Báo cáo hóa học: " Analysis of Spaceborne Tandem Conﬁgurations for Complementing COSMO with SAR Interferometry"

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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Analysis of Spaceborne Tandem Conﬁgurations for Complementing COSMO with SAR Interferometry

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EURASIP Journal on Applied Signal Processing 2005:20, 3304–3315 c 2005 A. Moccia and G. Fasano Analysis of Spaceborne Tandem Conﬁgurations for Complementing COSMO with SAR Interferometry A. Moccia Dipartimento di Scienza e Ingegneria dello Spazio “L.G. Napolitano,” Universita degli Studi di Napoli “Federico II,” ` Piazzale Tecchio 80, 80125 Napoli, Italy Email: antonio.moccia@unina.it G. Fasano Dipartimento di Scienza e Ingegneria dello Spazio “L.G. Napolitano,” Universita degli Studi di Napoli “Federico II,” ` Piazzale Tecchio 80, 80125 Napoli, Italy Email: g.fasano@unina.it Received 29 June 2004; Revised 22 December 2004 This paper analyses the possibility of using a ﬁfth passive satellite for endowing the Italian COSMO-SkyMed constellation with cross- and along-track SAR interferometric capabilities, by using simultaneously ﬂying and operating antennas. Fundamentals of developed models are described and potential space conﬁgurations are investigated, by considering both formations operating on the same orbital plane and on separated planes. The study is mainly aimed at describing achievable baselines and their time histories along the selected orbits. The eﬀects of tuning orbital parameters, such as eccentricity or ascending node phasing, are pointed out, and simulation results show the most favorable tandem conﬁgurations in terms of achieved baseline components, percentage of the orbit adequate for interferometry, and covered latitude intervals. Keywords and phrases: spaceborne SAR interferometry, multiplatform interferometry, cross-track interferometry, along-track interferometry, mission analysis. 1. INTRODUCTION tion of the main mission, in order to avoid both expensive redesign and checkout phases at this stage of COSMO devel- COSMO-SkyMed is the Italian constellation for high spatial opment, and degradations of its nominal performance. This and temporal resolution SAR imaging of the Earth [1, 2]. ﬁfth satellite, named BISSAT (BIstatic Sar SATellite), could COSMO stands for COnstellation of small Satellites for fulﬁll also interferometric applications, by selecting adequate Mediterranean basin Observation and, basically, it consists of tandem orbits, thus obtaining interferometric pairs without four satellites in sun-synchronous orbit, orbiting in the same time decorrelation [5]. plane and phased at 90◦ , each equipped with an advanced X - This paper analyses various tandem conﬁgurations of the band SAR (synthetic aperture radar). Constellation orbital proposed COSMO-BISSAT formation, aimed at cross-track parameters are reported in Table 1. (XTI) and along-track (ATI) interferometry. Several possi- The program has been approved and founded, the devel- ble orbits have been considered for BISSAT, evaluating their opment is carried out by Alenia Spazio as prime contractor, characteristics versus potential interferometric applications. under management of the Italian Space Agency (ASI), and Several authors have carried out performance evaluations the launch of the ﬁrst satellite is scheduled in 2006. of multistatic space conﬁgurations and have compared so- The possibility of ﬂying a passive satellite, that is lutions presented in the literature mainly in terms of range equipped with a receiving-only antenna, in formation with and azimuth ambiguities and geometric decorrelation [6, 7], COSMO-SkyMed for bistatic applications has been investi- whereas this paper is aimed at describing main aspects and gated in [3, 4]. The study has been conducted assuming that results of analytical models developed to compute the achiev- no modiﬁcations should be included in design and opera- able interferometric baselines, their time histories along the selected orbits and relevant ground coverages. In particular, XTI baselines are investigated accounting for system capa- This is an open access article distributed under the Creative Commons bility to attain satisfactory phase errors and, hence, height Attribution License, which permits unrestricted use, distribution, and measurement accuracy. Concerning ATI, a baseline study is reproduction in any medium, provided the original work is properly cited.
Tandem Conﬁgurations for Complementing COSMO with INSAR 3305 x Table 1: COSMO orbital parameters. Semimajor axis (a) 6997.940 km Eccentricity (e) 0.00118 90◦ Argument of perigee (ω) y COSMO 97.87◦ Inclination (i) B BISSAT Bz Bx By carried out to investigate radial velocity measurement capa- θ bilities. The ﬁrst part of the paper is devoted to the analysis of the R critical baselines for SAR interferometry since they constitute z the basic requirements to be fulﬁlled by the formation. Then, the model developed for propagating baseline components along the orbit is described in more detail. Finally, the pro- Towards Earth center posed tandem conﬁgurations are introduced and simulation results indicate achievable performance in terms of baseline components, percentage of the orbit adequate for interfer- ometry, covered latitude intervals. For the sake of presen- tation clarity, considered orbits have been divided in two groups: the coplanar tandem conﬁgurations, which apply when COSMO and BISSAT orbital planes are coincident Figure 1: Orbiting reference frame and interferometric baseline. and include the well-known cartwheel [8, 9], and pendu- lum tandem conﬁgurations [10, 11, 12], which refer to non- The interferometric phase can be expressed as coincident orbital planes. 2π Φ12 ∼ Bz cos θ − B y sin θ , = (1) 2. CRITICAL BASELINES FOR SAR INTERFEROMETRY λ Some values of baseline components, which are critical for where θ represents the look angle. interferometric processing, exist both in along- and cross- Thus, the above phase variation can be computed as track direction. A thorough analysis of these aspects can be found in [13, 14]. ∂Φ12 2π d Φ12 ∼ dθ = − Bz sin θ − B y cos θ d θ. Objective of this paragraph is to review and apply these = (2) ∂θ λ models to the system under study in order to deﬁne the base- line intervals which tandem on-orbit conﬁgurations must at- Since tain. Let xyz be a right-handed, Cartesian orbiting reference dRg cos θ dθ ∼ = , (3) frame (ORF) whose origin coincides with COSMO position, R with z-axis towards the geometrical center of the Earth, and y -axis opposite to the angular momentum vector, as shown where dRg is the ground-range resolution and R is the slant in Figure 1. The interferometric baseline components, Bx , B y , range, the condition to be satisﬁed is and Bz , coincide with BISSAT’s coordinates in the ORF: Bx is the along-track baseline, B y and Bz are the horizontal and λR Bz sin θ + B y cos θ ≥ σΦ . (4) vertical components of the cross-track baseline. 2πdRg cos θ As for cross-track interferometry is concerned, the min- imum baseline condition has been calculated by imposing To express interferometric phase uncertainty (σΦ ) as a func- that the interferometric phase variation dΦ12 between ad- tion of signal-to-noise ratio (SNR), it has been assumed [5] jacent targets is equal to the expected interferometric phase uncertainty σΦ . As shown in [15, 16, 17, 18], it is possible SNR γ0 = , (5) to set up a phase error model that accounts for baseline sep- 1 + SNR aration, in particular; interferometric phase uncertainty de- and the phase standard deviation as a function of coherence creases with baseline, due to an increasing capability in an- (γ0 ) has been numerically computed, on the basis of the sta- gular separation measurement. For the sake of consistency, tistical distributions reported in [19]. it has been assumed a theoretical limit in baseline reduction: Assuming θ = 33.5◦ , slant-range resolution equal to 5 m, when the unavoidable phase noise consequent to signal-to- SNR = 15 dB, and four-look processing, a value of 53.10 m noise ratio (σΦ ) is equal to the minimum phase measurement requirement, that is, the capability to detect the phase diﬀer- is obtained for the right-hand member in (4). In the case of coplanar orbits, B y = 0 and (4) becomes Bz ≥ 96.21 m. ence existing between adjacent targets at the same height.
3306 EURASIP Journal on Applied Signal Processing where Γ(t ) is the instantaneous interferometric data coher- The maximum-baseline conﬁguration is determined by ence and γ0 represents coherence for zero time lag (5), as- the phenomenon of baseline decorrelation, that is, the drop suming that ocean decorrelation time τs is equal to 15 mil- or even loss of the interferometric pair correlation because of excessively large antenna separation [5, 11, 14, 15]. Although liseconds, and accepting a coherence drop to 0.5. Of course, larger baselines allow a larger height measurement sensitivity, larger baselines could be adopted in other applications, when decorrelation determines a larger phase measurement noise. decorrelation is a less stringent constraint. It can be mitigated by making use of multilook processing The along-track baseline lower limit, instead, is related to and an optimal baseline can be identiﬁed, accounting for the achievable velocity measurement accuracy, and, above all, shorter baselines problems too. However, in the following to collision risk avoidance. In the considered case, the maxi- mum radial velocity (Vr max ) that can be measured avoiding only the theoretical limit on maximum-baseline consequent to decorrelation will be accounted for. the necessity of phase unwrapping is in the range [0.78, 1.56] From [5], in a single-pass case like that of COSMO- m/s, and the theoretical limit of measurement accuracy, that BISSAT tandem, the following expression for the spatial cor- is [21], relation coeﬃcient (ρ) can be taken: Vr max σφ , (12) cos Θ|δθ |dRg π ρ =1− , (6) λ is about 0.2 m/s adopting four looks. However, it can be greatly improved if processing is based on more looks, al- where δθ is the diﬀerence in look angle for the two antennas though causing a reduced geometric resolution [20]. and Θ is the local incidence angle, in the case of the above- assumed COSMO geometry and ﬂat terrain Θ = 37.3◦ . Introducing the so-called “eﬀective baseline” B⊥ , that is, 3. INTERFEROMETRIC BASELINES EVALUATION the baseline component normal to the direction of incidence In this section, the procedure for computing the interfero- [20], it results in metric baselines for any choice of the tandem orbital con- ﬁguration will be pointed out. The inertial position of both B⊥ |δθ | = (7) satellites is known if the instantaneous value of six orbital R parameters (right ascension of the ascending node Ω, in- clination i, semilatus rectum p, eccentricity e, argument of and so, considering a drop to 0.5 of the spatial correlation perigee ω, and true anomaly ν) are known. In the follow- coeﬃcient, making substitutions, we obtain ing the subscripts C and B will refer to COSMO and BIS- λR SAT. In practice, neglecting all perturbations and assigned B⊥ max = . (8) 2 cos ΘdRg the initial conditions, the true anomaly is the only param- eter which varies with time, according to Kepler’s equation In particular, for the above system characteristics, in the case [22]. Knowing COSMO’s and BISSAT’s orbital parameters of coplanar orbits at a given instant, it is possible to deﬁne two useful refer- ence frames. Let XC YC ZC be a geocentric, right-handed refer- λR ence frame with XC -axis directed towards COSMO ascending Bz max = = 2.98 km, (9) 2dRg sin θ cos Θ node and YC -axis opposite to its angular momentum vector, so that XC ZC plane is coincident with COSMO orbital plane, while for pendulum conﬁgurations (cross-track baseline while XB YB ZB is the same frame based on BISSAT position formed almost completely in horizontal direction), (Figure 2); xyz is the orbiting reference frame previously in- troduced (Figure 3). Obviously, in the case of coplanar con- λR ﬁgurations, XB YB ZB and XC YC ZC coincide. B y max = = 1.97 km. (10) 2dRg cos θ cos Θ BISSAT position in XB YB ZB is given by     Regarding along-track interferometry, a range of [75 m, cos ωB + νB XB    150 m] for Bx will be derived in the following, under the  YB  = rB ·  , 0 (13) assumption of performing oceanographic applications. Fur- sin ωB + νB ZB thermore, since when only one of the two antennas is a trans- mitting/receiving one, the eﬀective along-track baseline is where half the along-track physical separation between the anten- pB nas, the time lag between the antennas must be in the range rB = . (14) 1 + eB cos νB of about [5, 10] milliseconds. In more detail, the upper limit of Bx depends on the The transformation matrix from XB YB ZB to XC YC ZC can decorrelation of ocean echoes [20] and is obtained from be derived considering that the latter is obtained from the t former by applying the following sequence of Euler angles Γ(t ) = γ0 exp − , (11) 90◦ − iB , ΩC − ΩB , iC − 90◦ , with 90◦ − iB and iC − 90◦ around τs
Tandem Conﬁgurations for Complementing COSMO with INSAR 3307 This procedure allows propagation of the interferometric Z ZC baselines for any initial condition. In particular, inclusion of ZB orbital perturbations in propagating orbital parameters does not require any modiﬁcation of the procedure for baselines evaluation. Furthermore, it is worth noting that COSMO- BISSAT relative position within a single orbit can be de- scribed assuming unperturbed motion, while orbital pertur- bations help to foresee the long period evolution of the con- sidered formation. YB X 4. TANDEM FLIGHT IN COPLANAR ΩB ΩC CONFIGURATIONS YC iC iB XC To describe the kinematics of coplanar conﬁgurations, it is useful to consider the relative motion of a satellite moving XB Y on an orbit of given eccentricity and semimajor axis, with respect to a reference point describing a reference trajectory. The selected reference trajectory is a Keplerian circular or- Figure 2: Geocentric reference frames. bit lying in the satellite orbital plane, sharing the same mean ZC motion (n) of satellite elliptical orbit, hence the two orbits exhibit the same semimajor axes (i.e., the semimajor axis of x the elliptical orbit is equal to the circular orbit radius) and orbital periods (T ). Now, let xo zo be an orbiting reference frame (ORFO , in y COSMO the following the subscript O will refer to the reference or- bit) whose origin coincides with the position of the reference point (xo -axis directed as the velocity vector, zo -axis in nadir z direction); assuming that mean anomaly M initial value is ωC + νC XC M ≡ −ω + θ0 , (18) YC where θ0 is the initial value of the true anomaly of the refer- Figure 3: ORF and XC YC ZC . ence point with respect to the ascending node, the following equations can be derived by a series expansion in powers of the ﬁrst axis and ΩC − ΩB around the third axis, thus achiev- eccentricity [23]: ing ae2 MB→C = xo (t ) = 2ae sin nt + θ0 − ω + sin 2 nt + θ0 − ω   4 cos ∆Ω − sin iB sin ∆Ω − cos iB sin ∆Ω  sin iC cos iB cos ∆Ω+ , sin iC sin iB cos ∆Ω+ ae3    sin iC sin ∆Ω 7 sin 3 nt + θ0 − ω − 9 sin nt + θ0 − ω  + − cos iC sin iB   + cos iC cos iB 24   cos iC sin iB cos ∆Ω+ cos iC cos iB cos ∆Ω+  + O e4 , cos iC sin ∆Ω − sin iC cos iB + sin iC sin iB ae2 (15) zo (t ) = ae cos nt + θ0 − ω + 1 − cos 2 nt + θ0 − ω 2 where ∆Ω = ΩB − ΩC . 3 + ae3 cos nt + θ0 − ω − cos 3 nt + θ0 − ω Then, the passage from XC YC ZC to xyz is given by 8     + O e4 , x XC 0     (19)  y  =  0  + Mo YC  (16) z rC ZC where t is the time elapsed since initial instant. By truncating the series at ﬁrst order in eccentricity, the satellite trajectory with with respect to the reference point is an ellipse whose cen-   − sin ωC + νC 0 cos ωC + νC ter coincides with the reference point and with principal axes   Mo =  . 0 1 0 directions coincident with xo zo directions. In particular, hor- (17) − cos ωC + νC 0 − sin ωC + νC izontal and vertical semiaxes have length 2ae and ae, respec-
3308 EURASIP Journal on Applied Signal Processing xo Satellite Reference Satellite trajectory point Reference-point with respect to circular orbit the reference point zo Earth Satellite elliptical orbit Figure 4: Satellite and reference point motion, plotted with satellite at apogee (not in scale for clarity). tively. The angle in the ellipse plane varies with a constant axis. Multiple along-track and vertical baselines can be si- rate (coincident with the satellite mean motion), in oppo- multaneously achieved, although varying along the orbit. site direction with respect to the orbital motion, as shown in Obviously, increasing the number of microsatellites, forma- Figure 4. tion duty cycle can be greatly improved [24]. This is not ap- If the assumption (18) is discarded, the series expan- plicable to COSMO-BISSAT formation, since only two plat- sion is more complicated, but similar conclusions can be ob- forms are available. However, it is interesting to investigate tained, with the approximated ellipse translated and rotated limits and potentialities of cartwheel conﬁguration also in with respect to xo zo axes. this case. Obviously, the elliptical approximation is more and more First of all, considering that COSMO is in sun- inaccurate when orbit eccentricity increases [23], which is synchronous low-eccentricity orbit, a Keplerian circu- not our case. lar orbit with radius equal to COSMO semimajor axis (6997.940 km) has been selected as reference. Hence, 4.1. Cartwheel COSMO and BISSAT form a cartwheel around the circu- lar trajectory, as a consequence of their equal eccentricity The interferometric cartwheel, introduced and patented by (Figure 5). Massonnet [8, 9], is basically a formation of passive mi- From the linearized equations of motion (19), in order crosatellites forming an orbiting cartwheel in the orbital plane of an active one, thanks to adequate diﬀerences in to obtain that the two satellites occupy the same positions in the orbiting reference frame, with a time delay ∆t , we must perigee positions and true anomalies synchronization. All impose satellites exhibit the same orbit eccentricity and semimajor MC (0) = θ0 − ωC = MC (0) − MB (0) = ωB − ωC = γ = n · ∆t = f · π. ⇒ (20) MB (0) = θ0 − ωB conveniently for constellation tuning, as a fraction ( f ) of Considering the trajectories in the orbiting reference frame (for the sake of simplicity, it has been assumed that at t = π. 0, COSMO is at its perigee), γ is the angular separation As an example, Figure 7 reports cross-track (XTI, nec- between the satellites (Figure 6), and it can be expressed essarily vertical since orbits are coplanar) and along-track (ATI) interferometric baselines, for f = 0.0833 (γ = 15.0◦ , by multiplying the reference orbit mean motion times the ∆t = 243 seconds). required time separation between the satellites, or, more
Tandem Conﬁgurations for Complementing COSMO with INSAR 3309 Orbit fraction ZC xo 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 Reference zo 4 COSMO circular orbit BISSAT BISSAT perigee Baseline components (km) ωB = ωC + γ 2 XC 0 νB (0) = νC (T − ∆t ) −2 BISSAT orbit −4 COSMO orbit 0 1000 2000 3000 4000 5000 5826 Time (s) Along-track baseline Vertical baseline Figure 7: Interferometric baseline components for f = 0.0833 as a Figure 5: Geometry adopted to describe the COSMO-BISSAT cartwheel (plotted with COSMO at perigee and not in scale for clar- function of time along one orbit. ity). 1 10 BISSAT at t = 0 COSMO at t = 0 8 Orbit fractions adequate for XTI and ATI 0.9 6 0.8 γ 4 0.7 2 zo (km) 0.6 0 0.5 −2 0.4 −4 0.3 −6 −8 0.2 −10 0.1 −20 −15 −10 −5 0 5 10 15 20 0 xo (km) 0 20 40 60 80 100 120 140 160 180 Perigee separation (◦ ) Figure 6: Motion in the ORFO . XTI ATI Figure 8 and Table 2 summarize the main performances Figure 8: COSMO-BISSAT cartwheel performance as a function of achievable with this conﬁguration. γ, expressed in terms of fraction of orbit where satisfactory baselines It is worth noting that adequate vertical baselines can be are achieved. achieved almost along the whole orbit (only polar regions are excluded). trajectory for BISSAT which diﬀers from the COSMO one 4.2. Alternative coplanar conﬁgurations only in the argument of perigee (Figure 10). In these cases, the two satellites will not move on the Two alternative solutions for baseline formation, still based same ellipse, with respect to the ORFO . In the ﬁrst conﬁgura- on coplanar orbits, are achievable by orbiting COSMO and BISSAT with diﬀerent eccentricities and equal argument of tion, they will describe two concentric ellipses, as it is evident from (19), while in the second one the two ellipses, equal in perigee and mean anomaly (Figure 9), or by choosing a
3310 EURASIP Journal on Applied Signal Processing ZC ZC xo xo zo zo COSMO COSMO BISSAT BISSAT ωC ωB = ωC XC ωB XC Figure 10: Orbits with diﬀerent arguments of perigee (not in scale Figure 9: Orbits with diﬀerent eccentricities (not in scale for clar- for clarity). ity). dimensions, will be translated on xo -axis (this can be shown Orbit fraction performing the series expansion in powers of eccentricity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 without the assumption (18), and supposing M (0) − θ0 + ω small). 4 From the relative motion point of view, having two or- bits which diﬀer only in perigee argument is equivalent to Baseline components (km) put the two satellites on the same orbit, with a short time 2 separation: the ORFO coordinates of the ellipse origin are, in fact, a sin[M (0) + ω − θ0 ] and a − a cos[M (0) + ω − θ0 ]. The along-track baseline component is constant, and 0 to achieve an adequate vertical baseline, a large along-track separation (> 50 km) is needed. As a consequence, this conﬁguration can be used only for ATI; in particular, if ωB ∈ −2 [89.9988◦ , 89.9994◦ ], the entire orbit can be exploited. In the case of diﬀerent eccentricities, from the baselines’ point of view, a diﬀerent behavior from the cartwheel is ex- −4 hibited. In fact, as it is evident from Figure 11 (still supposing that at t = 0, COSMO is at its perigee), the vertical spacecraft 0 1000 2000 3000 4000 5000 5826 separation is maximum at the poles and minimum at the as- Time (s) cending/descending nodes, while the along-track baseline is maximum at the nodes. Along-track baseline The achievable performances are quantitatively similar to Vertical baseline the cartwheel ones, as shown in Figure 12 and Table 3, and it is interesting to point out the diﬀerence in useful latitude Figure 11: Interferometric baseline components along one orbit, for eB = 1.5 · 10−3 . intervals consequent to the diﬀerent trend of baseline com- ponents. In order to have “J2 -invariant” orbits [26, 27], it is appro- 5. TANDEM FLIGHT IN PENDULUM CONFIGURATION priate to choose equal inclinations. The pendulum relative kinematics can be better under- The wording “pendulum,” introduced in [7, 10, 25], refers to stood expanding the baseline components in a Taylor series orbits separated in the right ascension of the ascending node about the points ∆Ω = 0, ∆ν = 0, and truncating beyond the and, if required, with diﬀerent inclinations and true anoma- ﬁrst-order term in the approximation of circular orbits [21]; lies (Figures 2 and 13).
Tandem Conﬁgurations for Complementing COSMO with INSAR 3311 Table 2: Cartwheel phasings which maximize percentage of orbit adequate for XTI and ATI and consequent latitude intervals during as- cending/descending phases. Interferometric conﬁguration XTI ATI Perigee separation (◦ ) 20.74 0.52 Time separation (s) 335.6 8.41 Minimum distance (km) 2.97 0.0749 Orbit fraction (%) 97.94 66.64 [82.13, 78.49] desc. [82.13, 29.41] desc. Latitude intervals (◦ ) [75.56, −78.38] desc. [−30.02, −29.52] desc/asc. [−75.45, 82.13] asc. [29.91, 82.13] asc. 1 thus, recalling xyz as the orbiting reference frame whose ori- gin coincides with COSMO position, and assuming that at Orbit fractions adequate for XTI and ATI 0.9 t = 0, COSMO is at the ascending node, the following equa- 0.8 tion can be derived to express BISSAT position with respect 0.7 to COSMO:     0.6 a(∆ν + ∆Ω cos i) x (t )  ∼  0.5  y (t ) = a sin i∆Ω cos(nt ) . (21) 0.4 z (t ) 0 0.3 It is worth noting that cross-track baseline is formed in the 0.2 horizontal plane and depends only on ∆Ω, while along-track 0.1 baseline is constant and, for the considered value of sun- synchronous inclination, depends above all on ∆ν. In this 0 0.4 0.6 0.8 1.2 1.4 1.6 1.8 1 2 case, the two spacecrafts move along almost parallel trajecto- ×10−3 BISSAT eccentricity ries, for short orbital segments, whereas the horizontal base- line component varies as a function of latitude over longer XTI periods [28]. In particular, from the second component of ATI (21), the optimal value of ∆Ω for XTI can be estimated by imposing ymax = B y max , resulting ∆Ω = 0.0163◦ . The Figure 12: Conﬁguration performances as a function of BISSAT ec- centricity, expressed in terms of fraction of orbit where satisfactory numerical simulations, performed taking into account the baselines are achieved. slight eccentricities of the orbits, conﬁrmed this estimate, as shown in Figure 14. In order to get an adequate along-track separation, ∆ν = −5 · 10−3 ◦ has been assumed. As for along-track interferometry is concerned, with this and choosing ∆t = 10 milliseconds (corresponding to an conﬁguration we can achieve an ideal observation geometry. along-track baseline of about the order of 75 m, which allows In fact, it must be considered that Earth rotation prevents to evaluate a V rmax of the order of 1.56 m/s), the following two antennas, which move on the same orbit with a time sep- aration ∆t , from having the same viewing geometry. values are derived: As shown in [21], imposing the conditions (ΩE is Earth ΩB − ΩC = 4.18 · 10−5 ◦ , rotation rate) (24) νC − νB = 6.18 · 10−4 ◦ . ΩB − ΩC νC − νB ˙ = M + ω = ∆t , (22) ΩE − Ω ˙ ˙ Relevant results are summarized in Table 4, showing the excellent ATI performance achievable with pendulum tan- the two antennas will exhibit the same trajectory, with re- dem conﬁguration. spect to an Earth-ﬁxed, rotating reference frame Figure 15. Regarding cross-track interferometry, as previously Obviously, the two satellites will have the same ground track noted, in the presented conﬁgurations (cartwheel, ∆e, pen- too, thus allowing coverage geometry adequate for ATI. dulum) the diﬀerences in baseline trend lead to various use- Considering, for the sake of simplicity, the unperturbed ful latitude intervals. Moreover, in the common latitude in- case (n = µ/a3 ) tervals there is a diﬀerence in eﬀective baseline and so in the interferometric performance (phase ambiguity and DEM ΩB − ΩC νC − νB accuracy). In fact, cross-track separation is larger at the equa- = ∆t , = (23) ΩE n tor for pendulum and cartwheel, and at the poles in the case
3312 EURASIP Journal on Applied Signal Processing Table 3: BISSAT eccentricities which maximize percentage of orbit adequate for XTI and ATI and consequent latitude intervals during ascending/descending phases. Interferometric conﬁguration XTI ATI 7.62 · 10−4 1.169 · 10−3 BISSAT eccentricity Minimum distance (km) 2.89 0.0749 Orbit fraction (%) 97.92 66.61 [82.13, 1.94] desc. [59.02, −59, 03] desc. Latitude intervals (◦ ) [−1.77, −1.81] desc/asc. [−59.06, 58.99] asc. [1.90, 82.13] asc. 1 0.9 Orbit fraction adequate for XTI 0.8 0.7 0.6 0.5 0.4 0.3 Equator 0.2 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 Diﬀerence in right ascension of the ascending node (◦ ) Figure 13: Geometry of pendulum conﬁguration. Figure 14: Pendulum performance as a function of ∆Ω, expressed of diﬀerent eccentricities. So, if interferometric coverage is in terms of fraction of orbit where a satisfactory XTI baseline is requested at a particular latitude with a given eﬀective base- achieved. line, proposed orbits can fulﬁll it either by tuning the design parameters for a certain conﬁguration or by combining dif- estimate the long period evolution of designed conﬁgura- ferent conﬁgurations (e.g., combining cartwheel or pendu- tions. To this end, it must be considered the boundary con- lum with a diﬀerence in eccentricity). Of course, identiﬁed straint deriving from the fact that we are not dealing with orbital conﬁguration will be less eﬀective, or useless in the an original formation, on the contrary BISSAT strategies for worst case, at other latitudes. baseline control will strongly depend on the operative sched- On the other hand, one can think of using diﬀerent or- ule of the active satellite, which, as previously underlined, bital conﬁgurations to achieve diﬀerent interferometric pairs cannot be modiﬁed. on a given area, to combine the advantages of large and small In this context, it is evident that, being a dual-use constel- baseline (accuracy and easier phase unwrapping), though lation aimed at monitoring and surveillance for commercial, there will be some unavoidable temporal gap between the scientiﬁc, and military applications, COSMO will be char- acquisitions. To this end, optimal strategies for orbit trans- acterized by an accurate orbit control that counteracts the fer, accounting for spacecraft resources, will be addressed by eﬀects of drag, solar radiation pressure, and third-body ac- further studies. celerations. Regarding these perturbations, it can be foreseen As a matter of fact, ATI can be achieved in a more sta- that, for BISSAT, the ﬁring sequence and the propellant ex- ble fashion along the orbit, although with critically short pense should be similar to COSMO ones. It is expected that baselines, whereas XTI baselines can be achieved with safer this is particularly true if the same bus of COSMO will be orbital conﬁgurations, although criticality arises in this case adopted for BISSAT. from continuous baseline variations. In more detail, in the case of orbits with diﬀerent eccen- tricities, the diﬀerential drag is more consistent, while, for all 6. CONSIDERATIONS ON THE STABILITY OF the other conﬁgurations, the aerodynamic eﬀect, integrated COPLANAR AND PENDULUM CONFIGURATIONS over one orbital period, is the same for the two satellites, So far, orbit fractions, where satisfactory baselines are assuming that they are almost identical and neglecting at- mospheric randomness, attitude dynamics and diﬀerences in achieved, have been evaluated assuming unperturbed mo- tion. However, one must consider orbital perturbations to fuel consumption.
Tandem Conﬁgurations for Complementing COSMO with INSAR 3313 Table 4: Pendulum conﬁgurations which maximize percentage of orbit adequate for XTI and ATI and consequent latitude intervals during ascending/descending phases. Interferometric conﬁguration XTI ATI ∆Ω (◦ ) 4.18 · 10−5 ◦ 0.0162 ∆ν (◦ ) −5 · 10−3 −6.18 · 10−4 ◦ Minimum distance (km) 0.879 0.075 Orbit fraction (%) 97.93 100 [81.92, −81, 91] desc. Latitude intervals (◦ ) All the achievable latitudes [−81.92, 81.91] asc. the latitude intervals in which interferometry is possible, Z with certain baselines, would be altered. As previously stated, ΩE argument of perigee control is envisaged in COSMO opera- tive schedule because of strict repetitiveness requirements. As for BISSAT, only in the pendulum case, passive satellite or- BISSAT orbit bit is frozen; in the cartwheel case, for example, argument of perigee control will be more onerous, leading to a (presumi- COSMO bly slight) diﬀerence in fuel consumption. orbit To summarize, pendulum is the stablest conﬁguration, followed by cartwheel (not frozen) and ∆e (not J2 -invariant). Y ΩC i 7. CONCLUSIONS ΩB ∆Ω i This paper focused on orbital conﬁgurations adequate for complementing the Italian COSMO SAR constellation with ∆ν COSMO X interferometry. A ﬁfth satellite has been considered that, BISSAT thanks to expectable mass reductions consequent to a sim- pliﬁed passive payload, could oﬀer additional maneuvers ca- Figure 15: Along-track interferometer: positions at the ascending node. pabilities, thus allowing, as an example, mission changes from along-track to cross-track interferometry, depending on particular users’ requirements. To this end, further stud- As for nonspherical Earth eﬀects are concerned, COSMO ies will be addressed to characterize optimal strategies for or- bit transfer. Enlarged maneuvers capabilities could also al- argument of perigee will be kept in a certain range around 90◦ , by nullifying the J2 induced precession of the line of low ﬂight of the ﬁfth satellite in formation with a varying apsides. This eﬀect is moderate since COSMO orbit is frozen. COSMO spacecraft, thus achieving an overall reliability im- provement. In addition, it is worth noting that the proposed From formation keeping point of view, the conﬁguration with diﬀerent eccentricities is the only one that is not J2 - idea of a passive satellite can be fulﬁlled by considering only invariant. However, diﬀerential secular J2 eﬀects on the evo- recurrent costs or by using most of the engineering model of the spacecraft. On the other hand, the weight, volume, lution of the interferometric baseline are negligible in the and cost advantages connected with the use of a simpliﬁed, considered case, as it can be seen in Figures 16 and 17, where relative trajectory is reported for eB = 0.00140, and for 450 receiving-only payload could allow additional remote sens- COSMO nodal periods. In this simulation, only J2 secular ef- ing systems to be embarked, such as a laser altimeter or an atmospheric proﬁler which could take advantage of the fects have been considered (without any correction): it can COSMO terminator orbit too. be seen that the growth of baseline horizontal component is The paper described a general purpose model devel- so slow that, after one month, the secular value is still smaller oped for propagating tandem conﬁgurations and for evaluat- than 5 m. This is due to the fact that, for near circular orbits, Ω,˙ ing relevant geometric performance. The model was applied ˙ are much more sensitive to ∆i than to ∆e (in fact, ˙ ω, and M considering as input the most referenced orbital conﬁgura- ∂Ω/∂e|e=0 = ∂M/∂e|e=0 = ∂ω/∂e|e=0 = 0), so the diﬀerences ˙ ˙ ˙ tions for interferometric applications and accounting for ma- jor limiting factors in baseline time hystories along the orbit. in mean anomaly, argument of perigee, and right ascension Numerical simulation results pointed out that the most fa- of the ascending node are, in the considered case, of order 10−3 ◦ /y. vorable tandem conﬁguration for along-track interferome- try (allowing continuous coverage with constant along-track As for latitude coverage is concerned, it is obvious that if baseline) consists in pendulum tandem conﬁguration, with the precession of the line of apsides were not counteracted,
3314 EURASIP Journal on Applied Signal Processing 1.19 1.185 1.18 1.175 1.17 1.165 1.16 1.155 1.15 1.145 1.14 2 1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 1.5 z (radial separation) (km) 1st nodal period 1 450th nodal period 0.5 0 −0.5 −1 −1.5 −2 −4 −3 −2 −1 0 1 2 3 4 x (along-track separation) (km) Figure 16: In-plane relative trajectory, plotted for eB = 0.00140 and 450 nodal periods. 5 REFERENCES 4 [1] F. Caltagirone, G. Angino, A. Coletta, F. Impagnatiello, and A. Gallon, “COSMO-SkyMed program: Status and perspectives,” 3 y (horizontal distance) (m) in Proc. 3rd International Workshop on Satellite Constellations 2 and Formation Flying, pp. 11–16, Pisa, Italy, February 2003. [2] F. Caltagirone, P. Spera, G. Manoni, and L. Bianchi, 1 “COSMO-SkyMED: A dual use earth observation constella- 0 tion,” in Proc. 2nd International Workshop on Satellite Constel- lations and Formation Flying, pp. 87–94, Haifa, Israel, Febru- −1 ary 2001. −2 [3] A. Moccia, N. Chiacchio, and A. Capone, “Spaceborne bistatic synthetic aperture radar for remote sensing applications,” −3 50th 150th 250th 350th 450th International Journal of Remote Sensing, vol. 21, no. 18, nodal nodal nodal nodal nodal pp. 3395–3414, 2000. −4 period period period period period [4] M. D’Errico and A. Moccia, “Attitude and antenna pointing −5 design of bistatic radar formations,” IEEE Trans. Aerosp. Elec- Time tron. Syst., vol. 39, no. 3, pp. 949–960, 2003. [5] H. A. Zebker and J. Villasenor, “Decorrelation in interfer- Figure 17: Secular growth of baseline horizontal component, plot- ometric radar echoes,” IEEE Trans. Geosci. Remote Sensing, ted for eB = 0.00140. vol. 30, no. 5, pp. 950–959, 1992. [6] G. Krieger, H. Fiedler, J. Mittermayer, K. Papathanassiou, and A. Moreira, “Analysis of multistatic conﬁgurations for space- ascending node and true anomaly separations adequate to borne SAR interferometry,” IEE Proceedings - Radar, Sonar match Earth rotation. Regarding cross-track interferometry, and Navigation, vol. 150, no. 3, pp. 87–96, 2003. [7] G. Krieger, M. Wendler, H. Fiedler, J. Mittermayer, and A. developed model allowed identiﬁcation of several solutions Moreira, “Performance analysis for bistatic interferometric which enable coverage for more than 90% of the orbit. Fur- SAR conﬁgurations,” in Proc. IEEE International Geoscience thermore, it was shown that by tuning orbital parameters and Remote Sensing Symposium (IGARSS ’02), vol. 1, pp. 650– such as perigee, ascending node, anomaly separation, or or- 652, Toronto, Canada, June 2002. bit eccentricity, it is possible to set the latitude interval in [8] D. Massonnet, “Capabilities and limitations of the interfero- which cross-track SAR interferometry is carried out with se- metric cartwheel,” IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 3, pp. 506–520, 2001. lected horizontal or vertical baseline. [9] D. Massonnet, “The interferometric cartwheel: a constellation of passive satellites to produce radar images to be coherently ACKNOWLEDGMENT combined,” International Journal of Remote Sensing, vol. 22, no. 12, pp. 2413–2430, 2001. This paper has been carried out with ﬁnancial support from [10] H. Fiedler, G. Krieger, F. Jochim, M. Kirschner, and A. Mor- the Italian Space Agency and Ministry for Education, Univer- eira, “Analysis of bistatic conﬁgurations for spaceborne SAR sity and Research. interferometry,” in Proc. 4th European Conference on Synthetic
Tandem Conﬁgurations for Complementing COSMO with INSAR 3315 A. Moccia has been a Professor of aerospace Aperture Radar (EUSAR ’02), pp. 29–32, Cologne, Germany, June 2002. servosystems at the Faculty of Engineer- [11] H. A. Zebker, T. G. Farr, R. P. Salazar, and T. H. Dixon, “Map- ing, University of Naples, Naples, Italy, ping the world’s topography using radar interferometry: the since 1990. His research activities deal with TOPSAT mission,” Proc. IEEE, vol. 82, no. 12, pp. 1774–1786, aerospace high-resolution remote-sensing 1994. systems, mission analysis, design, and data [12] M. D’Errico, A. Moccia, and S. Vetrella, “High frequency ob- processing, as well as aerospace systems dy- servation by GTM antenna range beam steering,” EARSeL Ad- namics and control. He has been a mem- vances in Remote Sensing, vol. 3, no. 1-IX, pp. 60–69, 1994. ber of national and international commit- [13] R. Gens and J. L. van Genderen, “SAR interferometry— tees and working groups (ASI, NASA, ESA, issues, techniques, applications,” International Journal of Re- UE, CIRA, EARSA eL, MIUR) and principal or coinvestigator of mote Sensing, vol. 17, no. 10, pp. 1803–1836, 1996. several national and international research programs. He is the au- [14] P. A. Rosen, S. Hensley, I. R. Joughin, et al., “Synthetic aper- thor or coauthor of more than 100 scientiﬁc papers, and gained ture radar interferometry,” Proc. IEEE, vol. 88, no. 3, pp. 333– several references in international journals. Since 1975, he has held 382, 2000. several grants and contracts from the National Research Council, [15] A. Moccia, S. Esposito, and M. D’Errico, “Height measure- University of Naples, Italian Space Agency, ESA, Italian Ministry ment accuracy of ERS-1 SAR interferometry,” EARSeL Ad- for Education, University and Research, Alenia Spazio, Telespazio, vances in Remote Sensing, vol. 3, no. 1, pp. 94–108, 1994. and Technapoli. Since 2001, he has been the Chairman of the Ph.D. [16] F. K. Li and R. M. Goldstein, “Studies of multibaseline space- School in Aerospace Engineering at the University of Naples and borne interferometric synthetic aperture radars,” IEEE Trans. President of CORISTA, Consortium for Research on Aerospace Geosci. Remote Sensing, vol. 28, no. 1, pp. 88–97, 1990. Remote Sensing Systems, a nonproﬁt research consortium among [17] E. Rodriguez and J. M. Martin, “Theory and design of in- terferometric synthetic aperture radars,” IEE Proceedings F, Universities of Naples and Bari, Alenia Spazio, and Laben. Radar & Signal Processing, vol. 139, no. 2, pp. 147–159, 1992. G. Fasano received the M.S. degree in [18] A. Moccia and S. Vetrella, “A tethered interferometric syn- thetic aperture radar (SAR) for atopographic mission,” IEEE aerospace engineering in 2004 and is cur- Trans. Geosci. Remote Sensing, vol. 30, no. 1, pp. 103–109, rently a Ph.D. student, in aerospace engi- 1992. neering at the university of Naples “Fed- [19] R. Bamler and P. Hartl, “Synthetic aperture radar interferom- erico II.” His current research activities deal etry,” Inverse Problems, vol. 14, no. 4, pp. R1–R54, 1998. with interferometric and bistatic missions ¨ [20] R. Romeiser, M. Schwabisch, J. Schulz-Stellenﬂeth, et al., based on SAR constellations, formation ﬂy- “Study on concepts for radar interferometry from satellites ing dynamics, and automatic UAV sense- for ocean (and land) applications (KoRIOLiS),” Final Report, and-avoid systems. University of Hamburg, Hamburg, Germany, 2002. [21] A. Moccia and G. Ruﬁno, “Spaceborne along-track SAR inter- ferometry: performance analysis and mission scenarios,” IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 1, pp. 199–213, 2001. [22] V. A. Chobotov, Ed., Orbital Mechanics, AIAA Education Series, American Institute of Aeronautics and Astronautics, Washington, DC, USA, 1991. [23] A. Miele and M. D’Errico, “A relative orbital motion model oriented to formation design,” in Proc. 3rd International Workshop on Satellite Constellations and Formation Flying, Pisa, Italy, February 2003. ` [24] D. Massonnet, E. Thouvenot, S. Ramongassie, and L. Phalip- pou, “A wheel of passive radar microsats for upgrading exist- ing SAR projects,” in Proc. IEEE International Geoscience and Remote Sensing Symposium (IGARSS ’00), vol. 3, pp. 1000– 1003, Honolulu, Hawaii, USA, July 2000. [25] J. Mittermayer, G. Krieger, A. Moreira, and M. Wendler, “In- terferometric performance estimation of the interferometric cartwheel in combination with a transmitting SAR-satellite,” in Proc. IEEE International Geoscience and Remote Sensing Symposium (IGARSS ’01), vol. 7, pp. 2955–2957, Sydney, NSW, Australia, July 2001. [26] H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, “Spacecraft formation ﬂying control using mean orbit ele- ments,” in Proc. AAS/AIAA Astrodynamics Specialists Confer- ence, Girdwood, Alaska, August 1999, Paper No. AAS 99-310. [27] H. Schaub and K. T. Alfriend, “J2 invariant relative orbits for spacecraft formations,” Celestial Mechanics and Dynami- cal Astronomy, vol. 79, no. 2, pp. 77–95, 2001. [28] A. Moccia, S. Vetrella, and M. D’Errico, “Twin satellite or- bital and doppler parameters for global topographic map- ping,” EARSeL Advances in Remote Sensing, vol. 4, no. 2-X, pp. 55–66, 1995.