smooth and regionalized and it is generally this field to which we apply a proper operator transforming it into an estimate of the anomalus potential T«. As a final step we add back to T; the contribution of the global model, TM, that of the RTM, TI, to obtain a final estimate of the geoid through (3.6) Just to give an idea the model undulation NM is of the order of 45m ± 3m in the Western Mediterranean (but it goes down to 10m ± 5m in the Eastern part) while the topographic correction NI and the residual part N; are in the order of 1 m. B'í ) Stokes formula by FFT Two test computations have been performed by this method which is nothing but the application of the Stokes formula, to the window where we have data, computed by the FFT techniques exploiting its shape of a quasi-convolution (cfr. M.G. Sideris ,[1987], G. Strang van Hees, [1991]). The two geoids refer one to the Western Mediterranean (0° ::; A ::; 100j 37° ::; r/> ::; 50°), and it has been computed by the Milan group, the other is in the Eastern Mediterranean (14° ::; A ::; 25°j 34° ::; r/> ::; 40°) and it has been computed by the Thessaloniki group, both for the purpose of comparison with the results obtained by other techniques. The result of the experiment in the Eastern Mediterranean is displayed in Fig 9. Brove 111. and Sanso 7 B2) Collocation One of the drawbacks of this approach was till a few time ago its limited capacity of treating a number of points together (~ 3000 points) since the method implies the solution of a system of as many equations as points, with a completely filled in normal matrix. Fortunately enough we have now a technique (cfr. G.P. Bottoni and R. Barzaghi, [1992]) which allows a very fast solution even for very large systems of this kind on condition that the data be regoularly gridded, so that a suitable combination of Toeplitz and FFT methods can be applied. A large experiment with 17557points has been performed in the Western Mediter- ranean by the Copenhagen and Milan groups (cfr. Fig. 10), comparing the results with those obtained with the Stokes/FFT approach; the comparison is satisfactory, since the mean square difference between the two geoids is 13 cm as compared to 66 cm of signal. These numbers ignore the 45 cm of bias which is due to the fact the FFT techniques works with data referred to their average. B3) Pure collocation Since in open sea the topographic correction is not so strong and rough, in this case it is conceivable to perform a geoid computation by pure collocation, i.e. with no remove and restore of the topographic effects, with the main concern that the estimate will not be very accurate in coastal regions. When the data are treated in their originallocations no fast algorithm is available, so the computational burden has to be controlled by limiting the area of compu- tation; the Madrid group has estirnated a geoid in this way on the Mediterranean by splitting it into 330 (1° x 1°) zones on each of which the prediction was per- formed from a (2° x 2°) block covering the estimation area. The computed geoid is displayed in Fig.11. The prediction error is in most cases around 5 cm, apart from some coastal regions where it grows to tens of centimeters. C) An integrated approach This approach, pussed by the Thessaloniki group, is essentially a full collocation procedure, with adjustment of parameters, applied to the set of equations = ~T + (aA + b) + e = -(~~ + ~T)+r¡. (3.7) The interesting point in (3.7) is specially that the density of data referring to the anomalous potential, is extremely increased by the altimetric observations, thus filling the gaps of marine geodesy. Whence the accuracy of the estimate of the geoid should be better. On the other hand in (3.7) the sea surface topography is disappeared, which means that in part it will enter naturally in (aA + b), giving rise to a bilinear surface and in part it can deform T, i.e. the geoid. The first 8 KARE NOSTRUK 2 part is certainly the biggest and probably a bilinear model for a window like 26°:::; >-:::; 36°,31°:::; rP:::; 37°, is good for it, as one would infer from a global model of SST like the one by R. Rapp (cfr. R.Rapp, [1989a]); the second part, though smaller, is of interest but not available in this approach. In any way the internal consistency of the results is certainly very good as it has been tested by living 73 altimetric heights h out of the treatment and then com- paring them with quantities predicted in the processing; the differences between the two turned out to be zero in the average (as it ought) and have a s.d. of 4 cm. D) The sea surface topography One of the crucial questions of this project is: do we really believe that the accuracy of our data and the riliability of our models is sufficient to produce a significant estimate of the SST? In western Mediterranean we have computed a SST by subtracting the gravimetric geoid from the altimetric one; the result is shown in Fig. 12. As one can see we have a surface waving from -0.80 m along the African coast to -0.20 m along France and 0.20 m in Cataluña. These variations seem to be certainly higher than the noise we expect in each geoid, which is of the order of 5 cm for both of them. However, whether there are undetected systematic effects distorting our solutions we are not yet able to sayo 4. Comparisons There are two prossibilities of making external checks of our data; namely either we compare them with independent data of the same kind but coming from different sources, or we try to compare with other geophysical fields exploiting some mutual relation with the gravity field. Only little work has been done till now in this field, yet we like to mention: a) geoids comparisons: an external comparison has been performed between the Geomed geoid in the Western Mediterranean and another gravimetric geoid sup- plied by the Bureau Gravimetrique International. This last has been computed (J .P.Barriot, [1987]) over a large window (-150 :::; >- :::; 280,250 :::; rP < 550 ) by applying a truncated Stokes formula, with a 60 cap; terrain effects have not been considered (cfr. Fig 13). The difference, on 17557 points, shows that there is a bias of 0.49 m and an r.m.s. of 0.81 m. This suggest rather pessimistic conclusions; however by looking at the contour plot of these differences (cfr. Fig 14) we find they are quite flat in the marine area and they become very high and systematic (only positive signs) on land; this could be attributed to the different treatment of the contribution of topography. Brove II i and Sanso 9 In particular we believe that in Corse, where we have the maximum differences, there might be a problem with the gravity material; b) isostatic topographic corrections: some work has been done by applying dif- ferent isostatic-topographic corrections to the field of free air gravity anomalies, to verify which one would produce the best smoothing and homogeneization of the gravity field. The Moho depths used in the computation have been derived from the Airy- Heiskanen theory or an improved version of it; another Moho model was derived from the analysis of seismic data. By using the central Mediterranean as a test area, it has been proved by the Graz group that the Airy-Heiskanen corrections have a much higher performance, while the seismic Moho produced very large descrepancies in the southern Sicily and along the Calabrian are. 5. Discussion The following points represent the goals defined by the Geomed groups for the next period: a) new data: in particular it seems essential to acquire new gravity data, may be not at a high resolution, in north Africa and in France, Croatia, Turkey, etc. It also been decided to collect GPS, levelling data and deflections of the vertical, particularly along the coasts; b) validation: a project has been established to validate our marine gravity data sets with gravimetric profiles owned by DMA; very essential is the validation of ETOP05U by comparing it with Morelli's bathymetry as well as with national DTM's; mean-while the new ERSl data will be validated and included in the altimetric adjustments; c) new computations: the computation of the geoid and SST in the Central Mediterranen, will be completed in one year. After these intermediary goals will be reached we will have at least a geoid and a SST over the whole Mediterranean. The problems at that point will be to homogenize the solutions and to proceed to their interpretation at least in terms of geostrophic circulation. 10 MARE NOSTRUM 2 Figure captions: Fig. la - Distribution of the free air anomalies in the Eastern Mediterranean. Fig. lb - Distribution of the free air anomalies in the Central Mediterranean. Fig. le - Distribution of the free air anomalies in the Western Mediterranean. Fig. 2 - Available land gravity data. Fig. 3a - Bathymetry (on the left) and ETOP05U Model (on the right) in the Eastern Mediterranean. Fig. 3b - Bathymetry (on the left) and ETOP05U Model (on the right) in the Central Mediterranean. Fig. 3e - Bathymetry (on the left) and ETOP05U Model (on the right) in the Western Mediterranean. Fig. 4 - Seasat observations. Fig. 5 - Geosat observations. Fig. 6 - ERSl 35 days period observations. Fig. 7 - ERS1 3 days period observations. Fig. 8a - OSU91A model in the Eastern Mediterranean: geoid (on the left) and free air anomalies (on the right). Fig. 8b - OSU91A model in the Central Mediterranean: geoid (on the left) and free air anomalies (on the right). Fig. 8e - OSU91A model in the Western Mediterranean: geoid (on the left) and free air anomalies (on the right). Fig. 9 - Geoid eomputed by FFT teehniques in the Eastern Mediterranean. Fig. 10 - Geoid eomputed by fast eolloeation in the Western Mediterranean. Fig. 11 - Geoid eomputed by pure eolloeation in the Mediterranean sea. Fig. 12 - Sea surfaee topography in the Western Mediterranean sea. Fig. 13 - Gravimetrie geoid in the Western Mediterranean (J.P. Barriot, [1987]). Fig. 14 - Differenees between Geomed geoid and the geoid supplied by the BGI in the Western Mediterranean. Brove lit and Sanso 11 Fig. la 12 HARE NOSTRUH 2 Q o.."•• Q o. Q•• Q o· ~ Q o·."•• Q o· Q•• Q O· ~ Fig. lb Br-oveIli and Sanso 13 Q o· &1)••• Q o. Q••• Q o· ~ Q o· &1).•. Q o· Q.•. Q o· ~ Fig. le 40!O '"':1•.... ~ N 35!O -10!O -5!O O!O 5!O 10!O 45!O 3O!O 10!O 20!O 25!O 3O!O 35!O 15!O •.... ••• x>- Rl ~ ~ ~ '" ~ 3S!O \¡a?~~">t= "~ 1zí~~~S!O35. ' ~ _ __ •• '~~~~rl :~UOO··~~·I ~ '" Ul ••~ "o c;:," ~;JeV 000 N '0 U, 'TI C'. OQ 4S!O 40!O w P> 25!O 3O!O 3S!O 4S!O 45!O \) ~-'45!O 40!O 40!O40!O I ¡r~" ~&.: o C>o C><¡),~ O' O.,. Fig. 3b o o·-L--I'" o ~--------~---+----------~----------~~~~-----------¡~ o o· '"..• Fig. 3c 18 I!ARE NOSTRUI! 2 Fig. 4 Brovell! and Sanso 19 o..."...,. o.. o...,. o•• ~ o.. ~ Fig. 5 20 IlARE NOSTRUH 2 Fíg. 6 Brove 11 i and Sanso 21 a••a N a.. In a••a a••In a••a a••In I a••a a.. In~ a••a~ a•• ~ a.. ~ Fig. 7 35~O '"<:1 r-" ()Q co Il> 25~O 3O~O 35~O ·1"p~~I~' t71.040~O :S~O [\,) [\,) x,. ~ zoen>-i '"c: X t\J <:> o. LI>..•. c(;:)· o.. 11>... Fig. 8b 24 MARE NOSTRUM 2 o O O o' o· o' 11> O ~... ... O O O o' .. o. 11> O 11>... ... '" t"\D ~. ,,; o~~~~~M1~~~~~~~~ o o' 11>... o o. O... Fig. 8e Brovell! and Sanso 25 ~~~r-~'7~~ N Fig. 9 26 MARE NOSTRUM 2 o •• •• ~ Fig. la Brovelll and Sanso 27 Fig. 11 o~-J--------------~~~~~~~~wr~-t--~~--1---~~ 28 KARE NOSTRUK 2 o.. O..•. o•• ~ Fig. 12 o.. ~ o.. ~ Brove 11! and Sanso 29 Fig. 13 30 MARE NOSTRUM 2 Fig. 14 Brove 111 and Sanso 31 References: T.D. Allan and C. Morelli, 1971, A Geophysical Study of the Mediterranean Sea, Boll. Geofisica teorica e applicata, XIII, 50. D. Arabelos, 1980, Untersuchungen zur gravimetrischen Geoidbestimmung, dargestellt am Testgebiet Griechenland, Wissenschaftliche Arbeiten der Fachrich- tung Vermessungswesen der Universit"at Hannover, Nr. 98, XIV 151, Hannover. D. Arabelos and C.C. Tscherning, 1988, Gravity Field Mapping from Satellite Al- timetry, Sea-Gravimetry and Bathymetry in the Eastern Mediterranean, Geophys J., Int. 92, pp. 195-206. D. Arabelos, P. Knudsen, C.C. Tscherning, 1987, Covariance and Bias Treatment when Combining Gravimetry, Altimeter and Gradiometer Data by Collocation, Presented Intersection Symposium "Advances in Gravity Field Modelling", XIX General Assembly lAG, Vancouver, Canada, August 1987. Proceedings of the International Association of Geodesy (lAG) Symposia, Tome 11,pp. 443-454. D. Arabelos and I.N. Tziavos, 1989, The Hellenic geoid - new considerations and experiences". Ricerche di Geodesia Topografia e Fotogrammetria. Miscellanea per il 70' di Giuseppe Birardi, pp. 1-20, Milan. G. Balmino, 1992, Orbit Choice and the Theory of Radial Orbit Error for Al- timetry, Proceedings of the International Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanography", Trieste, May 25- July 6,1992, in print. J.P. Barriot , 1987, La determination du geoide par altimetrie oceanique et gravime- trie. Quelques aspects du traitement et interpretation geologique sur l'Ocean In- dien (partie Nord-Ouest) et Mediterranee Occidentale, PhD thesis, Academie de Montpellier, Universite des Sciences et Techniques du Languedoc. R. Barzaghi, M.A. Brovelli, F. Sansó, 1990, Altimetry rank deficiency in crossover adjustment, in "Determination of the geoid: present and future", lAG Symposya 106, Springer- Verlag, pp. 108-118. R. Barzaghi, M.A. Brovelli and P. Knudsen, 1992, Different crossover adjustments in the Mediterranean area, in print on Bollettino di Geofisica Teorica e Applicata. G.P. Bottoni, R. Barzaghi, 1992, Fast Collocation, in print on Bull. Geod. M.A. Brovelli and F. Migliaccio, 1992, The Direct Estimation of the Potential Coefficients by Biorthogonal Sequences, Proceedings of the International Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanogra- phy", Trieste, May 25- July 6, 1992, in print. R. Forsberg, 1985, Gravity field terrain effect computation, Bull. Geod n.59, pp. 342-360. 32 KARE NOSTRUK 2 P. Knudsen and M.A. Brovelli, 1991, Co-linear and Cross-over Adjustment of GEOSAT ERM and SEASAT Altimeter Data in the Mediterranean Area, Pro- ceedings of the EGS General Assembly, SE6, Wiesbaden, 22-26 April 1991, in print. C. Morelli, 1970, Physiography, Gravity and Magnetism of the Tyrrhenian Sea, Boll. Geofisica teorica e applicata, XIII, pp. 275-309. C. Morelli, M. Pisani, C. Gantar, 1975a, Geophysical Anomalies and Tectonics in the Western Mediterranean, Boll. Geofisica teorica e applicata, XVII, 67. C. Morelli, C. Gantar, M. Pisani, 1975b, Bathymetry, Gravity and Magnetism in the Strait of Sicily and the Jonian Sea, Boll. Geofisica teorica e applicata, XVII, pp. 39-58. C. Morelli, C. Gantar, M. Pisani, 1975c, Geophysical Studies in the Aegean Sea and in the Eastern Mediterranean, Boll. Geofisica teorica e applicata, XVII, pp. 128-168. R.H. Rapp, 1989a, The Treatment of the Permanent Tidal Effects in the Analysis of Satellite Altimeter Data for SST, MAnuscripta Geodaetica, n. 14 (6). R.H. Rapp, 1989b, Combination of Satellite, Altimetric and Terrestrial Gravity Data, on the Volume "Theory of Satellite Geodesy and Gravity Field Determina- tion" Lecture Notes in Earth Sciences, Springer-Verlag, 261-284. R.H. Rapp, 1992, Use of Altimeter Data in Estimating Global Gravity Models, Proceedings of the International Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanography", Trieste, May 25- July 6, 1992, in print. F. Sansó, 1992, Theory of Geodetic B.V.P.s Applied to the Analysis of Altimetric Data, Proceedings of the International Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanography", Trieste, May 25- July 6, 1992, in print. E.J.O Schrama, 1989, The role of orbit errors in processing of satellite altimetric data, Netherlands Geodetic Commission, Pubblication on Geodesy n. 33. M.G. Sideris, 1987, Spectral methods for the numerical solution of Molodensky's problem, UCSE Rep. n. 20024 - The University of Calgari. G. Strang van Hees, 1991, Stokes formula using Fast Fourier Techniques in "De- termination of the geoid: present and future", lag Symposia 106, Springer-Verlag, pp.405-408. C.A. Wagner, 1989, Summer School Lectures on Satellite Altimetry, on the Volume "Theory of Satellite Geodesy and Gravity Field Determination" Lecture Notes in Earth Sciences, Springer-Verlag, pp. 285-334. Y.M. Wang and R.H. Rapp, 1990, Revised Geosat Geophysical Data Records based on the OSU89 orbit improvement for the first year of the ERM, Internal Report, Dept. of Geod. Se. and Surveying, The Ohio State University, Columbus. Brove 111 and Sanso 33 C. Wunsch, 1992, Physics of the Ocean Circulation, Proceedings of the Interna- tional Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanography", Trieste, May 25- July 6, 1992, in print. V. Zlotnicki, 1992, Quantifying Time - Varying Oceanographic Signals with Al- timetry, Proceedings of the International Summer School of Theoretical Geodesy "Satellite Altimetry in Geodesy and Oceanography", Trieste, May 25- July 6,1992, in print. Altimeter Data from ERS-l in the Meditetranean Sea D. Arabelos, S.D. Spatalas, I.N. Tziavos Department ofGeodesy and Surveying University of Thessaloniki 540 06 Thessaloniki, Greece Abstract. New, five months about, repeat ERS-1 altimeter data have been preliminary processed in the Mediterranean Sea with respeet to seleetion eriteria in order to avoid data influeneed by errors eaused mainly by orbit errors and altimeter signal uneertainties, and of eourse due to presenee of land. After the removal of a number of erroneous observations and in order to assess the quality of the data a erossover analysis has been earried out and a erossover root-mean-square (rms) error diserepaney was found equal to 1.785 m. After a erossover adjustment model was applied introducing bias and tilt into the eomputations, the rms erossover diserepaney deereased to 0.046 m. Furthermore, a first eollinear analysis was performed and some indieative results were outlined. Introduction The ERS-1 satellite has been launehed during 1991 in order to investigate the environment. It has a sun-synehronous, near polar orbit, and operates in a 3 days, a 35 days and a 176 days cycle (ESRIN, 1992). Altimeter data for the Mediterranean Sea [300~ 4> ~ 50°, -5°~ A~ 40°] from ERS-1 mission were recently available to us in the frame of the partieipation of the first of the authors in the ERS-1 projeet DK2. The time period eovered by this data set is April4, 1992 to August 31, 1992 (four repeat periods). Data were de1ivered as reeords in aseii formato Eaeh of the reeords put in our disposal eontains the revolution number, time (UTC since January 1, 1985, midnight), latitude, longitude, eorreeted sea surfaee height He, standard deviation of He, signifieant wave height Hw and standard deviation of Hw. The eorreetion applied to sea surfaee height is given as follows (O. Andersen and c.c. Tseherning, personal communication): He = H - (ionosphere + dry + wet troposphere + solid Earth tide eorreetion ) It is worth mentioning here that the sea surfaee heights have not been eorreeted for ocean tide effeet. Coneerning the satellite orbits it is interesting to note that they have been eomputed using the GEM-T2 model instead the newer PG4491 model in order to enab1e mixing the altimeter measurements from GEOSAT and ERS-1. The ERS-1 orbital frequeney is HARE NOSTRUH 2, (1992). pp 35-42 36 MARE NOSTRUM 2 6035.9287 secs. The duration of each Exact Repeat Mission (ERM) is 35 days or correspondingly 501 revolutions. In our test area and for the five months mission period a number of 25757 subsatellite points (one second mean values) is available. These points are distributed in the four ERM periods mentioned above. From the altimeter data the influence ofthe OSU91A geopotential model complete to degree and order 360 and the influence of the OSU91 Sea Surface Topography (SST) complete to degree and order 10 have been subtracted. The results from the statistical analysis of the residual altimeter data are surnmarized in Table 1. In our preliminary analysis of the first ERS-1 altimeter data the following selection criteria have been adopted: - The test area has been restricted to the Mediterranean Sea omitting subsatellite points found above latitude 43° and westem of longitude 0°, as well as parts of the tracks located above latitude 40° and eastem oflongitude 27°. - Short tracks (having less than 5 points or equivalent length less than about 33.5 km) were eliminated. - Residual altimeter heights (ERS-1- OSU91A - OSU91SST) larger than 10 m were con- sidered as outliers and have not been taken into account in the computations. - Altimeter heights with a standard deviation larger than 0.25 m were also omitted from the further data elaboration. Table 1. Results of ERS·I aItimeter data reduced to OSU91AIF and the OSU91 SST (unit=m) ERM No. of data Min. vaIue Max. value Mean vaIue Standard dev. 1 6157 -4.163 6.110 0.367 1.618 2 6864 -4.416 6.953 0.353 1.483 3 6142 -4.660 6.170 0.420 1.675 4 6594 -35.479 7.112 -0.787 4.491 4- 6450 -5.210 7.112 0.161 1.588 - ERM 4 after the removaI of vaIues (ERS-l - OSU91A - OSU91 SST) > 10 m According to the criteria constructed above a number of 5992 observations was detected and removed. From these observation only a number of 144 outliers has been detected (i.e., reduced values > 10 m). A number of 21 short ares were detected and the 58 observations contained to these ares were neglected. The rest ones are located outside of our test area or were detected as erroneous data using the standard deviation of the satellite points. Following the above mentioned selection criteria a number of 19765 observations was resulted showing a mean value equal to 0.47 m and standard deviation equal to 1.364 m. These observations are distributed in 315 tracks (see Figure 1). From Figure 1 it is cIear that the ERS-1 35 days repeat period provides altimetry with a significantly improved coverage compared with GEOSAT 17 days ERM data (see, e.g. Arabelos and Tziavos, 1990). The distance between the tracks in the ERS-1 mission is about the half of the corresponding distance of GEOSAT mission in the Mediterranean area. i\rabelos, Spatalas and Tzlavos 37 42 40 38 44 -4 -2 O 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 42 36 40 38 36 34 32 34 32 -4 -2 O 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Figure 1. The distribution of ERS-l a1timeterdata. Data analysis Crossover analysis For the processing of the data selected by the method described above, a crossover analysis has been carried out. Owing to the large number of available data, we adopted crossover, when the distance between two neighbouring points of each of the two crossing tracks was less than 16.75 km. Finally, 842 crossovers were found belonging to 278 tracks (139 ascending and 139 descending). These crossovers were used in the crossover analysis, which the results are given in Table 2. From the results of Table 2 it seems that the ascending tracks present a small bias (offset equal to -0.188 m and a slightly larger mean tilt value (-0.271 m/lOO km). The crossover analysis of the descending tracks shows a larger bias (0.763m) in comparison with the ascending tracks and a smaller tilt (0.137 mi km) with respect again to the ascending tracks. Table 2. Resu!ts of the crossover ana!ysis of the ERS·! a1timeter data Ascending Descending AII Mean value (m) before the -0.906 R.M.S. (m) adjustrnent 1.775 Mean bias (m) ·0.188 0.775 0.294 Mean tilt (m/l00 km) ·0.271 0.125 ·0.073 Mean value (m) after the 0.000 R.M.S. (m) adjustrnent 0.046 When the crossovers are treated simultaneously (both the ascending and descending tracks) the crossovers were found with a rms crossover discrepancy 1.775 m and a mean 38 KARE NOSTRUK 2 value of differences -0.906 m. The mean value has been eliminated after the crossover analysis, while the rms crossover discrepancy has been considerably reduced (0.046 m). Concerning the bias (0.294 m) and tilt (-0.073 m/100 km) detected by the crossover analysis we could say that the descending tracks had larger bias (two times about) than the ascending ones. In Figures 2 and 3 the biases and tilts for both ascending and descending tracks are plotted. It is obvious from Figure 2 that the individual bias values are generally less than 3 m in absolute sense. From Figure 3 it is observed that tilt values rarely exceed 2 m/100 km. Collinear analysis Another treatment of the ERS-1 altimeter data has been carried out using observations from collinear tracks. A typical example of such an analysis is presented below,where three of the longest collinear tracks edited in this study are examined through a number of numerical tests. The starting point of these descending collinear tracks is (cp = 40°.5, A= 24°.7) and the end point is (cp = 32°.9, A= 22°.4). The track have a length of 128.4 secs, or 864.6 km. In Figure 4 the three descending collinear tracks are plotted by a solid line and the mean track is presented by not connected triangles. The long straight line parts between subsatellite points are due to the lack of altimeter data due mainly to presence of land. In Figure 5 the average resulted track denoted by crosses has been depicted simultaneously with the orbits derived by adding (or subtracting) the standard deviation of the mean subsatellite points from their original values. It is worth to comment here that the standard deviation of the subsatellite points from the three individual tracks is generally small, while the signal (reduced sea surface height) presents a large variation. In Figure 6 the differences of each subsatellite point from the corresponding mean value are plotted. The major part of these discrepancies is included between -6 cm and +6 cm. In Figure 7 the covariance function of the above mentioned discrepancies is presented with respect to the distance for each of the three individual collinear tracks as well as the covariance function of their mean values (denoted by x ). From this Figure we observe that the correlation length of the error covariances is considerably small. This means that the errors do not present a systematic behaviour and consequently these errors have rather a random character than a systematic one. This preliminary analysis has shown that the quality of the ERS-1 altimeter data is extremely high and the future analysis of more repeat orbits will give best information concerning the geoid and different oceanographic effects, as, e.g. SST, ocean tides, etc. This will lead to a more fine representation and structure of the gravity field in the Mediterranean Sea. Conclusions In this paper a preliminary analysis of five months ERS-1 altimeter observations has been carried out. More specifically, a preprocessing of the data has been done in order to validate the observations and to assess their quality. The altimeter data have been edited E~ (J) O orn -1 -2 -3 -4 Arabelos, Spatalas and Tzlavos 39 4 ~ l\ G 'G f!I: 1\ f--- f-- ~ r[ 13:111 ~ ¡-- I!J ¡-- 3 2 o 0.2 0.4 0.5 0.8 1 1.2 (Thousonds) Revolu\ion number x 1000 1.4 1.5 1.8 2 Figure 2. Biases of both ascending and descending tracks. o o "- E~ 5 .--,,-------------------------------------------------------, 5 4 3 2 o -1 -2 -3 o 0.5 0.8 1 1.2 (Thousonds) Revolu\ion number x 1000 1.4 1.5 1.8 20.2 0.4 Figure 3. Tilts of both ascending and descending tracks. 40 HARE NOSTRUM 2 250 240 230 220 210 E 200~ >-- 190 (/J 180(/J I 170 01 150:J (/J 150O I 140:c '" 130." 120s: " 110u ~ 100 ~ 90VJ o 80 "(/J 70 50 50 40 30 32 34 35 38 40 Lotitude (deg ree) Figure 4. Collinear tracks and their mean. 250 240 230 220 210 200 190 180 170 1()0 150 E 140~ 130 120 110 100 90 80 70 50 50 40 30 32 34 35 38 40 Lotitude (degree) Figure S. Mean (descending) satellite orbit. Upper line: mean-s.d., middle line: mean, lower line: mean-s.d. Arabelos, Spatalas and Tzlavos 41 15 14 12 ~ 10E ~ 8 v~ 6o > e 4 o v 2E .9 O v:S -2 Ee -4 ~ -6o~ :s> -8~ vt:t: -10 -12 -14 -15 32 34 35 38 40 Lo\i\ude (degree) Figure 6 Differences of the three collinear tracks from their mean. N,, E 30 ~ ~ g 20 v o v 10 u e o 'e Oo >o U -10 -20 -30 O 2 3 4 Dis\once (degree) 50 ,---,----------------------------------------------------------, 50 40 Figure 7. Error covariance functions of (a) the differences of the three collinear tracks frorn theír mean (solid lines), (b) the mean values of the differences. 42 MARE NOSTRUM 2 following a number of selection criteria in order to exclude erroneous data due to errors caused by orbit errors, altimeter signal uncertainties, and of course due to presence of land. In a second stage of the processing of the ERS-l altimeter data a crossover and collinear analysis were performed. The main conclusions from this analysis can be drawn in the following. The ERS-l mission provides altimetry with a significant1y improved coverage compared with GEOSA T 17 days ERM data. In the future, when the satellite enters more days repeat orbit, the coverage will be denser (Knudsen and Brovelli, 1991; Knudsen et al., 1992a; 1992b) and superior for a more accurate computation of mean sea surface heights, geoid heights and fine structure and representation of the gravity field. It is interesting to note here that the present coverage of ERS-l in Mediterranean Sea can not be used for tídal studies in the test area and we await that the improved coverage to aid the investigation of tidal effect in the basin of the Mediterranean Sea (Arabelos and Spatalas, 1992). The preliminary crossover and collinear analysis of the ERS-l altimetry in the Mediterranean Sea indicated the high quality of the data in comparison with the data from previous satellite missions (e.g., GEOSAT). The results from the above analysis clearly demonstrate the potential of these new ERS-l altimeter data which already permit the determination of the sea surface topography in the test area. Acknowledgement We would like to thank ESA for releasing the ERS-l altimeter data and the Geophysical Institute of the University of Copenhagen for making available to us these altimeter data files. References Arabelos D. and I.N. Tziavos: Sea surface heights in the Mediterranean sea from GEOSAT altimeter data. JGR Vol. 95, No ClO, pp. 17947-17956, 1990. Arabelos, D. and S. Spatalas: Tidal effects on satellite altimeter data in a closed sea area. Presented First Continental Workshop on the Geoid in Europe "Towards a Precise Pan-European Reference Geoid for the Nineties", Prague, May 11-14, 1992. ESRlN: ERS-l User Handbook, ESA SP-1148, May 1992. Knudsen P. and M. Brovelli, 1991 : Collinear and crossover adjustment of GEOSAT ERM and SEASA T altimeter data in the Mediterranean Sea. Proc. Euro. Geophys. SocoXVI Gen. Assembly, Wiesbaden, 22-26 April, in press. Knudsen, P., O.B. Andersen, and C.e. Tscherning, 1992a : Altimeter data from ERS-l. Presented Dansk Telemalingsdag, 4. March 1992, DTH, Lyngby. Knudsen P., O.B. Andersen and C.C. Tscherning, 1992b : Altimetric gravity anomalies in theNorwegian-Greenland Sea-Preliminary results from ERS-l 35 days repeat mission. Accepted for publication in Geophysical Research Letters. UNIVERSIDAD COMPLUTENSE DE MADRID CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS INSTITUTO DE ASTRONOMIA y GEODESIA MEDGED92 PROGRESS IN THE GRAVIMETRIC GEOID COMPUTA TIONS M.J. SEVILLA, G. RODRIGUEZ-CADEROT and A.J. GIL ABSTRACT The research developed by the Madrid GEOMEDGroup in the field of the Gravimetric Mediterranean Geoid computation is outlined. The incorporation of new gravity data and the analysis of the mergi ng zones s l l ow to complete the gravimetric geoid in the whole area. 1. INTRODUCTION Recently, new data has been added to the original dada bank which has been used to compute a geoid in the Mediterranean Sea following the same method as shown in Sevilla et al. (1992) and presented in the GEOMED Meeting held in Vienna in 1991. The new data has been provided by BGI corresponding to the area of limits 37<~<48 and 10<~<16 .There are 1104 free-air anomalies irregularly distributed. After having checked the new data, a comparison has been done to see their ·goodness which resulted in the same precision about 6 mgal. These gravi ty anomal ies have be en changed to IGSN71 and GRS80 systems and divided into several zones to be validated. Having validated the data a new geoid has been computed in the are a mentioned before and several analysis have been carried out as shown in the sequel. HARE NOSTRUH 2, (1992), PP 43-54 44 KARE NOSTRUK 2 2. SOURCE DATA BANK AND VALIDATION The updated available Medi terranean gravimetry data bank is formed as show Table 1 and Figure 1. TABLE 1. Source Data Bank FILE DESCRIPTION NUMBER OF DATA DISTRIBUTION G1MED Eastern Mediterranean 3652 irregular 31 <<{><37, 26<48, 10<48,10<48,-6<46, FREE AIR OSU81 FREE AIR-OSU81 PRED1CTION FREE AIR-PRED. G2MED 31<<1><46, FREE AIR OSU81 FREE AIR-OSU81 PRED1CTION FREE AIR-PRED. 10<46,10<46,10<45,-6<45,-6<45,-6,i\) ",G _ '" p í e ¡ k¡(,i\) G¡ (,i\) + m i\ = ",p - ",p le ¡ g¡(,i\) G¡ (,i\) + W¡ S(, i\) where m is the order of spherical harmonics, t he angular hour velocity of the correspondi ng harmonic the time zone W ¡ S G 1 the phase lag of the partial oceanic tide with respect to the equi l ibrium tide in Greenwich, the phase lag of the partial oceanic tide wi th respect to the equi l ibrium tide in the observation point, k¡ phase lag wi th Greenw i ch when in loc al time. equilibrium tide in the obser vation time respect we have to the expr e sed Vieira and de Toro 59 Provides the amplitude in centimeters and the various phase lag G, k, g f'or every one of the 60 harrnoníc included in the bank. Inf'orrns about the method of analysis used and the number of days used in such analysis. Provides the information about data variances and about residuals f'or every frequency bando These two statistics allow us to evaluate the ratio signal/noise and, ther-ef'or-e the standard deviation of the harmoníc constants. It includes as an output of the program some other complementary information which can be of interest f'or some applications of the tidal bank such as: • the altitude in centimeters of the mean level over the hydrographic zer-o of the local chart of a higher scale (Zo), • the link with the leveling network (ENR), • designation of the ref'erence and the height (So) above it of the mean sea level observed, • the Altitude Unity (UA) and • the Cornmon Establishment of the Port (EP). 4. APPLICATION PROGRAMS The following are thr-ee application programs we have created (f'Igureü): SAEDIF pr-ogr-am which allows complete or partial access to the information loaded in the bank with a bibliographic formato SANIM progr-am which provides the information about the mean levels Zo and the heights So over the reference of the observed mean level. SAMAR program, which allows the loading in CMS files of the information necessary to initiate the modeling pr-ocess of a given harmoníc. Some other programs are not included in the diagram of figure 3, as they do not affect the building up process of the data bank and its applications. However, they are interesting f'or the investigations like those r-elated to the analysis of ocean tidal observations. With this idea in mind, the author-s have carr-íed out in the I.A.G. two programs which are based on two different methodologies of analysis. They are: the M088 program, mainly based on the least squar-e technique and the LEMAGprogr-arn 60 KARE NOSTRUH which uses the Fourier analysis. Both programs are operative and with them we have analyzed some of the tidal data series incorporated in the BAMAGbank of the I.A.G. 5. MODELING The modeling process initiates from the information provided by the data base of the SAMARprogramo First of all, by starting with the DERAPprogram we proceed to study and elímínate, when necessary, the harmonic constants which we can consider aberrant, either as a consequence of mistakes or, more often, due to the singularities of given tide records. It is well know that many of the tide records of the coast are located in ports, bays, river estuaries, etc., places where the local phenomena produce amplitude and phase values which are representative of such place and its surroundings. For that r-eason, they are singular observations which we must eliminate from the process of regional or global modeling we are working on, although they may have a great importance at a local level. The criteria followed for this selection is to consider that in a radius of around 100 km the spatial distribution of the gradients from different parameters should be homogeneous and uniform except for places such as straits where such regimes may substantially vary over lesser distances. A second elimination program also carried out through the DERAPprogram consists of comparing every one and all of the parameters observed and calculated in one station (H, He, 1/1, I/Ie, G, k, g) to those of other stations located in an area of one degree in latitude and longitude. The LEMAG program allows to calculate the theoretical parameters such as amplitudes and phases of the equilibrium ti de for every one of the tidal stations (table 2). On figures 4 and 5 the spatial distribution of the harmonic constants which passed through the different tests for singularities detection can be seen. We have processed the values of the amplitudes and the phases through a graphic program and we have plotted the isoamplitude and isophase lines for the western Mediterranean are a, from the Gibraltar Strait till the natural barrier made by the Islands of Corsica and Sardinia and the group of small islands and bathymetric heights whích we can consider that shape Vieira and de Toro 61 and close the western area of the "Mare Nostrum". In parallel we have proceeded a subdivision of the area in trapeze shaped zones of 0.5 x 0.5 degrees and smaller ones in the coast band, in order to take into account, in this way, the real boundaries of this coast. The number and the dimensions of the spherical polygons for the area we are considering are: 197 - 0~5000 x 0~5000 81 -- 0~2500 x 0~2500 156 -- 0~1250 x 0~1250 2 -- 0~0625 x 0~0625 The central points of all polygons shape_ the digital network of the chart to which values are assigned by interpolation between the amplitude and phase lines next to that very center. The group of these values ordered by geographic coordinates forms the digital chart of the studied area (figures 6 and 7). The whole modeling process is carried out through the MOOELARMONIprogram which allows to create afile which contains for every polygon the average amplitude of the tide, measured in centimeters, the average differences with respect to the equilibrium tides in Greenwich meridian measured in degrees, and the geographic coordinates of the center and surface of the polygon calculated from the OESUPprogramo REFERENCES Admiralty Tide Tables, 1993. Volume 1: European Waters, including Mediterranean Sea. Pub. The Hydrographer of the Navy (UK), NP 201-93 438 pp. Cartwright, O.E. Zetler B.O., and Hamon, B.V., 1979. Pelagic Tidal Constant. lAPSOPub. SC. 30, 65 pp. Cartwright, O.E., Edden, A.C., Spencer, R. and Vassie, J.M., 1980. The tides of the Northeast Atlantic Ocean. Phil. Trans. R. Soco Lond. A., vol. 298. Cartwright, O.E. and Zetler ,B.O., 1985. Pelagic Tidal Constant 2. lAPSO. Pub. SC. 33, 59pp. García Lafuente, J., Castillejo; F. and García, M., 1987. Resultados de la red mareográfica del Estrecho de Gibraltar. Rev. de Geofísica (1987) 43, 37-56. Frutos Fernández, 1973. Constantes armónicas de marea de la zona del Estrecho de Gibraltar. Bol. lEO, 169. Frutos Fernández, 1973. Constantes arrnorucas de marea de las Islas Baleares, Canarias y Costa Occidental de Africa.Bol. lEO, 170. 62 KARE NOSTRUK Schwiderski, E.W., 1980. Ocean Tides, Parto 1: Global Ocean Tidal Equations. Marine Geodesy 3: 16l. Schwiderski, E.W., 1980. Ocean Tides, Parto II: A Hydrodinamical Interpolation Model. Marine Geodesy 3: 219. Schwiderski, E.W., 1980. On Charting Global Ocean Tides. Rev. Geophysics and Space Physics 18, 1, 243-268. Schwiderski, E.W., 1983. Atlas of Ocean Tidal charts and maps, semidiurnal lunar tide M2. Marine Geodesy, 6, 219-265. Toro, C., 1989. Determinación y evaluación de las variaciones periódicas de la gravedad y de las desviaciones de la vertical en la Península Ibérica producidas por las mareas oceánicas. Ph.D. Thesis, Fac. CC. Matemáticas. Universidad Complutense de Madrid, 378 pp. Vieira, R., Toro, C. and Sukhwani, P.K., 1983. Ocean effects on Gravity Tides in the Iberian Peninsula. In: J.T. Kuo (Sc.Ed.), Proc. 9th Int. Symp. Earth Tides, New York 1981. Schweizerbart'she Verlag., Stuttgart, 403-410. 1, The Vieira, R., Toro, C. and Megias, E., 1986. Ocean Tides in the nearby of the Iberian Peninsula. Part 1: M2 Iberia Map. In : R. Vieira (Sc. Ed.J, Proc. 10th Int. Symp. Earth Tides, Madrid 1985. Consejo Superior de Investigaciones Científicas, 679-696. Vieira, R., Toro, C. and Fernandez, J., 1986. Ocean Tides in the nearby of the Iberian Peninsula. Part II: S2 Iberia Map. In: R. Vieira (Sc. Ed.l, Proc. 10th Int. Symp. Earth Tides, Madrid 1985. Consejo Superior de Investigaciones Científicas, 697-706. Vieira, R., Fernandez, J., Toro, C. and Camacho, A.G., 1991. Structura1 and oceanic effects in the gravimetric tides observations in Lanzarote (Canary Islandsl. In: J. Kakkuri (Se, Ed.l, Proc. 11th Int. Symp. Earth Tides, Helsinki 1989. Schweizerbart'sche Verlag., Stuttgart, 217-230. Figure 1. Figure 2. Figure 3. Table 1. Table 2. Figure 4. Figure 5. Figure 6. INDEX OF THE ILLUSTRATIONS Mediterranean Sea. Modelingareas. Mareographic data. Block scheme of the mareographic data base. Mareographic data bank. Logical description of the source files. M2 component. Mareographic data located in the strait of of Gibraltar and Mediterranean Sea. Mediterranean Sea. Tidal constituent M2. Amplitude. Mediterranean Sea. Tidal Constituent M2. Phase lag. M2 Ocean tide Amplitudes. West Mediterranean Sea Model. Figure 7. M2 Ocean tide Greenwich Phases. West Mediterranean Sea Model. 4CC 30" ~~. ) ~ b. 2.6. .() . o~ O b. 2.1. b.2.4. b. 2.4. O" 10° 20° Figure 1. MEDITERRANEAN SEA. MODELING AREAS. ••5- Q. "..¡ o., o 0\ W OSTRUH64 MARE N 16006 PALMA DE MALLORCA 11.1. 2211.115 39.5500 2.6333 • O 2.8 210.2 5. 16007 ROSAS 11.1.2211.113 42.2333 3.2667 • O 5.5 250.4 Jl, "17001 ALHUCEMAS (1) 11.1. 2211.116 35.2333 -3.8833 • O 18.5 60.7 ...¡ 17002 ALHUCEMAS (2) 11.1. 2211.116 35.3333 -3.8667 • O 17.9 54.7 o., 17003 VILLA NADOR 11.1. 2211. 16 35.0900 -2.9167 • O 6.0 163.0 o 0\------------------------------------------------------------------------------------------- ...,¡ ------------------------------------------------------------------------------------------- 0\ ():) MAREOGRAPHIC STATION ZONE LATITUOE LONGITUOE o/s H(CM) G(GR) ~------------------------------------------------------------------------------------------- ~ 17005 ALGER 11.1. 2211.116 36.7833 3.0667 • O 2.4 219.3 :z: 17006 LA GOULETTE 11.1. 2222 .117 36.8167 10.3167 • O 8.0 249.0 o Ul 17007 GABES 11.1. 2244 .117 33.8833 10.1167 • O 48.0 121.9 ... 17008 HOUMT AOJIM 11.1.2244.117 33.7167 10.7500 • O 3l.0 103.0 ~ 17009 SFAX 11.1. 2244 .117 34.7333 10.7667 • O 42.0 77. O 17010 HOUMT SOUK 11.1. 2244 .117 33.8833 10.8667 • O 3l.0 104.0 17011 RAS TOURG-EN-NESS 11.1. 2244 .117 33.8167 11.0500 • O 27 .0 69.0 17012 ZARZIS 11.1.2244.117 33.5000 11.1167 • O 22 .0 77. O 17013 EL ABASSIA 11.1. 2244.11 7 34.7167 11.2500 • O 26.0 83.0 17014 TRIPOLI (TARABULUS) 11.1. 2244 .117 32.9000 13 .1833 • O 13 .0 65.0 17015 MISURATA 11.1. 2244 .117 32.3667 15.2167 • O 7.0 53.0 17016 MESA EL BREGA 11.1.2244.117 30.4167 19.5833 • O 5.0 60.0 17017 MERSA TOBRUK 11.1.2244.118 32.0833 23.9667 • O 1.0 285.0 17018 BAROIA 11.1. 2244 .118 31.7667 25.1667 • O 3.0 236.0 17019 ALEJANORIA 11.1.2244.118 31.1667 29.8500 • O 7.0 245.0 17020 PORT SAIO 11.1.2244.118 31.2667 32.3167 • O 12.0 240.0 18001 PORT VENORES 11.1. 2211.114 42.5167 3.1000 • O 5.0 288.0 18002 MARSELLA 11.1. 2211.114 43.3000 5.3667 • O 7.0 217.0 18003 TOLON 11.1.2211.114 43.1167 5.9333 • O 3.0 266.0 18004 NIZA 11.1. 2211.114 43.7000 7.2833 • O 7.0 244.0 18005 MONTE CARLO 11.1.2211.114 43.7333 7.4167 • O 4.0 259.0 18006 AJACCIO 11.1.2211.119 41.9333 8.7500 • O 7.0 250.0 18007 CAGLIARI 11.1.2222.119 39.2000 9.1000 • O 8.0 236.0 18008 CARLO FORTE 11.1.2212.114 39.1500 8.3000 • O 6.0 231.0 18009 IMPERIA 11.1.2211.120 43.8833 8.0167 • O 8.0 237.0 18010 GENOVA 11.1. 2211.120 44.4000 8.9000 • O 8.0 222.0 18011 LA ESPEZIA 11.1. 2211.120 44.0667 9.8500 • O 9.0 215.0 18012 LIVORNO 11.1. 2212 .120 43.5500 10.3000 • O 8.0 232.0 18013 CIVITAVECCHIA 11.1.2222.120 42.1000 11.7833 • O 11.0 239.0 18014 GAETA 11.1. 2222 .120 41.2167 13.5833 • O 11. O 234.0 18015 NAPOLES 11.1. 2222 .120 40.8333 14.2667 • O 11.0 237.0 18016 ISCHIA 11.1. 2222 .120 40.7333 13.9333 • O 12.0 232.0 ------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- MAREOGRAPHIC STATION ZONE LATITUDE LONGITUDE o/s H(CM) G(GRl ------------------------------------------------------------------------------------------- 18017 TROPEA 11.1.2222.120 38.6833 15.9000 • O 15.0 242.0 18018 VILLA SAN GIOVANNI 11.1.2224.120 38.1833 15.6333 • O 3.0 85.0 18019 REGGIO CALABRIA 11.1.2224.120 38.1167 15.6500 • O 6.0 62.0 18020 TAORMINA 11.1. 2244 .121 37.8167 15.2833 • O 9.0 57.0 18021 MESSINA 11.1.2224.121 38.2167 15.5667 • O 5.0 2.0 18022 CAPO PELORO 11.1. 2224 .121 38.2667 15.6500 • O 5.0 238.0 18023 LIPARI 11.1. 2222 .120 38.4833 14.9667 • O 12.0 232.0 18024 MILAZZO 11.1. 2222 .121 38.2167 15.2500 • O 12.0 234.0 18025 PALERMO 11.1.2222.121 38.1333 13.3333 • O 11. O 232.0 18026 MARSALA 11.1. 2224 .121 37.8000 12.4333 • O 7.0 207.0 18027 MAZARA DEL VALLO 11.1. 2224 .121 37.6333 12.5833 • O 4.0 128.0 18028 PORTO EMPEDOCLE 11.1.2244.121 37.2833 13.5333 • O 5.0 76.0 18029 CATANIA 11.1. 2244 .121 37.4833 15.1000 • O 6.0 61.0 18030 VALLETTA 11.1. 2244 .122 35.8833 14.5167 • O 7.0 48.0 18031 TARANTO 11.1. 2244 .120 40.4667 17.2167 • O 6.0 71. O 18032 OTRANTO 11.1. 2234 .120 40.1500 18.5000 • O 7.0 73. O 18033 BRINDISI 11.1. 2233 .120 40.6500 17.9667 • O 9.0 73. O 18034 VIESTE 11.1.2233.120 41.8833 16.1833 • O 8.0 61.0 18035 ORTONA 11.1. 2233 .120 42.3500 14.4167 • O 7.0 64.0 18036 ANCONA 11.1.2233.120 43.6167 13.5000 • O 7.0 303.0 18037 PESARO 11.1. 2233 .120 43.9167 12.9167 • O 13. O 288.0 18038 PORTO CORSINI 11.1. 2233 .120 44.5000 12.2833 • O 15.0 274.0 18039 CHIOGGIA 11.1.2233.120 45.2333 12.3000 • O 22.O 273.0 18040 MALAMOCCO 11.1. 2233 .120 45.3333 12.3167 • O 23.0 267.0 18041 VENECIA 11.1. 2233 .120 45.4333 12.3333 • O 24.0 285.0 18042 GRADO 11.1. 2233 .120 45.6833 13.3833 • O 23.0 276.0 18043 TRIESTE 11.1. 2233 .120 45.6500 13.7500 • O 26.5 302.5 ~ 18044 SAN GIULANO 11.1.2233.120 45.4667 12.2833 • O 24.0 306.0 :l '"18045 TORCELLO 11.1. 2233 .120 45.5000 12.4167 • O 19.6 345.8 e, <1> 18046 PALIAGAO 11.1. 2233 .120 45.5167 12.3833 • O 19.8 12.8 ...• 18047 TORSON DI SOTO 11.1.2233.120 45.5000 12.4167 • O 20.8 342.8 o., o 18048 MILLECAMPI 11.1. 2233 .120 45.3000 12.1833 • O 16.9 18.4 0'\------------------------------------------------------------------------------------------- \Q -..¡ o ------------------------------------------------------------------------------------------- ~ MAREOGRAPHIC STATION ZONE LATITUDE LONGITUDE D/S H(CM) G(GR) ¡;j ------------------------------------------------------------------------------------------- z:o 18049 PORTO PIAVE VECCHIA 11.1. 2233 .120 45.4867 12.5783 • O 22.3 307.2 Ul...¡ 18050 CAVALLINO 11.1. 2233 .120 45.5000 12.4167 • O 10.7 39.8 sx 18051 PULA 11.1. 2233 .123 44.8667 13.8333 • O 15.0 236.0 18052 RIJEKA (FIUME) 11.1.2233.123 45.3333 14!4333 • O 10.0 220. O 18053 SENJ 11.1. 2233 .123 45.0000 15.9000 • O 10.0 211. O 18054 MALI LOSINJ 11.1. 2233 .123 44.5333 14.4667 • O 8.0 211. O 18055 ZALIV PANTERA 11.1. 2233 .123 44.1500 14.8500 • O 4.0 165.0 18056 ZADAR 11.1. 2233 .123 44.1333 15.2000 • O 6.0 203.0 18057 SIBENIK 11.1. 2233 .123 43.7333 15.9000 • O 6.0 106.0 18058 ROGOZNICA 11.1. 2233 .123 43.5333 15.9833 • O 6.0 111.0 18059 SPLIT 11.1. 2233 .123 43.0500 16.0833 • O 8.0 100.0 18060 KOMIZA 11.1. 2233 ,123 43.0500 16.0833 • O 7.0 79.0 18061 OTOK SVTAC 11.1. 2233 .123 43.0333 15.7667 • O 7.0 93.0 18062 DUBROVNIK 11.1.2233.123 42.6667 18.0667 • O 9.0 86.0 18063 MELJINE 11.1.~233.123 42.4500 18.5500 • O 9.0 70.0 18064 BAR 11.1. 2233 .123 42.0667 19.0833 • O 9.0 80.0 18065 SHENGJIN 11.1. 2233 .123 41.8167 19.5833 • O 9.0 79.0 18066 DURRES 11.1. 2233 .123 41.3167 19.4500 • O 9.0 73.0 19001 NISOS LEROS 11.1. 2255 .124 37.1667 26.8333 • O 3.0 304.0 19002 NISOS ASTIPALAIA 11.1. 2255 .124 36.6333 26.3833 • O 3.0 295.0 19003 NISOS KOS 11.1. 2255 .124 36.8833 27.3167 • O 4;0 271. O 19004 NISOS SIMI 11.1. 2255 .124 36.6167 27.8667 • O 4.0 269.0 19005 RODHOS 11.1. 2244 .124 36.4500 28.2333 • O 5.0 250.0 19006 LINDHOS 11.1. 2244 .124 36.1000 28.1000 • O 6.0 249.0 19007 MEYISTI 11.1.2244.125 36.1500 29.6000 • O 7.0 245.0 19008 KYRENIA 11.1.2244.126 35.3333 33.3167 • O 10.1 8.6 19009 LIMAS SOL 11.1.2244.126 34.6667 33.0500 • O 10.0 235.0 19010 FAMAGUSTA 11.1.2244.126 35.1167 33.9500 • O 11. O 236.0-------------------------------------------------------------------------------------------- Figure 4. MEDITERRANEAN SEA. TIDAL CONSTITUENT M2. AMPLITUDE H (cm). ~ "., l> l>5- .,. "...¡ o., o -;¡•.... ...,¡ 1\.) ~ ill I Figure 5. MEDITERRANEAN SEA. TIDAL CONSTITUENT M2 J \ PHASE LAG G (degrees). ,e; and de Toro 73Vieira Eo / * .•.. ... -. ·i: . .. 1: ' .. *-~... ... ... ..• ..\p .•IHHI-* .•. .•* '*.•...... " 1" 'o .• .·. .. .. :1.. · ..* .•.1.. :1\;< " ** .. · :l .• *" • .• " ~ :I..• #-Jt* " 4f".. A •. -1: * -AA:•.... .o!t** .•. *. " " Figure 7. WEST MEDITERRANEAN SEA. TIDAL CONSTITUENT M2. PHASE LAG G (degrees). ~ ~ I 7' - * **"'*** lI< * * * * *.:* :+: :+: *\:*..J ~ . ,..! ~ * :" . * .~. * •• • • *1. ~ • l' ) . ,* *¡'*• • • * '" , • J:,.••*.***~: . *** * CROSSOVER ADJUSTMENT OF SATELLITE ALTIMETER DATA W. Fürst W. Hausleitner E. Hock W.-D. Schuh H. Sünkel Institute oí Mathematical Geodesy and Geoiníormatics Graz University oí Technology Steyrergasse 30, A-B01OGraz, Austria Abstract In the frame oí the GEOMED project various problems related to crossover adjustment are in- vestigated. Interpolation techniques which are currently in use, are dealt with numerically. The dependence oí the rank oí the linear system oí equations for crossover adjustrnent on the area size and on the geographical location is studied. Three methods to repair the rank deficiency are investigated and the pro's and con's are identified. 1 Introduction In order to exploit the information which is contained in satellite altimeter data from satellite missions such as SEASAT, GEOSAT, ERS-l or TOPEX-POSEIDON as much as possible, the satellite's orbit must be known with utmost accuracy, A method oí orbit correction which is being widely applied is the method oí crossover adjustment oí altimeter data. Such an ad- justment procedure requires altimeter data at the intersection oí northgoing and southgoing satellite groundtracks and a proper procedure to solve the singular linear system oí equations. The present contribution addresses these two issues. \ In particular interpolation techniques are compared against each other regarding the interpola- tion accuracy depending on the size oí the data gap, the smoothness oí the interpolator, and the MARE NOSTRUM2, (1992), pp 75-90 76 MARE NOSTRUK 2 smoothness properties oí the function to be interpolated. The system of linear equations which results írom the adjustment oí the crossover discrepancies is rank deficient with two eigenvalues being numerically zero in any case and one or two eigen- values being very close to zero, depending on the size and on the geographical location oí the data area in consideration. Three methods to repair the rank deficiency are compared against each other, the track fixing method, the geoid fitting method, and the singular value decomposition technique supplemented by surface fitting. The quality oí these methods is judged on numerical investigations which are based on GEOSAT exact repeat mission (ERM) data. 2 Interpolation of crossover points Before crossover adjustment oí satellite altimeter data can be pursued, crossover points and cor- responding crossover discrepancies must be available. The identification oí a crossover point is a simple matter oí intersection between a northgoing and a southgoing satellite groundtrack. The assignment oí proper altimeter data to such an intersection point is less trivial, unless altimeter measurements are performed exactly at the intersection point (which is indeed a very unlikely event). Thereíore, the data have to be interpolated at the crossover points using some kind oí interpolation technique. Numerical investigations with available satellite altimeter data such as SEASAT and GEOSAT data using existing software have revealed some weeknesses which are related to the applied interpolation technique. Since the quality of the crossover adjustment results depend to some extent on the input data (crossover data), an investigation oí simple interpolation procedures suggested itself. In particular two alternative interpolation methods have been investigated: linear interpolation as applied in existing satellite altimeter processing software, and altérnatively, cubic interpela- tion. Obviously interpolation results are not only dependent on the interpolator (linear, cubic, or more sophisticated) and on the data gap which is spanned by interpolation. Interpolation results depend also to a large extent on the properties of the underlying function to be interpolated (smooth versus rugged). In order to account for these facts, the two interpolators mentioned above have been used both for smooth and rugged geoid profiles with the interpolator spanning between 3 and 31 data points. The interpolated values were compared against the given data (which, oí course, were not used in the interpolation procedure). The respective results are summarized in Table 1 in terms of mean, rms and maximum inter- polation error. In Figure 1 mean interpolation errors are presented for two geoid profiles, a smooth one (lower curve) and a rugged one (upper curve). The histograms in Figure 2 provide a comparison oí interpolation errors between linear and cubic interpolation bridging a single data gap. The interpolation error s are broken down according to their size. Fürst, Hauslei tner, Hdck , Schuh and Sünkel 77 Qualitatively the results are in agreement with what one should expect: cubic interpolation is superior over linear interpolation and is therefore recommended to be used for the interpolation oí altimeter data in any case. A smooth profile may be bridged by cubic interpolation up to a gap oí about 10 data points with sufficient accuracy, while only about 7 points may be bridged in a rugged profile. The interpolation error increases with increasing length oí the data gap with a much higher rate for a rugged profile than for a smooth one. Quantitatively it is obvious that interpolation errors over a moderately small data gap are oí the order oí about 4 cm with an rms oí about 5 - 7 cm and maximum absolute values oí several decimeters, depending on the smoothness oí the geoid profile. linear cubic cubic cubic cubic Interpolation 3 points 5 points 7 points 13 points 31 points Mean 4 4 4 7 19 RMS 6 5 5 10 27 Maximum 52 32 39 83 173 Table 1: Interpolation errors depending on interpolation technique and size oí data gap Dala poinl s spanned by'inlerpolation Figure 1: Interpolation error depending on smoothness oí interpolated profile and on size oí data gap 78 MARE NOSTRUII 2 DISCREPANCIES OF RECOVERED DATA 35% ,------------------------------------------------------, 25% 15% 20% 10% 5% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 36 40 ~ linear inlerpolallon • O • 35%,------------------------------------------------------, 15% 30% 25% 10% 5% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 ~ cublc lntsrpolatlon • • O • • Figure 2: Interpolation error: linear versus cubic interpolation (1 data gap only) Fürst, Hausleitner, Hock, Schuh and Sünkel 79 3 Crossover adjustment - the principIe The crossover technique relies on the requirement that a measured geoid height at a given geographical Iocation must be independent oí the tracking instant. Crossover differences di; oí corrected altimeter data are thereíore essentially identical to the radial orbit error oí the respective satellite tracks: di; = h¡ - h; = d; - d; + €¡ - €; (1) with h¡, h; Measurement at the crossover location oí the satellite tracks i und j di, d; Orbit errors oí the satellite tracks at the crossover location €¡, €; Measurement error It is well known that in sufficiently small areas the radial orbit error may be modelled by either a time- or a distance-dependent polynomial, leading to the íollowing observation equations at the crossover points: le I Vi; = L a¡.(t¡ - tíO)" - L a;.(t; - tíO)" - tI.; + €¡ - €; e=O e=O (2) with k,l a (t - to) Degree oí chosen polynomials Polynomial coefficients Time parameter Since differences oí measurements are used as observations and those differences are invariant with respect to certain transformations, the corresponding adjustment problem is singular. The parameters A¡, used in the bias and tilt model, represent relative geographicallongitudes oí the crossover points, counted from the mean geographicallongitude oí the respective satellite track: llh¡; = (a¡ + b¡Aj) - (aj + bjA¡) + r¡j (3) with llh¡j a¡,b¡,aj,bj A¡,Aj rij Height difference at the crossover point oí the tracks i and j Bias and tilt parameters Relative geographical longitudes oí crossover point Residual 80 KARE NOSTRUK 2 The bias and tilt coefficients Ili and b¡ and aj and bj, respectively, have to be determined such that the residuals r¡j oí crossover height differences are minimized in the least squares sense. The observation equations ll.h = Az + r (4) with the vector oí unknown parameters z (bias, tilt) leads to a least squares adjustment prob- lem which is singular and has a rank deficiency to be dealt with in the next chapter. 4 Rank deficiency: global versus local The adjustment problem of crossover differences is known to be singular. The singularity is characterized by four vanishing eigenvalues in the planar case, while in the spherical case the rank deficiency is just two. In our particular situation oí the Mediterranean Sea we have neither a planar case nor a full spherical case. Therefore, two eigenvalues will be zero and two others very small, but numerically not zero. In order to understand the behaviour of the smallest eigenvalues, two investigations have been performed using simulated GEOSAT ERM tracks: a) The latitude dependence oí the smallest eigenvalues was studied for a diamond shaped system consisting of 6 x 6 crossover points. The diamond was located at various latitudes between 0° and 72° . The location of the diamonds is depicted in Figure 3. The behaviour of the eigenvalues no. 4 and 5 (eigenvalues are here ordered with increasing magnitude), depending on the geographicallocation oí the diamond, is presented in Figure 4. The first three eigenvalues are not presented here - they are practically zero. It is obvious that the 4th eigenvalue is almost zero in the neighborhood oí the equator, increasing towards the middle latitude range, and again approaching zero towards the pole (due to the degeneration oí the diamond). Also the 5th eigenvalue decreases when the diamond moves towards the poleo b) The dependence of the smallest eigenvalues on the size oí a diamond shaped crossover system, consisting oí between 10 x 10 and 80 x 80 crossover points, with the diamond centered at latitude

.....c -2.B0 al ...._ •....\ r ~•• --- ••• v, .•. ...¡ \ :. t· " / ~iQ):::r: - 3, ee -3.20 _3.40~~~~~~~~~~~~~~~~~~~~~~~~~~ 44.00 44.5121 45.0121 45.50 46.5121 Lo L L L lo u d El Figure 4. This is a graph comparing two tracks, 454 & 955 (full & dotted line). Two errors are clearly seen on track 454, namely the start of the graph at 44.30 and the spike at 46.20. ( Note that this example is situated in the Atlantic ) 3.0'121 2.50 .:» .....c 2.0121 al Q) I,·50 .- •••.• y. •••.,."' ••-,.,¿,,,.,/ •....•.....•.•__ / 1 .1210 \,...•..•, Figure 5. Shown here are the tracks 103 & 604 ( full & dotted lines ) It's the same track but for two different 35 day periods. It is seen that track 103 differs from track 604 by being more ragged and spiky, 100ks like there's a 10t of noise on the track. The track has not been included on the error list but maybe it should. Evaluation of the Ocean Tides in the Mediterranean Sea in a Collinear Analysis of Satellite Altimetry. Per Knudsen (National Survey and Cadastre (KMS), Geodetic Division, Rentemestervej 8, DK-2400 Copenhagen, Denmark. Fax: +45 35875052) Abstract. In this paper the ocean tides in the Mediterranean Sea have been evaluated. The tidalsignalwas expressed as H(t) =E(U,cos(w¡t+xJ +V,sin(w¡l+x)) where W¡and x¡ are the tidal frequencies and astronomical arguments respectively, associated with M2' S2' and N2. The estimated tides (except N~ look very coherent from track to track. The magnitudes of the M2' S]) and N2 tides are 2.9 cm, 2.4 cm, and 1.9 cm respectively. Along each track the collinear analysis method leaves unknown trends in the tide components, which may explain these low values. However, amplitudes around 10 cm are quite common on long tracks covering 2-3 ocean basins. The results display the complexity of the tides and may valuable in a future utilization of tide gauge data. INTRODUCTION Ocean tide corrections of altimeter data in the Mediterranean Sea have not been available on the standard altimetric data files. Compared with other region the tides in the Mediterranean are quite small. At Naples the amplitudes of the four major constituents: M2' S2' N2, and K¡ are 11.6, 4.3, 2.4, and 2.6 cm respectively (PALUMBO& MAZZARELLA,1982). With the high precision of altimetric products, however, such magnitudes may affect estimates of the mean sea surface and the sea surface height variability significantly (e.g. TROMAS& WOODWORTR, 1990). In this paper the ocean tide signals are analyzed in the Mediterranean Sea using Geosat altimetry from ERM 1-22 (NOV 1986 - NOV 1987) in the form of 2 seconds averages. COLLINEAR ANALYSIS An altimeter derived observation of the sea surface height, h, may be described in terms of the mean sea surface height, ho, the time varying sea surface topography, r" and an error, €, by h = ha + 't + € (1) where the error, e , accounts for the radial orbit errors and other altimetric errors. Along a set of collinear track segments (denoted as the j'th set) height observations hjkl are found at a distance along the tracks of P-k at a time t.. The distance P-k is related to time relative to the time ef e.g. the et}tlllt6f. erossing (-7 .km/s), while the limes t, depend on lhe repeat period 100 HARE NOSTRUH 2 of the satellite ( - 17 days) - e.g. the times of equator crossings. The major differences between the heights of different tracks are caused by radial orbit errors. In order to reduce the effects of the radial orbit errors the tracks may be merged. This was done by removing a trend from each of the tracks, so that the differences between the tracks, are minimized. In this case the trends were modelled using 1 cycle per revolution cosine and sine terms. Such a procedure leaves a common trend in the j'th set of collinear tracks, "i/J-l) , unsolved. Therefore the adjusted heights are expressed as ha = ho + {¡'t + (¡e + €iJ-l) (2) where Órt and ÓE are those parts of tt and E respectively, that have not been removed together with the trends. If the mean values of the heights, eq.(2), in each point along the tracks are subtracted both the mean sea surface and the unsolved trend cancel out. If, furthermore, the residual errors are ignored (except for the noise of the measurements, n) then sea surface height anomalies, Mt(t), are obtained. That is i1h(t) = {¡'t + n (3) AnaIysis of these height anomalies are usually carried out in a single points, where the height anomalies form a time series, or along a single track, where the height anomalies form a profile (FU & ZLOTNICKI,1989). Furthermore, the height anomalies may be analyzed in both dimensions simultaneously. Note (!) that the spectral contents of the data strongly depend on the lengths of the track segments, i.e. values of ótt and not rt are available. EXTRACTION OF OCEAN TIDE SIGNALS The variations of the sea surface heights due to ocean tide may be expressed as (e.g. CARTWRIGHT,1992, MAZZEGA, 1985, and WOODWORTH& THOMAS,1990) N H(t) = L [U¡cos(ú>,t+x¡) + V¡sin(w¡t+X¡)] ¡:1 (4) or N H(t) = L A,{cos(p¡)cos(ú>¡t+X¡)+sin(P)sin(ú>¡t+X¡)] ¡:1 N = L A¡cos(ú>,t+X¡-P) ¡:1 (5) where A¡ = J(U¡2 + v¡2), p¡ = tg -1( ~) (6) are amplitudes and Greenwich phase lags respectively, and Wi and Xi are the tidal frequencies and astronomical arguments associated with the tidal constituents. In an estimation of the ocean tide two surfaces, U;(4),A) and V¡(cJ>,A) in eq.(4), are estimated for each constituent. Knudsen 101 Then amplitudes and phase lags can be computed using eq.(6), so that the ocean tide can be expressed using eq.(5) in terms of amplitudes and phases. In this type of analysis it is assumed that the height anomalies are due to ocean tide. Note (again), the trends cannot be estimated, since they were removed, when the tracks were merged. That is using eq.(3) and eq.(4) 4h(t) = 4ho + aH(t) N = 4ho +L [a U¡eos(ú>l+x¡)+a V,sin(ú>/+x¡)] r=I where - -su, = u¡-UiJ.l), ay¡ = v¡-Y¡iJ.l) (7) (8) are the tidal components relative to their trends along the j'th set of eollinear traeks. In eq. (7) an additional parameter has been introduced. That is a correction term, ilho, which in this case reveals a possible aliasing of the mean sea surfaee height. Before the tidal residual s are estimated Table 1. Aliased periods of ocean tides due to from the Geosat ERM data it is important to the repeat period of the Geosat ground track eonsider the aliased periods of the tidal (from WOODWORTH& THOMAS,1990). constituents due to the sampling of the altimetry. In Table I such periods are listed. The tidal eonstituents treated in this paper (M2' S2' N2, and K¡)have aliased periods of 317 days, 169 days, 52 days, and 175 days respectively due to the repeat period of Geosat. Therefore, a separation of the S2 and the K¡ is impossible (ZLOTNICKI,1992). Here the K¡ tide is omitted, but together with a semi-annual variation it may affect the estimation of the S2tide. Furthermore an annual variation may affect an estimation of the M2 tide, when altimeter data covering one year on1y are used. RESULTS Constituent Aliased Period K¡ 175 days O¡ 113 days Q¡ 74 days p¡ 12.2 years M2 317 days S2 169 days N2 52 days K2 88 days The coverage of the observations were analyzed in order to determine selection eriteria, so that tracks with a too poor sampling (too few repeats) and short tracks can be eliminated. Figure 1 showing a histogram of the sampling shows that in average about one half only of the 22 repeats are available. Most locations are sampled between 9 and 15 times. This makes an estimation of the M2 tide, which has an aliased period of 317 days, uncertain. A histogram of the lengths of each set of collinear tracks is shown in Figure 2. The average length is about 1000 km. In this study sea surface height anomalies are computed at location, where at least 9 of the 22 repeat data are available and tracks shorter than 500 km were deleted. The results show that the estimated M2' S2' and N2 ocean tides have RMS amplitudes When these (U, V) vectors are evaluated >- ~ it is important to have in mind that most of the a tidal signal may have been removed together QJ..t200 with the orbit errors. The problem is sketched in Figure 3. Here one track segment crosses one ocean basin with one tidal (amphidrome) system. On the northern hemisphere the tidal wave (a Kelvin wave) moves with the coast (or \3 equator) on its right hand side. Hence, it \3 5 10 15 moves out of the paper in the left side of the N..urber of repeats profile shown in Figure 3. Since only one Fi 1 H' t f th li h b .. ed f h idal . al gure . IS ogram o e samp ing m eacocean asm IS cover most o t e tí sign . t will appear as a linear trend of the track and, pom . subsequently, most of the signal will be removed when the collinear tracks are merged. Therefore the remaining signal are much reduced and, furthermore, the characteristics have changed, so that both the amplitudes and the phases of the tide may change rapidly along the profile (3 times instead of 1). The above considerations may explain why the RMS values of the estimated tides are so small. A value around 10 cm for the M2 would be more suitable. Actually, on the long tracks covering 2-3 ocean basins several values around 10 cm are found. This is also the case for the S2 tide. The lack of signal is clearly seen on the short tracks in the eastern part of lErqth of tracks (m) the Mediterranean. Figure 2. Histogram of the lengths of the track segments. 102 KARE NOSTRUK 2 of 2.9, 2.4, and 1.9 cm respectively. The individual (U, V) parameters estimated in each point along ascending and descending tracks are shown in Figure 4-9. o -l~------------------------~o 100 '-L, L r }L lr ~ 25 15 r--'l L I n >- ~ 10 ..t 5 \3 \3 1fm 1:021 From track to track the M2 vectors (Figure 4-5) and the S2vectors (Figure 6-7) are coherent. However, some tracks do not correlate at all with its neighboring tracks. This may be due to different data coverage, but annual and semi-annual height variations may playa role too. (In fact only seasonal signals may have been interpreted as tidal signals, see Table 1). The N2 tides appear to be quite small everywhere in the Mediterranean Sea. Figure 3. Sketch showing the tide across a The spectrum of the temporal variations basin and the linear trend that is removed. is shown in Figure 10. Also the spectrum of 40~ 40~ 35~ 30~ ICnudsen 103 20~ 25~ \ 20~15! M2 Ocean Tide from Geosat. A 10 cm vector at (31.6). Figure 4. M2 ocean tide along ascending tracks: (U, V) vectors. M2 Ocean Tide from Geosat. A 10 cm vector at (31.6). Figure 5. M2 ocean tide along descending tracks: (U, V) vectors. l.. \., 104 MARE NOSTRUM 2 35? 30? 20? 25? \ \\. S2 Ocean Tide from Geosat. A 10 cm vector at (31.6). Figure 6. S2 ocean tide along ascending tracks: (U, V) vectors. S2 Ocean Tide from Geosat. A 10 cm vector at (31,6). Figure 7. S2 ocean tide along descending tracks: (U, V) vectors. 40~ 35~ 30~ 40~ 35~ 30~ ICnudsen 105 N2 Ocean Tide from Geosat. A 10 cm vector at (31.6). Figure 8. N2 ocean tide along ascending tracks: (U, V) vectors. 20~ 25~ r I / j .1,, { ! . j ! ( I l~ ~¿. ..•' 'f ! (~'i ¿ I i / l / 15~ 30~ N2 Ocean Tide from Geosat. A 10 cm vector at (31.6). Figure 9. N2 ocean tide along descending tracks: (U, V) vectors. 106 MARE NOSTRUM 2 the variations after the estimated tides have been removed from the height anomalies is shown. The frequencies associated with the aliased periods of M2' S2' and N2 are 1.15, 2.16, and 7.02 cycles/year respectively. The temporal covariance functions are shown in Figure 11. The RMS value of the height anomalies drops from 4.2 cm to 2.6 cm and the temporal correlations vanish. The spectra show that signals associated with 1 and 2 cycles/year approximately efficiently have been removed. Finally, preliminary ERS-l altimeter data were used in an estimation of the M2' which has an aliased period due to the 35 days sampling of about 94 days. The result is shown in Figure 12. This (preliminary) result shown many details that fully agree with the pattern obtained using the Geosat data (Figure 4-5), so the influence of the annual changes on the Geosat M2 estimate appears to be quite limited. DISCUSSION In this paper ocean tide signals have been evaluated in a collinear analysis of altimetry. The magnitudes of the M2' S2' and N2 tides are 2.9 cm, 2.4 cm, and 1.9 cm respectively. The estimated tides (except N2) look very coherent from track to track. However, along each track this analysis method leaves unknown trends in the tide components (eq.(8». Furthermore, annual and semi-annual changes may have aliased the M2' and S2 tides respectively. These problems, however, may in future be solved by integrating ERS-l data. Improved results may be obtained, if parametrizations of the surfaces Vi and V¡ are found, so that the tidal signal can be estimated taking spatial correlations into account, so that ERS-1 and tide gauge data can be integrated properly. Acknowledgement. This study is a contribution to the GEOMED project supported by EEC. The Geosat data were obtained from the Ohio State University and the ERS-l data merged with orbits from DUT/SOM, were obtained from NOAA. 8 G ~ 5 I 3 2 0 0 5 G 8 s 10 Freqo.JalC)l Iqdes/yea..- J Figure 10. Spectra of temporal variations before and after the tidal components have been removed. 20 ,--------------, 15 ~ ia ~ .~ 5 8 e -5~~~~~~~--~~--~~ o 20 ~ ~ 00 ~ m ~ ~ ~ ld;¡ IddysJ Figure 11. Covariance functions of temporal variations before and after the tidal components have been removed. K:nudsen 107 45~ 35~ 40~ l " ERS-1 M2 Tide Test - Mediterranean. Figure 12. Test: M2 estimation from 7 repeats of preliminary 35 days ERS-1 data. REFERENCES CARTWRIGHT,D.E.: Theory of Ocean Tides witñ Application to Altimetry. Lecture notes, lAG lnternational Summer School on Altimetry for Geodesy and Oceanography, Trieste, June, 1992. Fu, L.-L., ANOV. ZLOTNICKI:Observing Oceanic Mesoscale Eddies from Geosat Altimetry: Preliminary Results. Geophys. Res, Lett., Vol. 16, No. 5, 457-460, 1989. KNuDSEN,P., ANDM. BROVELLI:Collinear and Cross-over Adjustment of Geosat ERM and Seasat Altimeter Data in the Mediterranean Sea. Surveys in Geophysics, in press, 1992. MAZZEGA,P.: M2 Model ofthe Global Ocean Tide Derivedfrom SEASAT Altimetry. Marine Geodesy, Vol. 9, No, 3, 335-363, 1985. PALUMBO,A., ANO A. MAZZARELLA:Mean Sea Level Variations and Their Practical Applications. J. Geophys. Res" Vol. 87, No. C6, 4249-4256, 1982. THOMAS,J.P., ANDP.L. WOODWORTH:The Influence ofOcean Tide Model Corrections on Geosat Mesoscale Variability Maps of the North East Atlantic. Geophys. Res. Lett., Vol. 17, No. 13,2389-2392, 1990. WOODWORTH,P.L., ANOJ.P. THOMAS:Determination ofthe Major Semidiumal Tides ofthe Northwest European Continental Shelf From Geosat Altimetry. J. Geophys. Res., Vol. 95, No. C3, 3061-3068, 1990. ZLOTNICKI,V.: Measuring Oceanographic Phenomena with Altimetric Data. Lecture notes, lAG lnternational Summer School on Altimetry for Geodesy and Oceanography, Trieste, June, 1992. GEOID DETERMINATION WITH MASS POINT FREQUENCY DOMAIN INVERSION IN THE MEDITERRANEAN Martin Vermeer Finnish Geodetic Institute Helsinki, December 14, 1992 ABSTRACT We describe some tentative work done to show the possibility of geoid recovery in the Mediterranean using a frequency domain technique based on the representation of the geopotential by buried masses. The technique, which has been described in many earlier publications, is flexible in allowing data input of many types, e.g. gravity anomalíes, disturbances or satellite radar altimetry. In the present study, we use as input gravity anomaly data compiled for us by Prof. Miguel SEVILLA, of the Complutense University in Madrid. As the global reference model to be subtracted we used OSU86F produced by Prof. Richard H. RAPPof Ohio State University. 1. INTRODUCTION In the framework of the European Community financed international project GEOMED we undertook to do some computations on the geoid in the Mediterranean, using as input a 5' x 5' gravity anomaly grid, mostly based on shipboard gravimetry, generated by the group of Prof. SEVILLA (personal communication at the GEOMEDmeeting in Madrid in October 1992). HARE NOSTRUH 2, (1992), pp 109-119 110 KARE NOSTRUK 2 Due to severe time constraints and some difficulty in shifting the software to a different computer system in Helsinki, Finland, the results presented here are rather modesto We used the reference model OSU86F rather than the superior OSU91 A, because only the former was readily available in the form needed here in Helsinki. We decided to do the full computation on the FGI's VAXNMS cluster, even though this meant a rather slow progression, as we knew this system and its quirks and had experience running part of our software on it. Still, we had to Iimit the resolution of our solution to 15' x 30' (while we chose to keep the whole Mediterranean are a) in order to limit the CPU time requirement of the job. We chose to re-grid the data given to us directly to this coarser resolution, using software that is actually intended to be used with point data only. As gravity is a very "rough" quantity on the Earth's surface, this resulted in block average values that were also pretty rough, also because no terrain effects of any kind were used. Even subtracting the global reference field OSU86F cannot be expected to improve this much, as it is, by its nature, global and cannot remove localized, high frequency, variations. 2. THE AREA OF STUDY ANO THE DATA AREA The area of study was the Mediterranean without the Black Sea. A set of gravity data has been compiled by M. SEVILLA(personal comm.) and was made available to USo We prepared a control file for the FFT runs called grid. dsc which is depicted in Table 1. Interesting here are only the quantities MaxLat, MaxLon and Lat and Lon spacing (the latter in minutes of are) which together define the "frame of Table 1. Contents of the control file grid. dsc. This sample file actually is for satellite alti metry . o Surface type 0.000 Surface height 4 1 1 1 1 Obs.type, "UseParts", Attitude laws 5.0E-02 2.0E+00 1.0E+00 O.OE+OO O.OE+OO Obs. std. devs 6.0 1.0 o Months, Secs / Pts. / Diff.quant. 90.000 0.000 Inclination, Alpha 1 Regional solution 5000.000 Delta-H of grid sandwich 55.250 -14.750 MaxLat, MinLon 15.000 30.000 Lat, Ion spacing 128 Dimension of grid 8.000 Trunc. wavelength (degs) 128 Auxiliary grid dimension 1 No. of topographic models Vermeer 111 computation", as follows: Latitude: Longitude: 55.25 - 23.50 -14.75 - 48.75 The spacing is in the latitude direction 15', in the longitude direction 30'. It can be seen from these figures that the model we are working with has 128 node points in both latitude and longitude directions. In our software we have chosen to always use the power-of-two FFT transform, as this offers the greatest effectiveness. Inevitably there is a price to be paid for this: The choice of area bounds is constrained. We have however software to construct an area grid coverage meeting our requirements from arbitrarily distributed point data. 3. GENERATING THE GLOBAL REFERENCE FIELD We generated a reference field for the area of study using the spherical harmonic expansion OSU86F, produced by the group of Prof. R.H. RAPP of Ohio State University. In the mean time a better model, OSU91A, has become available, but within the time constraint of this study, we could only get OSU86F into a useable form quickly. For generating values of nine independent functionals of the geopotential, we used the program legOS. For a more detailed description of this software, cf. BALMINO et al. (1991), or VERMEER (1989). The legOS software produced a grid at a resolution in both directions of 0°.5. From this, a denser grid was produced at the working resolution of 15' x 30', implying a 2 x densification of the grid in the latitude dimension. For this purpose the software frefgrid was used, which densifies a grid by any power of two using a forward/backward FFT technique. More general software of this kind could certainly be written or adapted using cubic spline interpolation; this is one of those projects still in store for the future. The global reference field OSU86F, in the Mediterranean area, is depicted in Fig.1. 112 KARE NOSTRUK 2 Fig.1: The Mediterranean geoid generated by OSU86F. Unit: m. Contour interval: 5 m. 4. GENERATING THE DATA WORK GRID We used a crude technique, which nevertheless produced reasonable results quickly. The set of 5' x 5' values given by us was simply considered primary point data, measured at sea level. As the data was given only at sea, this fiction seems fairly realistic. We fed this data as a stream of point records through our gridding software fgridder, which not only binned the values into our 15' x 30' cells, but at the same time subtracted out the OSU86F gravity contribution, which was interpolated from the grid generated as discussed earlier. In the first working grid thus obtained, only part of the cells had received a value, all of them at sea. This grid was then completed using simple inversion-free prediction by the program ffilEn. Of the eight neighbours of every empty cell, those already containing values were averaged, giving the "diagonal" neighbours half weight. At the same time, these values were "depreciated" by multiplying with the value 0.8, or its square for diagonal neighbours. The value thus obtained was filled into the cell. Table 2: Statistics of the residual gravity anomalies calculated. Minimum: -97.514 mGal Maximum: +84.882 mGal RMS: ±12.115 mGal Vermeer 113 10.0 15.0 20.0 25.0 40! 30~ Fig.2 : Residual gravity in 15' x 30' blocks after subtracting OSU86F contribution. Unit: mGal, contour interval: 5 mGal. This procedure was repeated a number of times, until the "edge values" started approaching zero. After that, the remaining missing values were simply zeroed. We give the residual gravity values thus obtained with respect to OSU86F in Fig. 2. The statistics of these residuals, for the central quadrant of the area of study, are given in Table 2. 5. THE BURIED MASSES REPRESENTATION In our method, the external geopotential is represented by a grid of buried masses, one mass being located in the centre of every 15' x 30' grid cell. Note that the grid cel! centres are always in "even" locations with respect to the 0°.5 resolution grid nodes, which again are in latitudes and longitudes like 0°.25, 0°.75, etc.; this means that the grid cel! boundaries are in odd locations. This is again one of those things that will improve once we have a more flexible way of interpolating reference model values to data grid node locatlons. The depth of the mass points was kept constant at 60 km. This depth is proportional to the grid cell linear size, a principie that makes physical sense, cf. HEIKKINEN (1981). The coefficients between each mass point and each observation type at the Earth's surface can be derived by the application of Newton's law of gravitation, 114 KARE NOSTRUK 2 considering the nature of the observation type. The general coefficient is a function of the difference in longitude and (approximately) the difference in latitude only between mass point and data location, Le. it describes a (discrete) convolution. 6. GENERATINGTHE COEFFICIENTS 6.1. The transfer coefficients Our system is built in such a way, that first coefficient sets are generated for a set of nine standard observables: The disturbing potential, three components of the gravity disturbance vector, and five independent disturbing gravity gradient tensor components. For the true observables then, which are linear combinations of these nine, a set of transfer coefficients is then derived and sto red on ASCII file for future reference. Matrix multiplication then yields the final set of coefficients for the actual observables used. This approach has the elegance of generality. It makes it also easy, using the same transfer coefficients, to derive simulated true observables, or (as was employed by us) the contribution to the observable used by the OSU86F global reference model, which is used in both the remove (gravity anomaly) and restore (geoid undulation) step. More about the transfer coefficients, and a listing of those for gravity anomalies and for geoid undulations, is given in VERMEER(1992a). 6.2. Generating the observation equation coefficients The observation coefficient matrix is generated using the program fsyncoef, as diseussed using transfer coefficients from synth. The standard observable coefficient matrix was generated by fcoef, in five different locations (height, latitude combinations) to allow iterative interpolation taking both the topography and the Earth's curvature into account ("Spherical FFT", cf. VERMEER,FORSBERG (1992)). The possibility to perform such an iteration was not used, however, in order to save time; the error made in doing so is small. In this case fsyncoef can be made to extraet only the coefficients for reference latitude (mean latitude of area of computation) and height level O. Vermeer 115 7. APPL YING THE FAST FOURIER TRANSFORM We inverted the system of equations by, for every frequency domain constituent, dividing the data value by the coefficient value. In case there are several data values for every grid cell, this division takes the form of the solution of a small system of normal equations; here, with only one observable, it degenerates to a simple division. There are 1282 = 16384 such divisions to be made, Le. one for every point in the area. Note that this makes the operation cheaper than the FFT transform itself, which always requires n2 21nn operations, n = 128 being the linear size of the area. The simplicity of the technique is a direct consequence of the convolution theorem of Fourier theory, cf. Vermeer. 8. INVERSION WITH REGULARIZATION It is generally necessary to regularize the problem before a reasonable solution can be obtained. This is obvious in case of downward continuation, such as airborne gravimetry: The high frequency constituents of the geopotential cannot be recovered from measurements at altitude, and thus must be fixed more or less arbitrarily to find a solution and prevent the system of equations from going singular. Typically one uses a priori information on the allowed range of values of some functionals of the solution. However, also in inversions of our kind it is necessary to apply constraints of this kind, mainly beca use of the differing spectral behaviour of gravity and geoid undulations. Some parts of the spectrum of the geopotential are poorly estimable because of this. In our case we constrain both the mass point surface density and its horizontal gradient to geophysically "reasonable" values. We chose a value of ±1000 mGal for the a priori standard deviation of the mass surface density of our mass point "blanket" (remember, we work with values GM, which, divided by surface area o = lb, have the same dimension as Newton's acceleration GMr-2, and can thus be expressed in mGal!). For the horizontal gradient we chose ±108 E, Le. in practice infinite, so this constraint was not applied at all. We found for these parameters a "gain factor" (Le. the amount of original gravity signal propagating into the solution) of 56%, a little on the low side. 116 KARE NOSTRUK 2 9. TAPERING ANO FILTERING One other technique which must be used to produce acceptable results is tapering. With this is meant the smooth transition to zero of the residual input data by multiplying with a function window going smoothly to zero at its edges. We used a cubic taper with this behaviour for an edge width of 8°. At the same time, a frequency filter was applied to the residual solution, clipping off all frequency content below a limit corresponding to this 8°. Cf. VERMEER (1992a) for technicalities. This is not necessarily the best way to handle the problem of low frequency content: Others prefer to simply remove a bias term from the residual solution. The choice depends on a judgement on the quality of the high-frequency part of the global model before proceeding to apply Fourier techniques. 10. PREOICTING GEOIO UNOULATIONS We predict geoid undulations from the mass point solution found by multiplying with the coefficient matrix for satellite radar altimetry, with the program fpredict. In this way we obtain accurate geoid undulations expressed in m. Fig.3. Residual geoid undulations predicted from FFT solution. Unit: m, contour interval 0.25 m. Vermeer 117 We refer to Fig. 3 for our results. The statistics found for these residual geoid undulations are summarized in Table 3, again 10r the central quadrant 01the area of study. Table 3: Statistics of the residual geoid undulations predicted. Minimum: -2.961 m Maximum: +1.301 m RMS ±0.357 m 11. RESTORING THE TOTAL GEOID After computing the residual geoid, it should be added again to the geoid implied by the OSU86F global reference mode!. First we have to generate from the OSU86F standard observables grid file generated above, a grid file containing geoid undulations. We do this by multiplying with the satellite altimetry coefficient matrix in the program fsynobs. After that, the global and local contributions are added together by faddrhs. The result is shown in Fig. 4. • 0 Fig.4. Total geoid undulations, Le. predictions from FFT solution added on top 01the OSU86F contribution. Unit: m, contour interval5 m. 118 MARE NOSTRUM 2 12. RESULTS ANO OISCUSSION It can be seen from Fig. 3 that the residual geoid undulations found are large in some places, e.g. in the neighbourhood of Crete. They are expected to be large around Crete, where there is a very strong gravity signature with sharp edges having a strong high-frequency content. Then we see a number of extrema in the residual geoid located along the coast (Algeria) and the coasts of islands. This could also be the result of strong "edges" in the gravity field occurring preferentialiy in such places. It is also possible that, in extrapolating the sea-only gravity data into land, applied a too strong "depreciation factor" leading to too sharp edges. It is also seen that the effect of adding the local FFT solution to the OSU86F geoid is only barely visible in the figures. It wili be necessary to experiment further, both using better global reference models (OSU91 A), and using more refined data processing (gridding) techniques. At the very least we should be able to use the 5' x 5' data at its original resolution. Also proper regularization parameters for that resolution should be established by numerical experimentation. ACKNOWLEDGEMENTS We gratefuliy acknowledge the use of the gravity data compiled by M. SEVILLAof the Institute for Astronomy and Geodesy, Complutense University, Madrid, covering the Mediterranean at a resolution of 5'. We also gratefuliy acknowledge the use of OSU86F, a spherical harmonic expansion produced by R.H. RAPPof Ohio State University. The present work was done in the framework of European Community contract no. SC1 *CT92-0808 ("GEOMEO")and constitutes a first, preliminary demonstration of abilities. Financial support by the EC is gratefuliy acknowledged. Finally we are happy to acknowledge the support by the Danish National Survey and Cadastre, where the author developed part of the software used in this study and who also supported in part his travels in connection with GEOMEO. REFERENCES BALMINO, G. , J. BARRIOT, R.Kooe, B. MIDDEL, N.C.THONG, M. VERMEER (1991): Simulation of gravity gradients: a comparison study. Bulletin Géodésique 65:218-229. Vermeer 119 HEIKKINEN,M. (1981): Solving the shape 01 the Earth by using digital density models. Finnish Geodetic Institute, Report 81:2, Helsinki. VERMEER,M. (1989): FGI Studies on Satellite Gravity Gradiometry. 1. Experiments with a 5- degree buried masses grid representation. Finnish Geodetic Institute, Report 89:3, Helsinki. VERMEER,M. (1992a): FGI Studies on Satellite Gravity Gradiometry. 3. Regional high resolution geopotential recovery in geographical coordinates using a Taylor expansion FFT technique. Finnish Geodetic Institute, Report 92:1, Helsinki. VERMEER,M. (1992b): Exploiting Symmetry 10r Fast Inversion - The case 01 geophysical gravity inversion. Proc. Interdisciplinary Inversion Workshop 1, Aarhus, ed. B.H. Jakobsen, GeoSkrifter 41, 93-98. VERMEER,M., R. FORSBERG(1992): A generalised Strang van Hees approach to 1astgeopotential inversion. Manuscripta geodaetica 17: 302-314. M.A. Andreu, C. Simó Dept. Matemática Aplicada i Análisi Universitat de Barcelona Avda. Gran via de les Corts Catalanes, 585 E - 08007 Barcelona, Spain CATALAN GEOID 91: SUMMARY OF RESULTS In this paper we summarize some results of the computation of a gravimetric geoid in Cata- lonia [Andreu, Simó, 1992]. We have used the well known method of Least Squares Collocation (LSC) [Moritz, 1980] [Tscherning, 1984] and the remove-restore technique [Forsberg, Tscherning, 1981]. 1 Data used for the geoid computation 1) Spherical harmonic coefficients: as a first approximation of the gravity field, we took the spherical harmonic expansion corresponding to the coefficients set OSU89B up to degree 360 [Rapp, 1990]. This is a good approximation of the gravity field in Catalonia. The figure 1 shows a geoid computed using this expansion and Bruns formula. 2) Topographic data: presently, two digital terrain models (DTM) are availables: • A detailed grid of 15" x 15" from Institui Cartogrdfic de Catalunya (ICC). This model originally comes from the Defence Map Agency and it covers the area shown in the figure 2. • A coarse grid of 5' x 5' mean heights from NGDC. This grid is part of the ETOP05 model and it was provided by R.Forsberg. We found that this model needs to be shifted 7'.5 in longitude and -2'.5 in latitude. After this shift, both models have a good agreement. 3) Gravity anomaly data: in Catalonia there are a lot of gravity measurements covering all the land. We have used the data from Servei Geoloqic de Catalunya (SGC) as the main set and we have completed this set with data from Bureau Gravimétrique International. We have selected 1718 gravity anomalies (85 % from SGC) close to a 2'.5 x 2'.5 grid. The figure 3 shows the distribution of the selected gravity anomalies. 2 Terrain corrections and residual gravity anomalies In order to compute the terrain corrections we have used a residual terrain model (RTM). The reference surface was computed averaging blocks of 35' x 35' of the ETOP05 terrain model. In the computation of the effect of the RTM on the gravity anomaly it is only necessary to take into account the topography up to a distance of 40 Km from each station. The table 1 shows the statistics of the gravity anomaly when we subtract the contribution of the spherical harmonic expansion and the effect of the RTM. KARE NOSTRUK2, (1992), pp 121-131 122 HARE NOSTRUH 2 43.0 42.5 42.0 41.5 41.0 40.5 40.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 1: Geoid (m) using only spherical harmonic expansión OSU89B up to degree 360. Andreu and 51m6 123 4673700 4573700. 00 Figure 2: The detailed terrain model covers the area not marked. Latitude and longitude in UTM coordenates. 02 03 4473700 +- .-~--~~~~~~~-~~~~ 255050. 355050 455050 124 MARE NOSTRUI! 2 43.0 42.Q 41.0 40.0 0.0 1.0 2.0 3.0 4.0 Figure 3: 1718 gravity anomaly selected close to a 2/.5 X 2/.5 grid. Andreu and Slmó 125 Mean Var. Min. Max, mga/ mga/2 mga/ mga/ ó'gobs 10.39 980.63 -70.53 210.70 ó'gobs - ó'go -7.83 864.86 -122.34 143.55 ó'gobs - ó'gO - ó'gm 4.50 133.52 -27.85 43.01 Table 1: Statistics of the residual gravity anornaly. ó'gobs: observed gravity anomaly, ó'go: gravity anornaly using spherical harmonic expansión. ó'gm: contribution of RTM using detailed DTM up to 20 km and coarse DTM up to 40 km. But, we have observed that residual gravity anomaly and topographic height are correlated. The regression line (figure 4) is the following: ó'g = 0.60 + 8.39 . 10-3 H (ó'g in mgal and H in m). In order to avoid this trend, we modified the reference surface multiplying each height by a: m a=l--- 0.1119 ' where m is the slope of the regression line. The statistics of the new residual gravity anornalies are shown in the table 2. Mean Varo Min. Max. mga/ mgal2 mga/ mgal ó'gobs - ó'go - ó'gmm -0.62 103.82 -37.91 30.99 Table 2: Statistics of the last residual gravity anomaly. ó'9mm: contribution of RTM using the modified surface. 3 Covariance model After removing the contribution of spherical harrnonic coefficients and the effect of RTM, we have the residual gravity anomalies, We can compute the empirical covariance function in the usual way [Knudsen, 1987] averaging products of these gravity anomalies. The following step in our computation was to fit the covariance model to the empirical covariance function using the method exposed in [Knudsen, 1987]. The covariance model used was the following: 360 (R2)n+l 00 A J((P,Q)=a2:=(j~e) -, Pn(cos'I/J)+ 2:= ( )( )( ) n=2 r r n=361 n - 1 n - 2 n + 4 where (j~e) are the error degree-variances related to the potential coefficients set, a is a factor which represents the quality of the approximation of the potential coefficients set, R is the mean 126 KARE NOSTRUI! 2 S.A. 60. . , ~. ~. 30. H Figure 4: Relationship between ;",;f';;3í gravity anom aly and height. Gravity anomaly in mgal and topographic height (H) ir, t.r. o. -30. o. 1000. 2000. 3000. Andreu and 51m6 127 earth radius, r and r' are the radial distances of P and Q, RB is the radius of a Bjerhammar sphere, Pn are the Legendre polynomials and 1/J is the spherical distance between P and Q. The result of the estimation of the three parameters was the following: a = 0.3751, RB - R = -459 m, Varo = 103.90 mgal", 4 Predictions After fitting the covariance model, we were able to make predictions of sever al quantities re- lated with gravity potential as gravity anomalies, geoid undulations and deflections of the ver- tical. For this purpose, we used the Least Squares Collocation method (GEOCOL program, C.C.Tscherning). The input data were the selected gravity anomalies, and the covariance model was the one estimated in the preceding section. 1) Gravity anomaly: we compared gravity anomaly observed and predicted in 33 stations (not used as input data). The results of this comparison are shown in the table 3. Obs. Pred. Dií. Mean 10.93 10.65 0.28 St.Dv. 17.86 17.81 2.39 Table 3: Comparison oí observed ancl predicted gravity anomalies (mgal) in 33 stations. 2) Geoid undulations: we computed geoid heights on a grid in the zone oí latitude between 40° and 43°, and longitude between 0° and 3°.5. The geoid undulations were compu ted adding the contribution oí: • Spherical harmonic expansion OSU89B up to degree 360. • LSC (min.: -0.66 m, max.: 0.69 m). • RTM (min.: -1.56 m, max.: 2.32 m). This effect was computed using a fixed area: longitude [-2°,5°.5] and latitude [38°,45°]. The figure 5 shows the level curves ofthe geoid using the ellipsoid of reference WGS84. The figure 6 shows the absolute error oí the geoid estimated by LSC. But we were interested in the relative error, this means the error in the difference of the geoid undulations between two points. For this reason, we estimated the error using a fix point (see figure 7) and we found that the relative error is about 10 cm per 100 km. 3) Deflections oí the vertical: we also compared the computed value of 8 deflections of the vertical with a preliminary determination of this deflections. The differences between both values show: Mean 0".2, St.Dv. 0".8. 128 KARE NOSTRUK 2 43.0 42.5 42.0 41.5 41.0 40.5 47.00---------1 40.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 5: Catalan Geoid UB91 (m). Ellipsoid of reference WGS84. Min.: 46.12 m. Max.: 54.98 m. Andreu and 51mó 129 43.0 42.5 42.0 41.5 41.0 40.5 40.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 6: Error estimates of geoid heights computed by LSC. Min.: 0.13 m. Max.: 0.35 m. 130 KARE NOSTRUH 2 42.75 42.25 41.75 41.25 40.75 0.75 1.25 1.75 2.25 2.75 Figure 7: Error estimates of geoid heights using a fix point. Relative error: 10 cm per 100 km. 5 Conclusions and outlook We are able to make predictions of: • Gravity anomaly with an error of 2 mgal. • Geoid height with an absolut error of 13-20 cm and a relative error of 10 cm/IOO km. • Deílcctions of the vertical wi th and error of 0".8. lmprovements: The next step in our computations will probably iuelude: • DTM with higher resolution (perhaps 3" ~ 100 m). • Density data of the crust. • Gravimetric data at the sea near the coast. • Defiections of vertical. • GPS data. Andreu and Simó 131 Acknow ledgements The authors are grateful to several institutions and persons which make this work possible: ICC, C.C.Tscherning, R.Forsberg, A.Casas and G.Balmino. References [1] Andreu, M.A., Simó, C.: Determinació del geoide UB91 a Catalunya, Monografies técni- ques n. 1, Institut Cartográfic de Catalunya (1992). [2] Forsberg, R., Tscherning, C.C.: The use of Height Data in Gravity Field Approximation by Collocation, Journal of Geophysical Research, vol. 86, n. B9, pp. 7843-7854 (1981). [3] Knudsen, P.: Estimation and Modelling of the local empirical covariance function using gravity and satellite altimeter data, Bulletin Geodésique, vol. 61, pp. 145-160 (1987). [4] Moritz, H.: Advanced Physical Geodesy, 2°" ed., Wichmann (1989). [5] Rapp, R.H., Paulis, N.K.: The developement and analysis of geopotential coefficient mod- els to spherical harmonic degree 360, Journal of Geophysical Research (1990). [6] Tscherning, C.C.: Local approximation of the gravity potential by least squares colloca- tion. Summer School on Local Gravity Field Approximation, Beijing, China, ed. by K.P. Schwarz, University of Calgary (1984). PUBLICACIONES DEL INSTITUTO DE ASTRONOMIA y GEODESIA DE LA UNIVERSIDAD COMPLUTENSE - MADRID (Antes Seminario de Astronomía y Geodesia) l.-Efemérides de 63 Asteroides para la oposición de 1950 (1949). 2.-E. PAJARES:Sobre el cálculo gráfico de valores medios (1949). 3.-1. PENSADO:Orbita del sistema visual cf U Maj (1950). 4.-Efemérides de 79 Asteroides para la oposición de 1951 (1950). 5.-J. M. TORROJA:Corrección de la órbita del Asteroide 1395 "Aribeda" (1950). 6.-R. CARRASCOy J. M. TORROJA:Rectificación de la órbita del Asteroide 1371 "Resi" (1971). 7.-1. M. TORROJAy R. CARRASCO:Rectificación de la órbita del Asteroide 1560 (1942 XB) y efemérides para la oposición de 1951 (1951). 8.-M. L. SIEGRIST:Orbita provisional del sistema visual ;2 728-32 Orionis (1951). 9.-Efemérides de 79 Asteroides para la oposición de 1952 (1951). 10.-J. PENSADO:Órbita provisional de ;2 1883 (1951). 1l.-M. L. SIEGRIST:Orbita provisional del sistema visual ;2 2052 (1952). 12.-Efemérides de 88 Asteroides para la oposición de 1953 (1952). 13.-1. PENSADO:Orbita de ADS 9380 = ;2 1879 (1952). 14.-F. ALCÁZAR:Aplicaciones del Radar a la Geodesia (1952). 15.-J. PENSADO:Orbita de ADS 11897 = ;2 2438 (1952). 16.-B. RODRÍGUEZ-SALlNAS:Sobre varias formas de proceder en la determinación de perío- dos de las marcas y predicción de las mismas en un cierto lugar (1952). 17.-R. CARRASCOy M. PASCUAL:Rectificación de la órbita del Asteroide 1528 "Conrada" (1953). 18.-J. M. GONZÁLEZ-A.BOIN:Orbita de ADS 1709 = ;2 228 (1953). 19.-1. BALTÁ: Recientes progresos en Radioastronomía, Radiación solar hiperfrecuente (1953). 20.-J. M. TORROJAy A. VÉLEZ: Corrección de la órbita del Asteroide 1452 (1938 DZ,) (1953). 2 l.-J. M. TORROJA:Cálculo con Cracovianos (1953). 22.-S. AREND:Los polinomios ortogonales y su aplicación en la representación matemática de fenómenos experimentales (1953). 23.-J. M. TORROJAy V. BONGERA:Determinación de los instantes de los contactos en el eclipse total de Sol de 25 de febrero de 1952 en Cogo (Guinea Española) (1954). 24.-J. PENSADO:Orbita de la estrella doble ;2 2 (1954). 25.-1. M. TORROJA:Nueva órbita del Asteroide 1420 "Radcliffe" (1954). 26.-J. M. TORROJA:Nueva órbita del Asteroide 1557 (1942 AD) (1954). 27.-R. CARRASCOy M. L. SIEGRIST:Rectificación de la órbita del Asteroide 1290 "Alber- tine" (1954). 28.-J. PENSADO:Distribución de los períodos y excentricidades y relación período-excen- tricidad en las binarias visuales (1955). 29.-J. M. GONZÁLEZ-ABOIN:Nueva órbita del Asteroide 1372 "Harernari" (1955). 30.-M. DE PASCUAL:Rectificación de la órbita del Asteroide 1547 (1929 CZ) (1955). 31.-1. M. TORROJA:Orbita del Asteroide 1554 "Yugoslavia" (1955). 32.-1. PENSADO:Nueva órbita del Asteroide 1401 "Lavonne" (1956). 33.-1. M. TORROJA:Nuevos métodos astronómicos en el estudio de la figura de la Tierra (1956). 34.-D. CALVO:Rectificación de la órbita del Asteroide 1466 "Mündleira" (1956). 35.-M. L. SIEGRIST:Rectificación de la órbita del Asteroide 1238 "Predappia" (1956). 36.-1. PENSADO:Distribución de las inclinaciones y de los polos de las órbitas de las es- trellas dobles visuales (1956). 37.-J. M. TORROJAy V. BONGERA:Resultados de la observación del eclipse total de Sol de 30 de junio de 1954 en Sydkoster (Suecia) (1957). 38.--ST. WIERZBINSKI: Solution des équations normales par I'algorithme des cracoviens (1958). 39.-1. M. GONZÁLEZ-ABOIN:Rectificación de la órbita del Asteroide 1192 "Prisma" (1958). 40.-M. LóPEZ ARROYO: Sobre la distribución en longitud heliográfica de las manchas so- lares (1958). 41.-F. MÚGICA: Sobre la ecuación de Laplace (1958). 42.-F. MARTÍN ASÍN: Un estudio estadístico sobre las coordenadas de los vértices de la triangulación de primer orden española (1958). 43.-ST. WIERZBINSKI: Orbite améliorée de h 4530 = {'Cen = Cpd -48', 4965 (1958). 44.-D. CALVOBARRENA:Rectificación de la órbita del Asteroide 1164 "Kobolda" (1958). 45.-M. LóPEZ ARROYO: El ciclo largo de la actividad solar (1959). 46.-F. MÚGICA: Un nuevo método para la determinación de la latitud (1959). 47.-1. M. TORROJA: La observación del eclipse de 2 de octubre de 1959 desde El Aaiun (Sahara) (1960). 48.-J. M. TORROJA,P. JIMÉNEZ-LANDly M. SoLÍs: Estudio de la polarización de la luz de la corona solar durante el eclipse total de Sol del día 2 de octubre de 1959 (1960). 49.-E. PAJARES: Sobre el mecanismo diferencial de un celóstato (1960). 50.-1. M. GONZÁLEZ-ABOIN:Sobre la diferencia entre los radios vectores del elipsoide in- ternacional y el esferoide de nivel (1960). 51.-1. M. TORROJA: Resultado de las observaciones del paso de Mercurio por delante del disco solar del 7 de noviembre de 1960 efectuadas en los observatorios españoles (1961). S2.-F. MÚGlCA: Determinación de la latitud por el método de los verticales simétricos (1961). 53.-M. LÓPEZ ARROYO: La evolución del área de las manchas solares (1962). 54.-F. MÚGICA: Determinación simultánea e independiente de la latitud y longitud me- diante verticales simétricos (1962). 55.-P. DiEZ-PICAZO: Elementos de la órbita de la variable eclipsante V 499 Scorpionis (1964l. 56.-J. M. TORROJA: Los Observatorios Astronómicos en la era espacial (1965). 57.-F. MARTÍN ASÍN: Nueva aportación al estudio de la red geodésica