remote sensing Article
On Signal Modeling of Moon-Based Synthetic Aperture Radar (SAR) Imaging of Earth Zhen Xu 1,2 and Kun-Shan Chen 1, * 1 2
*
ID
State Key Laboratory of Remote Sensing Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100101, China;
[email protected] University of Chinese Academy of Sciences, Beijing 100049, China Correspondence:
[email protected]; Tel.: +86-139-1063-4643
Received: 26 December 2017; Accepted: 19 March 2018; Published: 20 March 2018
Abstract: The Moon-based Synthetic Aperture Radar (Moon-Based SAR), using the Moon as a platform, has a great potential to offer global-scale coverage of the earth’s surface with a high revisit cycle and is able to meet the scientific requirements for climate change study. However, operating in the lunar orbit, Moon-Based SAR imaging is confined within a complex geometry of the Moon-Based SAR, Moon, and Earth, where both rotation and revolution have effects. The extremely long exposure time of Moon-Based SAR presents a curved moving trajectory and the protracted time-delay in propagation makes the “stop-and-go” assumption no longer valid. Consequently, the conventional SAR imaging technique is no longer valid for Moon-Based SAR. This paper develops a Moon-Based SAR theory in which a signal model is derived. The Doppler parameters in the context of lunar revolution with the removal of ‘stop-and-go’ assumption are first estimated, and then characteristics of Moon-Based SAR imaging’s azimuthal resolution are analyzed. In addition, a signal model of Moon-Based SAR and its two-dimensional (2-D) spectrum are further derived. Numerical simulation using point targets validates the signal model and enables Doppler parameter estimation for image focusing. Keywords: Moon-Based Synthetic Aperture Radar; Doppler parameters; signal model; image focusing
1. Introduction Earth observation from remote sensing satellites orbiting in a low Earth orbit provides a continuous stream of data that can enable a better understanding of the Earth with respect to climate change [1]. Recently, the concept of observing the Earth from the Moon-based platform was proposed [2–4]. The Moon, as the Earth’s only natural satellite, is stable in periodic motion, making an onboard sensor unique in observing large-scale phenomena that are related to the Earth’s environmental change [5,6]. Synthetic aperture radar (SAR), an active sensor, provides effective monitoring of the Earth with all-time observation capabilities [7,8]. A SAR placed in the lunar platform was proposed [3], in which the configurations and performance of The Moon-Based SAR system was thoroughly investigated. Also, the concept of the Moon-based Interferometric SAR (InSAR) were analyzed by Renga and Moccia [4]. Later, the performance and potential applications of the Moon-Based SAR were characterized by Moccia and Renga [5], and the scientific and technical issues in the application of lunar-based repeat track and along track interferometry in [6]. Following this stream of development, an L-band Moon-Based SAR for monitoring large-scale phenomena related to global environmental changes was discussed [9] and the coverage performance of the Moon-based platform for global change detection [10]. The geometric modeling and coverage for a lunar-based observation by using Jet Propulsion Laboratory (JPL) ephemerides data were addressed in [11]. These studies are focused on the performance analysis and
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potential applications with some assumptions, such as a regular spherical Earth, an orbicular circular lunar orbit, a fixed earth’s rotational velocity, and a stationary Moon. By so doing, the SAR onboard Moon can be viewed as an inverse SAR (ISAR) or an equivalent sliding spotlight SAR [6,12]. However, when the effect of the lunar revolution was ignored in modeling the Doppler parameters, SAR image focusing is degraded to certain extent. Another aspect is that the imaging geometry of the Moon-Based SAR is quite different from that of LEO SAR systems. For an extremely long synthetic aperture, the signal path is a curved trajectory, and the common ‘stop-and-go’ assumption is no longer valid due to the long delay time in wave propagation. This paper aims at deriving a signal model of Moon-Based SAR imaging by considering the more complete, but more complex, geometry of SAR in Moon-Earth orbits. The Doppler parameters are first derived without assuming the ‘stop-and-go’ mode. The paper is organized as follows. In the next section, a right-handed Earth-centered reference coordinate is established to unify the ground target and Moon-Based SAR to model the Doppler parameters that are associated with the rotation and revolution of the Moon and Earth as given in Section 3. The signal model and its spectrum, based on a curved trajectory, are introduced in Section 4, followed by the imaging simulation based on a point target model. Finally, in Section 5, conclusions are drawn to close the paper. 2. The Reference Coordinate System Previous to the derivation of the Moon-Based SAR’s Doppler parameters, the Moon-Based SAR and the ground target shall be normalized to the same reference coordinate system. Accordingly, the coordinate reference frame that contains the moon-based SAR and ground target is first established in this section. The imaging of the SAR system depends on the relative motion between the earth target and Moon-Based SAR. In the relative motion, it is noted that the Moon revolves around the Earth with an average rotational velocity of 1023 m/s and a sidereal month of 27.32 days. The lunar orbit is elliptical, with an average semi-major axis of 384,748 km and an average eccentricity of 0.0549, which is close to a circular cycle. The lunar perigee is approximately 363,300 km, while the apogee can be up to 405,500 km. The angle between the lunar orbit plane and the Earth’s equatorial plane varies from 18.3◦ to 28.6◦ , with a period of 18.6 years [13]. To be more accurate in modeling, it is necessary to establish selenographic coordinates on the lunar surface to define the position of a Moon-based antenna. To derive this position in an inertial geocentric reference frame, a coordinate system that describes the Moon-Based SAR-ground targets geometric model is needed to unify the relative position of the Moon-Based SAR and the ground target position through a series of coordinate conversions. To derive the position in an inertial geocentric reference frame, four main reference frames are introduced [5]: (a)
(b) (c) (d)
a right-handed Earth-centered reference frame, which is a geocentric inertial reference frame with the x-axis from the Earth’s center to the true equinox of the date and the z-axis towards celestial North; a right-handed Moon-centered reference frame, with axes of XE and ZE that are parallel to the right-handed Earth-centered reference frame; a right-handed Moon-centered reference frame, with the XA axis from the Moon-center to the current center of the lunar disk and the ZA axis towards lunar north; and, a right-handed Moon-centered reference frame, with the XS axis from the Moon-center to the mean center of the apparent lunar disk and the ZS axis towards the lunar North Pole. The coordinate transformation matrix from frame (c) to frame (b) is − cos η M cos σM sin η M − cos η M sin σM TAE = − sin η M cos σM − cos η M − sin η M sin σM − sin σM 0 cos σM
(1)
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3 ofthe 24 are the right ascension and declination of the Moon, respectively; both of
variables are functions of time [14]. A ηrotation from frame (d) toascension frame (c) can achieved using matrix where the right andbe declination of thethe Moon, respectively; both of the M and σ M are variables are functions of time [14]. A rotation from frame (d) to frame (c) can be achieved using the matrix (2) − cos φm cos ϕm
where
− sin φm cos ϑm + cos φm sin ϕm sin ϑm
sin φm sin C + cos φm sin ϕm sin ϑm
TSA = − sin φm cos ϕm − cos φm cos ϑm + sin φm sin ϕm sin ϑm − cos l sin C + sin φm sin ϕm cos ϑm , , and are the three Euler angles of lunar libations [15] − sin ϕm
cos ϕm sin ϑm
(2)
cos ϕm cos ϑm
Consequently, the coordinates of a point l on the lunar surface in reference frame (d) can be relatedφto in the geocentric inertial reference frame[15]. (a), which takes the form of where ϕm ,coordinates and ϑm are the three Euler angles of lunar libations m , the Consequently, the coordinates of a point l on the lunar surface in reference frame (d) can be related to the coordinates in the geocentric inertial reference frame (a), which takes the form of (3) XM XSl Xl (3) YM = TSA TAE YSl + Yl l ZM ZS Zl where the superscript l refers to the point of the Moon-Based SAR and are the coordinates where the superscript l refers to the point of the inertial Moon-Based SAR and ( Xl , Yl , Zl ) are the coordinates of the Moon’s barycenter in the Earth-centered coordinate system. of theAs Moon’s barycenter in the Earth-centered inertial coordinate system. the selection of the lunar base-station site is still under study [16], the exact coordinates of the As the selection the lunar base-station site is site still is under studya [16], the exact geocentric coordinatesinertial of the Moon-Based SAR areofundecided. Once the station selected, right-handed Moon-Based SAR are undecided. Once the station site is selected, a right-handed geocentric inertial reference frame is defined, with the Z-axis towards the North Pole and the X-axis pointing to the true reference frame is defined, with the Z-axis theinNorth and the to and the equinox of the date. The coordinate systemtowards is shown FigurePole 1, where theX-axis lunarpointing ascension true equinoxare of the date. Theby coordinate system is shown in Figure 1, whereofthe lunar ascension and the longitude and latitude the ground targetand by declination designated declination are designated by ( am , δm ) and the longitude and latitude of the ground target by ( a g , δg ). . According to [17], when the ascending node of the lunar orbit points to the true equinox of According to [17], when the ascending node of the lunar orbit points to the true equinox of the date, the lunar date, the lunar declination reaches its maximum value under thisreference defined reference frame. declination reaches its maximum value under this defined frame. Now thatNow the that the Moon-Based and thetarget ground areunder unified same coordinate system. The Moon-Based SAR andSAR the ground aretarget unified theunder same the coordinate system. The Doppler Doppler parameters are discussed in the following parameters are discussed in the following sections. sections.
Figure Figure 1. 1. An unified unified right-handed right-handed geocentric geocentric inertial inertial reference reference coordinate coordinate system system for for Moon-Based Moon-Based synthetic (SAR), where point g represents the ground target, Rtarget, is the distance EM is the between distance syntheticaperture apertureradar radar (SAR), where point g represents the ground R EM the Earth and the Moon-Based SAR, which can be approximately regarded as the Earth and the Moon, between the Earth and the Moon-Based SAR, which can be approximately regarded as the Earth and and RE is the earth at point g. at (not to scale forto the sakefor of the clarity). the earth radius point g. (not scale sake of clarity). the Moon, and RE isradius
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3. The Doppler Parameters of Moon-Based SAR It is noted that the effect of the lunar revolution has been ignored for the simplicity in previous studies. On this basis, the Moon-Based SAR is assumed to be stationary and the synthetic aperture is realized simply by the rotation of the Earth [18]. However, the lunar revolution effect on the Doppler parameters is beyond negligible. To start with, the Doppler parameters for a stationary Moon-Based SAR are given for reference. 3.1. Doppler Parameters for a Stationary Moon-Based SAR Without loss of generality, we initialize the time to zero when the distance between the ground target and the Moon-Based SAR is smallest. The slant range between SAR and ground target at time t = η takes the expression R(η ) =
q
C1 + C2 cos a g (η ) + C3 sin a g (η )
(4)
where the coefficients Cs are explicitly given by C1 = R2E + R2EM − 2Zm R E sin δg
(4.a)
C2 = −2Xm R E cos δg
(4.b)
C3 = −2Ym R E cos δg
(4.c)
where ( Xm , Ym , Zm ) are the coordinates of the motionless Moon-Based SAR system, with Xm = R EM cos δm cos am
(4.d)
Ym = R EM cos δm sin am
(4.e)
Zm = R EM sin δm
(4.f)
Note that a g (η ) = a g0 + ωE η is the latitude at time η; a g0 the initial latitude at time zero; and, ωE the Earth’s rotation angular velocity. Note that that the initial latitude was set to zero in the previous studies [9,12]. The 1st-order derivative of the slant range reads R EM R E cos δg cos δm ωE sin a g (η ) − am ∂R(η ) = ∂η R(η )
(5)
According to [19], the Doppler frequency can be written as 2R EM R E cos δg cos δm ωE sin a g (η ) − am 2 ∂R(η ) f D (η ) = − =− λ ∂η λR(η )
(6)
where λ is the radar wavelength. It follows that the Doppler centroid takes the form [8] f ηc
2R EM R E cos δg cos δm ωE sin a g (ηc ) − am =− λR(ηc )
(7)
where ηc is the beam center crossing time. Then, the Doppler frequency modulation (FM) rate is written as [8,19] f dr
2R EM R E ωE2 cos δm cos δg cos am − a g (ηc ) 2R2EM R2E ωE2 cos2 δm cos2 δg sin2 am − a g (ηc ) = − (8) λR(ηc ) λR3 (ηc )
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The exposure time is defined as [20] Tsar =
λR(ηc ) L = VE ` a R E ωE cos δg
(9)
where VE is the linear velocity of ground target, defined as VE = R E ωE cos δg , and ` a is the aperture length of Moon-Based SAR in azimuthal direction. The total Doppler bandwidth during the target exposure is obtained by [21] BD = | f dr | Tsar =
2R EM ωE cos δm cos[ am − a g (ηc )] `a
−
2R2EM R E ωE cos2 δm cos2 δg sin2 [ am − a g (ηc )] ` a R2 ( ηc )
(10)
The 2nd term in the above equation is generally much smaller than the 1st term by two or three orders of magnitude and can be neglected. Though the exposure time of the Moon-Based SAR is in the order of hundreds of seconds, ωE η is very small. Consequently, cos am − a g (η ) = cos am − a g0 + sin am − a g0 ωE η + O(η 2 ) ≈ cos am − a g0 Then, the Doppler bandwidth rate is approximated to 2R EM ωE cos δm cos am − a g0 BD = `a
(11)
The azimuthal resolution is now given by ρa =
VE ` a R E cos δg 1 = BD 2 R EM cos δm cos am − a g0
(12)
From above equation, it is seen that the azimuthal resolution is no longer just ` a /2 [8], but is now modified by three terms: R E /R EM , cos δg /cos δm , and 1/cos( am − a g0 ). That is, the azimuthal resolution is determined by the aperture length, the relative position of the ground target and SAR, and the Earth’s radius and the distance between the Earth and the SAR. In general, for a specific position of the Earth’s target and the SAR, the Earth’s radius and the distance between the Earth and the Moon-Based SAR can be assumed to be constant. The azimuthal resolution given in Equation (12) is illustrated using the simulation parameters in Table 1. Table 1. Simulation parameters.
(1)
Parameter
Symbol
Quantity
Unit
Latitude of ground target Declination of Moon-Based SAR Inclination of lunar orbit to the equator of the Earth Earth radius Distance between the Earth and the Moon-Based SAR Earth’s rotation angular velocity Lunar revolution angular velocity Carrier frequency System bandwidth
δg δm ϑS Re Rem ωE ωM fc B
22.5 24.5 28.6 6371 389,408 7.292 × 10−5 2.662 × 10−6 1.2 50
degree degree degree km km rad/s rad/s GHz MHz
Effect of declination of Moon-Based SAR.
We investigate the impact of the declination of the Moon-Based SAR on the azimuthal resolution; the rest of the simulation parameters remain unchanged except for a variation in the declination of the lunar–based SAR. The Doppler bandwidth and azimuthal resolution are plotted as a function of the Moon-Based SAR’s declination in Figure 2a,b, respectively. In both cases, we set am − a g0 = 0.
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Remote Sens. 2018, 10, 486 the lunar–based SAR.
6 of 24 The Doppler bandwidth and azimuthal resolution are plotted as a function of the lunar–based SAR. The Doppler bandwidth and azimuthal resolution are plotted as a function of the Moon-Based SAR’s declination in Figure 2a,b, respectively. In both cases, we set . the Moon-Based SAR’s declination in Figure 2a,b, respectively. In both cases, we set . Asshown, shown,when whenthe thedeclination declinationincreases, increases,the theDoppler Dopplerbandwidth bandwidthdecreases, decreases,while whilethe theazimuthal azimuthal As As shown, when the declination increases, the Doppler bandwidth decreases, while the azimuthal resolutionbecomes becomescoarser, coarser,asasexpected. expected. resolution resolution becomes coarser, as expected.
(a) (a)
(b) (b)
Figure 2. (a) Relationship between Doppler bandwidth and Moon-Based SAR’s declination; (b) Figure Relationship between between Doppler Doppler bandwidth bandwidthand andMoon-Based Moon-BasedSAR’s SAR’sdeclination; declination; Figure2. 2. (a) Relationship (b) Azimuth resolution as a function of the Moon-Based SAR’s declination. (b)Azimuth Azimuthresolution resolutionasasa afunction functionofofthe theMoon-Based Moon-BasedSAR’s SAR’sdeclination. declination.
(2) Effects of ground target’s latitude. (2)(2) Effects Effects of of ground ground target’s target’s latitude. latitude. From the exposure time given in Equation (9), with the latitude increases, the linear velocity of From the exposure time given in Equation (9), with the latitude increases, the linear velocity of the Earth withtime a slight change in slant(9), range. effect is an increased exposure time, From decreases, the exposure given in Equation withThe thenet latitude increases, the linear velocity the Earth decreases, with a slight change in slant range. The net effect is an increased exposure time, which in turn affects thewith azimuthal show this, we exposure time and azimuthal of the Earth decreases, a slightresolution. change inTo slant range. Theplot net the effect is an increased exposure which in turn affects the azimuthal resolution. To show this, we plot the exposure time and azimuthal resolution, function theazimuthal ground target’s latitude Figure letting . From time, which as in aturn affectsofthe resolution. To in show this,3a,b weby plot the exposure time and resolution, as a function of the ground target’s latitude in Figure 3a,b by letting . From azimuthal resolution, as aresolution function ofbecomes the ground target’s latitude in Figure 3a,bground by letting am − latitude. a g0 = 0. Figure 3, the azimuthal finer with the increasing of the target’s Figure 3, the3,azimuthal resolution becomes finer with the increasing of the ground target’starget’s latitude. From Figure the azimuthal resolution becomes finer with the increasing of the ground This is contributed by the reduction of the linear speed, leading to increasing exposure time. This is contributed by the reduction of the linear leadingleading to increasing exposure time. time. latitude. This is contributed by the reduction of thespeed, linear speed, to increasing exposure
(a) (a)
(b) (b)
Figure 3. Variations of the (a) Exposure time and (b) Azimuthal resolution. The ascension is set to Figure Variations the Exposure time and Azimuthal resolution.The The ascensionis issetset Figure 3. 3.Variations ofof the (a)(a) Exposure time and (b)(b) Azimuthal resolution. ascension toto zero degree. zero degree. zero degree.
(3) Effect of azimuthal visible range. (3) Effect of azimuthal visible range. (3) Effect azimuthal visible range. From of Equation (12), a factor that is related to the azimuthal visible range, is From Equation (12), a factor that is related to the azimuthal visible range, is confined Fromby Equation (12), a factor cos am − a g0 that is related to the azimuthal visible range, confined by is confined by Ra (13) 0 < am − a g (η ) ≤ (13) (13) 2R E cos δg where the azimuthal visible range. whereR aRis is the azimuthal visible range. where Ra a is the azimuthal visible range.
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in (13) is bounded in To be more explicitly, the factor To be more explicitly, the factor cos am − a g0 in (13) is bounded in
Ra cos 2R E cos δg
(14) (14)
≤ cos am − a g0 < 1
Asseen, seen,the thelower lowerbound boundofofEquation Equation(13b) (13b)has hasaapositive positivecorrelation correlationwith withthe theazimuthal azimuthalvisible visible As rangeas aswell wellas asthe theabsolute absolutevalue valueofofthe theground groundtarget’s target’slatitude, latitude,which whichindicates indicatesthat thatwith withthe the range increasing of the azimuthal visible range, the Doppler bandwidth increases, resulting in a finer increasing of the azimuthal visible range, the Doppler bandwidth increases, resulting in a finer azimuthalresolution. resolution. azimuthal As an example,the themaximum maximumofofthe theazimuthal azimuthalvisible visiblerange rangeisisset settoto6000 6000km kmfor foraaground groundtarget target As an example, latitude of 22.5 degrees. The Doppler bandwidth and azimuthal resolution are plotted as a function latitude of 22.5 degrees. The Doppler bandwidth and azimuthal resolution are plotted as a function of of the azimuthal visible range in Figure the azimuthal visible range in Figure 4. 4.
(a)
(b)
Figure 4. Doppler bandwidth (a) and Azimuthal resolution (b) as a function of azimuth visible range. Figure 4. Doppler bandwidth (a) and Azimuthal resolution (b) as a function of azimuth visible range.
As evident from Figure 4, when the azimuthal visible range is limited to less than 1000 km with As evident Figure 4, when the azimuthal visible range is limited less thanfor1000 km target’s latitudefrom of 22.5 degrees, the effect of the visible range is quite small. to However, a larger with target’s latitude of 22.5 degrees, theincreases, effect of the visible range quite small. However,For for an a visible range, the Doppler bandwidth resulting a finerisazimuthal resolution. larger visible range, the Doppler bandwidth increases, resulting a finer azimuthal resolution. For an azimuthal visible range that is smaller than 1000 km, the factor is negligible; thus, azimuthal visible range that is smaller than 1000 km, the factor cos am − a g0 is negligible; thus, the azimuthal resolution can be approximated to the azimuthal resolution can be approximated to VE ` a R E cos δg (15) = (15) BD 2 R EM cos δm Nevertheless,in ingeneral, general, the the factor factor cos a − a should should takeninto intoaccount accountinindetermining determining Nevertheless, bebe taken m g0 theazimuthal azimuthalresolution, resolution,asasgiven givenininEquation Equation(12). (12). the ρa =
3.2. 3.2.Doppler DopplerParameters Parametersfor foraAMotional MotionalMoon-Based Moon-BasedSAR SAR In Inthe thepreceding precedingsub-section, sub-section,the theDoppler Dopplerparameters parametersare areevaluated evaluatedby byconsidering consideringthe theEarth’s Earth’s rotation only. However, because the Moon revolves around the Earth with an average rotational rotation only. However, because the Moon revolves around the Earth with an average rotational velocity 1023 m/s, the the Doppler parameters is much when both rotation and revolution velocityofof 1023 m/s, Doppler parameters is more muchinvolved more involved when both rotation and exerts. We need to consider the Moon revolution velocity, ω . The inclination between the plane of M revolution exerts. We need to consider the Moon revolution velocity, . The inclination between the Earth’s equator and the Moon revolution orbit is denoted by ϑS , which, typically, ranges from the plane of the Earth’s equator and the Moon revolution orbit is denoted by , which, typically, 18.3◦ to 28.6◦ , with a period of 18.6 years [22]. Since the variation of ϑS is extremely small during the ranges from to 28.6°, with a inclination period of 18.6 years [22]. Since the variation of is extremely exposure time,18.3° in what follows, the is assumed a constant.
small during the exposure time, in what follows, the inclination is assumed a constant. Figure 5 shows that the projected angle of the revolution velocity to the X-Y plane can be regarded as the inclination of that Moon-Based SAR, which varies to a limited extent during the exposure time. Consequently, the angular velocity of lunar revolution can be decomposed into two components along and perpendicular to the Z-axis as and ,
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Figure 5 shows that the projected angle of the revolution velocity to the X-Y plane can be regarded as the inclination thatPEER Moon-Based Remote Sens. 2018, 10, of x FOR REVIEW SAR, which varies to a limited extent during the exposure 8time. of 24 Consequently, the angular velocity of lunar revolution can be decomposed into two components along and perpendicular to thenoting Z-axis that as ωas = ωas cos ϑS and ωofMZ ϑS , respectively. is worth respectively. It is worth Moon-Based reaches the It maximum MZlong M declination ⊥ = ω M sin SAR noting that as long as declination of Moon-Based SAR reaches the maximum value or minimum value or minimum value, the sign of should change (+: for minimum to maximum;value, -: for the sign of ω should change (+: for minimum to maximum; -: for maximum to minimum). MZ ⊥ maximum to minimum).
Figure 5. Imaging geometry of Moon-Based SAR in consideration of lunar revolution. Figure 5. Imaging geometry of Moon-Based SAR in consideration of lunar revolution.
If the Moon shares the same orbit with the Earth’s self-rotation, the effect of the lunar revolution If Moon but shares the same orbit with theaEarth’s theby effect of the lunar revolution is quitethe limited, non-negligible. However, squint self-rotation, angle is caused , resulting in a more is quite limited, but non-negligible. However, a squint angle is caused by ω MZ⊥ , resulting in a more complicated imaging geometry due to an inclination between the lunar orbit and the orbit of the complicated imaging geometry due to an inclination between the lunar orbit and the orbit of the Earth’s Earth’s self-rotation. In addition, the presence of changes the position vector. When the Moonself-rotation. In addition, the presence of ϑ MZ changes the position vector. When the Moon-Based Based SAR around rotates an around X-Ytime plane 0 tothan timestationary, , ratherwhere than SAR rotates angle an of ϑangle X-Y planeinfrom 0 tofrom time time η, rather MZ in of θstationary, cos ϑS , the slant vector of, the Moon-Based SAR is totally different from of MZ = ω M ηwhere themotional slant vector of the motional Moon-Based SAR is that totally the stationary Moon-Based SAR. Accordingly, it is of vital importance to consider the effect of the lunar different from that of the stationary Moon-Based SAR. Accordingly, it is of vital importance to revolution on the Moon-Based SAR imaging. consider the effect of the lunar revolution on the Moon-Based SAR imaging.∗ Now that the motional Moon-Based SAR’s ascension can be rewritten as am (η ) = am + ω M η cos ϑS , Now that the motional Moon-Based SAR’s ascension can be rewritten as ∗ ( η ) = δ + ω η sin ϑ . As a result, the coordinates of the and its declination can be expressed as δm M S , and its declination canm be expressed as . As a Moon-Based SAR can be expressed as result, the coordinates of the Moon-Based SAR can be expressed as Xm (η ) = R EM cos[δm + ω M η sin ϑS ] cos[ am + ω M η cos ϑS ], (16) Ym (η ) = R EM cos[δm + ω M η sin ϑS ] sin[ am + ω M η cos ϑS ], Z (η ) = R sin[δ + ω η sin ϑ ]. (16) m m EM M S
We can obtain the slant range in the context of the motional Moon-Based SAR: (17)
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We can obtain the slant range in the context of the motional Moon-Based SAR: R EM (η )
= R2E + R2EM − 2Xm (η ) R E cos δg cos a g (η ) −2Ym (η ) R E cos δg sin a g (η ) − 2Zm (η ) R E sin δg
(17)
The Doppler frequency of the Moon-Based SAR in considering the lunar revolution can be modified as f DEM (η ) = f DE (η ) + f DM (η ) (18) where f DE (η ) is the Doppler frequency that is contributed by the Earth’s rotation: f DE (η ) = −
∗ (η )ω 2R EM R E cos δg cos δm E sin a g (η ) − a∗m (η ) EM λR (η )
(18.a)
f DM (η ) is the contributed by the lunar revolution: f DM (η )
2R EM R E ω M ∗ ( η ) sin a ( η ) − a∗ ( η ) cos ϑ cos δ cos δm g g S m λR EM (η ) ∗ ∗ ∗ − sin δm (η ) cos a g (η ) − am (η ) sin ϑS cos δg + cos δm (η ) sin ϑS sin δg
=
(18.b)
Noted that for ω M = 0, the Doppler frequency reduces to that of a stationary Moon-Based SAR in Equation (6). The Doppler FM rate can be approximated by the equation EM E M C f dr = f dr + f dr + f dr
(19)
E is the part of the Doppler FM rate mainly contributed by the Earth’s rotation where f dr E f dr
2 R EM R E 2 ∗ ∗ λ R EM (ηc ) ω E cos δg cos δm ( ηc ) cos a g ( ηc ) − am ( ηc ) R2EM R2E 2 2 2 2 ∗ a g (ηc ) − a∗m (ηc ) − λ2 EM 3 ω E cos δg cos δm ( η ) sin R (η )
=
[
c
(19.a)
]
M is the part of the Doppler FM rate that is mainly contributed by the lunar revolution and f dr M f dr
2 2 R EM R E ω M ∗ ( η ) cos a ( η ) − a∗ ( η ) cos δg cos δm c g c m c λ R EM (ηc ) ∗ ( η ) + sin 2ϑ sin δ∗ ( η ) sin a ( η ) − a∗ ( η ) − sin2 ϑS sin δg sin δm c g c S m c m c R R E ω 2M 2 ϑ cos2 δ cos2 δ∗ ( η ) sin2 a ( η ) − a∗ ( η ) − λ2 EM cos g g S 3 m m [ REM (ηc )] ∗ ( η ) cos a ( η ) − a∗ ( η ) − sin ϑ sin δ cos δ∗ ( η ) − cos ϑS sin ϑS cos δg sin δm g g o m S m ∗ ( η ) cos a ( η ) − a∗ ( η ) − sin δ cos δ∗ ( η ) 2 + sin2 ϑS cos δg sin δm g g m m
=
(19.b)
C is the coupling term of the Doppler FM rate between Earth’s rotation (ω ) and lunar and fdr E revolution (ω M ): C f dr
RE ωE ω M ∗ ( η ) sin a ( η ) − a∗ ( η ) = − λ4 REM cos δg sin ϑS sin δm c g c m c R EM (ηc ) ∗ ( η ) cos a ( η ) − a∗ ( η ) + cos δg cos ϑS cos δm c g c m c RE ωE ω M ∗ ( η ) sin2 a ( η ) − a∗ ( η ) −4 cos ϑS cos2 δg cos2 δm − REMEM g c 3 m c λ [ R ( ηc ) ] ∗ ( η )) sin 2a ( η ) − 2a∗ ( η ) +ωE ω M sin ϑS cos2 δg cos(2δm g c m c ∗ ( η ) sin a ( η ) − a∗ ( η ) −2ωE ω M sin ϑS sin 2δg cos2 δm g c m c
(19.c)
(19.c)
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For , Equation (19) reduces to Equation (7). For ω = 0, Equation (19) reduces to Equation (7). M It is interested to compare the Doppler frequency and Doppler FM rate for stationary and It is interested to compare the Doppler frequency and Doppler rate for stationary and in motional motional Moon-Based SAR. For a numerical illustration, we useFM the parameters shown Table 1 Moon-Based SAR. For a numerical illustration, we use the parameters shown in Table 1 with two with two cases of visible ranges: and . From Figure 6, for a small cases of visible ranges: am − a g0 = 0 and a g0 − am = π/6. From Figure 6, for a small visible range in visible range in azimuth, the motional SARDoppler produces larger Doppler but not azimuth, the motional Moon-Based SARMoon-Based produces larger frequency, but notfrequency, much, compared much, compared to the stationary one. The Doppler FM rate is little smaller in comparison, as to the stationary one. The Doppler FM rate is little smaller in comparison, as seen from Figureseen 6b. from Figure 6b. For a large visible range in azimuth, the difference of Doppler between the For a large visible range in azimuth, the difference of Doppler frequency betweenfrequency the motional and the motional and the stationary now much Figure 6c. Again, the stationary Moon-Based SARsMoon-Based is now muchSARs larger,isas shown inlarger, Figureas 6c.shown Again,inthe Doppler FM rate Doppler FM rate is not changed much, as seen from Figure 6d. is not changed much, as seen from Figure 6d.
(a)
(b)
(c)
(d)
Figure6.6.Comparison Comparisonbetween betweenthe the Doppler parameters stationary Moon-Based SAR (superscript Figure Doppler parameters of of stationary Moon-Based SAR (superscript S) S) motional and motional Moon-Based SAR (superscript with a small rangeand in aazimuth and and Moon-Based SAR (superscript M) with aM) small visible rangevisible in azimuth − a = 0: m g0 : (a) Doppler frequency; (b) Doppler frequency with a large (a) Doppler frequency; (b) Doppler frequency modulation (FM) modulation rate; with a (FM) large rate; visible range in azimuth a g0 − am = π/6; (c) Doppler frequency; (d) Doppler FM rate. visible range in azimuth ; (c) Doppler frequency; (d) Doppler FM rate.
It is noted that the change of Doppler frequency leads to extra quadratic phase error [19], which affects the image focusing. To this end, in Figure 7a,b we plot the quadratic phase error that is caused by the lunar revolution, using the same parameters as in Figure 6b,d. From Figure 7, we see that quadratic phase error exceeds the threshold of π/4. Thus, the imaging defocusing, mainly the image shift, is expected and should be corrected
It is noted that the change of Doppler frequency leads to extra quadratic phase error [19], which affects the image ToREVIEW this end, in Figure 7a,b we plot the quadratic phase error that is caused Remote Sens. 2018, 10,focusing. x FOR PEER 11 of 24 by the lunar revolution, using the same parameters as in Figure 6b,d. From Figure 7, we see that It is noted theexceeds change of frequency leads to extra quadraticdefocusing, phase errormainly [19], which quadratic phasethat error theDoppler threshold of . Thus, the imaging the affects the image focusing. To this end, in Figure 7a,b we plot the quadratic phase error that is11caused Remote Sens. 2018, 10, 486 of 24 image shift, is expected and should be corrected by the lunar revolution, using the same parameters as in Figure 6b,d. From Figure 7, we see that quadratic phase error exceeds the threshold of
. Thus, the imaging defocusing, mainly the
image shift, is expected and should be corrected
(a)
(b)
Figure 7. Quadratic phase error caused by the lunar revolution within an exposure time of 100 s: (a) Figure 7. Quadratic phase error caused by the lunar revolution within an exposure time of 100 s: (b)a g0 −(a) (a) a g0 − am = 0 (b) am = π/6. . (b) Figure 7. Quadratic phase error caused by the lunar revolution within an exposure time of 100 s: (a)
TheDoppler Dopplerbandwidth bandwidthisisthe theproduct productof ofthe theabsolute absolutevalue valueof ofthe theDoppler DopplerFM FMrate rateand andthe the The (b) . exposuretime. time.ItItisisinstructive instructivetotocompare comparethe theDoppler Dopplerbandwidth bandwidthbetween betweenaamotional motionalMoon-Based Moon-Based exposure SARand andaastationary stationaryMoon-Based Moon-BasedSAR, SAR,and andthe thecorresponding correspondingazimuthal azimuthalresolutions. resolutions.The TheDoppler Doppler SAR The Doppler bandwidth is the product of the absolute value of the Doppler FM rate and the bandwidthand and the the azimuthal azimuthal resolution with 0 to 80 bandwidth with varying varyingground groundtarget targetlatitudes, latitudes,ranging rangingfrom from 0 to exposure time. It is in instructive to compare the Doppler between a motional Moon-Based degrees are respectively, where the declination the Moon-Based SARisis 80 degrees areshown shown inFigure Figure8a,b 8a,b respectively, where bandwidth of the Moon-Based SAR SAR and a stationary Moon-Based SAR, andand the azimuthal corresponding azimuthal resolutions. Doppler assumed 24.5 degrees. The Doppler bandwidth and azimuthal resolution as of functions of the assumed 24.5 degrees. The Doppler bandwidth resolution as functions the The declination bandwidth and the azimuthal resolution with varying ground target latitudes, ranging from 0 to 80 declination with a ground at (0, 0)inare plotted in Figure 8c,d. with a ground target locatedtarget at (0, located 0) are plotted Figure 8c,d. degrees shown in Figure 8a,b respectively, where the declination of theMoon-Based Moon-Based SAR Fromare Figure 8 a,c can observe that Doppler bandwidth a motional Moon-Based SAR From Figure 8a,c wewe can observe that thethe Doppler bandwidth of aofmotional SAR is is assumed 24.5 degrees. The Doppler bandwidth and azimuthal resolution as functions of the is slightly different from that of a stationary Moon-Based SAR. The azimuthal resolutions of the slightly different from that of a stationary Moon-Based SAR. The azimuthal resolutions of the motional declination withand a ground target at (0, 0) Moon-Based are plotted inSAR Figure 8c,d. motional Moon-Based and oflocated theMoon-Based stationary aresame, almost same, as shown in Moon-Based SAR ofSAR the stationary SAR are almost the as the shown in Figure 8b,d, 8 a,c we canvelocity observe that the Doppler bandwidth of atodecreases motional SAR FigureFrom 8b,d, because both of strip and Doppler bandwidth due to the lunar because bothFigure the velocity of the strip and Doppler bandwidth decreases due the lunarMoon-Based revolution. is slightly different from that of a stationary Moon-Based SAR. The azimuthal resolutions of the revolution. At this point, the Doppler parameters of the motional Moon-Based SAR are discussed. It is found motional Moon-Based SARand andthe of Doppler the stationary Moon-Based SAR are almost theSAR same, shown in that the Doppler frequency FM rate of the motional Moon-Based areasdifferent Figure 8b,d, because both the velocity of strip and Doppler bandwidth decreases due to the lunar from those of the stationary Moon-Based SAR, indicating that the lunar-revolution must be considered. revolution. In the next section, the issue of signal propagation time delay for Moon-Based SAR is discussed.
(a)
(b)
(a)
(b) Figure 8. Cont.
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(c)
(d)
Figure 8. With the increasing ground target latitude, variations of the (a) Doppler bandwidth and (b) Figure 8. With the increasing ground target latitude, variations of the (a) Doppler bandwidth and Azimuth resolution. With the increasing declination of the lunar-based SAR, the variations of the (c) (b) Azimuth resolution. With the increasing declination of the lunar-based SAR, the variations of Doppler bandwidth and (d) Azimuth resolution. The superscript S indicates the stationary Moonthe (c) Doppler bandwidth and (d) Azimuth resolution. The superscript S indicates the stationary Based SAR, while superscript M represents the motional Moon-Based SAR. Moon-Based SAR, while superscript M represents the motional Moon-Based SAR.
At this point, the Doppler parameters of the motional Moon-Based SAR are discussed. It is found 3.3. Doppler Parameters for a Motional Moon-Based SAR that the Doppler frequency and the Doppler FM rate of the motional Moon-Based SAR are different motional is disturbed the error that isthe caused by the ‘stop-and-go’ fromThe those of theMoon-Based stationary SAR Moon-Based SAR,byindicating lunar-revolution must be assumption be more complete, context of the total time delay, earth’s rotation, considered. [23]. In theTonext section, the issueinofthe signal propagation time delay for the Moon-Based SAR is and lunar revolution should be considered. The total time delay consists of two terms: discussed. TD = TSAR 3.3. Doppler Parameters for A Motional Moon-Based 1 + T2
(20)
Moon-Based is disturbed bytothe thattarget, is caused ‘stop-and-go’ whereThe T1 ismotional the arriving time of theSAR transmitted signal theerror ground whichbyis the given by assumption [23]. To be more complete, in the context of the total time delay, the earth’s rotation, and lunar revolution should be considered. The time T1 total = R EM (η )delay /c consists of two terms: (20.a) T2 is the received time of a backscattered signal, which can be approximated by [24]
(20)
where T1 is the arriving time of the transmitted the ground target, which is given by(20.b) T = R EMsignal (η + Tto)/c 2
1
(20.a) where R EM (η ) is the slant range under lunar revolution; c the speed of light. The slant range of a motional Moon-Based SAR without assuming the ‘stop-and-go’ is given as T2 is the received time of a backscattered signal, which can be approximated by [24] RSEM (η ) =
R EM (η ) + R EM (η + TDEM ) 2
(21) (20.b)
Under this condition, therange Doppler frequency can be expressed where is the slant under lunar revolution; the as speed of light.
The slant range of a motional Moon-Based SAR without assuming the ‘stop-and-go’ is given as (22) f DEM (η ) = 0.5 f DEM (η ) + 0.5 f DEM (η + TD ) TD where f DEM (η ) is given in Equation (18), and f DEM (η + TD ) takes the same form as f DEM (η ) with η +(21) TD . The Doppler FM rate of a motional Moon-Based SAR without the ‘stop-and-go’ assumption can Under this condition, the Doppler frequency can be expressed as be approximated as E E E f dr = 0.5 f dr (ηc ) + 0.5 f dr (ηc + TD ) TD E ( η ) is given in Equation (19). where f dr c
where
is given in Equation (18), and
takes the same form as
(23) (22)
with
. The Doppler FM rate of a motional Moon-Based SAR without the ‘stop-and-go’ assumption can be approximated as
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(23) Remote Sens. 2018, 10, 486
where
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is given in Equation (19).
TheDoppler Doppler parameters a motional Moon-Based SAR without stop-and-go The parameters of a of motional Moon-Based SAR without assumingassuming stop-and-go becomes becomes more complex, but more accurate. To illustrate this, in Figure 9, we plotted the Doppler more complex, but more accurate. To illustrate this, in Figure 9, we plotted the Doppler frequency and frequency and the Doppler FM rate after removal of ‘stop-and-go’ assumption with the same the Doppler FM rate after removal of ‘stop-and-go’ assumption with the same simulation parameters, parameters, asthe in Doppler Figure 6.frequency The errorand in the Doppler Doppler FM rate assimulation in Figure 6. The error in Doppler FM frequency rate causedand by the ‘stop-and-go’ caused by the ‘stop-and-go’ assumption is relatively small for a smaller visible range in azimuth; assumption is relatively small for a smaller visible range in azimuth; is modified by it. With increase ofis modified it. With increase of the azimuth visible range, the error is azimuth large enough to cause image the azimuthby visible range, the error is large enough to cause image shift in direction. However, shift in azimuth direction. However, the variation of the Doppler FM rate is relatively small, the variation of the Doppler FM rate is relatively small, as evident in Figure 9b,d, implying that theas evident inresolution Figure 9b,d, implying thatthe thesame. azimuthal resolution remains almost the same. azimuthal remains almost
(a)
(b)
(c)
(d)
Figure 9. Comparison between the motional Moon-Based SAR with the ‘stop-and-go’ assumption and Figure 9. Comparison between the motional Moon-Based SAR with the ‘stop-and-go’ assumption and thatwithout withoutthe the‘stop-and-go’ ‘stop-and-go’assumption; assumption; equals (a) Doppler frequency; (b) Doppler that a g0 − am equals 0: (a)0:Doppler frequency; (b) Doppler FM rate; with a g0 − am equal to π/6; (c) Doppler frequency; (d) Doppler FM rate. FM rate; with equal to ; (c) Doppler frequency; (d) Doppler FM rate.
To further assess the influence of the change of Doppler FM rate on imaging focusing, we plot the To further assess the influence of the change of Doppler FM rate on imaging focusing, we plot quadratic phase error due to the total time delay in Figure 10a,b. As evident in Figure 10, the quadratic the quadratic phase error due to the total time delay in Figure 10a,b. As evident in Figure 10, the phase error exceeds its threshold of π/4 during an exposure time of 100 s. Consequently, in estimating quadratic error the exceeds its threshold ofthe total during an exposure of 100 ifseconds. the Doppler phase parameters, error that is caused by time delay should betime considered image Consequently, in estimating the Doppler parameters, the error that is caused by the total time delay shift in azimuth is to be corrected. should be considered if image shift in azimuth is to be corrected.
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(a) (a)
(b) (b)
Figure 10. Quadratic phase error caused by the total time delay during an exposure time of 100 Figure 10. Quadratic phase error caused the total time delay during an exposure time of 100 seconds (a) phase (b) by by Figure 10.with: Quadratic error caused the total time .delay during an exposure time of 100 s with: seconds (b) . (a) a g0 − with: am = (a) 0 (b) a g0 − am = π/6.
The azimuthal resolution, with and without propagation time delay, as a function of the latitude The azimuthal resolution, with and and without propagation time delay,11a,b, as aa function function ofMoon-Based the latitude latitude of the ground target and declination are respectively plottedtime in Figure with the The azimuthal resolution, with without propagation delay, as of the of theplaced ground(0, target declination are respectively plotted in0).Figure 11a,b, the Moon-Based SAR 28.5and °).and Indeclination Figure 11b, the target is located atin(0,Figure From Figure 11a,b, it demonstrates of the ground target are respectively plotted 11a,b, withwith the Moon-Based SAR SAR placed (0, 28.5 ° ). In Figure 11b, the target is located at (0, 0). From Figure 11a,b, it demonstrates ◦ that the azimuthal resolutions with and without the propagation time delay are almost the same, placed (0, 28.5 ). In Figure 11b, the target is located at (0, 0). From Figure 11a,b, it demonstrates that that the azimuthal resolutions with and without the propagation time delay are almost the same, since the Doppler bandwidth and the exposure time are only slightly modified by the propagation the azimuthal resolutions with and without the propagation time delay are almost the same, since the since Doppler bandwidth and thetime exposure time are only slightlybymodified by the propagation time the delay. Doppler bandwidth and the exposure are only slightly modified the propagation time delay. time delay.
(a) (a)
(b) (b)
Figure 11. Dependence of azimuthal resolution on (a) ground target latitude; (b) declination of the Figure 11. Dependence of azimuthal resolution on (a) ground target latitude; (b) declination of the Figure 11. Dependence of azimuthal resolution on (a) ground target (b) declination of the lunar-based SAR. Superscript M denotes motional Moon-Based SAR,latitude; and the additional superscript lunar-based SAR. Superscript M denotes motional Moon-Based SAR, and the additional superscript lunar-based Superscript M denotes TD indicatesSAR. the propagation time delay.motional Moon-Based SAR, and the additional superscript TD indicates the propagation time delay. TD indicates the propagation time delay.
From above, The Doppler parameters can be affected by the lunar revolution and the ‘stop-and– From The parameters can be lunar revolution thethe ‘stop-and-go’ From above, above,even TheDoppler Doppler parameters can beaffected affected bythe the lunar revolution ‘stop-and– go ’assumption, though the azimuthal resolution isby little impacted by both and ofand them. On the other assumption, even though the azimuthal resolution is little impacted by both of them. On go ’assumption, thoughofthe resolution little impacted both ofisthem. the other other hand, the rangeeven resolution theazimuthal Moon-Based SAR, asisin common SARby system, givenOn bythe hand, hand, the the range range resolution resolution of of the the Moon-Based Moon-Based SAR, SAR, as as in in common common SAR SAR system, system, isis given givenby by ρr =
c 2B
(24) (24) (24)
where is the system bandwidth. where is the system bandwidth. whereInB what is the system bandwidth. follows, evaluation of the imaging properties of the Moon-Based SAR is presented. In what follows, evaluation In what follows, evaluation of of the the imaging imaging properties properties of of the the Moon-Based Moon-Based SAR SARisispresented. presented. 4. Imaging Properties of Moon-Based SAR 4. Imaging Properties of Moon-Based SAR In this section, we investigate the characteristics of Moon-Based SAR imaging. Before proceeding In this section, investigate the characteristics of Moon-Based SARonimaging. Before proceeding to the analysis, we we need a signal model for the Moon-Based SAR based the curve trajectory, where to the analysis, we need signal model for the based on the curve trajectory, where the relative positions ofathe Earth’s target andMoon-Based the Moon areSAR accounted for. the relative positions of the Earth’s target and the Moon are accounted for.
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4. Imaging Properties of Moon-Based SAR In this section, we investigate the characteristics of Moon-Based SAR imaging. Before proceeding to the analysis, a signal model for the Moon-Based SAR based on the curve trajectory, Remote Sens.we 2018,need 10, x FOR PEER REVIEW 15 ofwhere 24 the relative positions of the Earth’s target and the Moon are accounted for. 4.1. Moon-Based SAR Signal Model
4.1. Moon-Based SAR Signal Model
Before presenting the Moon-Based SAR signal model, we examine the relations between the
exposure time of the the Moon-Based Moon-Based SAR the model, azimuthal by relations combining Equations and Before presenting SARand signal weresolution, examine the between the(9) exposure (12), which is given: time of the Moon-Based SAR and the azimuthal resolution, by combining Equations (9) and (12), which is given: (25) λRSEM (ηc ) Tsar = (25) 2ρ a R EM (ωE − ω M cos ϑs ) cos δm cos am − a g0 Notethe that the ω term is due to the lunar revolution. We plot the exposure time of Note that term E − ω M cos ϑs is due to the lunar revolution. We plot the exposure time of the the Moon-Based SAR as aoffunction of the azimuthal 11. The simulation Moon-Based SAR as a function the azimuthal resolutionresolution in Figure in 11.Figure The simulation parameters parameters remain the same as in Table 1, with . remain the same as in Table 1, with am − a g0 = 0. 12 shows the exposure time theMoon-Based Moon-Based SAR SAR is ofof hundreds of of FigureFigure 12 shows that that the exposure time of of the isin inthe theorder order hundreds seconds, or even thousands of seconds. For a Moon-Based SAR, the large-scale phenomenon is of the seconds, or even thousands of seconds. For a Moon-Based SAR, the large-scale phenomenon is of most interest in observations, and thus it is preferable to design the azimuthal resolution on the the most interest in observations, and thus it is preferable to design the azimuthal resolution on the decametric level. decametric level.
Figure 12. Relationship between the exposure time and azimuthal resolution. Here, the declination of
Figure the 12. moon Relationship between the exposure time and azimuthal resolution. Here, the declination of is set to 28.5 degrees. the moon is set to 28.5 degrees.
Note that the straight path assumption that is used in traditional SAR imaging is no longer applicable in Moon-Based SARassumption imaging, butthat instead, a curve should beimaging used. By expanding Note that the straight path is used intrajectory traditional SAR is no longer the slant range into a Taylor series about to order, we have applicable in Moon-Based SAR imaging, but instead,up a curve trajectory should be used. By expanding the slant range into a Taylor series about η = 0 up to 6th order, we have (26)
where
5 6 RisSEM R η + R η 2 + R3 η 3 + R4of η 4 the + Rexpansion + O(η 7 ) is tedious but(26) ( η ) = R0 + 5 η + R6 ηcoefficients the shortest slant1 range.2 The derivation
straightforward. Appendix A lists these coefficients up to 3rd order.
where R0 is the shortest slant range. The derivation of the expansion coefficients is tedious but In Figure 13, plot the phase error as a function of exposure time, with parameters given in Table straightforward. Appendix A lists these coefficients up to 3rd order. 1 and . It is clear that the 4th order curve trajectory is sufficient to meet the In Figure 13, plot the phase error as a function of exposure time, with parameters given in Table 1 an trajectory exposure time of 500 s to andmeet an azimuth visible and amrequirements − a g0 = π/6.ofItMoon-Based is clear thatSAR the imaging 4th orderwith curve is sufficient the requirements range that is larger than 6000 km. However, when an azimuthal resolution in the range of meters is of Moon-Based SAR imaging with an exposure time of 500 s and an azimuth visible range that is required, the corresponding exposure time ranges from 200 to 2000 s. For an exposure time longer larger than 6000 km. However, when an azimuthal resolution in the range of meters is required, than 700 s, the slant range expansion up to order is no longer enough, because the phase error the corresponding exposure time ranges from 200 to 2000 s. For an exposure time longer than 700 s, exceeds threshold of andorder an expansion of the slant range is need to error meet the metric the the slant rangethe expansion up to 4th isorder no longer enough, because the phase exceeds level Moon-Based SAR imaging for a larger visible range. threshold of π/4 and an 6th order expansion of the slant range is need to meet the metric level Moon-Based SAR imaging for a larger visible range.
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102
π/4
100
10-2
10
π/4 π/4
100
10-2
10-2
10-4
102
100
π/4
100
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102
Magnitude of phase Magnitude of phase error error
Magnitude of phase Magnitude of phase error error
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0
-4
0
500
1000
1500
Exposure time in seconds
500
1000
1500
(a)
2000
10-2
10-4
0
10-4
2000
0
500
1000
1500
Exposure time in seconds
500
1000
1500
(b)
2000 2000
Exposure time in seconds
Exposure time in seconds
(a)error with range expansion up to (a) 4th order and (b)(b) 6th order. Figure 13. Phase Figure 13. Phase error with range expansion up to (a) 4th order and (b) 6th order. Figure 13. Phase error with range expansion up to (a) 4th order and (b) 6th order.
In Figure 14, we plot the azimuthal resolution dependence on the phase error in terms of the
Magnitude of phase in radian Magnitude of phase errorerror in radian
In Figure 14, we plot the azimuthal resolution dependence on the phase error in terms of the expansion orders. the phase threshold of , we see that the 6th-order expansion ofofthe In Figure 14, Within we plot the azimuthal resolution dependence on the phase error in terms the expansion orders. Within phase threshold of π/4, of we meter see that the 6th-order expansion of the slant range is valid, eventhe when an azimuthal resolution is the required. In Table 2, we show expansion orders. Within the phase threshold of , we1 see that 6th-order expansion of the slantthe range is valid, even when an azimuthal resolution of 1 m is required. In Table 2, we show expansion orders and their corresponding azimuth resolution. An azimuthal resolution larger the slant range is valid, even when an azimuthal resolution of 1 meter is required. In Table 2, we show expansion their corresponding azimuth resolution. An For azimuthal than 3.0orders m can and be achieved by a 4th order expansion curve trajectory. a smallerresolution azimuthal larger visible than the expansion orders and their corresponding azimuth resolution. An azimuthal resolution larger the azimuthal resolution becomes coarser within the valid For expansion orders. 3.0 mrange, can be achieved by a 4th order expansion curve trajectory. a smaller azimuthal visible range, than 3.0 m can be achieved by a 4th order expansion curve trajectory. For a smaller azimuthal visible the azimuthal resolution becomes coarser within the valid expansion orders. range, the azimuthal resolution becomes coarser within the valid expansion orders. 101
10
1
π/4
100 10
π/4
0
2nd expansion 3rdndexpansion 2 expansion 4thrdexpansion 3 expansion 5ththexpansion 4 expansion 6ththexpansion 5 expansion 6th expansion
10-1 10-1
10-2 10-1 10-2 10-1
100
101
102
103
Azimuthal resolution in meter 100 101 102
103
in meter Figure 14. Relationship between Azimuthal the azimuthalresolution resolution and phase error caused by different expansion orders. Figure 14. Relationship between the azimuthal resolution and phase error caused by different Figure 14. Relationship between the azimuthal resolution and phase error caused by different expansion orders. Table 2. Relationship between azimuthal resolution and valid expansion orders. expansion orders. Table 2. Relationship between resolution and valid expansion orders. Expansion Orderazimuthal Azimuthal Resolution Unit
Table 2. Relationship between azimuthal resolution and valid 2 >52.9 m expansion orders. Expansion Order 3 2 Expansion4Order 3 5 2 4 6 3 5 4 6
5 6
Azimuthal Resolution Unit >10.2 m >52.9 m >3.0Resolution m Unit Azimuthal >10.2 m >1.5 m >3.0 m m >52.9 >0.85 m >1.5 m m >10.2 >0.85 m m >3.0
>1.5 >0.85
m m
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A Moon-Based SAR signal model with a 4th-order expansion curve trajectory is adopted in the following analysis. The received signal of the Moon-Based SAR from a single point target can be expressed as [19] n o sr (τ, η ) = wr [τ − 2RS (η )/c]wa (η ) exp jπKr [τ − 2RS (η )/c]2 exp{− j4π f c RS (η )/c}
(27)
where wr and wa are window functions in the fast time and slow time domains, respectively. Kr is the chirp rate, and f c is the carrier frequency. The two-dimensional (2-D) signal spectrum of Equation (28) can be obtained by using the method of series reversion (MSR) [25] and the principle of stationary phase (POSP) [26], sr ( f τ , f η ) = wr ( f τ )wa ( f η ) exp jΨ( f τ , f η )
(28)
where the phase spectrum takes the form Ψ( f τ , f η )
2 ( fc + fτ ) 2 1 c = −π Kf τr − 2π c P0 ( f c + f τ ) + 2π 4P2 · f c + f τ · f η + P1 c 2 3 9P32 −4P2 P4 ( fc + fτ ) 3 ( fc + fτ ) 4 P3 c c · +2π 8P f + P + 2π f + P η η 1 1 3 · 5 c c fc + fτ fc + fτ 64P 2
(28.a)
2
where P0 = 2R0 , P1 = 2R1 , P2 = 2R2 , P3 = 2R3 , P4 = 2R4 . From the above Equation (28.a), the range and azimuth frequency are highly coupled in the phase term Ψ( f τ , f η ). Detailed derivation is given in Appendix B. To process the signal of the Moon-Based SAR, Equation (28.a) is further expanded as Ψ( f τ , f η ) = Ψ a ( f η ) + Ψr ( f τ ) + Ψrcm ( f τ , f η ) + Ψsrc ( f τ , f η ) + Ψres
(29)
where Ψ a ( f η ) is in connection with the azimuth compression, and its specific expression is given by Ψa ( f η )
3P12 P3 9P32 −4P2 P4 3 P1 c = 2π 2P + + P f + η 1 4P2 f c + 8P23 16P25 2 2 9P32 −4P2 P4 P3 c2 2 f 3 + 9P3 −4P2 P4 c3 f 4 + 8P + P c 1 3 f2 η η 16P5 f 2 64P5 f 3 2 c
2 c
3P1 P3 c 8P23 f c
+
9P32 −4P2 P4 3P12 c 32P25 f c
f η2 (29.a)
2 c
Ψr ( f τ ) is related to the range compression, which can be written as P3 P3 P2 9P2 − 4P2 P4 4 P0 Ψr ( f τ ) = 2π − + 1 + 1 3 + 3 P1 c 4P2 c 8P2 c 64P25 c
! fτ −
1 2 f 2Kr τ
(29.b)
Ψsrc ( f τ , f η ) is related to the secondary range compression given by Ψsrc ( f τ , f η )
= 2π
c 4P2 f c3
9P2 −4P P
+
3P1 P3 c 8P23 f c3
+
9P32 −4P2 P4 3P12 c 32P25 f c3
f η2
+
3P3 c2 8P23 f c4
+
9P32 −4P2 P4 3P1 c2 16P25 f c4
f η3
9P2 −4P P
2 4 2 4 3 4 2 3 3 1 P3 c + 32P − 4Pc f 4 − 3P 3P12 c f η2 5 f 5 3c f η f τ + 3 4 − 8P 32P25 f c4 c 2 c 2 fc 2 2 9P32 −4P2 P4 P3 c2 2 f 3 − 9P3 −4P2 P4 5c3 f 4 f 3 + − 2P P c 1 3 f5 − 5 5 η η τ 4P f 32P5 f 6 2 c
2 c
(29.c)
2 c
Ψrcm ( f τ , f η ) concerns the range cell migration and takes the form: Ψrcm ( f τ , f η ) = 2π − 4Pc f 2 + 2 c
3P1 P3 c 8P23 f c2
+
9P32 −4P2 P4 3P12 c 32P25 f c2
f η2 −
9P32 −4P2 a4 P1 c2 8P25 f c3
+
P3 c2 4P23 f c3
f η3 −
9P32 −4P2 P4 3 4 3c f η 64P25 f c4
fτ
(29.d)
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(29.d)
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Finally,
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is the residual phase term that has a bearing on the polarization:
Finally, Ψres is the residual phase term that has a bearing on the polarization: (29.e) # P13 P3 P12 9P32 − 4P2 P4 4 P0 fc + fc + f + P1 f c Ψ =2 π− (29.e) 3 c on the 5 trajectory At this point, res the signal model curve are derived, and c and 2-D 4cPspectrum 8cPbased 64cP 2 2 2
"
the Moon-Based SAR can be achieved by using the Range Doppler algorithm [27]. The characteristics of this Moon-Based SAR imaging areand studied in the nextbased sub-section. At point, the signal model 2-D spectrum on the curve trajectory are derived, and the
Moon-Based SAR can be achieved by using the Range Doppler algorithm [27]. The characteristics of 4.2. Properties of Moon-Based SAR Imaging Moon-Based SAR imaging are studied in the next sub-section. From Equation (14), the azimuthal resolution is a function of the ground target’s position, the Moon-Based SAR’s position the aperture length in the azimuth. We plot the point target image 4.2. Properties of Moon-Based SARand Imaging in azimuth and range directions in Figure 15, for three declination angles: 18 degrees, 22 degrees, and
From Equation (14), the azimuthal resolution is a function of the ground target’s position, 28 degrees, respectively, shown in 1st(a1,b1,c1,d1), 2nd(a2,b2,c2,d2), and 3rd (a3,b3,c3,d3) columns. the Moon-Based position the azimuth. thecrucial point factor target in image As we can SAR’s see from Figureand 15, the theaperture latitude length of the in ground target isWe theplot most in azimuth and range directions in Figure 15, for three declination angles: 18 degrees, 22 degrees, determining the azimuthal imaging of Moon-Based SAR. The quantitative values of azimuthal and 28resolutions degrees, respectively, shown are listed in Table 3. in 1st (a1,b1,c1,d1), 2nd (a2,b2,c2,d2), and 3rd (a3,b3,c3,d3) columns. As we can see from Figure 15, the latitude of the ground target is the most crucial factor in determining Table 3. Azimuthal Resolution of Simulation the azimuthal imaging of Moon-Based SAR. The quantitative valuesResults. of azimuthal resolutions are listed in Table 3. Declination of Moon-based ( ) Simulation ( )Results. ( Table 3. Azimuthal Resolution of Latitude of SAR Ground Target
( Latitude of Ground Target
)
Declination of Moon-Based SAR ◦ (δmm= 18◦ ) 27.8(δm m = 22 ) ) 25.8 m 26.5
(δg = 0, am −( a g = 0) (δg = 0, am − a g = 30◦ ) (δg = 40◦ , am (− a g = 0) (δg = 70◦ , am − a g = 0) (
)
29.8 m
)
19.8 m
)
8.8 m
30.6 25.8 m m
26.5 m 32.1 m
29.8 m 20.3 19.8 m m 8.8 m 9.1 m
30.6 m 21.3 m 20.3 m 9.1 m 9.5 m
(a1)
(a2)
(a3)
(b1)
(b2)
(b3)
Figure 15. Cont.
(δm = 28◦ ) 27.8 m 32.1 m 21.3 m 9.5 m
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(c1)
(c2)
(c3)
(d1)
(d2)
(d3)
Figure 15. Comparison of azimuthal imaging with different positions, where the 1st(a1,b1,c1,d1),
Figure 15. Comparison of azimuthal imaging with different positions, where the 1st (a1,b1,c1,d1), 2nd(a2,b2,c2,d2), and 3rd (a3,b3,c3,d3) columns show declination angle of 18 degrees, 22 degrees, and 2nd (a2,b2,c2,d2), and 3rd (a3,b3,c3,d3) columns show declination 18 degrees, 22 degrees, 28 degrees, respectively. The first row signifies a latitude of 0 degrees, angle with noofdifference between the and 28ascension degrees,and respectively. The first row signifies a latitude of 0 degrees, with no difference between longitude. The second row denotes a latitude of 0 degrees, with a difference of 30 the ascension and longitude. Theand second row denotes a latitude 0 degrees, withofalatitude difference degrees between the ascension longitude. The 3rd row and 4th of row show the case of of 30 degrees between ascension and longitude. Theno3rd row and 4th row show the case of latitude of 40 degrees and the 70 degrees, respectively, both with difference between ascension and longitude. 40 degrees and 70 degrees, respectively, both with no difference between ascension and longitude. 5. Conclusions
5. Conclusions The signal model for Moon-Based SAR imaging is investigated using a complex, but more complete, geometry of Moon-Based SAR and Earth, where the rotations and revolutions are
The signal model for Moon-Based SAR imaging is investigated using a complex, but more considered. A unified right-handed geocentric inertial reference coordinate system for the Mooncomplete, geometry of Moon-Based SAR Earth,the where the rotations and revolutions considered. Based SAR is established, followed by and analyzing Doppler parameters under different are conditions. A unified right-handed geocentric inertial reference coordinate system for the Moon-Based SAR is It is found that the lunar revolution and the error that is caused by the ‘stop-and-go’ assumption are established, followed by analyzing the Doppler parameters under different conditions. It is found potentially significant to affect the Doppler parameters, and thus must be taken into account in the image focusing. The Doppler exposure on the other hand, are much less that the lunar revolution and thebandwidth error thatand is caused bytime, the ‘stop-and-go’ assumption areaffected, potentially and as resultthe theDoppler azimuthal resolution remains unaffected. The signal model and itsfocusing. 2-D significant toaaffect parameters, and thusalmost must be taken into account in the image spectrum of a motional moon-based SAR without imposing the ‘stop-and-go’ assumption are derived. The Doppler bandwidth and exposure time, on the other hand, are much less affected, and as a considering azimuthal resolutions are analyzed. result The the expansion azimuthalorders resolution remains almost unaffected. The signal model and its 2-D spectrum of a Nevertheless, the Moon-Based SAR imaging is influenced from several salient features, such as motional moon-based SAR without imposing the ‘stop-and-go’ assumption are derived. The expansion atmospheric effect and ionospheric effect [28,29] and tropospheric effects [30], which should be orders considering azimuthal resolutions are analyzed. examined in great detailed. Nevertheless, the Moon-Based SAR imaging is influenced from several salient features, such Acknowledgments: This ionospheric work was supported National Science Foundation of China under the be as atmospheric effect and effect by [28,29] andNatural tropospheric effects [30], which should grants 41580853 and 41531175, and the Director General’s Innovative Funding-2015. examined in great detailed. Author Contributions: Kun-Shan Chen and Zhen Xu conceived and developed the theory and signal model;
Acknowledgments: Thisthe work was supported by National Natural Science Foundation of China under the grants Zhen Xu performed numerical simulations; Zhen Xu and Kun-Shan Chen wrote the paper. 41580853 and 41531175, and the Director General’s Innovative Funding-2015. Conflicts of Interest: The authors declare no conflict of interest.
Author Contributions: Kun-Shan Chen and Zhen Xu conceived and developed the theory and signal model; Zhen Xu performed the numerical simulations; Zhen Xu and Kun-Shan Chen wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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Appendix A The derivation of the expansion coefficients appearing in Equation (26) is tedious but straightforward. For reference and for saving space, coefficients up to 3rd order are given below. R0
1/2 = 0.5 R2E + R2EM −2R EM R E cos δm cos δg cos am cos a g0 − 2R EM R E cos δm cos δg0 sin am sin a g0 − 2R EM R E sin δm sin δg0 2 +0.5 R E + R2EM − 2R EM R E cos(δm + ω M TD sin ϑS ) cos δg cos( am + ω M TD cos ϑS ) cos a g0 + ωE TD −2R EM R E cos(δm + ω M TD sin ϑS ) cos δg sin( am + ω M TD cos ϑS ) sin a g0 + ωE TD 1/2 −2R EM R E sin(δm + ω M TD sin ϑS ) sin δg
R1
R2
RE = REM ωE cos δg cos δm sin a g0 − am − cos δm ω M cos ϑS sin a g0 − am 2R0 + sin δm ω M sin ϑS cos a g0 − am + cos(δm + ω M TD sin ϑS )ωE sin a g0 − am + ωE TD − ω M TD cos ϑS − cos(δm + ω M TD sin ϑS )ω M cos ϑS sin a g0 − am + ωE TD − ω M TD cos ϑS + sin(δm + ω M TD sin ϑS )ω M sin ϑS cos a g0 − am + ωE TD − ω M TD cos ϑS sin δg ω M sin ϑS [cos δm + cos(δm + ω M TD sin ϑS )]
(A2)
R EM R E ωE2 2R0 ωE2 − 2ωE ω M
− 2ωE ω M cos ϑS + ω 2M cos δg cos δm cos a g0 − am cos ϑS + ω 2M cos δg cos(δm + ω M TD sin ϑS ) cos a g0 − am + ωE TD − ω M TD cos ϑS −2(ωE − ω M cos ϑS )ω M sin ϑS cos δg sin δm sin a g0 − am
=
−2(ωE − ω M cos ϑS )ω M sin ϑS cos δg sin(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω 2M sin2 ϑS sin δg [sin(δm + ω M TD sin ϑS ) + sin δm ] R2 R2E − EM (ωE − ω M cos ϑS ) cos δg cos δm · sin a g0 − am 2R30 2 +ω M sin ϑS cos δg sin δm cos a g0 − am + g cos δm + (ωE − ω M cos ϑS ) cos δg cos(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω M sin ϑS cos δg sin(δm + ω M TD sin ϑS ) cos a − am + ωE TD − ω M TD cos ϑS o g0 2 −ω M sin ϑS sin δ cos(δm + ω M TD sin ϑS )] R3
(A1)
R EM R E 2R0
−3ω M sin θS ωE2 − 2ωE ω M cos ϑS − sin θS ω 3M − 2 sin θS cos2 ϑS ω 3M cos δg sin δm cos a g0 − am − (ωE − ω M cos θS ) ωE2 − 2ωE ω M cos ϑS + ω 2M + 2ω 2M sin2 ϑS cos δg cos δm sin a g0 − am −3ω M sin θS ωE2 − 2ωE ω M cos ϑS − sin θS ω 3M − 2 sin θS cos2 ϑS ω 3M cos δg sin(δm + ω M TD sin ϑS ) cos a g0 − am + ωE TD − ω M TD cos ϑS − (ωE − ω M cos θS ) ωE2 − 2ωE ω M cos ϑS + ω 2M + 2ω 2M sin2 ϑS cos δg cos(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω 2M sin2 ϑS sin δg ω M sin θS [cos δm + cos(δm + ω M TD sin ϑS )] R2 R2E − EM (ωE − ω M cos ϑS ) cos δg cos δm · sin a g0 − am R30 +ω M sin ϑS cos δg sin δm cos a g0 − am − ω M sin ϑS sin δg cos δm + 2 2 ωE − 2ωE ω M cos ϑS + ω 2M cos δg cos δm cos a g0 − am −2(ωE − ω M cos ϑS )ω M sin ϑS cos δg sin δm sin a g0 − am +ω 2M sin2 ϑS sin δg sin δm + (ωE − ω M cos ϑS ) cos δg cos(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω M sin ϑS cos δg sin(δm + ω M TD sin ϑS ) cos a g0 − am + ωE TD − ω M TD cos ϑS −ω M sin ϑS sin δg cos(δm + ω M TD sin ϑS ) + 2 2 ωE − 2ωE ω M cos ϑS + ω 2M cos δg cos(δm + ω M TD sin ϑS ) cos a g0 − am + ωE TD − ω M TD cos ϑS −2(ωE − ω M cos ϑS )ω M sin ϑS cos δg sin(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω 2M sin2 ϑS sin δg sin(δm + ω M TD sin ϑS ) 3R3EM R3E + 2R (ωE − ω M cos ϑS ) cos δg cos δm sin a g0 − am 5 0 3 +ω M sin ϑS cos δg sin δm cos a g0 − am −ω M sin ϑS sin δg cos δm (ωE − ω M cos ϑS ) cos δg cos(δm + ω M TD sin ϑS ) sin a g0 − am + ωE TD − ω M TD cos ϑS +ω M sin ϑS cos δg sin(δm + ω M TD sin ϑS ) cos a g0 − am + ωE TD − ω M TD cos ϑS 3 o −ω M sin ϑS sin δg cos(δm + ω M TD sin ϑS )
=
(A3)
(A4)
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Appendix B The slant distance, considering the lunar revolution and without assuming the “stop and go”, is expanded about the slow time η up to 4th order: R S ( η ) = R0 + R1 η + R2 η 2 + R3 η 3 + R4 η 4 + O ( η 5 )
(A5)
To apply MSR, the linear phase term of the received signal need to be removed: n o sr_a (τ, η ) = wr (τ − R∗ (η )/c)wa (η ) · exp{− j2π f c R∗ (η )/c} exp jπKr (τ − R∗ (η )/c)2
(A6)
where R∗ (η ) = P0 + P2 η 2 + P3 η 3 + P4 η 4 , P0 = 2R0 , P1 = 2R1 , P2 = 2R2 , P3 = 2R3 , P4 = 2R4 The Fourier transform in range direction is applied to Equation (A6) by using principle of stationary phase method: h i sr_a ( f τ , η ) = wr ( f τ )wa (η ) exp − jπ f τ2 /Kr exp[− j2πR∗ (η )( f c + f τ )/c]
(A7)
Next, the Fourier transform is carried out in azimuth direction: h iZ sr_a ( f τ , f η ) = wr ( f τ )wa ( f η ) exp − jπ f τ2 /Kr
∞
−∞
exp{ jφ(η )}dη
(A8)
The phase term in the above Fourier integral is φ(η ) = −
2π ∗ R (η )( f c + f τ ) − 2π f η η c
(A9)
Taking the derivative of φ(η ) with respect to η, we have φ0 (η ) = −
2π (2P2 η + 3P3 η 2 + 4P4 η 3 )( f c + f τ ) − 2π f η c
(A10)
By way of POSP, we obtain
−
c fη = (2a2 η + 3a3 η 2 + 4a4 η 3 ) fc + fτ
(A11)
The slow time η can be expressed in terms of f η via MSR 2 3 ηmsr ( f η ) = − A1 c f η /( f c + f τ ) + A2 c f η /( f c + f τ ) − A3 c f η /( f c + f τ )
(A12)
where the coefficients are A1 = 1/2P2 , A2 = −3P3 /8P23 , A3 = 9P32 − 4P2 P4 /16P25 . Substituting the slow time (A12) into (A9), we arrive at the two-dimensional (2D) frequency spectrum of Equation (A6): sr_a ( f τ , f η ) = wr ( f τ )wa ( f η ) exp jϕ( f τ , f η ) where the exponent is given by ϕ( f τ , f η ) = −π f τ2 /Kr − 2π f η ηmsr ( f η ) − 2πR∗ ηmsr ( f η ) ( f c + f τ )/c
(A13)
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According to the skew and shift properties of the Fourier transform, f η is substituted by f η + ( f c + f τ ) Pc1 in Equation (A13) and the corresponding 2-D spectrum is given as P sr ( f τ , f η ) = wr ( f τ )wa f η + ( f c + f τ ) 1 exp jΨ( f τ , f η ) c
(A14)
Keeping up to 4th order, the phase term can be written as and the phase can be written as 2 = −π f τ2 /Kr − 2πP0 ( f c + f τ )/c + 2π 4P1 2 · f c +c f τ · f η + P1 ( f c +c f τ ) 2 3 9P32 −4P2 P4 ( fc + fτ ) 3 ( fc + fτ ) 4 P3 c c +2π 8P f + P + 2π · f + P η η 1 1 3 · 5 c c fc + fτ fc + fτ 64P
Ψ( f τ , f η )
2
(A15)
2
The range and azimuth frequencies are highly coupled in the phase term. Equation (A15) is further expanded as follows:
To proceed,
Ψ( f τ , f η ) = Ψ0 ( f η ) + Ψ1 ( f η ) f τ + Ψ2 ( f η ) f τ2 + Ψ3 ( f η ) f τ3 + Ψ4 ( f η ) f τ4
(A16)
P3 P P2 3P12 P3 9P2 −4P P 9P32 −4P2 P4 3 P1 = 2π − Pc0 f c + 4P12 c f c + 8P1 3 c3 f c + 364P5 c2 4 P14 f c + 2P + + P 1 fη 8P23 16P25 2 2 2 2 9P32 −4P2 P4 9P32 −4P2 P4 P3 c2 2 f 3 + 9P3 −4P2 P4 c3 f 4 2c f 2 + 1 P3 c + P c + 3P + 4Pc2 f c + 3P 1 η η η 1 8P3 f 32P5 f 8P3 f 2 16P5 f 2 64P5 f 3
(A16a)
where Ψ0 ( f η )
2 c
2 c
2 c
h P3 P P2 9P2 −4P P = 2π − Pc0 + 4a12 c + 8P1 3 c3 + 364P5 c2 4 P14 + − 4Pc f 2 − 2 c 2 2 9P32 −4P2 P4 3 4 9P32 −4P2 P4 P3 c2 3 2 − P1 c 8P5 f 3 + 4P3 f 3 f η − 64P5 f 4 3c f η
Ψ1 ( f η )
2 c
Ψ2 ( f η )
2 c
= 2π +
Ψ3 ( f η )
− 2K1 r
+
9P32 −4P2 P4 3 3 3c f η 32P25 f c5
= 2π −
Ψ4 ( f η )
h
2 c
c 4P2 f c3
+
2 c
3P1 P3 c 8P23 f c2
−
9P32 −4P2 P4 3P12 c 32P25 f c2
f η2
(A16b)
2 c
3P1 P3 c 8P23 f c3
+
9P32 −4P2 P4 3P12 c 32P25 f c3
f η2
+
3P3 c2 8P23 f c4
+
9P32 −4P2 P4 3a1 c2 16P25 f c4
− 4Pc f 4 2 c
−
3P1 P3 c 8P23 f c4
−
9P32 −4P2 P4 3a21 c 32P25 f c4
f η2
P3 c2 − 2P 3 5 2 fc
−
9P32 −4P2 P4 a1 c 4P25 f c5
5P3 c2 8P23 f c6
9P32 −4P2 P4 5P1 c2 16P25 f c6
+
= 2π +2π
c 4P2 f c5
+
3P1 P3 c 8P23 f c5
9P32 −4P2 P4 15c3 f η4 64P25 f c7
+
9P32 −4P2 P4 3a21 c 32P25 f c5
f η2
+ 2π
+
(A16c)
f η3 (A16d)
9P32 −4P2 P4 3 4 5c f η 32P25 f c6
f η3
f η3
(A16e)
References 1. 2.
3.
Campbell, J.B.; Wynne, R.H. Introduction to Remote Sensing; Guilford Press: New York, NY, USA, 2011. Johnson, J.; Lucey, P.; Stone, T.; Staid, M. Visible/Near-Infrared Remote Sensing of Earth from the Moon. NASA Advisory Council Workshop on Science Associated with the Lunar Exploration Architecture White Papers; 2007. Available online: http://www.lpi.usra.edu/meetings/LEA/whitepapers/Johnson_etal_v02.pdf (accessed on 19 July 2017). Renga, A. Configurations and Performance of Moon-based SAR Systems for very high resolution Earth remote sensing. In Proceedings of the AIAA Pegasus Aerospace Conference, Naples, Italy, 12–13 April 2007.
Remote Sens. 2018, 10, 486
4.
5. 6. 7. 8. 9. 10.
11.
12. 13. 14.
15. 16. 17. 18.
19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
23 of 24
Renga, A.; Moccia, A. Preliminary analysis of a Moon-based Interferometric SAR system for very high resolution Earth remote sensing. In Proceedings of the 9th ILEWG International Conference on Exploration and Utilisation of the Moon, Sorrento, Italy, 22–26 October 2007. Moccia, A.; Renga, A. Synthetic aperture radar for earth observation from a lunar base: Performance and potential applications. IEEE Trans. Aerosp. Electron. Syst. 2010, 46, 1034–1051. [CrossRef] Fornaro, G.; Franceschetti, G.; Lombardini, F.; Mori, A.; Calamia, M. Potentials and limitations of moon-borne sar imaging. IEEE Trans. Geosci. Remote Sens. 2010, 48, 3009–3019. [CrossRef] Purkis, S.J.; Klemas, V.V. Remote Sensing and Global Environmental Change; John Wiley & Sons: Chichester, West Sussex, UK, 2011. Chen, K.-S. Principles of Synthetic Aperture Radar Imaging: A System Simulation Approach; CRC Press: Boca Raton, FL, USA, 2016. Guo, H.D.; Ding, Y.X.; Liu, G.; Zhang, D.W.; FU, W.X.; Zhang, L. Conceptual study of lunar-based sar for global change monitoring. Sci. China Earth Sci. 2014, 57, 1771–1779. [CrossRef] Ding, Y. Moonborne Earth Observation Synthetic Aperture Radar and Its Application in Global Change. Ph.D. Thesis, Signal and Information Processing, University of Chinese Academy of Sciences, Beijing, China, 2014. Ren, Y.; Guo, H.; Liu, G.; Ye, H. Simulation study of geometric characteristics and coverage for moon-based earth observation in the electro-optical region. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2017, 10, 2431–2440. [CrossRef] Guo, H.D.; Liu, G.; Ding, Y.X. Moon-based earth observation: Scientific concept and potential applications. Int. J. Digit. Earth 2017, 1–12. [CrossRef] Ouyang, Z.Y. Introduction to Lunar Science; China Astron Publ House: Beijing, China, 2005. Standish, E. Jpl Planetary and Lunar Ephemerides: De 405/le 405, Jet Propulsion Laboratory Interoffice Memorandum, IOM 312. F-98-048. August 1998. Available online: http://iau-comm4.jpl.nasa.gov/de405iom/ de405iom.pdf (accessed on 23 September 2017). Meeus, J. Explanatory supplement to the astronomical ephemeris. Ciel et Terre 1962, 78, 376. Jia, Y.Z.; Zhou, Y.L. Research on lunar site selection for lunar based earth observation. Spacecr. Eng. 2016, 25, 116–121. Goldreich, P. History of the lunar orbit. Rev. Geophys. 1966, 4, 411–439. [CrossRef] Renga, A.; Moccia, A. Moon-based synthetic aperture radar: Review and challenges. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Beijing, China, 10–15 July 2016; pp. 3708–3711. Cumming, I.G.; Wong, F.H. Digital Signal Processing of Synthetic Aperture Radar Data: Algorithms and Implementation; Artech House: Boston, MA, USA, 2004. Skolnik, M. Radar Handbook, 2nd ed.; McGraaw-Hill Book Company: New York, NY, USA, 1990. Ulaby, F.T.; Long, D.G.; Blackwell, W.J.; Elachi, C.; Fung, A.K.; Ruf, C.; Sarabandi, K.; Zebker, H.A.; Van Zyl, J. Microwave Radar and Radiometric Remote Sensing; University of Michigan Press: Ann Arbor, MI, USA, 2014. Xi, X.N.; Zeng, G.Q.; Ren, X. Orbit Design of Lunar Probe; National Defence Industry Press: Changsha, China, 2001. Curlander, J.C.; Mcdonough, R.N. Synthetic Aperture Radar: Systems and Signal Processing; John Wiley & Sons: New York, NY, USA, 1991. Hu, C.; Long, T.; Zeng, T.; Liu, F.; Liu, Z. The accurate focusing and resolution analysis method in geosynchronous sar. IEEE Trans. Geosci. Remote Sens. 2011, 49, 3548–3563. [CrossRef] Neo, Y.L.; Wong, F.; Cumming, I.G. A two-dimensional spectrum for bistatic SAR processing using series reversion. IEEE Geosci. Remote Sens. Lett. 2007, 4, 93–96. [CrossRef] Key, E.L.; Fowle, E.N.; Haggarty, R.D. A method of designing signals of large time-bandwidth product. IRE Int. Conv. Rec. 1961, 4, 146–155. Wu, C.; Liu, K.Y.; Jin, M. Modeling and a correlation algorithm for spaceborne SAR signals. IEEE Trans. Aerosp. Electron. Syst. 1982, AES-18, 563–575. [CrossRef] Ishimaru, A.; Kuga, Y.; Liu, J.; Kim, Y.; Freeman, T. Ionospheric effects on SAR at 100 MHz to 2 GHz. Radio Sci. 1999, 34, 257–268. [CrossRef]
Remote Sens. 2018, 10, 486
29. 30.
24 of 24
Xu, Z.; Wu, J.; Wu, Z. A survey of ionospheric effects on space-based radar. Waves Random Media 2004, 14, S189–S273. [CrossRef] Quegan, S.; Lamont, J. Ionospheric and tropospheric effects on synthetic aperture radar performance. Int. J. Remote Sens. 1986, 7, 525–539. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
