Ocean Surface Topography Altimetry by Large Baseline Cross-Interferometry from Satellite Formation

remote sensing

Article Ocean Surface Topography Altimetry by Large Baseline Cross-Interferometry from Satellite Formation

Weiya Kong 1,2, Bo Liu 2,*, Xiaohong Sui 2, Running Zhang 3 and Jinping Sun 1 1 School of Electronic and Information Engineering, Beihang University, Beijing 100191, China; [email protected] (W.K.); [email protected] (J.S.) 2 Qian Xuesen Laboratory of Space and Technology, Beijing 100094, China; [email protected] 3 Beijing Institute of Spacecraft System Engineering, Beijing 100094, China; [email protected] * Correspondence: [email protected]; Tel.: +86-010-6811-3401

Received: 21 September 2020; Accepted: 26 October 2020; Published: 27 October 2020

Abstract: Imaging Radar Altimeter (IRA) is the current development tendency for ocean surface topography (OST) altimetry,which utilizes Synthetic Aperture Radar (SAR) and interferometry to improve the spatial resolution of OST to several kilometers or even better. Meanwhile, centimetric altimetry accuracy should be guaranteed for applications such as geostrophic currents or marine gravity anomaly inversion. However, the baseline length of IRA which determines the altimetric sensitivity is confined by the satellite platform, in consideration of baseline vibration and payload capability. Therefore, the baseline length from a single satellite can extend to only tens of meters, making it difficult to achieve centimetric accuracy. Referring to the successful experience from TerraSAR-X/TanDEM-X, satellite formation can easily extend the baseline length to hundreds or thousands of meters, depending on the helix orbit. Therefore, we propose the large baseline IRA (LB-IRA) from satellite formation for OST altimetry: the carrier frequency shift (CFS) is brought in to compensate for the severe baseline decorrelation, and the helix orbit is carefully selected to prevent severe time decorrelation from along-track baseline. The numerical results indicate that the LB-IRA, whose cross-track baseline ranges between 629~1000 m and along-tack baseline ranges between 0~40 m, can achieve ~1 cm relative accuracy at 1 km resolution.

Keywords: satellite formation; carrier frequency shift (CFS); ocean surface topography (OST) altimetry; cross-interferometry

1. Introduction As one of the most fundamental elements in marine dynamic environment, ocean surface topography (OST) is of great signiﬁcance in climatology, meteorology, and oceanography, such as geostrophic current and marine gravity anomaly inversion [1,2]. OST is mostly measured by satellite radar altimeter, which sends a special shaped pulse to the nadir and times the round-trip delay to calculate the instantaneous range between ocean surface and the satellite, and after applying several instrumental and geophysical corrections, OST can be precisely extracted [3]. From the ﬁrst generation of altimeter on Geosat, to the latest generation on Jason-CS, satellite altimeter has evolved into ~3 cm of altimetric accuracy [4]. Up to now, all the operational altimetry satellites, such as TOPEX/Poseidon, Jason series, and HY-2A/B, can only measure one-dimensional OST along the orbital track, and the cross-track measurement is realized by combining orbits. Even from satellite constellation, the best achievable grid resolution of interpolated OST is no better than 100~200 km [5]. To improve the spatial resolution of OST, the most promising instrument is the so-called Interferometric Synthetic Aperture Radar (InSAR), which records the interferometric phase, instead of timing the round-trip delay for altimetry [6]. So far, InSAR has been successfully applied in terrain

Remote Sens. 2020, 12, 3519; doi:10.3390/rs12213519 www.mdpi.com/journal/remotesensing Remote Sens. 2020, 12, 3519 2 of 19 altimetry, however, none of them can be directly used for OST altimetry due to the coarse accuracy, which is no better than a few meters. Therefore, the concept of Imaging Radar Altimeter (IRA) was first proposed by Rodriguez [7], which can be considered as the special version of InSAR, dedicated for OST altimetry with an altimetric accuracy that should be at least comparative to traditional radar altimeters [8]. Another distinguishable characteristic of IRA is its wide swath, which resides along one side of the nadir track and extends to tens of kilometers wide. After aperture synthesis and pulse compression [9], the intrinsic resolution of IRA is only a couple of meters. The most prominent IRA plan is the Surface Water and Ocean Topography (SWOT) mission [10,11]. The primary payload of SWOT satellite is the Ka-band Radar Interferometer (KaRIn), whose dual transmit/receive antennas are connected by a deployable rigid mast. The interferometric baseline, which is the distance between the phase centers of two antennas, is about 10 m [12]. The baseline length of KaRIn is much shorter than the 60 m baseline of Shuttle Radar Topography Mission (SRTM), which was dedicated for terrain topography mapping [13]. To a certain extent, longer baseline results in higher altimetric sensitivity, and furthermore, higher altimetric accuracy. However, SRTM suffers from serious baseline vibration [14], which should be strictly constrained by KaRIn as well. To compensate for the insufficient baseline length, the radar look angle of KaRIn is set to be 0.7~4◦. SWOT is expected to achieve ~2.5 cm relative accuracy at 1 km resolution (under 1σ demand, a swath averaged result among the 50 km swath width [15]). To fulfill the rigorous accuracy and resolution goal, the SWOT team are facing several technical and engineering problems, and hence the launching date has been postponed several times; recently, SWOT has been scheduled to launch in 2022. On 15 September 2016, China launched the Tiangong-2 space laboratory, and the Interferometric Imaging Radar Altimeter (InIRA) was one of the experimental payloads on it. The dual Ku-band antennas were mounted outside the capsule, and the baseline length was only 2.3 m, limited by the capsule size. The radar look angle ranges from 1◦ to 8◦, to ensure a swath width of at least 40 km. Based on the acquired data, Kong et al. has evaluated the relative altimetric accuracy of InIRA to be ~3 cm at 10 km resolution [16]. Though the baseline of InIRA is even shorter than KaRIn, it has proved the great potential in OST altimetry. Since SWOT KaRIn and Tiangong-2 InIRA both suffer from short baseline deficiency, in reference to the successful practice from TerraSAR-X/TanDEM-X (TSX/TDX) [17], we propose the Large Baseline Imaging Radar Altimeter (LB-IRA) from satellite formation, whose baseline length is no longer restrained by a single platform. However, large baseline also aggravates the signal decorrelation, preventing altimetric accuracy from being further improved. We utilize the cross-interferometry method, of which a small carrier frequency shift (CFS) is introduced in between two transmitted signals to compensate for the severe baseline decorrelation. The principle of cross-interferometry was first proposed by Gatelli [18], and further verified by ERS-ENVISAT terrain datasets [19–22]. As for LB-IRA, a helix orbit is carefully selected to prevent severe time decorrelation from along-track baseline while maintaining a minimum safety distance. Furthermore, in order to prevent radio interference while maintaining a sufficient swath width, the idea of matching CFS with signal bandwidth is introduced for the first time, innovated by the principle of frequency division multiplexing. The numerical results indicate that the LB-IRA system can achieve ~1 cm relative accuracy on a 1 km resolution grid. The paper is organized as follows: Section2 briefly introduces the IRA altimetry principle and defines the concept of relative/absolute altimetric accuracy, and Section3 discusses the helix orbit configuration in consideration of arranging proper cross-track/along-track baseline. The principle of cross-interferometry, and the design principle of several key parameters in consideration of CFS, is discussed in Section4, and the methodology for system parameter selection and phase noise suppression are explained in Section5. Section6 presents and analyzes the numerical results of the altimetric accuracy based on the system parameters of LB-IRA, and the last section (Section7) summarizes the whole paper. RemoteRemote Sens.Sens.2020 2020,,12 12,, 3519x FOR PEER REVIEW 33 ofof 1920

2. Measurement Principle 2. Measurement Principle 2.1. Altimetry Principle 2.1. Altimetry Principle The altimetry principle of IRA is basically triangulation, inherited from InSAR. As shown in FigureThe 1, altimetrytwo antennas principle measure of IRA the is ranges basically to the triangulation, target at slightly inherited different from InSAR.geometry As due shown to the in Figure1, two antennas measure the ranges to the target at slightly di ﬀerent geometry due to the existence of baseline, B , the range difference, Δr, related to the target height is recorded by the existence of baseline, B, the range diﬀerence, ∆r, related to the target height is recorded by the interferometric phase [6]. Based on triangulation, the height of the target is expressed as: interferometric phase [6]. Based on triangulation, the height of the target is expressed as: =− θ hHr1 cos (1) h = H r cos θ (1) − 1 r wherewhere HHis the is the altitude altitude of of antenna antenna phase phase center center on on Sat1, Sat1,r 1is is the the range range between between the the target target andand thethe θ 1 phasephase center,center, andandθ is is the the radar radar look look angle angle from from Sat1 Sat1 to to the the target. target.

FigureFigure 1.1. Schematic diagramdiagram ofof triangulationtriangulation altimetryaltimetry ofof ImagingImaging RadarRadar AltimeterAltimeter (IRA).(IRA).

Altitude, H, and range, r1,r can be precisely measured by other means, only leaving the look angle, Altitude, H, and range, 1 , can be precisely measured by other means, only leaving the look θ, whichθ varies with the target height unknown. However, the interferometric phase is directly related toangle, the look, anglewhich as: varies with the target height unknown. However, the interferometric phase is 2 2 2 directly related to the look angle as: 16π B + 8πr1φλ (φλ) θ = α + arcsin − (2) 2 32π Br1 16π22Br+− 8 π φλ ( φλ ) 2 where φ is the so-called interferometricθα=+arcsin phase, i.e., the phase1 diﬀerence between two complex images π 2 (2) of IRA, B is the baseline length, α is the baseline inclination32 Br1 angle, and λ is the radar wavelength determinedφ by center carrier frequency. whereBased is on the Equations so-called (1) interferometric and (2), the target phase, height i.e., can the be phase expressed difference as [23]: between two complex images of IRA, B is the baseline length, 𝛼 is the baseline inclination angle, and 𝜆 is the radar 2 2 2 wavelength determined by center carrier frequency.16 π B +8πr1∆φλ (∆φλ) h = H r1 cos α + arcsin 2 − − 32π Br1 (3) Based on Equations (1) and (2), then target height∆φλ ocan be expressed as [23]: H r cos α + arcsin ≈ − 1 4πB 16ππφλφλ22Br+Δ−Δ 8 ( ) 2 the approximation onhHr the=− right sidecos ofα the + equation arcsin is true when1B << r1. The baseline length of IRA 1 π 2 is no larger than a few kilometers, which meets the approximation32 Br1 condition. The approximated (3) equation can be used to derive altimetric error transferΔφλ functions for simpliﬁcation, however, only the ≈−α + non-approximation equationHr should1 cos be used arcsin to calculate the target height, since any approximation 4π B may introduce major altimetric errors. Br<< the approximation on the right side of the equation is true when 1 . The baseline length of IRA is no larger than a few kilometers, which meets the approximation condition. The approximated equation can be used to derive altimetric error transfer functions for simplification, however, only

Remote Sens. 2020, 12, x FOR PEER REVIEW 4 of 20 the non-approximation equation should be used to calculate the target height, since any Remote Sens. 2020, 12, 3519 4 of 19 approximation may introduce major altimetric errors. The altimetric accuracy is determined by all kinds of errors, the most significant ones are baseline lengthThe error, altimetric baseline accuracy inclination is determinederror, and interf by allerometric kinds ofphase errors, error. the Based most on significant Equation ones (3), the are altimetricbaseline length error transfer error, baseline functions inclination are derived error, from and partial interferometric differentiation: phase error. Based on Equation (3), the altimetric error transfer functions are derived from partial differentiation: HXtanθ σσσB ≈⋅= ⋅ B hBBH tan θθα−− θαX (4) σ BBcot( )σB = cot( ) σB (4) h ≈ B cot(θ α) · B cot(θ α) · σθσσα−≈⋅⋅− h HXtanαα = (5) α = σh H tan θ σα X σα (5) φ HX≈tanθλ· · λ σσσ≈⋅= ⋅ φ h H tan θλ φφXλ (6) σ 4cos()πθαπθαBB−−σ = 4cos() σ (6) h ≈ 4πB cos(θ α) · φ 4πB cos(θ α) · φ − − wherewhereXX is the target position along the range direction. σ EquationEquation (4) indicatesindicates thatthat the the altimetric altimetric error error introduced introduced by baselineby baseline length length error, error,σB, is affBected, is affectedby the baseline by the baseline length, baselinelength, baseline inclination, inclination, altitude, altitude, and radar and look radar angle. look The angle. selection The selection of altitude of altitudeand baseline and baseline inclination inclination has limited has limited degrees degrees of freedom; of freedom; therefore, therefore, small look small angle look and angle large and baseline large baselineshould beshould set to be reduce set to thereduce altimetric the altimetric error. error. EquationEquation (5) (5) indicates indicates that that smal smalll look look angle angle decreases decreases the the altimetric altimetric error error induced induced by by baseline baseline inclinationinclination error, error, σαα,, andand thethe altimetricaltimetric errorerror increasesincreases linearlylinearly with target position position along along the the range range direction.direction. Different Different from from single-platform single-platform IRA, IRA, whose whose baseline baseline is usually is usually formed formed by rigid by rigid mast, mast, the baselinethe baseline of LB-IRA of LB-IRA is quite is quite flexible, flexible, therefore therefore the the baseline baseline inclination inclination error error is ismostly mostly from from baseline baseline lengthlength error error instead instead of of attitude attitude jitter jitter of of th thee platform platform [24], [24], as as shown shown in in Figure Figure 22..

FigureFigure 2. 2. BaselineBaseline inclination inclination error of Large BaselineBaseline ImagingImaging Radar Radar Altimeter Altimeter (LB-IRA) (LB-IRA) introduced introduced by bybaseline baseline length length error. error.

AccordingAccording to Figure 2 2,, the the baseline baseline inclination inclination error error between between two two satellites satellites can can be expressedbe expressed as [ 24as]: [24]: σB σ σα = (7) σ B Bxt α = (7) Bxt where Bxt is the cross-track baseline length, and Bestimated is the baseline length estimated from the orbit B B wheretrack. Apparently,xt is the cross-track for LB-IRA, baseline large baselinelength, and can relativelyestimated is decrease the baseline the altimetric length estimated error. from the orbit track.Equation Apparently, (6) indicates for thatLB-IR largeA, baselinelarge baseline or small can look relatively angle leads decrease to low the altimetric altimetric error error. introduced by interferometricEquation (6) indicates phase error. that Interferometriclarge baseline or error small includes look phaseangle leads bias and to low random altimetric phase error noise: introducedthe phase bias by partinterferometric can usually phase be calibrated, error. Interf and therefore,erometric theerror phase includes noise ultimatelyphase bias determines and random the phasealtimetric noise: accuracy the phase level bias [25 part]. can usually be calibrated, and therefore, the phase noise ultimately determinesFrom thethe abovealtimetric analysis, accuracy we have level noticed [25]. that large baseline can improve the altimetric accuracy. However,From largethe above baseline analysis, can also we deteriorate have noticed the phase that noise large level, baseline and thus can prevent improve further the improvingaltimetric accuracy.altimetric However, accuracy fromlarge increasingbaseline can the also cross-track deteriorate baseline the phase length. noise The level, contradiction and thus canprevent be solved further by improvingcross-interferometry, altimetric accuracy which will from be addressedincreasing laterthe cross-track in Section4 baseline. length. The contradiction can be solved by cross-interferometry, which will be addressed later in Section 4.

Remote Sens. 2020, 12, 3519 5 of 19 Remote Sens. 2020, 12, x FOR PEER REVIEW 5 of 20 2.2. Relative Altimetric Accuracy 2.2. Relative Altimetric Accuracy For marine applications such as geostrophic current inversion or gravity anomaly inversion, the For marine applications such as geostrophic current inversion or gravity anomaly inversion, the relative altimetric accuracy is being more concerned than absolute accuracy, since the physical variables relative altimetric accuracy is being more concerned than absolute accuracy, since the physical are inversed from the OST gradient instead of the absolute value. Apart from phase noise, all parameter variables are inversed from the OST gradient instead of the absolute value. Apart from phase noise, errors of IRA will introduce spatial correlated altimetric error along the range direction, as indicted by all parameter errors of IRA will introduce spatial correlated altimetric error along the range direction, variable X in Equations (4)–(6). Figure3 gives the example of altimetric error introduced by baseline as indicted by variable X in Equations (4)–(6). Figure 3 gives the example of altimetric error inclination error. The error increases linearly along the range direction, the absolute altimetric error is introduced by baseline inclination error. The error increases linearly along the range direction, the defined as the difference between True OST and OST measured from IRA, and the relative altimetric absolute altimetric error is defined as the difference between True OST and OST measured from IRA, error is defined as the difference between two adjacent points, A and B. and the relative altimetric error is defined as the difference between two adjacent points, A and B.

FigureFigure 3. 3.Schematic Schematic diagramdiagram ofof relativerelative andand absolute absolute altimetric altimetric error. error.

TheThe relative relative altimetric altimetric error error related related to to baselinebaseline lengthlength error,error, baseline baseline inclination inclination error, error, and and phase phase noise, are as follows: noise, are as follows: ∆X δB = σ (8) h (Δ ) B δσBB cot θX α ·⋅ hB= − (8) α Bcot(θα− ) δ H tan θ σα = ∆X σα (9) h ≈ · · α δθσσ≈⋅Δ⋅HXtan∆Xλ αα = (9) φ =h δh σφ (10) 4πB cosΔ(θλ α) · δσφ X − ⋅ h = φ (10) where ∆X is the horizontal distance between two4cos()πθαB points;− in this paper, it stands for the grid resolution after multi-looking. Comparing Equations (8)–(10) to Equations (4)–(6), it appears that the relative altimetricwhere ∆𝑋 errors is the are horizontal usually much distance smaller between than the two absolute points; errors in sincethis paper, most of it the stands common for partthe hasgrid beenresolution cancelled after out multi-looking. by differentiation. Comparing Therefore, Equations the relative (8)–(10) altimetric to Equations accuracy (4)–(6), of IRA it appears is determined that the byrelative systematic altimetric parameter errors errorsare usually and the much final smaller resolution than of the the absolute OST product. errors since most of the common part has been cancelled out by differentiation. Therefore, the relative altimetric accuracy of IRA is 3.determined Satellite Formation by systematic and Interferometryparameter errors Baseline and the final resolution of the OST product. 3.1. Helix Orbit Configuration 3. Satellite Formation and Interferometry Baseline The main objective of LB-IRA from satellite formation is to extend the baseline length, which is essential3.1. Helix for Orbit improving Configuration altimetric accuracy. Different from single-platform interferometry, the baseline formedThe by main satellite objective formation of LB-IRA is flexible from without satellite any formation mast, thus is to the extend baseline the lengthbaseline is length, determined which by is theessential orbit configuration. for improving In altimetric order to obtain accuracy. a cross-track Different baseline, from single-platform a tiny angle is introduced interferometry, between the twobaseline orbit planes,formed asby shown satellite in Figureformation4a. To is provide flexible passive without safety any inmast, case thus of a vanishingthe baseline along-track length is separation,determined the by suggestedthe orbit configuration. safety formation In order distance to shouldobtain a be cross-track larger than baseline, 150 m, a hence tiny aangle small is eccentricityintroduced isbetween introduced two orbit which planes, yields as the shown radial in separationFigure 4a. To at theprovide poles, passive as shown safety in in Figure case 4ofb. a Thisvanishing type of along-track orbit configuration separation, is referred the suggested to as helix safety orbit formation by the TSX distance/TDX teamshould [24 be]. larger than 150 m, hence a small eccentricity is introduced which yields the radial separation at the poles, as shown in Figure 4b. This type of orbit configuration is referred to as helix orbit by the TSX/TDX team [24].

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((aa)) ((bb)) ((cc))

FigureFigureFigure 4.4. LB-IRALB-IRALB-IRA satellitesatellite satellite formationformation formation andand and thethe the helixhelix orbitorbit configuration.configuration. configuration. ((a (aa)) ) MaximumMaximum Maximum separationseparation separation atat at equator,equator,equator, intersectsintersects intersects nearnear near thethe the poles,poles, poles, ((b (bb)) ) maximummaximum maximum separationseparation separation atat at equator,equator, equator, thethe the smallsmall small eccentricityeccentricity eccentricity offsetoffset offset causescausescauses differentdifferent different heightsheights heights ofof of perigee,perigee, perigee, sndsnd snd ((c (cc)) ) optionaloptional optional rotationrotation rotation ofof of thethe the perigeeperigee perigee toto to achieveachieve achieve properproper proper baselinebaseline baseline configurationconfigurationconfiguration nearnear near thethe the high-latitudehigh-latitude high-latitude region.region. region. Under the helix orbit configuration, the variation cycle of cross-track/along-track baseline length UnderUnder the the helix helix orbit orbit configuration, configuration, the the variation variation cycle cycle of of cross-track/along-track cross-track/along-track baseline baseline length length could be adjusted by thrusters at the initial stage, and the cross-track baseline reaches its maximum couldcould bebe adjustedadjusted byby thrustersthrusters atat thethe initialinitial stage,stage, andand thethe cross-trcross-trackack baselinebaseline reachesreaches itsits maximummaximum length while the along-track baseline is about zero at the equator; after that, along-track baseline length lengthlength whilewhile thethe along-trackalong-track baselinebaseline isis aboutabout zerozero atat thethe equator;equator; afterafter that,that, along-trackalong-track baselinebaseline starts to increase, and reaches its maximum near the poles. Since ocean surface is randomly changing, lengthlength startsstarts toto increase,increase, andand reachesreaches itsits maximummaximum nearnear thethe poles.poles. SinceSince ococeanean surfacesurface isis randomlyrandomly OST altimetry near the high latitude region will suffer from worse time decorrelation due to larger changing,changing, OSTOST altimetryaltimetry nearnear thethe highhigh latitudelatitude regionregion willwill suffersuffer fromfrom worseworse timetime decorrelationdecorrelation duedue along-track separation. Therefore, the optional rotation of the argument of perigee can be adjusted toto largerlarger along-trackalong-track separation.separation. Therefore,Therefore, thethe optioptionalonal rotationrotation ofof thethe argumentargument ofof perigeeperigee cancan bebe to achieve proper baseline configuration [24], as shown in Figure4c; however, at the same time, the adjustedadjusted toto achieveachieve properproper baselinebaseline configurationconfiguration [24][24],, asas shownshown inin FigureFigure 4c;4c; however,however, atat thethe samesame low-latitude region will no longer be suitable for OST altimetry. time,time, thethe low-latitudelow-latitude regionregion willwill nono longerlonger bebe suitablesuitable forfor OSTOST altimetry.altimetry. 3.2. Along-Track Baseline and Time Decorrelation 3.2.3.2. Along-TrackAlong-Track BaselineBaseline andand TimeTime DecorrelationDecorrelation Different from single-platform interferometry, the triangulation of satellite formation is formed by DifferentDifferent fromfrom single-platformsingle-platform interferometry,interferometry, thethe triangulationtriangulation ofof satellitesatellite formationformation isis formedformed two different time-space planes (except when the satellite formation is at the equator), as shown in byby twotwo differentdifferent time-spacetime-space planesplanes (except(except whenwhen ththee satellitesatellite formationformation isis atat thethe equator),equator), asas shownshown Figure5. At t ∆−Δt moment, Sat1 is ahead of Sat2, right in the triangulation area (the gray shaded area); inin FigureFigure 5.5. AtAt − tttt−Δ moment, moment, Sat1Sat1 isis aheadahead ofof Sat2,Sat2, rightright inin thethe triangulationtriangulation areaarea (the(the graygray shadedshaded after ∆t, Sat2Δ reaches to the same area, therefore the along-track baseline of LB-IRA is inevitable during area);area); afterafter Δtt,, Sat2Sat2 reachesreaches toto thethe samesame area,area, thereforetherefore thethe along-trackalong-track baselinebaseline ofof LB-IRALB-IRA isis OST altimetry. inevitableinevitable duringduring OSTOST altimetry.altimetry.

Figure 5. Interferometry by LB-IRA. FigureFigure 5.5. InterferometryInterferometry byby LB-IRA.LB-IRA. The orbital velocity of ocean surface wave will aggravate the time decorrelation considerably, The orbital velocity of ocean surface wave will aggravate the time decorrelation considerably, eventuallyThe orbital decreasing velocity the of altimetric ocean surface accuracy. wave The will echoes aggravate received the by time two antennasdecorrelation of LB-IRA considerably, are from eventually decreasing the altimetric accuracy. The echoes received by two antennas of LB-IRA are eventuallya combination decreasing of multiple the wavelikealtimetric scatterers accuracy. within The echoes the same received speciﬁc by region, two antennas however of with LB-IRA diﬀerent are fromfrom aa combinationcombination ofof multiplemultiple wavelikewavelike scatterersscatterers withinwithin thethe samesame specificspecific region,region, howeverhowever withwith differentdifferent time time moments moments due due to to the the along-track along-track sepa separation.ration. Because Because of of the the randomness randomness of of wave wave orbital orbital velocityvelocity ofof thethe scatterers,scatterers, thethe dopplerdoppler spectrumspectrum ofof twotwo receivedreceived echoesechoes statisticallystatistically deviatedeviate fromfrom eacheach

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other [26]. The ocean coherence time can be used to quantify the statistics of doppler spectrum of timeocean moments surface wave, due to and the along-trackhigher sea state separation. (usually Because determined of the by randomness surface wind of wave velocity) orbital and velocity radar ofwaveband the scatterers, lead to the shorter doppler ocean spectrum coherence of two time. received The time echoes decorrelation statistically can deviate then be from expressed each other as [27]: [26]. The ocean coherence time can be used to quantify the statistics of doppler spectrum of ocean surface wave, and higher sea state (usually determined by surfaceΔt wind2 velocity) and radar waveband lead to γ =−exp shorter ocean coherence time. The time decorrelationtime  canτ 2 then be(11) expressed as [27]: 2 c ! ∆t 2 γ = exp (11) time Bat 2 Δ=t −2τc τ V where c is the ocean coherence time, sat is the time-lag determined by the along-track Bat where τ is theV oceansat coherence time, ∆t = is the time-lag determined by the along-track baseline, baseline,c and is the satellite velocity. InV satconsideration of other technical and engineering issues, andthe wavebandVsat is the satellite of LB-IRA velocity. is chosen In consideration to be Ku-band of other (center technical carrier and frequency engineering is about issues, 13.5~13.6 the waveband GHz). ofThe LB-IRA surface is wind chosen velocity to be Ku-bandis set to be (center 7.4 m/s carrier (1𝜎 result) frequency based is on about one 13.5~13.6 year of statistical GHz). The data surface from windCross-Calibrated velocity is set Multi-Platform to be 7.4 m/s (1 σ[25].result) According based on to one the year evaluation of statistical method data fromfrom Cross-Calibrated Reference [28], Multi-Platformassuming surface [25 ].wave According propagates to the evaluationalong the methodorbit track, from the Reference ocean [coherence28], assuming time surface is about wave 8 propagatesmilliseconds along at the the Ku-band orbit track, and the under ocean 7.4 coherence m/s of wind time isvelocity. about 8 According milliseconds to atEquation the Ku-band (11), andthe underinterferometric 7.4 m/s of coherence wind velocity. will decrease According to to0.79 Equation due to time (11), decorrelation, the interferometric which coherence corresponds will decreaseto about to40 0.79m of duethe along-track to time decorrelation, separation whichat the altitude corresponds of 891 to km. about Since 40 IRA m of interferometric the along-track data separation will suffer at thefrom altitude all kinds of 891 of km.decorrelations, Since IRA interferometricwe decided that data 0.8 will should suﬀer frombe the all kindslower ofbound decorrelations, for time wedecorrelation. decided that 0.8 should be the lower bound for time decorrelation.

3.3. Baseline Variation Cycle 3.3. Baseline Variation Cycle Under the helix orbit conﬁguration, in consideration of the along-track baseline limitation, Under the helix orbit configuration, in consideration of the along-track baseline limitation, the the angle between orbit plane, satellite altitude, and orbit inclination is set to be 7.88 m , 891 km, angle between orbit plane, satellite altitude, and orbit inclination is set to be 7.88 m °, 891◦ km, and and 78 , respectively. The orbit center deviates ~53 m from the earth center, to maintain a minimum 78°, respectively.◦ The orbit center deviates ~53 m from the earth center, to maintain a minimum safety safety distance of 150 m during the orbital cycle, as shown in Figure6a. The variation cycle of cross-track distance of 150 m during the orbital cycle, as shown in Figure 6a. The variation cycle of cross-track and along-track baseline length are shown in Figure6b,c, respectively. The cross-track baseline length and along-track baseline length are shown in Figure 6b,c, respectively. The cross-track baseline length reaches its maximum at the equator and decreases as latitude increases. The along-track baseline length reaches its maximum at the equator and decreases as latitude increases. The along-track baseline reaches its minimum at the equator and increases as latitude increases, and the positive/negative sign length reaches its minimum at the equator and increases as latitude increases, and the stands for the relative position change of two satellites. In order to avoid severe time decorrelation, positive/negative sign stands for the relative position change of two satellites. In order to avoid severe the suggested interferometry region should be within 44.6 northern/southern latitude, where the time decorrelation, the suggested interferometry region should◦ be within 44.6° northern/southern along-track baseline length is shorter than 40 m, and the across-track baseline length ranges between latitude, where the along-track baseline length is shorter than 40 m, and the across-track baseline 629~1000 m, as indicated by the blue shaded area in Figure6b,c. The baseline inclination angle is small length ranges between 629~1000 m, as indicated by the blue shaded area in Figure 6b,c. The baseline and poses a minor inﬂuence to OST altimetry within the 44.6 latitude range. inclination angle is small and poses a minor influence to OST altimetry◦ within the 44.6° latitude range.

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Figure 6. Cont.

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-60

-80 -60 -40 -20 0 20 40 60 80 latitude[deg]

FigureFigure 6. 6.Real-timeReal-time distance distance and baseline and baseline variation variation versus versuslatitude latitude during half during an orbit half cycle. an orbit (a) Real- cycle. a b c d time( ) Real-timedistance, distance,(b) cross-track ( ) cross-track baseline baseline length, length, (c) along-track ( ) along-track baseline baseline length, length, and and (d () baseline) baseline inclination angle. inclination angle. According to Figure6c, under the helix orbit configuration, OST altimetry near the high-altitude According to Figure 6c, under the helix orbit configuration, OST altimetry near the high-altitude region will suffer from severe time decorrelation. In order to improve the altimetric accuracy, the orbit region will suffer from severe time decorrelation. In order to improve the altimetric accuracy, the perigee can be adjusted for proper baseline length near the latitude region of interest. orbit perigee can be adjusted for proper baseline length near the latitude region of interest. 4. CFS for Baseline Decorrelation Compensation 4. CFS for Baseline Decorrelation Compensation The cross-track baseline length of LB-IRA ranges between 629~1000 m, which is far larger than thatThe of SWOT.cross-track Though baseline SWOT length has chosenof LB-IRA the rang Ka-bandes between and near-nadir 629~1000 lookm, which angle is for far short larger baseline than thatdefection, of SWOT. its altimetricThough SWOT sensitivity has ischosen still much the Ka-band lower than and that near-nadir of LB-IRA. look However, angle for too-large short baseline baseline defection,also decreases its altimetric the signal sensitivity coherence dueis still to baselinemuch lower decorrelation, than that preventing of LB-IRA. altimetric However, accuracy too-large from baselinefurther beingalso decreases improved. the In signal this section, coherence we introducedue to baseline the carrier decorrelation, frequency shiftpreventing (CFS) methodaltimetric for accuracybaseline from decorrelation further being compensation, improved. hence In this LB-IRA section, will we work introduce in cross-interferometry the carrier frequency mode shift to achieve(CFS) methodmuch higher for baseline altimetric decorrelation accuracy than compensation, normal interferometry. hence LB-IRA will work in cross-interferometry mode to achieve much higher altimetric accuracy than normal interferometry. 4.1. Principle of Baseline Decorrelation 4.1. Principle of Baseline Decorrelation The interferometry of microwave also follows the optical diffraction grating principle from physical optics,The therefore, interferometry the essential of microwave reason for also baseline follows decorrelation the optical is thediffraction radar look grating angle diprinciplefference, from which physicalintroduces optics, a relative therefore, shift the between essential the targetreason spectrum for baseline received decorrelation by two antennas is the radar [18,29 look]. The angle dual difference,antennas inwhich normal introduces interferometry a relative mode shift transmit between pulses the oftarget the same spectrum carrier received frequency by andtwo bandwidth,antennas [18,29].or, in otherThe words,dual antennas the transmitted in normal spectrum interferometry window mode for target transmit observing pulses is identical,of the same as shown carrier in frequencythe upper and left bandwidth, corner of Figure or, in other7. The words, transmitted the transmitted pulse eventually spectrum measureswindow for the target target observing in space isdomain, identical, and as theshown spatial in the frequency upper left ofthe corner target of isFigure usually 7. The defined transmitted as wavenumber, pulse eventually i.e., the reciprocal measures of thethe target wavelength. in space Duedomain, to the and tiny the di spatialfference frequenc of looky angle,of the target the transmitted is usually spectrumdefined as window wavenumber, will be i.e.,shifted the reciprocal and stretched of the after wavelength. being projected Due toto thethe wavenumber tiny difference domain of look (the angle, stretching the transmitted effect can be spectrumomitted sincewindow radar will carrier be shifted frequency and stretched is much higherafter being than projected signal bandwidth), to the wavenumber as shown domain in the bottom (the stretchingof Figure 7effect. Only can the be overlapped omitted since region radar of carrier wavenumber frequency can is be much used higher for interferometry, than signal bandwidth), leaving the asnon-overlapped shown in the bottom part to decayof Figure to phase 7. Only noise. the overlapped region of wavenumber can be used for interferometry, leaving the non-overlapped part to decay to phase noise.

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Figure 7. Schematic diagram of the principle of baseline decorrelation.

Since wavenumber and frequency are basically thethe same physical quantity in space and timetime domain, therefore, thethe wavenumberwavenumber shiftshift cancan bebe equallyequally expressedexpressed byby frequencyfrequency shiftshift[ [18]:18]:

B cosθθ Δ≈ B⊥ cos ∆ f fff0 0 ⊥ (12) (12) ≈ HHtantan((θθ -ββ )) − where f0f is the radar carrier frequency, B =BBB=−cos(cos(θ θαα) is )the vertical component of cross-track where 0 is the radar carrier frequency,⊥ ⊥ − is the vertical component of cross- baseline, and β is theβ average slope of ocean surface which changes the local incidence angle of microwave.track baseline, According and is to the Equation average (12), slope longer of ocean vertical surface baseline which andchanges smaller the local look incidence angle result angle in aof bigger microwave. non-overlapped According partto Equation of the received (12), longer target vertical spectrum, baseline and and since smaller the envelope look angle of result the target in a spectrumbigger non-overlapped is relatively flat, part the of baseline the received decorrelation target spectrum, can be expressed and since as [ 18the]: envelope of the target spectrum is relatively flat, the baseline decorrelation can be expressed as [18]: ∆ f f0B cos θ = = ⊥ γB 1 Δf 1 fB cosθ (13) γ =−−11Wb =−− Wb0tan⊥ (θ β)H B θβ− (13) WWbbtan( - ) H From Equation (13), we learn that enlarging the bandwidth can relatively restrain baseline decorrelation.From Equation However, (13), we a learn larger that bandwidth enlarging leads the to bandwidth lower signal-to-noise can relatively ratio (SNR),restrain as baseline well as heavierdecorrelation. data transmitting However, a volume,larger bandwidth which should leads be to avoidedlower signal-to-noise to facilitate extensive ratio (SNR), OST as altimetry. well as Besides,heavier thedata cross-track transmitting baseline volume, of LB-IRA which can should extend be to avoided 1000 m orto evenfacilitate longer extensive and enlarging OST altimetry. the signal bandwidthBesides, the cannot cross-track be of baseline much help of sinceLB-IRA the can frequency extend to shift 1000 is farm largeror even than longer the signaland enlarging bandwidth. the Figuresignal 8bandwidth shows the frequencycannot be shift of much versus help radar since look th angle:e frequency the cross-track shift is baseline far larger length than is 1000the signal m in thisbandwidth. case. The Figure frequency 8 shows shift the can frequency reach to shift 180~760 versus MHz radar among look angle: the angle the cross-track range of 0.7~4 baseline◦, which length is faris 1000 bigger m in than this thecase. bandwidth The frequency of most shift of can the reach spaceborne to 180~760 SAR MHz/InSAR. among It appearsthe angle that range larger of 0.7~4°, look anglewhich can is far alleviate bigger thethan demand the bandwidth for wider of bandwidth, most of the however, spaceborne even SAR/InSAR. if the look angleIt appears increases that tolarger 20◦, thelook maximum angle can coherencealleviate the is onlydemand 0.69, for aff ectedwider by bandwi baselinedth, decorrelation however, even alone, if the under look theangle 100 increases MHz of bandwidthto 20°, the maximum setting. coherence is only 0.69, affected by baseline decorrelation alone, under the 100 MHz of bandwidth setting. 4.2. Design Princile of CFS According to the principle of baseline decorrelation, a carrier frequency shift (CFS) can be introduced in between two transmitted signals, in order to shift back the non-overlapped part of the received spectrum as much as possible, and eventually compensate for the severe baseline decorrelation [18]. The CFS method in this paper is also named as cross-interferometry, first verified by the ERS-ENVISAT dataset, however, the dataset cannot be used to verify OST altimetry since the minimum interferometry time-lag is 28 min, during which the signals are no longer correlated due to time decorrelation.

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Figure 8. Frequency shift versus radar look angle.

4.2. Design Princile of CFS According to the principle of baseline decorrelation, a carrier frequency shift (CFS) can be introduced in between two transmitted signals, in order to shift back the non-overlapped part of the received spectrum as much as possible, and eventually compensate for the severe baseline decorrelation [18]. The CFS method in this paper is also named as cross-interferometry, first verified by the ERS-ENVISAT dataset, however, the dataset cannot be used to verify OST altimetry since the minimum interferometry time-lagFigure 8. isFrequencyFrequency 28 min, shiftduring versus which radar the look signals angle. are no longer correlated due to time decorrelation. 4.2. DesignTheThe along-trackalong-track Princile of CFS baselinebaseline lengthlength ofof LB-IRALB-IRA isis 4040 mm atat mostmost inin casecase ofof severesevere timetime decorrelation,decorrelation, however,however, the the dual dual antennas antennas both both send send pulses pulses for for interferometry, interferometry, and and therefore, therefore, the the received received signals signals According to the principle of baseline decorrelation, a carrier frequency shift (CFS) can be maymay su sufferffer from from radio radio interference interference with with each each other. other. In orderIn order to prevent to prevent radio radio interference, interference, the CFS the andCFS introduced in between two transmitted signals, in order to shift back the non-overlapped part of the bandwidth should be compatible with each other. As shown in Figure9, the CFS is set to be ∆ f basedΔf receivedand bandwidth spectrum should as muchbe compatible as possible, with eachand other.eventually As shown compensate in Figure for 9, the the CFS severe is set baselineto be on Equation (12) to compensate for the baseline decorrelation. The bandwidth of the transmitted pulses decorrelationbased on Equation [18]. The (12) CFS tomethod compensate in this paperfor th ise alsobaseline named decorrelation. as cross-interferometry, The bandwidth first verified of the is W , the bandwidth shouldW be smaller than the CFS, so that the transmitted spectrum windows will bytransmitted theb ERS-ENVISAT pulses is dataset, b , the however, bandwidth the shoulddataset becannot smaller be usedthan tothe verify CFS, OST so that altimetry the transmitted since the not overlap with each other, as shown in the upper left corner of Figure9. The received signal spectrum minimumspectrum interferometrywindows will nottime-lag overlap is 28 with min, each during other, which as shown the signals in the areupper no longerleft corner correlated of Figure due 9. of both antennas are exactly the same, after being demodulated with the carrier frequency of f and toThe time received decorrelation. signal spectrum of both antennas are exactly the same, after being demodulated0 with f0 + ∆ f , respectively. Therefore, only+Δ the corresponding part of the spectrum can be demodulated The along-track baselinef0 lengthff0 of LB-IRA is 40 m at most in case of severe time decorrelation, correctlythe carrier as afrequency baseband of signal. and After bandpass, respectively. filtering, Therefore, the signal only spectrum the corresponding overlaps with eachpart otherof the however, the dual antennas both send pulses for interferometry, and therefore, the received signals veryspectrum well owing can be to demodulated CFS, besides, thecorrectly radio interferenceas a baseband is avoided signal. sinceAfter the bandpass band-pass filtering, filter is the designed signal mayspectrum suffer overlapsfrom radio with interference each other withvery welleach owingother. toIn CFS,order besides, to prevent the radioradio interferenceinterference, is the avoided CFS according to the signal bandwidth. Δf andsince bandwidth the band-pass should filter be compatibleis designed withaccording each other.to the Assignal shown bandwidth. in Figure 9, the CFS is set to be based on Equation (12) to compensate for the baseline decorrelation. The bandwidth of the W transmitted pulses is b , the bandwidth should be smaller than the CFS, so that the transmitted spectrum windows will not overlap with each other, as shown in the upper left corner of Figure 9. The received signal spectrum of both antennas are exactly the same, after being demodulated with f ff+Δ the carrier frequency of 0 and 0 , respectively. Therefore, only the corresponding part of the spectrum can be demodulated correctly as a baseband signal. After bandpass filtering, the signal spectrum overlaps with each other very well owing to CFS, besides, the radio interference is avoided since the band-pass filter is designed according to the signal bandwidth.

FigureFigure 9. 9.Proper Proper design design of carrier of carrier frequency frequency shift (CFS) shift and (CFS) signal and bandwidth signal bandwidth to avoid radio to interference.avoid radio Theinterference. * along with the circle with a multiplication sign in it, stands for complex multiplication, which is a standard operation for SAR signal processing.

Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 20

Cross-interferometry or CFS compensation is easy to implement since the carrier frequency and bandwidth are programmable by Field-Programmable Gate Array (FPGA) nowadays [30]. The Remotecritical Sens. part2020 is, 12how, 3519 to properly design the CFS to achieve both centimetric accuracy and kilometer11 of 19 resolution. According to Equation (12), the CFS is determined by several key parameters of LB-IRA, most of which are correlated or mutually restricted with each other. Therefore, the optimal design shouldprinciple be of addressed CFS should for be LB-IRA. addressed The followingfor LB-IRA. focuses The following on the design focuses principle on the ofdesign CFS, principle in terms ofof radarCFS, in look terms angle, of radar ocean look surface angle, slope, ocean and surface the cross-track slope, and baseline the cross-track length. baseline length. 4.2.1. Radar Look Angle 4.2.1. Radar Look Angle During the selection of CFS, the swath width of LB-IRA and SNR should be taken into consideration During the selection of CFS, the swath width of LB-IRA and SNR should be taken into while choosing the proper radar look angle. The look angle varies within a certain range to acquire consideration while choosing the proper radar look angle. The look angle varies within a certain a swath width of 50 km, at least. According to Figure8, a smaller center look angle leads to wider range to acquire a swath width of 50 km, at least. According to Figure 8, a smaller center look angle variation range of look angle in order to cover up the swath width, and eventually, larger CFS variation leads to wider variation range of look angle in order to cover up the swath width, and eventually, range since the frequency shift is determined by radar look angle. larger CFS variation range since the frequency shift is determined by radar look angle. However, the CFS method can only select a constant frequency shift for the entire swath. However, the CFS method can only select a constant frequency shift for the entire swath. The The selected CFS may perfectly compensate for the baseline decorrelation at the swath center, however, selected CFS may perfectly compensate for the baseline decorrelation at the swath center, however, it leaves the other part of the swath only partially compensated. Figure 10 shows the spectrum overlap it leaves the other part of the swath only partially compensated. Figure 10 shows the spectrum before and after CFS; around the center range where CFS is selected accordingly, the spectrum perfectly overlap before and after CFS; around the center range where CFS is selected accordingly, the overlaps with each other after CFS. However, due to the variation of radar look angle, the spectrum spectrum perfectly overlaps with each other after CFS. However, due to the variation of radar look only partially overlaps with each other around the near or far range of the swath. angle, the spectrum only partially overlaps with each other around the near or far range of the swath.

FigureFigure 10.10. Spectrum overlapoverlap beforebefore andand afterafter CFSCFS alongalong thethe rangerange direction.direction.

FigureFigure 1111a,ba,b showsshows thethe coherencecoherence beforebefore andand afterafter CFSCFS compensation,compensation, thethe cross-trackcross-track baselinebaseline lengthlength andand bandwidthbandwidth areare setset toto bebe 10001000 mm andand 4040 MHz,MHz, respectively.respectively. TheThe looklook angleangle rangesranges ofof 9~129~12°◦ andand 12~1512~15°◦ cover up nearlynearly thethe samesame swathswath widthwidth ofof 5050 km,km, howeverhowever thethe CFSCFS variationvariation ofof 9~129~12°◦ isis largerlarger thanthan thatthat ofof 12~1512~15°.◦. Before CFS compensation,compensation, as shownshown inin FigureFigure 1111a,a, thethe coherencecoherence isis zerozero fromfrom thethe swath center center to to the the near near swath swath since since the the sp spectrumectrum has has no nooverlap overlap part part due dueto a tosmaller a smaller look lookangle. angle. Only a Only small a part small of partthe sp ofectrum the spectrum overlaps overlaps with each with other each near other the far near range the as far the range radar as look the radarangle lookincreases, angle due increases, to limited due signal to limited bandwidth. signal bandwidth.After CFS compensation, After CFS compensation, as shown in Figure as shown 11b, inalthough Figure 11bothb, althoughof the coherence both of are the high coherence at the areswat highh center at the (corresponds swath center to (correspondscenter look angle), to center the lookcoherence angle), of the 9~12° coherence decreases of 9~12more◦ decreasessharply than more 12~15° sharply due than to larger 12~15 CFS◦ due variation, to larger as CFS indicated variation, by asthe indicated dashed line by the in Figure dashed 11a. line Apparently, in Figure 11a. 12~15° Apparently, is more 12~15 preferable◦ is more since preferable the coherence since the is higher coherence and ismore higher consistent and more among consistent the entire among swath. the entire swath. However, too-large look angle may decrease the SNR signiﬁcantly since the normalized radar cross-section (NRCS) decreases as radar look angle increases, and the decorrelation by thermal noise will get worse. Referring to the 1σ NRCS from Global Precipitation Mission (GPM) [31], 12~15◦ of radar look angle is selected for LB-IRA, which compensates for the baseline decorrelation well among the entire swath and maintains a reasonable SNR level as well.

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(a) (b)

Figure 11. Coherence determined by baseline decorrelation, (a) before CFS compensation, and (b) after CFS compensation.

However, too-large look angle may decrease the SNR significantly since the normalized radar cross-section (NRCS) decreases(a) as radar look angle increases, and the decorrelation(b) by thermal noise will get worse. Referring to the 1𝜎 NRCS from Global Precipitation Mission (GPM) [31], 12~15° of radarFigure look angle 11. Coherence is selected determineddetermined for LB-IRA, by by baseline whichbaseline decorrelation,compensates decorrelation, ( afor) ( beforea the) before baseline CFS CFS compensation, decorrelationcompensation, and well (andb) after (amongb) the entireafterCFS compensation. CFSswath compensation. and maintains a reasonable SNR level as well. 4.2.2. Cross-Track Baseline Length 4.2.2.However, Cross-Track too-large Baseline look Length angle may decrease the SNR significantly since the normalized radar cross-sectionAlthough (NRCS) the decorrelation decreases as from radar large look baseline angle increases, can be well-compensated and the decorrelation by CFS, bylarger thermal baseline noise Although the decorrelation from large baseline can be well-compensated by CFS, larger baseline willalso get leads worse. to smaller Referring altimetric to the ambiguity,1𝜎 NRCS from and furtherGlobal leadsPrecipitation to more Mission severe volume(GPM) [31], decorrelation. 12~15° of also leads to smaller altimetric ambiguity, and further leads to more severe volume decorrelation. radarThe volume look angle decorrelation is selected offor ocean LB-IRA, surface which is compensates different from for terrain the baseline [32], introduced decorrelation by thewell random among The volume decorrelation of ocean surface is different from terrain [32], introduced by the random thedistribution entire swath of surface and maintains wave height a reasonable as well as SNR theslightly level asdi well.fferent measurement geometry. As shown distribution of surface wave height as well as the slightly different measurement geometry. As shown in Figure 12, the received echoes by LB-IRA antennas are from a combination of multiple scatterers in Figure 12, the received echoes by LB-IRA antennas are from a combination of multiple scatterers 4.2.2.within Cross-Track the same resolution Baseline Length grid. LB-IRA records an echo as a complex number, the phase of which is within the same resolution grid. LB-IRA records an echo as a complex number, the phase of which is directly related to satellite-ocean surface range, and because of the randomness of surface wave height, directlyAlthough related the to decorrelationsatellite-ocean from surface large range, baseline and can because be well-compensated of the randomness by CFS, of larger surface baseline wave the range-associated phase recorded by two antennas deviate from each other, as indicated in Figure 12 height,also leads the torange-associated smaller altimetric phase ambiguity, recorded and by two further antennas leads deviate to more from severe each volume other, asdecorrelation. indicated in by blue and yellow arrows within the complex plane, the summation of which suffer from a certain FigureThe volume 12 by decorrelationblue and yellow of oceanarrows surface within isthe diff complexerent from plane, terrain the summation [32], introduced of which by thesuffer random from degree of randomness, or equally speaking, the signal decorrelation. adistribution certain degree of surface of randomness, wave height or asequall welly as speaking, the slightly the differentsignal decorrelation. measurement geometry. As shown in Figure 12, the received echoes by LB-IRA antennas are from a combination of multiple scatterers within the same resolution grid. LB-IRA records an echo as a complex number, the phase of which is directly related to satellite-ocean surface range, and because of the randomness of surface wave height, the range-associated phase recorded by two antennas deviate from each other, as indicated in Figure 12 by blue and yellow arrows within the complex plane, the summation of which suffer from a certain degree of randomness, or equally speaking, the signal decorrelation.

Figure 12. SchematicSchematic diagram diagram of of wave wave volume volume decorrelation. decorrelation.

To quantify the wave volume decorrelation, suppose surface wave height follows the Gaussian distribution, then, the coherence decided by wave volume decorrelation can be expressed as [33]:

1 σ 2 γwave = e− 2 · φ (14) Figure 12. Schematic diagram of wave volume decorrelation.

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To quantify the wave volume decorrelation, suppose surface wave height follows the Gaussian distribution, then, the coherence decided by wave volume decorrelation can be expressed as [33]:

−⋅1 σ 2 Remote Sens. 2020, 12, 3519 γ = 2 φ (14) 13 of 19 wave e

2cos()πσB B( ) θ− α σ 2=πσh cosh θ α =φ − where σφ H tan θλ θλis the standard deviation of interferometric phase, and σh is theσ standard where H tan is the standard deviation of interferometric phase, and h is the deviation of surface wave height of ocean surface, related to the signiﬁcant wave height (SWH) standard deviation of surface wave height of ocean surface, related to the significant wave height as SWH = 4σ . Figure 13a,b shows the wave volume decorrelation under diﬀerent SWH settings, SWh H = 4 σ where(SWH) the as cross-track baselineh . Figure length 13a,b of each shows is629 the andwave 1000 volume m, respectively. decorrelation The under SWH different= 2 m situation SWH correspondssettings, where to 1σ theof globalcross-track ocean baseline wave height length distribution. of each is 629 From and 1000 Figure m, 13 respectively., we learn that The though SWH = CFS2 m situation corresponds to 1𝜎 of global ocean wave height distribution. From Figure 13, we learn can compensate for the baseline decorrelation, the wave volume decorrelation still gets worse as the that though CFS can compensate for the baseline decorrelation, the wave volume decorrelation still baseline length increases. In order to avoid severe wave volume decorrelation, the cross-track baseline gets worse as the baseline length increases. In order to avoid severe wave volume decorrelation, the length is restricted below 1000 m for the LB-IRA. cross-track baseline length is restricted below 1000 m for the LB-IRA.

(a) (b)

FigureFigure 13. 13.Coherence Coherence determined determined byby wavewave volumevolume decorrelation. (a (a) )Cross-track Cross-track baseline baseline length length of of 629629 m, m, and and (b ()b cross-track) cross-track baseline baseline length length of of 10001000 m.m. 4.2.3. Ocean Surface Slope 4.2.3. Ocean Surface Slope The slope here refers to the ocean surface slope that extends beyond at least one resolution grid of The slope here refers to the ocean surface slope that extends beyond at least one resolution grid 1 km. The influence of the wave slope variation within one resolution grid causes the signal decorrelation of 1 km. The influence of the wave slope variation within one resolution grid causes the signal ratherdecorrelation than frequency rather shift than and frequency can be decomposed shift and can into be wave decomposed volume or into time wave decorrelation, volume etc.or time decorrelation,The OST variation etc. is normally less than a few decimeters over a spatial extension of tens or hundredsThe of OST kilometers. variation Suppose is normally a sub-mesoscale less than a few eddy decimeters of 50 km over diameter a spatial and 50extension cm height of tens anomaly or located in the middle of the swath, then the slope introduced by the eddy is merely 2.1 arcsec, and its hundreds of kilometers. Suppose a sub-mesoscale eddy of 50 km diameter and 50 cm± height anomaly inﬂuencelocated toin the frequency middle shiftof the can swath, betotally then the omitted. slope introduced Under rough by the condition eddy is merely where 2.1 surges arcsec, may reside,and theits slope influence could to be frequency large enough shift can that be it causestotally omitted. a non-negligible Under rough frequency condition shift, where however, surges the may OST altimetryreside, the of LB-IRA slope could is restricted be large toenough work that below it caus leveles 4 a of non-negligible sea state (corresponds frequency toshift, SWH however, smaller the than 2 m),OST since altimetry OST ofof highLB-IRA sea is state restricted is considered to work below unreliable. level 4 of sea state (corresponds to SWH smaller than 2 m), since OST of high sea state is considered unreliable. 5. Methodology for Altimetric Accuracy Analysis

5. Methodology for Altimetric Accuracy Analysis

5.1. Parameters Setting The altimetric errors of LB-IRA are mainly introduced from interferometric phase noise, baseline inclination, and length error. The interferometric phase noise is determined by the signal coherence, therefore, LB-IRA works in cross-interferometry mode which introduces the CFS to compensate for baseline decorrelation. Apart from baseline decorrelation, time decorrelation, thermal decorrelation, and wave volume decorrelation should all be taken into consideration while selecting the system parameters. The signal coherence can also be improved by averaging neighboring resolution cells, Remoteusually Sens. 2020referred, 12, 3519 to as multi-looking by the InSAR community, which will be explained later. 14 of 19 The altimetric error from baseline inclination or length error can only be decreased by longer cross-track, higher radar band, or smaller radar look angle. Therefore, apart from the CFS method, proper system parameters must be selected for LB-IRA. Figure 14 shows the ﬂowchart of the proper system parameters must be selected for LB-IRA. Figure 14 shows the flowchart of the methodology for parameters selection. Since the LB-IRA is mainly focused on geostrophic current or methodology for parameters selection. Since the LB-IRA is mainly focused on geostrophic current or marine gravity anomaly inversion, whether the systems parameters are properly selected is determined marine gravity anomaly inversion, whether the systems parameters are properly selected is if thedetermined centimetric if the relative centimetric accuracy relative is achieved accuracy at is 1achieved km resolution. at 1 km resolution.

FigureFigure 14. 14.Flowchart Flowchart ofof thethe methodologymethodology for for parameters parameters selection. selection.

TheThe main main system system parameters parameters for for numerical numerical analysis analys areis are listed listed in Table in Table1. Since 1. Since the baseline the baseline length thatlength determines that determines the CFS variesthe CFS with varies latitude, with latitude, the CFS the needs CFS to needs be accommodated to be accommodated with varying with varying latitude duringlatitude the during orbit cycle, the orbit besides, cycle, the besides, bandwidth the bandwidth needs to beneeds accommodated to be accommodated with varying with varying CFS in caseCFS of radioin case interference. of radio interference. Therefore, theTherefore, CFS and the its CFS corresponding and its corresponding radar carrier radar frequency carrier frequency and bandwidth and varybandwidth within a vary recommended within a recommended range rather range than beingrather a than ﬁxed being value, a fixed as listed value, in as Table listed1. in Table 1.

TableTable 1. 1. SystemSystem parameters.

ParameterParameter Value Value Altitude 891 km Altitude 891 km CarrierCarrier frequency frequency of of Sat1 Sat1 13.5 13.5GHz GHz CarrierCarrier frequency frequency of of Sat2 Sat2 13.5~13.6 13.5~13.6 GHz GHz CFS 30~60 MHz CFS 30~60 MHz Bandwidth 30~60 MHz Radar look angle 12~15 Peak transmit power 3000 W Pulse length 5 µs Pulse repeat frequency 4000 Antenna length 5 m Antenna width 0.4 m Cross-track baseline length 629~1000 m Along-track baseline length 0~40 m

5.2. Multi-Looking The total coherence which ultimately determines the phase noise level of LB-IRA can be expressed as:

γ = γuBγthermalγwaveγtimeγothers (15) Remote Sens. 2020, 12, x FOR PEER REVIEW 15 of 20

Bandwidth 30~60 MHz Radar look angle 12~15 Peak transmit power 3000 W Pulse length 5 𝜇𝑠 Pulse repeat frequency 4000 Antenna length 5 m Antenna width 0.4 m Cross-track baseline length 629~1000 m Along-track baseline length 0~40 m

5.2. Multi-Looking The total coherence which ultimately determines the phase noise level of LB-IRA can be expressed as:

Remote Sens. 2020, 12, 3519 γγγ= γ γγ 15 of 19 uB therm al w ave tim e others (15)

γ γ uB thermal where γuB is theis the baseline baseline decorrelation decorrelation in cross-interferometry in cross-interferometry mode, γthermal mode,is the thermal is decorrelation,the thermal γ γ decorrelation,γwave is the wave wave volume is the decorrelation, wave volume and decorrelation,γtime is the time and decorrelation. time is the In time addition decorrelation. to the above In additiondecorrelation to the factors, above quantization, decorrelation ambiguities, factors, quanti and processing,zation, ambiguities, etc., all may and lead processing, to signal decorrelation, etc., all may leadhowever, to signal they decorrelation, can be considered however, minor they comparing can be considered to those major minor ones. comparing The total to coherence those major is shown ones. Thein Figure total coherence15, along with is shown all major in Figure decorrelation 15, along factors with all as well,major and decorrelation the parameters factors used as forwell, numerical and the parametersanalysis are used from for Table numerical1. Figure analysis 15a,b correspondsare from Table to the1. Figure results 15a,b of 629 corres andponds 1000 m to of the cross-track results of 629baseline and 1000 length, m of respectively. cross-track Thebaseline total coherencelength, respec of 629tively. m is The above total 0.5, coherence and as the of cross-track629 m is above baseline 0.5, andlength as increasesthe cross-track to 1000 baseline m, the total length coherence increases decreases to 1000 duem, the to more total severecoherenc wavee decreases volume decorrelation, due to more severehowever, wave it is volume still above decorrelation, 0.4 throughout however, most it of is the still swath. above 0.4 throughout most of the swath.

0.9 0.9

0.7 0.7

0.5 0.5 Total Total Thermal Thermal 0.3 Baseline 0.3 Baseline volume volume timelag timelag 0.1 0.1 12 13 14 15 12 13 14 15 Radar Look angle[deg] Radar Look angle[deg]

(a) (b)

Figure 15. Total coherence alongside other major decorrelation factors of LB-IRA. (a) Cross-track Figure 15. Total coherence alongside other major decorrelation factors of LB-IRA. (a) Cross-track baseline length of 629 m, and (b) cross-track baseline length of 1000 m. baseline length of 629 m, and (b) cross-track baseline length of 1000 m. The phase noise due to coherence loss can be considered as white noise, hence spatial averaging, The phase noise due to coherence loss can be considered as white noise, hence spatial averaging, or, in other words, multi-looking, can be utilized to suppress phase noise in order to increase the or, in other words, multi-looking, can be utilized to suppress phase noise in order to increase the altimetric accuracy [6,16]. Diﬀerent from terrain topography, OST varies extremely mildly, therefore, altimetric accuracy [6,16]. Different from terrain topography, OST varies extremely mildly, therefore, large numbers of independent pixels of intrinsic resolution can be multi-looked. The phase noise large numbers of independent pixels of intrinsic resolution can be multi-looked. The phase noise standard deviation after multi-looking is [23]: standard deviation after multi-looking is [23]: s 1 γ2 σ = − (16) φ 2 2NaziNrgγ

ρgrid ρgrid where γ is the total coherence before multi-looking, Nrg = , N = are the number of ρrg azi ρazi independent resolution cells along the range and azimuth direction respectively, ρazi and ρrg are the intrinsic resolution along range and azimuth direction respectively, and ρgrid is the grid resolution after multi-looking.

6. Numerical Results of LB-IRA The altimetric accuracy of LB-IRA is evaluated and discussed based on the selected parameters from Table1. Figure 16a, b shows the absolute altimetric accuracy determined by phase noise alone, under a cross-track baseline length of 629 and 1000 m, respectively. Although the total coherence of LB-IRA is relatively low, as shown in Figure 15, owing to multi-looking, which considerably suppresses the phase noise, the absolute altimetric accuracy still meets the centimetric accuracy requirement at 1 km resolution. Assuming the phase noise of two neighboring grids are totally uncorrelated, then, the relative accuracy will be decreased by a factor of √2, which is still better than 1 cm for most cases. Remote Sens. 2020, 12, x FOR PEER REVIEW 16 of 20

−γ 2 σ = 1 φ γ 2 (16) 2NNazi rg

ρ ρ N = grid N = grid γ rg ρ azi ρ where is the total coherence before multi-looking, rg , azi are the number of ρ ρ independent resolution cells along the range and azimuth direction respectively, azi and rg are the ρ intrinsic resolution along range and azimuth direction respectively, and grid is the grid resolution after multi-looking.

Remoterequirement Sens. 2020, 12at, 35191 km resolution. Assuming the phase noise of two neighboring grids are totally16 of 19 uncorrelated, then, the relative accuracy will be decreased by a factor of √2, which is still better than 1 cm for most cases.

(a) (b)

FigureFigure 16. 16.The The absolute altimetric altimetric accuracy accuracy determined determined by phase by phase noise, noise, at 1 km at resolution. 1 km resolution. (a) Cross- (a) Cross-tracktrack baseline baseline length length of 629 of m, 629 and m, (b and) cross-track (b) cross-track baseline baseline length lengthof 1000 ofm. 1000 m.

AccordingAccording to to Reference Reference [24], [24], the the baseline baseline length length error error of TSX/TDX of TSX/ TDXis smaller is smaller than 1 mm than by 1 DGPS mm by DGPSevaluation. evaluation. Figure Figure 17a shows 17a shows the absolute the absolute accuracy accuracy determined determined by a baseline by a baselinelength error length of 1 error mm, of 1 mm,the absolute the absolute accuracy accuracy is a few is a centimeters few centimeters among among the variation the variation range rangeof cross-track of cross-track baseline. baseline. The Therelative relative accuracy accuracy determined determined by bythe the same same baseline baseline length length error error is isshown shown in inFigure Figure 17b. 17 b.Due Due to to diﬀdifferentiationerentiation which which cancels cancels out out most most of of thethe commoncommon pa partsrts of of error, error, the the relative relative accuracy accuracy is isfar far better better than the centimetric requirement. thanRemote the Sens. centimetric 2020, 12, x FOR requirement. PEER REVIEW 17 of 20

(a) (b)

FigureFigure 17. 17.The The altimetric altimetric accuracy accuracy determineddetermined by a baseline length length error error of of 1 1mm. mm. (a ()a The) The absolute absolute accuracyaccuracy and and (b ()b the) the relative relative accuracy. accuracy.

AccordingAccording to to Equation Equation (7), (7), a a baseline baseline lengthlength error of 1 1 mm mm will will result result in in an an equivalent equivalent inclination inclination errorerror of of 0.2~0.3 0.2~0.3 arcsec, arcsec, corresponding corresponding toto thethe baselinebaseline length of of 629~1000 629~1000 m. m. Figure Figure 18a 18 indicatesa indicates that that thethe absolute absolute centimetric centimetric accuracy accuracy cannot cannot be be met met without without further further calibrationcalibration fromfrom thethe ground, which which is alsois also the samethe same dilemma dilemma faced faced by by SWOT SWOT [34 [34].]. However, However, thethe relativerelative accuracy accuracy is is as as high high as as 1~2 1~2 mm, mm, asas shown shown in Figurein Figure 18b, 18b, which which is quite is quite suﬃ sufficientcient for applicationsfor applications such such as geostrophic as geostrophic current current or marine or gravitymarine anomaly gravity inversion.anomaly inversion.

(a) (b)

Figure 18. The altimetric error determined by an equivalent baseline inclination error. (a) The absolute accuracy and (b) the relative accuracy.

Although the relative altimetric accuracy of LB-IRA can be as high as a few millimeters, the absolute accuracy still cannot fulfill the centimetric accuracy requirement at present. The biggest altimetric error comes from the baseline length error, due to limited ranging accuracy between two satellites. If the ranging accuracy can be improved by an order of magnitude by calibration from the ground, or utilizing other advanced ranging techniques, for example, the K-band ranging system on

Remote Sens. 2020, 12, x FOR PEER REVIEW 17 of 20

(a) (b)

Figure 17. The altimetric accuracy determined by a baseline length error of 1 mm. (a) The absolute accuracy and (b) the relative accuracy.

According to Equation (7), a baseline length error of 1 mm will result in an equivalent inclination error of 0.2~0.3 arcsec, corresponding to the baseline length of 629~1000 m. Figure 18a indicates that the absolute centimetric accuracy cannot be met without further calibration from the ground, which is also the same dilemma faced by SWOT [34]. However, the relative accuracy is as high as 1~2 mm, Remoteas shown Sens. 2020 in, 12Figure, 3519 18b, which is quite sufficient for applications such as geostrophic current17 or of 19 marine gravity anomaly inversion.

(a) (b)

FigureFigure 18. 18.The The altimetric altimetric error error determineddetermined by an equivalent baseline baseline inclination inclination error. error. (a) ( aThe) The absolute absolute accuracyaccuracy and and (b ()b the) the relative relative accuracy. accuracy.

AlthoughAlthough the the relative relative altimetric altimetric accuracyaccuracy of LB- LB-IRAIRA can can be be as as high high as as a afew few millimeters, millimeters, the the absoluteabsolute accuracy accuracy still still cannot cannot fulﬁll fulfill thethe centimetriccentimetric accuracy requirement requirement at at present. present. The The biggest biggest altimetricaltimetric error error comes comes from from the the baseline baseline lengthlength error, due to to limi limitedted ranging ranging accuracy accuracy between between two two satellites.satellites. If If the the ranging ranging accuracy accuracy cancan bebe improvedimproved by an an order order of of magnitude magnitude by by calibration calibration from from the the ground,ground, or or utilizing utilizing other other advanced advanced rangingranging techniques,techniques, for for example, example, the the K-band K-band ranging ranging system system on on Gravity Recovery And Climate Experiment (GRACE) satellite whose accuracy is only a couple of microns [35], the altimetric capability of LB-IRA will be improved remarkably.

7. Conclusions This paper proposed the LB-IRA from satellite formation which utilizes CFS to compensate for large baseline decorrelation. The simulation results indicate that ~1 cm relative accuracy can be achieved at 1 km resolution. The LB-IRA is currently a possible solution rather than a specific satellite mission for OST altimetry. Comparing to single satellite altimetry, the advantages of satellite formation are:

The CFS method can release the baseline length limitation from baseline decorrelation, • also increasing the freedom of selecting other interferometric parameters. The altimetric error associated to phase noise is only a few millimeters, which is quite extraordinary • owing to much larger baseline length than that of the single platform. It poses relatively easier or fewer technical or engineering challenges, such as the rigorous • attitude control accuracy, or the mast vibration or dilation error that should be addressed by the SWOT team. The relative altimetric accuracy is quite consistent throughout the entire swath, facilitating marine • application based on the OST gradient.

However, one major issue to be addressed is the limited observation coverage of LB-IRA; due to time decorrelation, the suggested altimetry region is within 44.6◦ N/S latitude at an orbit inclination of 78◦. The possible solution is to adjust the orbit perigee so that the along-track baseline is short at the high-latitude region, as shown in Figure4c. An alternative solution for the insu ﬃcient coverage is to launch two sets of LB-IRAs, however, this is at the cost of a higher budget. Another issue is the restricted condition for conducting OST altimetry. The SWH should be smaller than 2 m for higher reliability of the altimetry result, which is the same dilemma faced by any other IRA as well. However, since one of the primary applications for LB-IRA is ocean bathymetry Remote Sens. 2020, 12, 3519 18 of 19 inversion (which requires centimetric relative accuracy of OST at least), a large volume of data within a long time range may compensate for the limited observation eﬃciency.

Author Contributions: Conceptualization, W.K. and B.L.; funding acquisition, B.L.; investigation, W.K. and B.L.; methodology, W.K.; supervision, B.L. and J.S.; writing—original draft, W.K.; writing—review and editing, W.K., B.L., J.S., R.Z. and X.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Qian Xuesen Lab.—DFH Sat. Co Joint Research and Development Fund (Grant No. M-2017-006), the National Defense Science and Technology Innovation Special Zone Project (Grant No. Y-SYS-GZZ-SK-002), and the National Natural Science Foundation of China (Grant No. 42074225). Acknowledgments: We would like to thank the support from the “National defense Basic research program” and the “China Aerospace Science and Technology Corporation Limited independent research and development project”. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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