AD-A276 899 N PAGEN -- Im EEEEEEE m~~~dn~~~d (cid:127)~.8 emnMlpUid UICm Bm. WiieaNroltndV.t l ftM3mb oden or W1,t C =aeenp thwi aos0m40U1n al WWk0W Wvbmn,c kduclnqle agw Im m. umulon wojenp asn ad Repons. 1216 Jsllmmw DarviA IVmay. Suls 120e. Astilton. VA 2=432. anti 10 the O!t Natniwi. DC 20606 1. Agency Use Only (Leave blank). 2. Report Date. 3. Report Type and Dates Covered. 20 January 1994 Contractor Report 4. Title and SubiUte. 5. Funding Numbers. The Effects of Altimeter Sampling Characteristics: Some Geosat Examples Co, ad N00014-92-J-6004 ProgramE /wrenr No. 0601153N 6. Author(s). ProjecAN. 3208 Michael E. Parke* and George Bom* Task NO. 31-03-4A Accession No. DN250131 Wo,* Unit No. 73505603 7. Performing Organization Name(s) and Address(es). 8. Performing Organization *The Regents of The University of Colorado Report Number. 380 Administrative Center Boulder, CO 80309-0019 9. Sponsoring/Monitoring Agency Name(s) and Address(es). 10. Sponsoring/Monitoring Agency Naval Research Labot itory Report Number. Ocean Technology Branch D wfl C C.9 Stennis Space Center, MS 39529-5004 NRIJCR/7320--93-0004 11. Supplementary Notes. 12s. DistrlbutlowAvallabillty Statement. " 12b. Distribution Code. Approved for public release; distribution is unlimited. 13. Abstract (Maximum 200 words). Altimetric satellites have characteristic sampling patterns in both space and time. This paper looks at the distortions of oceanographic and atmospheric signals as seen in geosat altimeter data. Because of the pattern of ground tracks lain down by Geosat, measurements of oceanographic and atmospheric phenomena can be distorted spatially as well as temporally. As a result, phenomena measured by altimetric measurements can appear as propagating waves with both wavelength and wavenumber different from the original phenomena. These changes, if not understood, can result in misinterpretation of results from altimeter data. Also, in some parts of the world, distorted signals satisfy the dispersion relation of other oceanographic waves, or two phenomena can both have the same distorted dispersion relations. A paper describing the results of this research is in preparation and will be submitted to the Journal of Geophysical Research -Oceans. 14. Subject Terms. 15. Number of Pages. Altimetry, mesoscal oceanography, ocean forecasting 28 16. Price Code. 17. Security Classification 18. Security Classification 19. Security Classification 20. Umltatlon of Abstract. of Report. of This Page. of Abstract. Unclassified Unclassified Unclassified SAR NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) Prescrbed by ANSI Sid. Z39-18 296-102 The Effects of Altimeter Sampling Characteristics: Some Geosat Examples Final Report to the Naval Research Office Stennis Space Center, Mississippi ArcceT-jOzi For Michael E. Parke NTIS ICRA&I- U I'IC iAlB George Born U d~i:ou..ced 0 .j,,titicatiofl .. ... ..-... - By ........... CCAR, Universityj of Colorado D~jýt. ibution I AvailabilitY Codes Avail and I r January 20, 1994 Dist special Approved for public release; distribution is unlimited. 94--07990 94 3'10 055 Abstract Altimetric satellites have characteristic sampling patterns in both space and time. This paper looks at the distortions of oceanographic and atmospheric signals as seen in Geosat al- timeter data. Because of the pattern of ground tracks lain down by Geosat, measurements of oceanographic and atmospheric phenomena can be distorted spatially as well as temporally. As a result, phenomena measured by altimetric measurements can appear as propagating waves with both wavelength and wavenumber different from the original phenomena. These changes, if not understood, can result in misinterpretation of results from altimeter data. Also, in some parts of the world, distorted signals satisfy the dispersion relation of other oceanographic waves, or two phenomena can both have the same distorted dispersion rela- tions. A paper describing the results of this research is in preparation and will be submitted to the Journal of Geophysical Research - Oceans. 1 Introduction Despite the phenomenal coverage supplied by altimetric satellites, the measurements of a single satellite are not sufficient to provide a synoptic picture of many atmospheric and oceanographic features. Temporal aliasing (particularly of tides) at a single point has been discussed in other papers (see e.g. Parke et al. 1987, Cartwright and Ray, 1990). In fact, much of the discussion of sampling characteristics has been in terms of temporal aliasing. However, the sampling characteristics of altimetric sampling produce spatial distortions that do not occur in traditional aliasing. The differences in how an altimetric satellite samples the world occur because the world is not sampled as a snapshot, but rather as a sequence of individual tracks lain down from turning latitude to turning latitude. Thus, in addition to the time between repeats (important for traditional aliasing), the time between adjacent tracks becomes important. This will be discussed more in the following paper. It is important to note, however, that the effects of measuring a given phenomena will be different for different orbit parameters, and in particular can be radically different even for satellites with the same repeat period. The spatial distortions introduced by altimetric sampling mean that measured phenomena can satisfy different dispersion relations than the underlying phenomena. Because of this, it is possible for some aliased phenomena to satisfy the dispersion relation of other non-aliased oceanographic phenomena or of other aliased phenomena in some parts of the world. Thus, in addition to having aliased phenomena sometimes exhibit unexpected wavelengths, it is possible that in some parts of the world analysis for one type of phenomena will incorporate energy from another type of phenomena, even using knowledge of the dispersion relation. The purpose of this paper, therefore, is to discuss the spatial as well as temporal charac- teristics of aliased oceanographic and atmospheric signals, using the Geosat Exact Repeat Mission (ERM) as an example. These results will be used to illustrate some of the possi- ble confusions that can arise and in what parts of the world that the measured dispersion relations of different phenomena coincide. The following discussion is divided into six sections: the pattern of ground tracks for Geosat, three examples of the effect of sampling generic aliased signals, discussion of the effect of sampling non-aliased signals, discussion of sampling some specific oceanographic signals, examples of the possible confusions between measured phenomena, and a general discussion. ! P 1 2 Geosat Ground Tracks ERM ground tracks are lain down in a pattern that is primarily a function of the orbit inclination and the length of the repeat period. The purpose of this section is to describe the time sequence by which independent ground tracks within a repeat period are filed in. On each day, ground tracks will be lain down at evenly spaced intervals (the fundamental interval) toward the west. Ground tracks on successive days will fill in the intervals between the the tracks lain down the first day. For repeating orbits such as the orbit of the ERM, the longitudinal spacing (S) between sequential orbit tracks is equal to the nodal period (P,,) times the difference between the inertial rotation rate of the Earth (WE) and the precession rate of the line of nodes (a); 27rD (1) where (N) is the integer number of orbital revolutions and (D) the integer number of nodal days before the ground track repeats. A nodal day is that period between recurrence of the ascending node of the satellite orbit over the same Earth-fixed meridian. Because the precession of the node is much slower than the Earth's rotation rate, a nodal day differs only slightly from a calendar day. All references to repeat days in the rest of this paper will refer implicitly to nodal days. Note that an east positive coordinate system has been adopted, e.g. the nodal precession rate 1 is positive (easterly) for retrograde orbits. During a repeat period, a complete grid of ground tracks will be produced. The time sequence by which this grid of ground tracks is filled in is determined by the decimal part, q, of the ground track parameter, Q, defined by Q = NID = I + q where I is an integer and q < 1. For repeating orbits, q must be a rational number. If the fundamental interval is divided into k equal parts (called sub-intervals) where k is the irreducible denominator of q (in this case k represents the repeat period in nodal days), then subsequent tracks within the interval will occur at the sub-intervals with an order determined by the numerator of q. If the numerator is one, then each subsequent track in the interval (at time intervals of approximately one day) will be immediately to the east of the preceding track for bath prograde and retrograde orbits. If the numerator is two, then the second track will be two sub-intervals to the east. If it is three, the second track will be three sub-intervals and so forth. Subsequent tracks are filled in in a cyclical fashion. For the GRM mission, D = 17, while N = 244. Thus Q = 141 and q = This leads to -. the ground tracks being filled in as in Table I. As can be seen neighboring ground tracks are lain down successively to the east approximately every three days. This pattern is important to the way many sampled phenomena will appear in ERM data. This will be discussed more in the next section. 2 3 Three Examples of Geosat Sampling The effect of sampling oceanographic and atmospheric phenomena can be illustrated by looking at three special cases. First suppose that the underlying phenomena is composed solely of a sinusoidal variation in time, but constant phase horizontally, i.e. h = elfft where h is the amplitude of the underlying phenomena while a is the angular frequency of the underlying phenomena. This situation corresponds roughly to the aliasing of very broad scale phenomena where the distance between adjacent ground tracks is much less than the wavelength of the measured phenomena. Because the amount of time for a satellite to measure a complete ascending or descending track (,-50 minutes for the ERM) is small compared to the time scales of typical oceanographic variations, adjacent ground tracks may be considered to be sampled at a constant time difference apart for purposes of this discussion. Thus if at a given point on a given ground track the underlying phenomena is sampled at times tj, the equivalent point on the next track to the east will be sampled at times tj + At. In other words, the measurements at the given point will be e't' while the measurements on the track to the east will be eW'(ti+At). If one adopts a local coordinate system aligned with the ground tracks such that y is aligned with the ground tracks while x is normal to the ground tracks, then given a ground track separation of d, the measurements at the adjacent track can be considered as ewt+jikRX where kR = aAt M. Thus, the result of sampling a non-propagating phenomena may be considered as a propagating wave with horizontal wavelength AR. There are then two basic parameters that need to be determined in order to determine the result of sampling a simple signal like this; the change in phase of the underlying phenomena between successive neigboring tracks, Ar-R, and the change in phase between successive repeat tracks, AKT. Because ground tracks effectively represent the shape of wave crests, the direction of propagation will be at an angle P to the meridion that is normal to the ground tracks, and thus be a function of latitude and whether one is considering ascending or descending ground tracks. The direction of propagation of the measured wave may be calculated simply by considering that the normal to the ground track is perpendicular to the velocity vector at the measured location. Consider the angle, -y, between a satellite ground track and north (i.e. a meridian) as shown in Figure 1. The angle, y, is the angle between the meridian and the satellite velocity vector seen by an observer fixed to the Earth. This angle is found by subtracting the Earth's surface rotational velocity vector from the satellite velocity vector projected on the Earth's surface. For a circular orbit the angle, -y, is given by [Parke et al. (1987)] as -y =tan-1 abs (VssinVa cos )Cos (2) 3 where the two terms in the numerator are added for retrograde orbits and subtracted for prograde orbits, VS is the spacecraft velocity VS LE(3) and sina = abs (csi (4) where i is the inclination of the satellite orbit and phi is latitude. The quantity Vg is the equatorial rotation velocity of the Earth given by REWs. The wavelength, X, of the apparent wave is determined by the distance between ground tracks in the normal direction, and the phase change between adjacent tracks, AxR. The distance between ground tracks, d, is approximately given by d = S sin(/3) (5) and so the wavelength of the apparent wave is given by [7r AR=abs d] (6) Figure 2 shows the variation in d with latitude. As one follows a ground track towards the turning latitude, the direction of propagation of the apparent wave tends towards north- south, and the wavelength tends towards zero. The direction of propagation of the apparent wave may be either along the easterly normal or along the westerly normal. This may be determined as follows. Given the east positive coordinate system adopted here, if ALIR and Ar-T are of the same sign then the measured phenomena will appear as a westwardly propagating wave. If AL.R and Ar.T are of opposite signs then the measured phenomena will appear as an eastwardly propagating wave. These situations are illustrated in Figure 3 using the periods of the M2 and N2 tides. Second, assume that the underlying phenomena is a plane wave travelling in a crosstrack direction with wavelength AU, i.e. h = eikux-art where ku = XI. When AU < 2d then the 4 phenomena will be aliased spatially. For this example, only Au > 2d will be considered. As before, if at a given point on a ground track the measurements are etkuz - atj then the measurements on the adjacent track to the east will be e'ku(cid:127) - a(tj + At). These measurements may be considered as ei(kI+kR)x-ftj = e O' as before. The direction of propagation of the aliased wave will be normal to the ground track with a wavelength given by A= = 2w = - The direction of propagation will be determined by the signs of AKR + AKu and AKT where Aru= 27rr (7') Since limAU_.. ru = 0 then lim.u_..o xR + xu = KR and the limit is case 1 as before. Note that if KR and ru are of opposite signs, then the direction of propagation will switch as Au becomes larger. Figure 4 shows an example of the change in propagation of the measured wave as lambdau varies. Note that when AU and AR are equal and of opposite sign, that the apparent wavelength of the measured phenomena is infinite. Third, assume that the underlying phenomena is a plane wave travelling in an along track direction, i.e. h = ei(-'u-Ut). As before, if the measurements at a point on a given track are ei(lu1I-Oti), then the measurements on the adjacent track to the east will be ei(1UVy-C'(ti+At)) - ei(kAR+lUWP -o'tj) In this case, the direction of propagation of the aliased signal will be at an angle, v, to the ground track given by [ v= tan'_ A] (8) Note that at the extremes, lim,\__oV = 0 and 1irnu,..v = ±7r/2. Thus the lirrtU_., represents case 1 as might be expected. As the wavelength of the underlying phenomena trends from zero to infinity, the direction of propagation of the aliased wave rotates from the along track to the cross track direction. Figure 5 shows an example of the change in propagation of the measured phenomena as Au varies. Any plane wave may be considered as being composed of along track and cross track com- ponents, and thus its aliased behavior can be discussed in terms of cases 2 and 3 above. An arbitrary wave will have a direction of propagation that rotates to one of the normals to the ground track as Au tends to infinity. If one considers a wave for which rR and Ku are of opposite signs, then this rotation will cover an angle greater than 90 degrees. Thus it is 5 possible for a plane wave with finite wavelength to have an alias that propagates at right angles to the underlying wave. These considerations will be used to describe the behavior of aliased oceanographic and atmosperic waves in the next sections. 4 Sampling of Unaliased Phenomena The purpose of this section is to discuss the sampling of unaliased equatorially trapped kelvin waves as an example of the results of sampling unaliased phenomena. Figure 7 shows the dispersion relation of measured unaliased kelvin waves. Note that for a given frequency, the wavelength of the apparent wave is shortened, and that the distortion increases linearly as the frequency increases. It should be noted that the speed of propagation of the measured phenomena will not change, despite the change in slope of the inferred dispersion relation. This is because the inferred dispersion relation has no dynamical significance. However, distortions such as this could lead to problems in interpreting phenomena whose underlying dynamics are not known, even for cases in which the underlying phenomena is not aliased. 5 Some Examples of Sampling Oceanographic Phe- nomena It should be noted that the problems of sampling discussed in this paper affect all phe- nomena that appear in altimeter data, whether oceanographic, atmospheric, related to orbit determination, instrumental, or from other causes. It is the intent of this paper to limit the discussion to oceanographic signals, but similar problems will occur in the analysis of altimeter data due to other sources, such as the effect of sampling atmospheric contributions to the altimeter data. These may well be important in a given analysis. Because of the complexity of the ocean, its stratification and topography, it would be a daunting task to discuss the result of sampling even all known oceanographic phenomena. It is the intent here to discuss some examples that are either of known importance or thought to be illustrative. The cases chosen here are tides, Rossby waves, and equatorially trapped waves. 6 5.1 Tides Tidal variations are pervasive in the deep ocean. Even when the best present solutions are applied to altimeter data, there are still significant residual errors. As discussed below, the result of sampling the tides is a wide variety of frequencies and wavenumbers; often radically different from the original tidal signal. The temporal aliasing of tides at a single location has been discussed extensively in other publications (see e.g. Parke et al 1987 and Cartwright and Ray, 1990). The purpose here is to investigate the spatial characteristics of these aliases. In regions where the amplitude of a given tidal constituent is locally large, it is often the case that the phase of the tide is slowly varying in space (such locations are often called anti-amphidromes). In these cases, the alias of the tide behaves much like case 1 of section 3. Thus it is meaningful to discuss the sampling of an idealized tide with no spatial variation. Table 2 gives the frequency and wavelength of the alias for the limit AU tends to infinity for every tidal constituent for which the equilibrium forcing amplitude is greater than 0.1 cm. Note that the resulting aliased wave can be propagating either along the eastwards normal or the westwards normal. When the apparent wavelength listed in Table 2 is small, the resulting wavelength will appear througout much of the deep water. When the apparent wavelength listed in Table 2 is large, the underlying wavelength of the tide will dominate, and the sampled tide will appear as if it were simply aliased. It should be noted that techniques for removing orbit error will transfer tidal errors along track. When this occurs, the observed distribution of tidal errors will not match the original distribution of tidal error, but will still show the same imposed sampling pattern. It is the spatial pattern rather than geographic distribution that is the most sensitive indicator of the source of an observed error. Because the effect of tidal error can become "globalized" in this fashion, the resulting pattern can be mistaken for having another source, such as orbit error. 5.2 Rossby Waves The apparent frequency and wavenumber of the M2 tidal alias has been shown to satisfy the dispersion relation of first mode baroclinic Rossby waves at mid-latitudes in the Pa- cific (Jacobs, et al., 1992). It is important to note that the latitude where this occurs is geographically dependent. Figure 8 shows the frequency that barotropic and first mode baroclinic Rossby waves would have to match the wavelength and direction of propagation of the aliased M2 tide. Figure 8 differs slightly from the equivalent figure of Jacobs et al. in that figure 8 uses the internal and external Rossby radii of Emery et al., 1984 for the 7