# Analysis for Resolution of Bistatic SAR Configuration with Geosynchronous Transmitter and UAV Receiver

Authors: Yicheng Jiang; Zhuoqun Wang
Journal: International Journal of Antennas and Propagation
 (2013)

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## Abstract

Bistatical SAR with geosynchronous illuminator and unmanned aerial vehicle receiver (GEO-UAV BiSAR) has significant potential advantages in the field of continuous local observation under a dangerous environment within nearly 24 h. Due to the extreme platform velocity differences, the ellipse orbital movement of GEOSAR makes this BiSAR configuration not like the conventional spaceborne BiSAR. In this paper, based on the orbital kinetic characteristic of GEOSAR, we theoretically analyze the variations of bistatic configuration effect on common azimuth coverage and coherent accumulated time. In addition, two-dimension the resolution is deduced by geometrical configuration on the basis of gradient method. The simulations show that the appropriate selection of initial bistatic configuration can restrain from the appearance of the dead zone in common coverage. And the image results are obtained by frequency domain RD based on Method of Series Reversion (MSR). It is shown that GEO-UAV BiSAR has the high resolution ability.


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## Body

## 1. Introduction

Spaceborne synthetic aperture radar (SAR) has been applied to the wide fields such as landform measurement, ocean observation, earthquake monitor, and digital elevation model (DEM) [1]. For decades, lots of low-orbit (LEO) satellite emitted have high-resolution capability. However, beam irradiation extent of LEO is limited in dozens of kilometers near the nadir, and the revisit time is much longer. To this aspect, geosynchronous SAR (GEOSAR) can overcome the shortcomings aforementioned [2–4].Geosynchronous orbits have the unique characteristic that their orbital period is nearly 24 h. It makes GEOSAR suitable for the continuous imaging on the specific partial region within 24 h [5]. And the ground coverage provided by a GEOSAR can be nearly one third of the globe for a highly inclined orbit, and it can be as small as few hundred kilometers for a SAR placed in a slightly inclined orbit [6].Nevertheless, it requires a larger antenna to overcome the energy attenuation of the ionosphere and stratosphere [7–9]. There exists difficulties to hardware realization. In addition, imaging accumulation time, which reaches a few hours, makes the imaging quality quite easier to be affected by unstable factors in larger ground coverage. To solve these problems, the literature [10] derives a space-based radar surveillance concept employing geosynchronous illumination and bistatic reception on either unmanned aerial vehicles (UAVs) or LEO. Due to the wide beam coverage of GEOSAR, it is easy to realize distributed imaging configuration with illuminator GEOSAR and multiple receivers (aircraft, UAV, LEO, and MEO satellites). Not only the wide imaging area is achieved, but also the requirements for the transmitted power and antenna size are reduced [11, 12].In particular, bistatic geometrical configuration of GEOSAR illuminator and UAVSAR receiver (GEO-UAV BiSAR) may realize local observation in a dangerous environment. However, since the orbital characteristics of GEOSAR, we need to control the attitude of UAVSAR in the ground station for high resolution. Hence, it is different from other configurations, for example, GEO illumination and LEO receiver; GEO illumination and airborne receiver.A key factor in determining the performance of GEO-UAV BiSAR systems is the two-dimensional resolution, but performance analysis of BiSAR characterized for any configuration is usually complex. The literature [13] describes comprehensive knowledge regarding the resolution of BiSAR with geostationary illuminator and UAV receiver, which can be obtained from the gradient method in terms of time delay and the Doppler shift [14, 15]. Yet, it does not consider the variation of geometrical configuration as a function of time effect on resolution. In this case, the geostationary orbit is just a special one of geosynchronous orbit, and it ignores orbital motion and earth rotation effect on GEOSAR velocity, when orbit inclination is not equal to zero.Before the imaging study for GEO-UAV BiSAR, its two-dimensional spectrum is demanded to discuss. Currently, the traditional method has two kinds, Loffeld’s bistatic formula (LBF) [16, 17] and Method of Series Reversion (MSR) [18, 19]. The former mainly applies the Taylor expansion on phase histories of transmitter and receiver individually. The latter considers a power series method to count the stationary point of bistatic phase histories, and its accuracy is scalable in a sense. To this end, MSR is more applicable to the imaging study of GEO-UAV BiSAR.This paper is organized as follows. It begins with a description of GEOSAR ground track in local ground coordinate system. Section3 discusses the relative movement of two platforms effect on common coverage and coherent accumulated time owing to the extreme platform velocity differences. In Section 4, the formulas of two-dimensional resolution are deduced from the geometrical relationship of GEOSAR and UAVSAR based on gradient method. In Section 5, we analyze GEO-UAV BiSAR configuration influences on range and azimuth resolution by simulations. And proper initial selecting configuration can avoid the appearance of dead zone in the swath. Then, the imaging results by frequency domain RD method based on MSR are presented and analyzed.


## 2. Analyses to Movement Model of GEOSAR

### 2.1. Movement Track of GEOSAR

Suppose that GEOSAR runs on the ellipse orbit and the earth is uniform sphere, orbital inclination isi, right ascension of ascending node is Ω0, and argument of perigee is W0. The satellite position equation can be expressed in earth fixed coordinate system:
(1)[Xs(t0+ta)Ys(t0+ta)Zs(t0+ta)]=RWB(t0+ta)WAWC(t0+ta),
where
(2)WA=[cosΩ0-sinΩ00sinΩ0cosΩ00001][1000cosi-sini0sinicosi]×[cosW0-sinW00sinW0cosW00001],WB(t0+ta)=[cos(ωe·(t0+ta))sin(ωe·(t0+ta))0-sin(ωe·(t0+ta))cos(ωe·(t0+ta))0001]WC(t0+ta)=[cosf(t0+ta)-sinf(t0+ta)0sinf(t0+ta)cosf(t0+ta)0001]t0 is the center time point of time range, ωe is the earth rotation angular velocity and R is the geocenter distance of the satellite.In the time range foregoing, the velocity of GEOSAR related to the earth is given by(3)V⇀sT_EF(t0+ta)=μaWA(t0+ta)WB[-sinf(t0+ta)cosf(t0+ta)0]+[ωeYs(t0+ta)-ωeXs(t0+ta)0],
where a is a semimajor axis, μ is a gravitation constant, Re is the earth radius, and HT is a GEOSAR distance of ground.Thus, wheni=0, GEOSAR is the stationary state and GEO-UAV BiSAR images in the local area near the equator. With the increase of i, V⇀sT goes up gradually, and the variation of latitude is [-i,i] in “Figure 8” imaging area. For maximum irradiation of Chinese territory, we choose i=60°. Based on GEOSAR basic parameters (see Table 1), the ground track of GEOSAR (i≠0) is shown in Figure 1.Table 1
GEOSAR transmitter parameters.
GEOSAR parameters
Value
Orbit altitude
36000 km
Argument of perigee
90°
Right ascension of ascending node
30°
Orbit inclination
60°
Azimuth beam width
0.2°
Angle of elevation
0.6°Figure 1
Ground track of 60° inclined geosynchronous orbit.Due to higher altitude of GEOSAR [20], the rate V⇀sT at which the spacecraft moves along its orbital path is unequal to the rate V⇀eT at which the footprint of the antenna beam moves along the surface of the earth: V⇀eT=ReV⇀sT/(Re+HT).


### 2.2. GEOSAR Velocity in Local Ground Coordinate System

In general, GEOSAR is described in earth-fixed coordinate system as well as UAVSAR in local ground coordinate system. We will convert GEOSAR into the local ground coordinate system in which the original pointO represents the intersection between the satellite line of sight and the ground, the Ox-axis corresponds with track direction of GEOSAR, the Oz-axis corresponds with the connection between satellite and subpoint, and the Oy-axis is determined according to the right-hand rule (see Figure 2).Figure 2
The GEOSAR and UAVSAR are in the local ground coordinate system, where the red region represents beam irradiation coverage of UAVSAR, and the blue is GEOSAR.GEOSAR runs around the earth within a periodical time,Ts. In Ts, the range (Oy-axis) and the azimuth (Ox-axis) acceleration of GEOSAR ground track is shown in Figure 3.The range acceleration (a) and the azimuth acceleration (b) of GEOSAR ground track in local ground coordinate system.
(a)
Range acceleration
(b)
Azimuth accelerationThe trend of range acceleration is similar to a cosine curve whose period isTs/2. The peak value is 2.236×10-2m/s2, and the valley value is 3.9×10-4m/s2. Furthermore, the trend of azimuth acceleration is similar to a sine curve whose period is Ts/4. The maximum is 9.0088×10-3m/s2, and the minimum is 5.395×10-4m/s2.The range coverage of GEOSAR ground trackCtr=Rtcθtran/sin(θtin)=3.75×105m and the azimuth coverage Cta=Rtcθtaim/cos(φtsq)=1.25×105m where θtran and θtaim are the range and azimuth bandwidth of GEOSAR separately; θtin is an incidence angle, φtsq is a squint angle, when range acceleration aa reaches peak, the surface coverage will deviate from the Ox-axis for Tv>4.0952×103s (Tv represents common irradiation time, of two platforms; the limited time length Tvlim is calculated by the formula Ctr/2=V⇀eTTvlim+1/2aaTvlim2). Besides, for the azimuth acceleration is much smaller, the ground track velocity of GEOSAR is considered uniform motion as V⇀eT(t0) in the condition of Tv<4.0952×103s.


## 2.1. Movement Track of GEOSAR

Suppose that GEOSAR runs on the ellipse orbit and the earth is uniform sphere, orbital inclination isi, right ascension of ascending node is Ω0, and argument of perigee is W0. The satellite position equation can be expressed in earth fixed coordinate system:
(1)[Xs(t0+ta)Ys(t0+ta)Zs(t0+ta)]=RWB(t0+ta)WAWC(t0+ta),
where
(2)WA=[cosΩ0-sinΩ00sinΩ0cosΩ00001][1000cosi-sini0sinicosi]×[cosW0-sinW00sinW0cosW00001],WB(t0+ta)=[cos(ωe·(t0+ta))sin(ωe·(t0+ta))0-sin(ωe·(t0+ta))cos(ωe·(t0+ta))0001]WC(t0+ta)=[cosf(t0+ta)-sinf(t0+ta)0sinf(t0+ta)cosf(t0+ta)0001]t0 is the center time point of time range, ωe is the earth rotation angular velocity and R is the geocenter distance of the satellite.In the time range foregoing, the velocity of GEOSAR related to the earth is given by(3)V⇀sT_EF(t0+ta)=μaWA(t0+ta)WB[-sinf(t0+ta)cosf(t0+ta)0]+[ωeYs(t0+ta)-ωeXs(t0+ta)0],
where a is a semimajor axis, μ is a gravitation constant, Re is the earth radius, and HT is a GEOSAR distance of ground.Thus, wheni=0, GEOSAR is the stationary state and GEO-UAV BiSAR images in the local area near the equator. With the increase of i, V⇀sT goes up gradually, and the variation of latitude is [-i,i] in “Figure 8” imaging area. For maximum irradiation of Chinese territory, we choose i=60°. Based on GEOSAR basic parameters (see Table 1), the ground track of GEOSAR (i≠0) is shown in Figure 1.Table 1
GEOSAR transmitter parameters.
GEOSAR parameters
Value
Orbit altitude
36000 km
Argument of perigee
90°
Right ascension of ascending node
30°
Orbit inclination
60°
Azimuth beam width
0.2°
Angle of elevation
0.6°Figure 1
Ground track of 60° inclined geosynchronous orbit.Due to higher altitude of GEOSAR [20], the rate V⇀sT at which the spacecraft moves along its orbital path is unequal to the rate V⇀eT at which the footprint of the antenna beam moves along the surface of the earth: V⇀eT=ReV⇀sT/(Re+HT).


## 2.2. GEOSAR Velocity in Local Ground Coordinate System

In general, GEOSAR is described in earth-fixed coordinate system as well as UAVSAR in local ground coordinate system. We will convert GEOSAR into the local ground coordinate system in which the original pointO represents the intersection between the satellite line of sight and the ground, the Ox-axis corresponds with track direction of GEOSAR, the Oz-axis corresponds with the connection between satellite and subpoint, and the Oy-axis is determined according to the right-hand rule (see Figure 2).Figure 2
The GEOSAR and UAVSAR are in the local ground coordinate system, where the red region represents beam irradiation coverage of UAVSAR, and the blue is GEOSAR.GEOSAR runs around the earth within a periodical time,Ts. In Ts, the range (Oy-axis) and the azimuth (Ox-axis) acceleration of GEOSAR ground track is shown in Figure 3.The range acceleration (a) and the azimuth acceleration (b) of GEOSAR ground track in local ground coordinate system.
(a)
Range acceleration
(b)
Azimuth accelerationThe trend of range acceleration is similar to a cosine curve whose period isTs/2. The peak value is 2.236×10-2m/s2, and the valley value is 3.9×10-4m/s2. Furthermore, the trend of azimuth acceleration is similar to a sine curve whose period is Ts/4. The maximum is 9.0088×10-3m/s2, and the minimum is 5.395×10-4m/s2.The range coverage of GEOSAR ground trackCtr=Rtcθtran/sin(θtin)=3.75×105m and the azimuth coverage Cta=Rtcθtaim/cos(φtsq)=1.25×105m where θtran and θtaim are the range and azimuth bandwidth of GEOSAR separately; θtin is an incidence angle, φtsq is a squint angle, when range acceleration aa reaches peak, the surface coverage will deviate from the Ox-axis for Tv>4.0952×103s (Tv represents common irradiation time, of two platforms; the limited time length Tvlim is calculated by the formula Ctr/2=V⇀eTTvlim+1/2aaTvlim2). Besides, for the azimuth acceleration is much smaller, the ground track velocity of GEOSAR is considered uniform motion as V⇀eT(t0) in the condition of Tv<4.0952×103s.


## 3. Coherent Integration Time and Azimuth Common Coverage

Based on the theoretical research of resolution [21], the coherent integration time and azimuth common coverage influence azimuth resolution capability of GEO-UAV BiSAR.Coherent integration timeTsar and azimuth coverage Ca are relative to GEOSAR azimuth coverage LT, UAVSAR azimuth coverage LR, and antenna footprint velocity of two platforms. Movement of GEOSAR is described as the uniform motion in Tv under the aforementioned study. While UAV is difficult to control motion stability in aerodynamic interference (roll angle, pitch angle, and heading angle of UAVSAR are variable). In this case, the accurate calculation is very troubling. To simplify calculation, we consider that roll angle, pitch angle and heading angle of UAVSAR are all constants, and UAV is a uniform motion in a straight line, V⇀R. Therefore, the antenna footprint velocity ratio k can be shown as
(4)k=e⇀·V⇀eTe⇀·V⇀R,
where  e⇀=[100]T is the azimuth unit vector.(1)
Ifk=0, GEOSAR is the stationary station.(2)
Ifk>0, UAV flies the same direction with transmitter. When VeT<VR,k∈(0,1), otherwise k∈(1,+∞).(3)
Ifk<0, UAV flies the opposite direction with transmitter.Assume that footprint middle points of GEOSAR and UAVSAR are given byxT and xR, respectively, on the Ox-axis in Figure 2. Left and right edges of GEOSAR is PA and PB; PA=[xA00]Tand PB=[xB00]T. If two platforms intersect at PA, xT=xT(0) and xR=xR(0); if they intersect at PB, xT=xT(1) and xR=xR(1). Then, xT(0)=xA+LT/2,xR(0)=xA-LR/2,xT(1)=xB-LT/2,andxR(1)=xB+LR/2.For the subsequent simulation data showVeT>VR (see Tables 1 and 2), we primarily discuss Ca and Tsar in the case of k∈(1,+∞). When xR=xR(0),  Ca=0, while xR=xR(1) and Ca achieve the maximum. Suppose that l(x) represents the distance between the corresponding right edge points of the transmitter and the receiver in the Ox-axis. The change trend of l(x) is LR⇒0⇒LT with the relative movement of two platforms (see Figure 2). Thus, Tsar by (23) and (24) in the Appendix is calculated. Besides, the common irradiation time of two platforms is Tv=Ca/VR.Table 2
UAVSAR receiver parameters.
UAVSAR parameters
Value
Wavelength
0.03 m (X-band)
Bandwidth
150 MHz
Altitude
8 km
Incidence angle
1°~80°
Angle betweenRT(t0,O) and RR(t0,O)
0°~360°
Antenna length
1 mTaking into account the difference value ofk, we count Ca and Tsar, respectively (see the Appendix), when k=0andCa=LT. While in the case that VeT,VR,LT, and LR are constants, Ca is inversely proportional to |VeT-VR| if k>0 (|VeT+VR| if k<0). And the maximum Tsar depends on the synthetic aperture time of receiver. If the coherent accumulated time is less than the maximum Tsar, the imaging quality is fuzzy in the zone which is known as “dead zone1”. Conversely, it is called as “clear zone”.


## 4. Resolution of GEO-UAV BiSAR

GEO-UAV BiSAR geometry in local ground coordinate system is shown in Figure4. The center of the scene is O point, GEOSAR locates PT(t0+ta); UAVSAR locates PR(ta). The time-domain matched filter is constructed by forming an instantaneous slant range to a point target Pm=[xmymzm]T referred to as the range equation
(5)R⇀(ta,Pm)=R⇀T(t0+ta,Pm)+R⇀R(ta,Pm),
where
(6)R⇀T(t0+ta,Pm)=PT(t0+ta)-Pm,R⇀R(ta,Pm)=PR(ta)-Pm=(PR(0)-Pm)+V⇀Rta.Figure 4
GEO-UAV BiSAR geometry.Because GEOSAR shows a variable motion asV⇀sT(t0+ta), the gradient of time delay ∇t⇀ and the gradient of the Doppler shift ∇fa⇀(ta,Pm) are defined as (4.a) and (4.b):
(4.a)∇t⇀=1c(R⇀T(t0+ta,Pm)|R⇀T(t0+ta,Pm)|+R⇀R(ta,Pm)|R⇀R(ta,Pm)|).Note that∇fa⇀(ta,Pm) depends on the angular rate of two platforms. For |V⇀sT|/|R⇀T(ta,Pm)|=O(10-4) and |V⇀R|/|R⇀R(ta,Pm)|=(10-1), UAVSAR dominates ∇fa⇀(ta,Pm):
(7)∇fa⇀(ta,Pm)≈12π∂∇Φ(ta,Pm)∂ta=-1λ(1|R⇀R(ta,Pm)|(V⇀R-(V⇀RR⇀R(ta,Pm)|R⇀R(ta,Pm)|)=-1λ=-1λ-1λ×R⇀R(ta,Pm)|R⇀R(ta,Pm)|)),(4.b)∇fa⇀(ta,Pm)=12π∂∇Φ(ta,Pm)∂ta=-12π×2πλ∂∂ta(R⇀T(t0+ta,Pm)|R⇀T(t0+ta,Pm)|+R⇀R(ta,Pm)|R⇀R(ta,Pm)|)=-1λ(1|R⇀T(t0+ta,Pm)|×({R⇀T(t0+ta,Pm)|R⇀T(t0+ta,Pm)|}V⇀sT(t0+ta)-(V⇀sT(t0+ta)R⇀T(t0+ta,Pm)|R⇀T(t0+ta,Pm)|)×R⇀T(t0+ta,Pm)|R⇀T(t0+ta,Pm)|)+1|R⇀R(ta,Pm)|×(V⇀R-(V⇀RR⇀R(ta,Pm)|R⇀R(ta,Pm)|)R⇀R(ta,Pm)|R⇀R(ta,Pm)|)).The maximum delay time for movement in the ground plane is along the projection of∇t⇀ into xOy plane, ∇te⇀. Similarly, the maximum change in the Doppler frequency moves along the projection of ∇fa⇀ into xOy plane, ∇fe⇀. Thus, the range resolution Dr and the azimuth resolution Da are given by
(8)Dr=1Br|∇te⇀|·∇te⇀|∇te⇀|,Da=1|∇fe⇀|Tsar·∇fg⇀|∇fe⇀|.BiSAR resolution has the properties of time varying and spatial varying. On the basis of the GEO-UAV BiSAR geometry,  (8) is redescribed as
(9)Dr=C2Brcos(Ψ/2)·∇te⇀|∇te⇀|,Da≈HRλTsar(t0+ta)VRcosθrin1-sin2θrincos2(Θr)·∇fe⇀|∇fe⇀|,
where Ψ is the illuminator out-of-plane angle (with respect to the receiver and centered in the origin), Θr is the angle between the Ox-axis and the projection of UAVSAR position into xOy plane, and Θr is determined by Ψ.The range resolution mainly depends onΨ through (9). It attains the maximum for Ψ=180° and the minimum values for Ψ=0°. In other words, the gradient of time delay has no component onto xOy plane for Ψ=180°, GEO-UAV BiSAR construction has the worst range resolution.The azimuth resolution is relative to coherent accumulated time and the attitude change of UAV. With the shorter coherent accumulated timeTsar(t0+ta), the imaging quality of GEO-UAV BiSAR will decline gradually (dead zone1). Whereas, the azimuth resolution can further simplify as Da≈D/1-sin2θrincos2Θr in the clear zone. When θrin reaches an enough value, Da is the maximum for Θr=0°,180°,and360°. And the gradient of the Doppler shift is the smallest component onto xOy plane.For aforesaid reasons, the proper selection of original UAVSAR attitudes,θrin and Ψ, can make GEO-UAV BiSAR exhibit high resolution. To achieve this aim in clear zone, we will avoid the appearance of the worst construction in mutual movement process of transmitter and receiver. (Worst construction illuminates the area called dead zone2.)In a periodical time of GEOSAR, the relative movement of GEOSAR and UAVSAR works in finite time[t0-Tv/2,t0+Tv/2] (the local ground coordinate system is different for t0 varying). Suppose that the original value of Ψ (corresponding t0) is in CCI with regard to good imaging quality. For the sake of platform velocity differences, the variation range of  Ψ, Ψvar is still in CCI:
(10)[Ψ-max(Ψvar-Ψ),Ψ+max(Ψvar-Ψ)]∈CCI,whereΨvar is related to the velocity ratio of the two platforms, k. Considering k>0, Ψvar is given by
(11)Ψvar=acos((Ps_LS(t0+ta)-V⇀R×ta)|Ps_LS(t0+ta)-V⇀R×ta|·R⇀R(ta,Pm)|R⇀R(ta,Pm)|),
where Ps_LS(t0+ta) is the GEOSAR subpoint. And (11) can be further simplified as
(12)Ψvar≈acos((PsLS(t0)+|V⇀eT-V⇀R|×ta)|PsLS(t0)+|V⇀eT-V⇀R|×ta|·R⇀R(ta,Pm)|R⇀R(ta,Pm)|).
Since
(13)max(ta)=Tv2=VRLT-VeTLR2VR|V⇀eT-V⇀R|,wheremax(Ψvar) is expressed as
(14)max(Ψvar)=acos((Ps_LS(t0)+(V⇀RLT-V⇀eTLR)/2)|Ps_LS(t0)+(V⇀RLT-V⇀eTLR)/2|·R⇀R(ta,Pm)|R⇀R(ta,Pm)|).To meet the condition of (10), max(Ψvar) will be in CCI also. Note from (14) that Ψvar is related to V⇀eT, LR, V⇀R, and LT.Whenk<0, max(Ψvar) is the same to (14). However, GEOSAR runs on a circle orbit if k=0. Under this circumstance, Ψ is meaningless for θtin=0. Hence, the range resolution is given by
(15)Dr=C2Br·∇te⇀|∇te⇀|.


## 5. Simulation Analysis

In order to study the resolution of GEO-UAV BiSAR, we discuss the influence of coherent accumulated time and attitude change of UAV on the two-dimensional resolution. And the imaged results are disposed through using frequency domain RD method based on MSR.


### 5.1. Effect of Coherent Accumulated Time on Azimuth Resolution

Within a periodical time of GEOSAR, the velocityVeT is a variable. The bistatic configuration changes with the different time instant t0 selected (origin O is different with a function t0 in local ground coordinate system). According to the parameters of GEOSAR and UAVSAR (see Tables 1 and 2), we can gain k∈(1,+∞). In the condition of θrin=10° and Ψ=311°, the azimuth coverage and the coherent accumulated time are calculated through (23) and (24) in the appendix. GEOSAR beam surface velocity, clear-zone, dead zone1, and coherent accumulated time for t0=Ts/8,Ts/4,andTs/2 are shown in Table 3.Table 3
Azimuth coverage and coherent accumulated time of GEOSAR at different time instants.
Timet0 (s)
T
s
/
8
T
s
/
4
T
s
/
2
V
e
T (m/s)
366.54
464.15
232.08
Clear zone (×104 m)
[
-
2.09,2.09
]
[
-
1.49,1.49
]
[
-
3.95,3.95
]
Dead zone1 (×104 m)
[
-
2.12
,
2.09
] & [2.09, 2.12]
[
-
1.52
,
-
1.49
] & [1.49, 1.52]
[
-
3.98
,
-
3.95
] & [3.95, 3.98]
max
(
T
sar
) (s)
2.71
2.71
2.73Table 4
Space-time analysis of GEO-UAV BiSAR.
Velocity ratio
Azimuth maximum coverage(UAVSAR position in Ox axis)
Coherent accumulated time
T
sar
=
x
-
x
A
V
R
,
x
∈
[
x
A
,
L
R
+
x
A
],
k
=
0
C
a
=
L
T (xR=xR(0))
T
sar
=
L
R
V
R
,
x
∈
[
L
R
+
x
A
,
L
T
-
L
R
+
x
A
],
(20)
T
sar
=
x
-
L
T
+
L
R
-
x
A
V
R
,
x
∈
[
L
T
-
L
R
+
x
A
,
L
T
+
x
A
].
k
∈
(
0,1
)
C
a
=
V
R
L
T
+
V
e
T
L
R
V
R
-
V
e
T
(
x
R
=
x
R
(
0
)
)
l
(
x
)
=
L
T
-
V
R
-
V
e
T
V
R
(
x
-
x
A
)
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
x
A
,
V
R
L
T
(
V
R
-
V
e
T
)
+
x
A
],
(21)
l
(
x
)
=
V
R
-
V
e
T
V
g
T
(
x
-
x
A
)
-
V
R
V
e
T
L
T
,
T
sar
=
min
(
L
R
V
R
,
L
R
-
l
(
x
)
V
R
)
,
x
∈
[
V
R
L
T
(
V
R
-
V
g
T
)
+
x
A
,
V
R
L
T
+
V
e
T
L
R
(
V
R
-
V
e
T
)
+
x
A
].
(22)
k
∈
(
1
,
+
∞
)
C
a
=
V
R
L
T
+
V
e
T
L
R
V
e
T
-
V
R
(
x
R
=
x
R
(
1
)
)
l
(
x
)
=
L
R
-
(
V
e
T
-
V
R
)
(
x
-
x
B
)
V
e
T
,
T
sar
=
L
R
-
l
(
x
)
V
R
,
x
∈
[
x
B
,
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
],
(23)
l
(
x
)
=
(
V
e
T
-
V
R
)
(
x
-
x
B
)
V
R
-
L
R
V
e
T
V
R
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
,
V
R
L
T
+
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
].
(24)
k
<
0
C
a
=
V
R
L
T
+
V
e
T
L
R
V
R
+
V
e
T
(
x
R
=
x
R
(
1
)
)
l
(
x
)
=
L
R
-
V
R
+
V
e
T
V
e
T
(
x
-
x
B
)
,
T
sar
=
min
(
L
R
V
R
,
L
R
-
l
(
x
)
V
R
)
,
x
∈
[
x
B
,
V
e
T
L
R
2
(
V
R
+
V
e
T
)
+
x
B
],
(25)
l
(
x
)
=
V
R
+
V
e
T
V
R
(
x
-
x
B
)
-
V
e
T
V
R
L
R
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
V
e
T
L
R
(
V
R
+
V
e
T
)
+
x
B
,
V
e
T
L
R
+
V
R
L
T
(
V
e
T
+
V
R
)
+
x
B
].
(26)Simulations that demonstrate the azimuth coverage of GEO-UAV BiSAR are inversely proportional to|VeT-VR|. At a constant LR, the size of dead zone1 shows a decrease with VeT going up. The change of max(Tsar) follows the squint angle of receiver that varies owing to the orbital movement of GEOSAR.Takingt0=Ts/8 as an example, the influence of coherent accumulated time on azimuth resolution is drawn in Figure 5. In the clear zone, the azimuth resolution will achieve 0.99 m, while the azimuth resolution capability reduces with coherent accumulated time dropping gradually in the dead zone1.Coherent accumulated time (a) and azimuth resolution (b) (t0=Ts/8).
(a)
Coherent accumulated time
(b)
Azimuth resolution


### 5.2. Effect of the Initial Attitudes for Two Platforms on Resolution

Based on resolution analysis in Section4, the appropriate attitude control of UAVSAR enables to avoid the appearance dead zone2 in the clear zone. As far as of range and the Doppler resolution are concerned, Figure 6 shows their variations as two functions of θrin and Ψ for GEO-UAV BiSAR (t0=Ts/8).Range resolution (a) and azimuth resolution (b) as two functions ofθrin and Ψ for GEO-UAV BiSAR.
(a)
Range resolution
(b)
Azimuth resolutionΨ has a great effect on the range resolution, to which θrin is meaningless. If Ψ∈[0°,90°] and Ψ∈[276°,360°], BiSAR has higher resolution and Dr≤1.4m. If Ψ∈[91°,275°], BiSAR resolution is much lower and Dr>1.4m. When Ψ=180°, it is the worst condition that gradient of time delay is equal to zero such that the range resolution reaches the maximum value (see Figure 6(a)).For the azimuth resolution, bothθrin and Ψ have certain mount of influences. If θrin≤45°, the azimuth resolution of GEO-UAV BiSAR is lower than 1.4 m (critical value of resolution). And if θrin>45°, the azimuth resolution is still impacted by Ψ (see Figure 6(b)). 								(16)Ψ∈[0°,14°]&[44°,198°]&[230°,360°]⟹Da≤1.4m,Ψ∈[15°,43°]&[199°,229°]⟹Da>1.4m.To assure the high resolution of range and azimuth, the range value ofΨ is drawn in Figure 7. A unit circle represents the position projection of transmitter and receiver in xOy plane. If θrin≤45°, GEO-UAV BiSAR configuration can show high-resolution ability in the content of Ψ∈area 2 and area 3, whereas we need to avoid Ψ that appears in area 1 and area 3  if θrin>45°.Figure 7
The unit location distribution of GEOSAR and UAVSAR.Figure 8
Max
(
Ψ
var
-
Ψ
) in an orbital period of GEOSAR.With the mutual movement of two platforms (V⇀eT(t0+ta)≠V⇀R,  and V⇀eT(t0+ta) is time varying), the variational range of  Ψvar is different in the periodical time of GEOSAR. To avoid the appearance of the dead zone2 in the clear zone, we demand to meet (10). The variational range of max(Ψvar-Ψ) in the whole period of GEOSAR is shown in Figure 8. And max(Ψvar-Ψ) obtains the maximum 15° when t0=Ts/20. Hence, the proper extent of initial UAV attitude can be written as
(17)Ψ∈[0°,75°]&[291°,360°](θrin≤45°),Ψ∈[59°,75°]&[291°,358°](θrin>45°).Based on the aforementioned condition, we selectθrin=10° and Ψ=311°. Figure 9 shows the range and azimuth resolution distributions of GEO-UAV BiSAR in xOy plane (t0=Ts/8). It is worth noting that the coherent accumulated time (2.71 s in Table 3) is considered for all point target in the bistatic scene area. The bistatic scene dimensions are 400 m in range and 244 m in azimuth. And the range resolutions from 1.07 m to 1.14 m are expected along with azimuth resolutions ranging from 0.99 m to 1.00 m. These values are deduced by  (8) and (9). Thus, GEO-UAV BiSAR is suitable to the image in the local area.Range resolution in meters (a) and azimuth resolution in meters (b) of GEO-UAV BiSAR inxOy plane (t0=Ts/8). The red region represents the bistatic scene area.
(a)
Range resolution (m)
(b)
Azimuth resolution (m)


### 5.3. Frequency Domain RD Based on MSR

To evaluate the resolution of GEO-UAV BiSAR, we will focus on an image with a frequency-domain SAR processor. The corresponding processing approach has been developed based on MSR by means of the four-level Taylor expansion [18, 19]. RD method in frequency domain is performed by four steps, that is, range compression, secondary range compression, range cell migration, and azimuth compression.If the attitude parameters of UAVSAR areθrin=10° and Ψ=311°, the impulse response output at the scene center target (t0=Ts/8) is drawn in Figure 10. The corresponding measured −3 dB resolutions are 1.08 m and 1.08 m, respectively, in comparison to the theoretical values of 1.08 m and 0.99 m (in Figure 9). It can be seen that the resolution has range and azimuth broadening of about 0.3% and 9%.Range impulse response output (a) and azimuth impulse response output (b) for the point target (t0=Ts/8).
(a)
Range impulse response output
(b)
Azimuth impulse response output


## 5.1. Effect of Coherent Accumulated Time on Azimuth Resolution

Within a periodical time of GEOSAR, the velocityVeT is a variable. The bistatic configuration changes with the different time instant t0 selected (origin O is different with a function t0 in local ground coordinate system). According to the parameters of GEOSAR and UAVSAR (see Tables 1 and 2), we can gain k∈(1,+∞). In the condition of θrin=10° and Ψ=311°, the azimuth coverage and the coherent accumulated time are calculated through (23) and (24) in the appendix. GEOSAR beam surface velocity, clear-zone, dead zone1, and coherent accumulated time for t0=Ts/8,Ts/4,andTs/2 are shown in Table 3.Table 3
Azimuth coverage and coherent accumulated time of GEOSAR at different time instants.
Timet0 (s)
T
s
/
8
T
s
/
4
T
s
/
2
V
e
T (m/s)
366.54
464.15
232.08
Clear zone (×104 m)
[
-
2.09,2.09
]
[
-
1.49,1.49
]
[
-
3.95,3.95
]
Dead zone1 (×104 m)
[
-
2.12
,
2.09
] & [2.09, 2.12]
[
-
1.52
,
-
1.49
] & [1.49, 1.52]
[
-
3.98
,
-
3.95
] & [3.95, 3.98]
max
(
T
sar
) (s)
2.71
2.71
2.73Table 4
Space-time analysis of GEO-UAV BiSAR.
Velocity ratio
Azimuth maximum coverage(UAVSAR position in Ox axis)
Coherent accumulated time
T
sar
=
x
-
x
A
V
R
,
x
∈
[
x
A
,
L
R
+
x
A
],
k
=
0
C
a
=
L
T (xR=xR(0))
T
sar
=
L
R
V
R
,
x
∈
[
L
R
+
x
A
,
L
T
-
L
R
+
x
A
],
(20)
T
sar
=
x
-
L
T
+
L
R
-
x
A
V
R
,
x
∈
[
L
T
-
L
R
+
x
A
,
L
T
+
x
A
].
k
∈
(
0,1
)
C
a
=
V
R
L
T
+
V
e
T
L
R
V
R
-
V
e
T
(
x
R
=
x
R
(
0
)
)
l
(
x
)
=
L
T
-
V
R
-
V
e
T
V
R
(
x
-
x
A
)
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
x
A
,
V
R
L
T
(
V
R
-
V
e
T
)
+
x
A
],
(21)
l
(
x
)
=
V
R
-
V
e
T
V
g
T
(
x
-
x
A
)
-
V
R
V
e
T
L
T
,
T
sar
=
min
(
L
R
V
R
,
L
R
-
l
(
x
)
V
R
)
,
x
∈
[
V
R
L
T
(
V
R
-
V
g
T
)
+
x
A
,
V
R
L
T
+
V
e
T
L
R
(
V
R
-
V
e
T
)
+
x
A
].
(22)
k
∈
(
1
,
+
∞
)
C
a
=
V
R
L
T
+
V
e
T
L
R
V
e
T
-
V
R
(
x
R
=
x
R
(
1
)
)
l
(
x
)
=
L
R
-
(
V
e
T
-
V
R
)
(
x
-
x
B
)
V
e
T
,
T
sar
=
L
R
-
l
(
x
)
V
R
,
x
∈
[
x
B
,
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
],
(23)
l
(
x
)
=
(
V
e
T
-
V
R
)
(
x
-
x
B
)
V
R
-
L
R
V
e
T
V
R
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
,
V
R
L
T
+
V
e
T
L
R
(
V
e
T
-
V
R
)
+
x
B
].
(24)
k
<
0
C
a
=
V
R
L
T
+
V
e
T
L
R
V
R
+
V
e
T
(
x
R
=
x
R
(
1
)
)
l
(
x
)
=
L
R
-
V
R
+
V
e
T
V
e
T
(
x
-
x
B
)
,
T
sar
=
min
(
L
R
V
R
,
L
R
-
l
(
x
)
V
R
)
,
x
∈
[
x
B
,
V
e
T
L
R
2
(
V
R
+
V
e
T
)
+
x
B
],
(25)
l
(
x
)
=
V
R
+
V
e
T
V
R
(
x
-
x
B
)
-
V
e
T
V
R
L
R
,
T
sar
=
min
(
L
R
V
R
,
L
T
-
l
(
x
)
V
e
T
)
,
x
∈
[
V
e
T
L
R
(
V
R
+
V
e
T
)
+
x
B
,
V
e
T
L
R
+
V
R
L
T
(
V
e
T
+
V
R
)
+
x
B
].
(26)Simulations that demonstrate the azimuth coverage of GEO-UAV BiSAR are inversely proportional to|VeT-VR|. At a constant LR, the size of dead zone1 shows a decrease with VeT going up. The change of max(Tsar) follows the squint angle of receiver that varies owing to the orbital movement of GEOSAR.Takingt0=Ts/8 as an example, the influence of coherent accumulated time on azimuth resolution is drawn in Figure 5. In the clear zone, the azimuth resolution will achieve 0.99 m, while the azimuth resolution capability reduces with coherent accumulated time dropping gradually in the dead zone1.Coherent accumulated time (a) and azimuth resolution (b) (t0=Ts/8).
(a)
Coherent accumulated time
(b)
Azimuth resolution


## 5.2. Effect of the Initial Attitudes for Two Platforms on Resolution

Based on resolution analysis in Section4, the appropriate attitude control of UAVSAR enables to avoid the appearance dead zone2 in the clear zone. As far as of range and the Doppler resolution are concerned, Figure 6 shows their variations as two functions of θrin and Ψ for GEO-UAV BiSAR (t0=Ts/8).Range resolution (a) and azimuth resolution (b) as two functions ofθrin and Ψ for GEO-UAV BiSAR.
(a)
Range resolution
(b)
Azimuth resolutionΨ has a great effect on the range resolution, to which θrin is meaningless. If Ψ∈[0°,90°] and Ψ∈[276°,360°], BiSAR has higher resolution and Dr≤1.4m. If Ψ∈[91°,275°], BiSAR resolution is much lower and Dr>1.4m. When Ψ=180°, it is the worst condition that gradient of time delay is equal to zero such that the range resolution reaches the maximum value (see Figure 6(a)).For the azimuth resolution, bothθrin and Ψ have certain mount of influences. If θrin≤45°, the azimuth resolution of GEO-UAV BiSAR is lower than 1.4 m (critical value of resolution). And if θrin>45°, the azimuth resolution is still impacted by Ψ (see Figure 6(b)). 								(16)Ψ∈[0°,14°]&[44°,198°]&[230°,360°]⟹Da≤1.4m,Ψ∈[15°,43°]&[199°,229°]⟹Da>1.4m.To assure the high resolution of range and azimuth, the range value ofΨ is drawn in Figure 7. A unit circle represents the position projection of transmitter and receiver in xOy plane. If θrin≤45°, GEO-UAV BiSAR configuration can show high-resolution ability in the content of Ψ∈area 2 and area 3, whereas we need to avoid Ψ that appears in area 1 and area 3  if θrin>45°.Figure 7
The unit location distribution of GEOSAR and UAVSAR.Figure 8
Max
(
Ψ
var
-
Ψ
) in an orbital period of GEOSAR.With the mutual movement of two platforms (V⇀eT(t0+ta)≠V⇀R,  and V⇀eT(t0+ta) is time varying), the variational range of  Ψvar is different in the periodical time of GEOSAR. To avoid the appearance of the dead zone2 in the clear zone, we demand to meet (10). The variational range of max(Ψvar-Ψ) in the whole period of GEOSAR is shown in Figure 8. And max(Ψvar-Ψ) obtains the maximum 15° when t0=Ts/20. Hence, the proper extent of initial UAV attitude can be written as
(17)Ψ∈[0°,75°]&[291°,360°](θrin≤45°),Ψ∈[59°,75°]&[291°,358°](θrin>45°).Based on the aforementioned condition, we selectθrin=10° and Ψ=311°. Figure 9 shows the range and azimuth resolution distributions of GEO-UAV BiSAR in xOy plane (t0=Ts/8). It is worth noting that the coherent accumulated time (2.71 s in Table 3) is considered for all point target in the bistatic scene area. The bistatic scene dimensions are 400 m in range and 244 m in azimuth. And the range resolutions from 1.07 m to 1.14 m are expected along with azimuth resolutions ranging from 0.99 m to 1.00 m. These values are deduced by  (8) and (9). Thus, GEO-UAV BiSAR is suitable to the image in the local area.Range resolution in meters (a) and azimuth resolution in meters (b) of GEO-UAV BiSAR inxOy plane (t0=Ts/8). The red region represents the bistatic scene area.
(a)
Range resolution (m)
(b)
Azimuth resolution (m)


## 5.3. Frequency Domain RD Based on MSR

To evaluate the resolution of GEO-UAV BiSAR, we will focus on an image with a frequency-domain SAR processor. The corresponding processing approach has been developed based on MSR by means of the four-level Taylor expansion [18, 19]. RD method in frequency domain is performed by four steps, that is, range compression, secondary range compression, range cell migration, and azimuth compression.If the attitude parameters of UAVSAR areθrin=10° and Ψ=311°, the impulse response output at the scene center target (t0=Ts/8) is drawn in Figure 10. The corresponding measured −3 dB resolutions are 1.08 m and 1.08 m, respectively, in comparison to the theoretical values of 1.08 m and 0.99 m (in Figure 9). It can be seen that the resolution has range and azimuth broadening of about 0.3% and 9%.Range impulse response output (a) and azimuth impulse response output (b) for the point target (t0=Ts/8).
(a)
Range impulse response output
(b)
Azimuth impulse response output


## 6. Conclusion

Bistatic geometrical configuration using GEOSAR as a transmitter and UAVSAR as a receiver is analyzed. From a theoretical level, firstly, the variation of bistatic geometrical configuration which has influence on azimuth common coverage and coherent accumulated time is studied according to the ellipse orbital movement of GEOSAR; then, the gradient method based on GEO-UAV SAR configuration (GEOSAR exhibits the characteristics of high altitude and periodical motion) is proposed to deduce two-dimensional resolution.Simulations illustrate that the resolution is accurately obtained from the geometrical simplification of gradient method which consists of two impact factors on the coherent accumulated time and the UAV attitude. The azimuth resolution capacity of GEO-UAV BiSAR shows a gradual decrease with coherent accumulated time reducing the relative motion of two platforms. Moreover, the proper initial selection of UAV attitude can avoid the appearance of dead zone in the swath. Such theoretical analysis, along with simulation results, further demonstrates the potential performance of GEO-UAV BiSAR that is mainly affected by the UAV attitude design when GEOSAR follows the orbital operation. Besides, the feasibility of GEO-UAV BiSAR configuration for high resolution imaging in the local field is verified.The foregoing works in this paper inspire us to research on the existing bistatic particularities. However, the effect of nonstationary movement related to phase center for GEOSAR and UAVSAR, which reduce the two-dimensional resolution, is ignored in this paper. In this case, the further experiments based on the nonstationary will be developed, and the resolution will be analyzed.



---

*Source: 237463-2013-04-03.xml*