The Staring Mode Properties and Performance of Geo-SAR Satellite with Reflector Antenna

Abstract

Geosynchronous synthetic aperture radar (Geo-SAR) with a short revisit time can obtain wide-area images. This paper advances a new two-dimensional pitch and roll squint controlling (2D-PRSC) method that can make satellites continuously stare at any scene in the whole orbital period. The maximum attitude steering angle is less than ±7.6 degrees, and the attitude controlling time can be greatly shortened compared with the yaw steering method. Furthermore, a Geo-SAR staring mode model is illustrated and compared with that of low earth orbital SAR (Leo-SAR). Finally, Geo-SAR’s ambiguity property is discussed. The simulation results illuminate that the cross-term ambiguity to signal ratio (CASR) also needs to be considered in addition to the azimuth and range ambiguity to signal ratio (AASR, RASR), and the whole orbital ergodic analysis should be carried out. To ensure that RASR, AASR, and CASR meet the requirement of −20 dB, it is necessary to select an appropriate PRF in the range of a few hundred Hertz.

1. Introduction

In 1978, K. Tomiysu firstly proposed the concept of Geo-SAR satellite [1]. Since then, it has gradually become a hot research topic. It has the advantages of short revisit cycle, fast maneuver monitoring capability, large imaging width, and long staring time. Therefore, it has a wide application prospection in large-scale disaster monitoring, marine observation, geological mapping, and so on [2,3,4,5,6,7,8]. The China Academy of Space Technology(CAST) is currently developing Geo-SAR for disaster monitoring.

Geo-SAR’s orbit altitude is about 60 to 70 times higher than that of Leo-SAR, and the former’s satellite speed is about 1/60 to 1/10 of the latter’s. As a result, there are significant differences between them regarding the signal model, system design method, imaging algorithm, etc. So far, a variety of imaging modes have been advanced and developed for Leo-SAR, such as stripmap mode, ScanSAR mode, slide spotlight mode, TOPSAR mode, mosaic mode, and so on [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

Spotlight mode is classical, and its beam is fixed pointing. In the Leo-SAR system, its beam points to a scene center through antenna electronical scanning or satellite attitude controlling in order to obtain higher azimuth resolution than the stripmap mode. Many SAR satellites have spotlight mode, including the Cosmo-Skymed satellite [27,28], Sentinel-1 satellite [29,30], CAST’s GaoFen-3 (GF-3) satellite [31], etc. A staring spotlight mode is proposed in 2014 by DLR. This test mode extends the TerraSAR-X phased array antenna azimuth scanning angle to ±2.2 degrees. Then, an azimuth resolution of 0.2 m is obtained in a 3 dB azimuth range [32,33]. The staring mode is suitable for Geo-SAR. Similar to Leo-SAR, its beam boresight direction invariably points to the scene center. This study discusses the staring mode of an L-band Geo-SAR with a reflector antenna. Its characteristics are presented as follows:

◼

Staring mode imaging range mainly depends on a beam’s 3 dB coverage scale. Geo-SAR can constantly obtain a scene range of 320 km × 320 km in the whole orbit through a 25 m aperture circular antenna.

◼

The Geo-SAR staring mode aims to obtain constantly an azimuth resolution of 25 m in different orbit positions through partial synthetic aperture.

◼

A new two-dimensional pitch and roll squint controlling (2D-PRSC) method is proposed. The beam can quickly point to any area of 320 km × 320 km distributed in a zone of 5000 km × 5000 km. The satellite’s maneuver monitoring capability is promoted due to its attitude controlling angles being less than ±7.6 degrees. The time and energy expended for attitude controlling can be significantly saved.

◼

Due to the distinct variation of the beam footprint direction angle (BFPA) caused by the satellite’s periodic large yaw angle, CASR also needs to be considered as well as AASR and RASR. Furthermore, it is necessary to analyze its ambiguity in the whole orbital period because of Geo-SAR’s strong time-varying velocity.

These above aspects are mainly caused by the differences of orbit parameters and attitude controlling technology between Geo-SAR and Leo-SAR. Generally, yaw steering (YS) technology is used to make Leo-SAR’s residual Doppler centroid zero for broadside imaging. This method has gone through several stages. SEASAT-A, as the first spaceborne SAR, was launched by NASA in 1978. It had a periodically changing squint angle because YS technology was not used [34]. Tomiyasu illustrated the relationship between yaw angle and squint angle in 1983, which can be regarded as the rudiment of YS theory [2]. Raney proposed a one-dimensional YS method in 1986, which can accurately calculate the yaw angle of the SAR satellite with an orbital eccentricity of zero. However, the residual Doppler centroid cannot completely achieve zero when the orbital eccentricity is not zero. This technology was firstly employed for ERS-1 launched by ESA in 1991, and its residual Doppler centroid could be kept within ±500 Hz most of the time [35]. Fiedler advanced a two-dimensional yaw steering(2D-YS) approach in 2005 [36], which is suitable for spaceborne SAR with certain orbital eccentricity. The yaw angle calculation expression is consistent with that in [35], and the pitch angle is calculated with an approximate formula. TerraSAR-X, with an eccentricity of 0.0011, launched by DLR in 2006 used this method, and its residual Doppler centroid could be kept to less than 50 Hz [36]. A more accurate 2D-YS algorithm is presented in [37]. The analytical calculation formulas of yaw angle and pitch angle are derived through vector equations. If the engineering error, such as attitude controlling error and measuring error, are not considered, the residual Doppler centroid can be controlled to zero theoretically. The GF-3 satellite launched by CAST in 2016 adopted this method.

The residual Doppler centroid of Geo-SAR can also be kept to zero theoretically by using the method in [37]. The maximum yaw angle of Leo-SAR is about ±4 degrees; however, that of Geo-SAR is much bigger, as shown in Figure 1. If this technology is still used, Geo-SAR’s attitude control system, power system, and thermal control system will be very complicated. At present, there are also relevant papers to discuss the attitude steering technology of Geo-SAR, which is based on 2D-YS technology, and they carry out the research of improved methods [38,39].

The two-dimensional pitch and roll steering (2D-PRS) technology for Geo-SAR broadside imaging is presented in [40]. A new 2D-PRSC method for squint staring mode is proposed in this paper. The maximum attitude angle to be controlled is less than 7.6 degrees, and the attitude controlling time could be greatly reduced to be shorter than 400 s.

The ambiguity to signal ratio (ASR) is one of the important parameters of spaceborne SAR. Generally, only AASR and RASR are needed to be considered for Leo-SAR. CASR is generally ignored because it is much smaller than the former two parameters [41,42,43,44]. This study discusses the ASR properties of Geo-SAR. Firstly, the earth surface curvature must be considered to accurately calculate the signal and ambiguity distribution due to having a large field of view. Secondly, due to Geo-SAR’s large yaw angle, the BFPA changes periodically with latitude argument and is much larger than that of Leo-SAR. So, it will cause great error if the antenna pattern is simplified into one-dimensional sinc functions in elevation or azimuth directions. It is necessary to calculate the 2D antenna pattern intensity distribution on the earth’s surface. Thirdly, as mentioned above, CASR cannot be ignored. Therefore, all these three kinds of ASR must meet the threshold constraint requirements in the process of satellite system designing.

The contents are arranged as follows. The new 2D-PRSC method is described in the Section 2. In the Section 3, the model of Geo-SAR staring mode is analyzed. Furthermore, the antenna size, peak transmitting power, and NESZ are also analyzed in this section. The ASR calculation approach is illuminated in the Section 4. Finally, the conclusion is presented in the Section 5.

2. Attitude Controlling Technology

2.1. Satellite Attitude Characteristic

Due to the earth rotation, the satellite’s actual flight direction will deviate from the orbital plane when it flies along the orbit, resulting in the so-called yaw angle. If there is eccentricity in the orbit, the satellite velocity direction will not coincide with the orbit tangent, resulting in the so-called pitch angle. The two angles above produce a squint angle together, which makes the Doppler centroid not zero. To solve this problem, the simplest way is to calculate these two angles and then compensate for them directly by satellite’s attitude controlling or antenna electrical scanning. This is the so-called 2D-YS technology, which is widely used for Leo-SAR satellites [36,37]. In order to minimize the satellite weight, Geo-SAR is equipped with a reflector antenna, which does not have two-dimensional beam scanning ability. As a result, it is necessary to control the satellite attitude to achieve the residual Doppler centroid of zero.

After 2D-YS, the residual Doppler centroid can be expressed as follows.

$$f_{dc}=-\frac{2r^{\prime }}{\lambda }=-\frac{2}{\lambda }\frac{\left(R_s-R_t\right)\left(V_s-V_t\right)}{r}$$

(1)

where R and V, respectively, represent the position vector and velocity vector, the subscript s means satellite and t means target, r is the range between the satellite and target, λ is the wavelength, and f_dc is the Doppler centroid. The formulas in [37] are adopted to calculate the two angles above for the highest accuracy. The yaw and pitch angles of GF-3 and Geo-SAR in one orbital period are compared in Figure 2. The longitude of the ascending node is ignored here because it does not affect the analysis results. The simulation parameters are shown in

Table 1. Simulation parameters of GF-3 and Geo-SAR.

Name	GF-3	Geo-SAR
Semi-major axis (km)	7126.4	42,163.5
Inclination angle (°)	98.4	28
Eccentricity	0.001	0.001
Incidence angle (°)	45.5	45.5
Argument of perigee (°)	90	90
Wave length (m)	0.06	0.24
Earth model	WSG84	WSG84

.

The incidence angle of 45.5 degrees is taken as an example here for simulation. In fact, the simulation results of all different incidence angles are consistent

The solid blue line and dotted red line in Figure 2a,b, respectively, represent the yaw angle and pitch angle. There will be both small pitch angles for the two satellites due to the small eccentricity of 0.001. The pitch angle scale of GF-3 in one period is (−0.06°, 0.06°), and that of Geo-SAR is (−0.12°, 0.12°). The yaw angle scale of GF-3 in one period is (−3.9°, 3.9°), and that of Geo-SAR is (−76.1°, 76.1°).

2.2. 2D-PRSC Technology for Squint Staring Mode

2.2.1. Large Instantaneous Field of View for Geo-SAR

As described in [7,21], the angle between the beam center and zero Doppler plane is defined as the squint angle (SA), and its projection on the ground is defined as the ground squint angle (GSA). Their relationship is illustrated as follows.

$${\theta }_s=\arcsin \left(\sin \gamma \sin {\theta }_{ds}\right)$$

(2)

where γ is the look angle, θ_s is the SA, and θ_ds is the GSA. Supposing that the GSA of GF-3 and Geo-SAR are both 60 degrees and their IAs are both 45.5 degrees, their SA in one orbit period are compared, as shown in Figure 3. It can be seen that Geo-SAR has the characteristics of large GSA and small SA due to its orbital height. This means that Geo-SAR can cover a wide range of ground areas through small attitude-controlling angles.

As depicted in Figure 4, Geo-SAR’s instantaneous field of view can exceed about 5000 km × 5000 km in this condition. As a result, the satellite’s revisit time can be greatly shortened, and its maneuvering imaging capability can be significantly improved if the squint mode is used.

2.2.2. 2D-PRSC Theory Illumination

We can obtain the Geo-SAR staring mode attitude-controlling angle calculating formulas for a scene of any longitude and latitude. It is named as the 2D-PRSCmethod, and its steps are presented as follows.

(a)

Calculating the original GSA.

(b)

Calculating the target direction angle. It is defined as the angle between the meridian and the line between the nadir and scene center.

(c)

Pitch and roll angle’s expressions are derived. The satellite can stare at a fixed scene by controlling these two angles.

(d)

Whether the current orbital segment meets the imaging conditions or not is confirmed according to the GSA and incidence angle (IA) threshold constraints.

The geometric model is shown in Figure 5.

The spherical geometry model based on a satellite local coordinate system (SLCS), depicted as O_sXYZ, is shown in Figure 5a. O_s is the satellite centroid, the X-axis points to the flight tangent direction, the Z-axis points to the earth’s core depicted by O_e, and the Y-axis obeys the Cartesian right hand rule. N is nadir, O_sD₁ is the beam direction in the zero Doppler plane when the squint angle is zero, D₁ is the intersection of beam center and earth surface, and γ₁ is the look angle. D₁₀ means the intersection of O_sD₁ and the tangent plane of the earth’s surface. O_sD₂ is the beam direction in no attitude steering condition, D₂ is the intersection of the beam center and the earth’s surface, and γ₂ is the look angle. D₂₀ means the intersections of O_sD₂ and the tangent plane of the earth’s surface. R₀₁ is the range between O_s and D₁₀, R₀₂ is the range between O_s and D₂₀. In order to simplify the analysis process, this spherical geometry model can be transformed into plane solid geometry by projection, as shown in Figure 5b.

In Figure 5b, O_sND₁₀ is the zero Doppler plane, O_sND₂₀ is the plane with the beam center without attitude steering, <D₁₀ND₂₀ is the GSA, and <D₁₀O_sD₂₀ is the squint angle. <D₁₀ND₂₀ and <D₁₀O_sD₂₀ change periodically if the satellite is in the state of no-attitude steering. H is the satellite height. θ_yj and θ_pj, as simulated in Figure 1, respectively, represent the yaw and pitch angle offset zero Doppler plane in the state of no attitude steering. θ_dj is the original GSA, which consists of θ_ydj from θ_yj and θ_pdj from θ_pj. P represents the range between N and S₁₀, T represents the range between N and D₁₀, and G represents the range between D₁₀ and S₁₀. To simplify the analysis process, this paper assumes that the beam center direction is consistent with the Z-axis of SLCS, and they both point to the earth core in the state of no attitude steering.

A satellite adjusts its beam direction on the zero Doppler plane through matrix A_ypj. Furthermore, its beam can point to the predetermined target by A_kj.

$$A_{\mathrm{ypj}}=\left[\begin{array}{ccc}\cos {\theta }_{pj}& \sin {\theta }_{pj}& 0\\ -\sin {\theta }_{{}_{pj}}& \cos {\theta }_{{}_{pj}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\theta }_{yj}& \sin {\theta }_{yj}\\ 0& -\sin {\theta }_{yj}& \cos {\theta }_{yj}\end{array}\right]$$

(3)

$$A_{\mathrm{kj}}=\left[\begin{array}{ccc}\cos {\theta }_{rj}& 0& -\sin {\theta }_{rj}\\ 0& 1& 0\\ \sin {\theta }_{rj}& 0& \cos {\theta }_{rj}\end{array}\right]\cdot A_{\mathrm{ypj}}$$

(4)

where θ_rj is the roll angle, which is equal to the look angle, and θ_pj and θ_yj are the original pitch and yaw angle. Expressions about these two angles are presented in [37]. The look angle is assumed to be consistent before and after attitude steering.

$$R_{01}=H/\cos {\gamma }_1$$

(5)

The original GSA θ_dsj can be calculated as:

$${\theta }_{dsj}=u\left({\theta }_{ydj}+{\theta }_{pdj}\right)$$

(6)

$$\{\begin{array}{l}{\theta }_{ydj}={\theta }_{yj}\\ {\theta }_{pdj}={\sin }^{-1}\left(H\tan {\theta }_{pj}/R_{01}\sin {\gamma }_1\right)\end{array}$$

(7)

where u = 1 means right look, and u = −1 means left look. Furthermore, we can get:

$$\{\begin{array}{l}{\theta }_{p0}={\tan }^{-1}\left(R_{01}\sin {\gamma }_1\sin {\theta }_{dsj}/H\right)\\ {\theta }_{r0}={\tan }^{-1}\left(\frac{R_{01}\sin {\gamma }_1\cos {\theta }_{dsj}}{\sqrt{H^2+{\left(R_{01}\sin {\gamma }_1\sin {\theta }_{dsj}\right)}^2}}\right)\\ {\theta }_{y0}=0\end{array}$$

(8)

where θ_p0, θ_r0, and θ_y0, respectively, indicate the pitch, roll, and yaw angles to be compensated. It can be seen that we can realize zero Doppler centroid by steering pitch and roll angles without any controlling on the yaw angle. Its attitude steering matrix can be written as:

$$A_{\mathrm{k}0}=\left[\begin{array}{ccc}\cos {\theta }_{r0}& 0& -\sin {\theta }_{r0}\\ 0& 1& 0\\ \sin {\theta }_{r0}& 0& \cos {\theta }_{r0}\end{array}\right]\left[\begin{array}{ccc}\cos {\theta }_{p0}& \sin {\theta }_{p0}& 0\\ -\sin {\theta }_{p0}& \cos {\theta }_{p0}& 0\\ 0& 0& 1\end{array}\right]$$

(9)

The 2D-PRSCtheory can be further analyzed through a geometric model in an earth-centered inertial (ECI) coordinate, as shown in Figure 6.

Where O_e is the earth’s core, N is nadir, R_ne is the earth radius at nadir, T is the scene center, NC represents meridians, C and T have the same latitude, and NB represents the intersection of the zero Doppler plane and the earth’s surface. NS represents the beam center projection on the ground without attitude controlling. NT_r1 and NT_r0, respectively, represent the nadir track with and without the earth’s rotation. The deviation angle between NT_r0 and NC is α₁, and that angle between NT_r1 and NC is α₂. They are presented as follows.

$${\alpha }_1=\arccos \left[\tan \left({\delta }_s\right)\cot \left({\theta }_{na}\right)\right]$$

(10)

$${\alpha }_2=\arctan \left[\frac{\left(H{V}_s\sin {\alpha }_1\right)/R_s-{\omega }_e{R}_{ne}\cos {\delta }_s}{\left(H{V}_s\cos {\alpha }_1\right)/R_s}\right]$$

(11)

Furthermore, the deviation angle between NB and NC, expressed by α_b, is written as:

$$\begin{array}{cc}{\alpha }_b={\alpha }_2+u\pi /2,& u=\pm 1\end{array}$$

(12)

where θ_na is the latitude argument, δ_s is the nadir’s latitude, V_s is the satellite’s velocity in ECI, and R_s is the range between the satellite and the earth core. ω_e (ω_e = 7.292115 $\times$ 10⁻⁵ rad/s) is the earth rotation angular velocity [21]. According to the Kepler orbit theory:

$$R_{ne}=\sqrt{\frac{R_e^2{R}_p^2}{{\left[R_p\cos {\delta }_s\right]}^2+{\left[R_e\sin {\delta }_s\right]}^2}}$$

(13)

$$R_s=\left[a\left(1-e^2\right)\right]/\left(1+e\cos f\right)$$

(14)

$$V_s=\sqrt{{\left[\frac{R_s^{2}e\sin f}{a\left(1-e^2\right)}{\omega }_s\right]}^2+\frac{\mu a\left(1-e^2\right)}{R_s^2}}$$

(15)

where e is the orbital eccentricity, f is the true anomaly, a is the semi-major axis, μ (μ = 3.98696 $\times$ 10¹⁴ m³/s²) is the constant of earth’s gravitation, R_e and R_p (R_e = 6378.137 km, R_p = 6356.752 km), respectively, represent the equatorial semi-axis and polar semi-axis of ellipsoid earth [21].The original GSA, θ_dsj, has been written in (6), and the deviation angle between NS and NC, expressed by α_s, is written as:

$${\alpha }_s={\alpha }_b-{\theta }_{dsj}$$

(16)

Furthermore, the deviation angle between NT and NC, expressed by α₃, is written as:

$${\alpha }_3=\mathsf{\Psi }-\mathsf{\Theta }$$

(17)

where

$$\mathsf{\Psi }=\arccos \left\{\tan {\delta }_s\cot \left[\arcsin \left(\frac{\sin {\delta }_s}{\sin {\varphi }_{im}}\right)\right]\right\}$$

(18)

$$\mathsf{\Theta }=\arctan \chi -\arctan \varsigma$$

(19)

$$\mathsf{\Omega }=\arctan \chi +\arctan \varsigma$$

(20)

where (18) can be derived from (21), and (19) and (20) can be derived from (22):

$$\{\begin{array}{l}{\varphi }_{im}=\arccos \left(D_{im1}/D_{im0}\right)\\ D_{im1}=R_{ne}\cos {\delta }_s\cos \left[\left(\left|{\xi }_s-{\xi }_t\right|\right)/2\right]\\ D_{im0}=\sqrt{{\left(R_{ne}\sin {\delta }_s\right)}^2+D_{{}_{im1}}^2}\end{array}$$

(21)

$$\{\begin{array}{l}\chi =\frac{\cos \left[\left(H_t-H_n\right)/2\right]}{\cos \left[\left(H_t+H_n\right)/2\right]}\cot \frac{\mathsf{\Psi }}{2}\\ \varsigma =\frac{\sin \left[\left(H_t-H_n\right)/2\right]}{\sin \left[\left(H_t+H_n\right)/2\right]}\cot \frac{\mathsf{\Psi }}{2}\\ H_t=2\arcsin \left[\cos {\delta }_s\sin \left(\left|{\xi }_s-{\xi }_t\right|/2\right)\right]\\ H_n=\left|{\delta }_s-{\delta }_t\right|\end{array}$$

(22)

where ξ_s and ξ_t represent the longitude of the nadir and scene center, and δ_t is the scene center latitude. H_t is the arc length of the great circle between N and C and H_n is the arc length of the great circle between T and C. H_c is the arc length of the great circle between T and N, and it is equal to geocentric angle α_t. Its corresponding look angle is γ_t.

$${\alpha }_t=2\arctan \left\{\frac{\cos \left[\left(\mathsf{\Omega }+\mathsf{\Theta }\right)/2\right]\tan \left[\left(H_t+H_n\right)/2\right]}{\cos \left[\left(\mathsf{\Omega }-\mathsf{\Theta }\right)/2\right]}\right\}$$

(23)

$${\gamma }_t=\arcsin \left[\left(R_{te}\sin {\alpha }_t\right)/R_{st0}\right]$$

(24)

where R_te is the earth’s radius at the scene center, R_st0 is the range between the satellite and scene center. They can be presented as:

$$R_{te}=\sqrt{\frac{R_e^2{R}_p^2}{{\left(R_p\cos {\delta }_t\right)}^2+{\left(R_e\sin {\delta }_t\right)}^2}}$$

(25)

$$R_{st0}=\sqrt{R_s^2+R_{te}^2-2R_s{R}_{te}\cos {\alpha }_t}$$

(26)

Based on (6), (12), (17), (24), and (26), the expressions of pitch and roll angles need to be controlled, which are derived as follows.

$$\{\begin{array}{l}{\theta }_{pt0}=\arctan \left[R_{st0}\sin {\gamma }_t\sin \left({\theta }_{dsj}+{\alpha }_3-{\alpha }_b\right)/H\right]\\ {\theta }_{rt0}=\arctan \left(\frac{R_{st0}\sin {\gamma }_t\cos {\theta }_{dsj}}{\sqrt{H^2+{\left(R_{st0}\sin {\gamma }_t\sin {\theta }_{dsj}\right)}^2}}\right)\\ {\theta }_{yt0}=0\end{array}$$

(27)

where θ_pt0, θ_rt0, and θ_yt0, respectively, represent the pitch, roll, and yaw angles, which need to be controlled. According to (27), the attitude controlling matrix A_kt0 can be expressed as follows.

$$A_{\mathrm{kt}0}=\left[\begin{array}{ccc}\cos {\theta }_{rt0}& 0& -\sin {\theta }_{rt0}\\ 0& 1& 0\\ \sin {\theta }_{rt0}& 0& \cos {\theta }_{rt0}\end{array}\right]\left[\begin{array}{ccc}\cos {\theta }_{pt0}& \sin {\theta }_{pt0}& 0\\ -\sin {\theta }_{pt0}& \cos {\theta }_{pt0}& 0\\ 0& 0& 1\end{array}\right]$$

(28)

Using (28), a satellite’s beam can always point to a scene center through the pitch and roll attitude controlling from the original state. Furthermore, according to (12), (17), (23), and (24), the corresponding expressions of IA(θ_int0) and GSA(θ_dt0) are presented as follows.

$${\theta }_{int0}={\gamma }_t+{\alpha }_{\mathrm{t}}$$

(29)

$${\theta }_{dt0}={\alpha }_b-{\alpha }_3$$

(30)

Finally, when a satellite keeps staring at some scene of any latitude and longitude, considering the threshold constraints of GSA and IA, the imageable orbit segments in one orbital period can be calculated.

2.2.3. Simulation

This manuscript supposes that the constraint scale of GSA is (−60°, 60°), and that of IA is (18°, 60°). Taking Beijing (116.41°E, 39.90°N) and Hong Kong (114.10°E, 22.39°N) as examples, the simulation results are shown in Figure 7. Five subgraphs, in Figure 7a,b, from top to bottom separately show the variation curves of roll angle(RA), pitch angle(PA), IA, GSA, and Doppler central frequency(FR) in one orbital period. The blue solid line indicates the imageable track segment. The red dotted line indicates that the beam points to the scene center; however, these orbit segments’ parameters do not meet the constraints of IA or GSA, so the satellite cannot produce an image.

In fact, both pitch and roll attitude controlling scale will not exceed ±7.6 degrees in the whole orbit if the 2D-PRSC method is employed for Geo-SAR to stare at any ground scene. This indicates that the satellite can promptly complete the attitude controlling and achieve squint staring. The obtained result from the two cases above indicates that the Geo-SAR satellite has a great advantage of revisiting. In addition, Figure 7 shows that the absolute Doppler frequency maybe higher than the PRF, and thus, the data can be aliased. As a result, this should be considered for system parameter designing.

3. Staring Imaging Mode

3.1. Geo-SAR Staring Mode Model and Characteristics

The Leo-SAR geometric model is depicted in Figure 8a. T_s, T₀, and T_e, respectively, represent the start time, middle time, and end time of imaging. W_r is the range swath width, and W_a is the azimuth imaging scale. The staring and stripmap modes are marked by different subscripts. The stripmap mode is marked by a black line, and the staring mode is marked by a blue line. The rotation center is marked by a red dot. In addition, the satellite trajectory can be approximated as a straight line, and the earth surface can be approximated as a plane due to its short integration time. Leo-SAR’s different modes can be realized by only changing the beam’s azimuth direction through the attitude controlling or antenna electrical scanning. Its azimuth resolution is determined by the beam scanning scale and rate [21,22,23,32,33].

Similarly, Figure 8b shows the Geo-SAR geometric model of different imaging modes. The curvature of orbital trajectory and earth surface cannot be ignored due to the orbital height, long integration time, and large beam coverage. The staring mode differences between Geo-SAR and Leo-SAR are illuminated as follows. All the simulation plots in this section correspond to the parameters presented in Table 1.

◼

Geo-SAR’s stripmap mode is similar to Leo-SAR’s slide spotlight mode. Since Geo-SAR satellite’s speed is about seven times its beam’s moving speed, the rotation center is not at infinity, as Leo-SAR is in the stripmap mode condition. However, both of their rotation centers are in the scene center in the staring mode condition, as shown in Figure 8.

◼

Geo-SAR’s azimuth imaging scale is mainly determined by its beam coverage. The staring mode is recommended to select, and the azimuth imaging scale is always equal to the beam coverage during its orbital period.

◼

Leo-SAR just needs one-dimensional azimuth beam direction controlling; however, Geo-SAR needs a more complicated two-dimensional beam direction controlling algorithm to realize the staring mode. The theory in Section 2 has illustrated the 2D-PRSC method for squint staring (broadside is a special case of squint).

◼

The Geo-SAR Doppler FM rate, positive or negative, always changes during the whole orbit. Its staring mode Doppler history is also strongly time-varying. The Leo-SAR Doppler FM rate, almost unchanged, is negative in the whole orbit, and its Doppler history hardly changes. Figure 9a compares the Doppler FM rate of Geo-SAR and GF-3 in the same IA of 45.5 degrees. The Doppler FM rate of Geo-SAR is (−0.11, 0.1) Hz/s, and that of GF-3 is (−1680, −1618) Hz/s. Furthermore, considering the orbital symmetry, a latitude argument of 23.4 degrees and 75.7 degrees, marked by P₁ and P₂, are selected to compare the two satellites’ staring mode Doppler history. Figure 9b–f show the simulation results. The Doppler history of the starting, middle, and terminal point targets of the staring area are, respectively, represented by Ta-S, Ta-M, and Ta-E, which are marked by different colors and lines. Unlike GF-3, Geo-SAR Doppler history has a strong time-varying property, as listed in

Table 2. Doppler history comparison between GF-3 and Geo-SAR.

Name	GF-3		Geo-SAR
Latitude argument (°)	23.4	75.7	23.4	75.7
Azimuth resolution (m)	1	1	25	25
Integration time (s)	0.57	0.58	115.8	311.2
Doppler bandwidth (Hz)	1075.2	1043.5	46.1	400.7

. In addition, the Doppler frequency interval between Ta-S, Ta-M, and Ta-E of Geo-SAR is not uniform, and it changes obviously with the latitude argument. This is also quite different from Leo-SAR.

In summary, the Geo-SAR staring mode is selected to improve the satellite’s flexible maneuver imaging capability and to obtain a constant azimuth imaging scale.

3.2. Antenna and Power

Large antenna and large radiation power are necessary to achieve the required 25 m resolution and −18 dB NESZ for Geo-SAR.

3.2.1. Antenna Selection

The reflector antenna is smaller when gathered together and is lighter than a phased-array antenna. Considering the weight and volume constraints of the rocket and satellite, the reflector antenna is a better choice for Geo-SAR. Therefore, all the analyses in this paper are based on the reflector antenna.

3.2.2. Antenna Shape and Area

CAST’s Geo-SAR satellite adopts the L-band, and its expected range swath width is no less than 320 km in the IA scale of (18°, 60°). So, its elevation antenna size should not exceed 25 m. Its azimuth antenna size should be about 25 m to match the range swath width in the staring mode. Furthermore, affected by the periodic yaw angle, the beam foot-print direction angle (BFPA) on the ground also has a periodic direction angle if the satellite adopts a 2D-PRSC algorithm. The BFPA will be illustrated in Section 4. For the reasons above, a circular reflector antenna of 25 m diameter is chosen to ensure the same imaging range in all directions during the whole orbital period.

3.2.3. NESZ

To cover the required IA range, thirteen swaths of 320 km are designed for Geo-SAR. Considering the ambiguity and other constraints, each swath’s PRF should be selected in the range of (160, 430) Hz. The satellite has a peak radiation power of 20,000 W, pulse width of 400 us, signal width of 50 MHz, and resolution of 25 m × 25 m.

The system’s NESZ will change with the latitude argument due to the satellite time-varying velocity. The NESZ simulation results in one orbit period and the whole IA range are shown in Figure 10. The system losses are assumed as 3 dB in this part.

Therefore, an L band Geo-SAR satellite adopts a circular reflector antenna with a diameter of 25 m. Its resolution is 25 m × 25 m, and the peak radiation power is 20,000 W. The system’s worst NESZ will be better than −18.4 dB.

4. Ambiguity Analysis

4.1. Geo-SAR Staring Mode Ambiguity Model

ASR is an important parameter for spaceborne SAR. Its expression can be written as follows [41,42].

$$ASR\left(\tau ,f_d\right)=\frac{{\displaystyle \sum _{m,n=-\infty }^{\infty }{\displaystyle {\int }_{-B_p/2}^{B_p/2}S\left(f_d+mPRF,\tau +n/PRF\right)d{f}_d-{\displaystyle {\int }_{-B_p/2}^{B_p/2}S\left(f_d,\tau \right)d{f}_d}}}}{{\displaystyle {\int }_{-B_p/2}^{B_p/2}S\left(f_d,\tau \right)d{f}_d}}$$

(31)

$$S\left(f_d,\tau \right)=\epsilon \frac{G^2\left(f_d,\tau \right){\sigma }_0\left(f_d,\tau \right){\rho }_a{\rho }_{gr}}{R_{st}^4\left(f_d,\tau \right)\sin {\theta }_{int}}$$

(32)

where B_p is the Doppler bandwidth, G²(f_d,τ) is the dual-way far field antenna pattern, σ₀(f_d,τ) is the backscattering coefficient, ρ_a is the azimuth resolution, ρ_gr is the ground range resolution, R_st(f_d,τ) is the range between the signal or ambiguity point and radar, f_d is the Doppler frequency, τ is the echo delay time, PRF is the pulse repetition frequency, m is the ambiguity number in the Doppler frequency domain, and n is that in the time domain. The ambiguities are divided into three types [41].

◼

RASR: those with the same Doppler frequency as signal (i.e., m = 0 and n ≠ 0 in (31));

◼

AASR: those along the same slant range as signal (i.e., m ≠ 0 and n = 0 in (31));

◼

CASR: the remaining terms (i.e., m ≠ 0 and n ≠ 0 in (31)).

Neither the satellite trajectory nor earth surface curvature can be ignored. Figure 11 shows the signal and ambiguity point distribution on the earth’s surface. In Figure 11, the signal, azimuth ambiguity, range ambiguity, and cross-term ambiguity are, respectively, marked by a blue solid point, green hollow point, red hollow point, and purple hollow point. θ_ds is the GSA.

It can be seen that the ambiguity position and its corresponding calculation accuracy of the antenna pattern gain determine the ASR calculation accuracy. The angle between the profile of a one-dimensional antenna pattern in the elevation direction and the zero Doppler plane, represented by θ_BFPA, is defined as BFPA. Figure 12 compares its periodic variation with the latitude argument between GF-3 and Geo-SAR.

Leo-SAR, such as GF-3, uses 2D-YS technology, so θ_{BFPA =} θ_{ds j} − θ_dyj. Generally, Leo-SAR adopts near-circular orbit, that is, its orbital eccentricity is less than 3‰, so θ_BFPA ≈ 0. It shows that the profile of a one-dimensional antenna pattern in the elevation direction is perpendicular to the satellite’s track, and the profile of a one-dimensional antenna pattern in the azimuth direction is parallel to the satellite’s track. The range or azimuth ambiguity point will just be located on the elevation or azimuth profile of the antenna pattern. Therefore, RASR and AASR can be calculated by using the corresponding one-dimensional antenna pattern, respectively. In addition, CASR could be ignored [42].

Geo-SAR uses 2D-PRSC technology, so θ_BFP = θ_dsj. This indicates that the ambiguity properties will be quite complicated and different from Leo-SAR. It is necessary to employ the geometry model depicted in Figure 11. Furthermore, a two-dimensional antenna pattern and its distribution variety should be considered for ASR calculation.

This paper uses an ideal sinc function to characterize the antenna pattern. The dual-way two-dimensional antenna pattern is shown as follows.

$$G_{ea}={\sin \mathrm{c}}^2\left[\frac{D_r}{\lambda }\sin \left({\xi }_e-{\xi }_{e0}\right)\right]{\sin \mathrm{c}}^2\left[\frac{D_a}{\lambda }\cos \left({\xi }_e-{\xi }_{e0}\right)\sin \left({\zeta }_a-{\zeta }_{a0}\right)\right]$$

(33)

where D_r and D_a, respectively, indicate the elevation and azimuth antenna size. The antenna pattern elevation and azimuth angles, represented by ξ_e0 and ζ_a0, are calculated based on the reference line from the satellite to the earth center.

$$\{\begin{array}{l}{\xi }_{e0}=\arcsin \left[\sin {\gamma }_t\cos \left({\theta }_{dt0}+{\theta }_{BFP}\right)\right]\\ {\zeta }_{a0}=\arctan \left[\tan {\gamma }_t\sin \left({\theta }_{dt0}+{\theta }_{BFP}\right)\right]\end{array}$$

(34)

where γ_t is the beam center look angle shown in (24), and θ_dt0 is the GSA. It is zero in the broadside condition; otherwise, it is as shown in (30) in the squint staring condition. Compared with Leo-SAR, the ASR of Geo-SAR has several different characteristics:

◼

It is necessary to use a two-dimensional antenna pattern to calculate ASR; otherwise, the error will be very large;

◼

The ASR of Geo-SAR is obtained in 2D-PRSC conditions. This calculation result is quite different from that in 2D-YS conditions.

◼

Due to the influence of the large BFPA, the CASR of Geo-SAR cannot be ignored for system designing;

◼

Geo-SAR usually works in the squint condition for emergency missions. So, it is necessary to consider both GSA and BFPA to obtain the ambiguity point location. This further increases the ASR calculation complexity.

◼

RASR, AASR, and CASR of Geo-SAR are all strongly time-varying. As a result, whole orbital ergodic analysis is necessary in system designing process.

4.2. RASR

Similar to (34), the elevation angle and azimuth angle of the target signal or range ambiguity point can be expressed as

$$\{\begin{array}{l}{\xi }_e=\arcsin \left[\sin \gamma \cos \left({\theta }_{dt0}+{\theta }_{BFP}\right)\right]\\ {\zeta }_a=\arctan \left[\tan \gamma \sin \left({\theta }_{dt0}+{\theta }_{BFP}\right)\right]\end{array}$$

(35)

where γ is the look angle of the target signal or range ambiguity point.

$$\gamma =\arcsin \left(R_{te}\sin \alpha /R_{st}\right)$$

(36)

where R_te is the earth radius at the scene center as expressed in (25), and R_st is the range between the satellite and target signal or range ambiguity point.

$$R_{st}=\sqrt{R_s^2+R_{te}^2-2R_s{R}_{te}\cos \alpha }+n\left(c/2PRF\right)$$

(37)

where R_s, expressed in (14), is the range between the satellite and the earth core; n ∈ (−N,N). When the parameter n equals to zero, it means target signal; otherwise, it is the range ambiguity signal. The corresponding geocentric angle is represented by α.

$$\alpha =\arcsin \left(R_{st}\sin {\gamma }_t/R_{te}\right)-{\gamma }_t+q\left[W_r/\left(2R_{te}\right)\right],q\in \left(-1,1\right)$$

(38)

where W_r is the range swath width. The variation of backscattering coefficient is ignored. RASR could be presented as follows.

$$RASR=\frac{G_{ea\_Abm}^2/\left(R_{st\_Abm}^4\sin {\theta }_{in\_Abm}\right)}{G_{ea}^2/\left(R_{st}^4\sin {\theta }_{in}\right)}$$

(39)

where θ_in is the incidence angle.

4.3. DT-AASR

Considering Geo-SAR’s large coverage field of view, azimuth ambiguity points are distributed on a circle of equal squint range, as shown in Figure 11. Furthermore, it is necessary to calculate the azimuth Distributed Targets AASR (DT-AASR) for the staring mode due to the antenna pattern gain variety [26,32]. The total pulse number during the integration time is k_p (k_p = PRF × T_sar). T_sar is the integration time. Similar to (34) and (35), the elevation angle and azimuth angle of the azimuth distributed targets or corresponding azimuth ambiguity point can be expressed as follows.

$$\{\begin{array}{l}{\xi }_e\left(k,m\right)=\arcsin \left\{\sin \gamma \left(k\right)\cos \left[{\theta }_{dt0}\left(k\right)+{\theta }_{amb}\left(m,k\right)+{\theta }_{BFPA}\left(k\right)\right]\right\}\\ {\zeta }_a\left(k,m\right)=\arctan \left\{\tan \gamma \left(k\right)\sin \left[{\theta }_{dt0}\left(k\right)+{\theta }_{amb}\left(m,k\right)+{\theta }_{BFPA}\left(k\right)\right]\right\}\end{array}$$

(40)

where k is the pulse number (k ∈ (1, k_p)). m ∈ (−M, M). When the parameter m equals to zero, it means target signal; otherwise, it is the azimuth ambiguity signal. γ(k)is the look angle of the distributed targets or the azimuth ambiguity points. Since they are distributed on a circle with an equal squint range, γ(k)is a fixed value (γ(k) = γ_t). θ_dt0(k) means the GSA. It is almost constant during the integration time, so it equals to θ_dt0 as expressed in (30). θ_BFPA(k) is almost constant during the integration time, so it could be written as θ_BFPA. θ_amb(m,k) represents the angle between the target or azimuth ambiguity point and beam center ground projection, and it can be presented as:

$${\theta }_{amb}\left(m,k\right)=\arcsin \left\{\sin {\gamma }_t\sin \left[\arcsin \left(\frac{\lambda \left(f_{dc0}+mPRF+B_{ta}\right)}{2V_s\left(k\right)}\right)\right]\right\}$$

(41)

where B_ta is the Doppler frequency offset of the azimuth distribution target and the corresponding ambiguity (B_ta ∈ (−B_a/2, B_a/2)), subscript ta represents the azimuth distributed target, and B_a is the Doppler bandwidth. V_s(k) means the satellite velocity. It is almost constant during the integration time, so it equals to V_s, as expressed in (15). θ_amb(m,k) represents the signal angle when m equals to zero; otherwise, it is the azimuth ambiguity angle. Equation (40) could be rewritten as:

$$\{\begin{array}{l}{\xi }_e\left(k,m\right)=\arcsin \left\{\sin {\gamma }_t\cos \left[{\theta }_{dt0}+{\theta }_{amb}\left(m\right)+{\theta }_{BFPA}\right]\right\}\\ {\zeta }_a\left(k,m\right)=\arctan \left\{\tan {\gamma }_t\sin \left[{\theta }_{dt0}+{\theta }_{amb}\left(m\right)+{\theta }_{BFPA}\right]\right\}\end{array}$$

(42)

Finally, according to (31), (32), (33), (41), and (42), the DT-AASR for the Geo-SAR staring mode is expressed as follows.

$$AAS{R}_{ta}=\frac{{\displaystyle \sum _{k=1}^{k_p}{\displaystyle \sum _{m=\pm 1}^{\pm M}{G}_{ea}^2\left[{\xi }_e\left(k,m\right),{\zeta }_a\left(k,m\right)\right]}}}{{\displaystyle \sum _{\begin{array}{l}k=1\\ m=0\end{array}}^{k_p}{G}_{ea}^2\left[{\xi }_e\left(k,m\right),{\zeta }_a\left(k,m\right)\right]}}$$

(43)

4.4. DT-CASR

Similar to (42), the elevation angle and azimuth angle of the azimuth distributed targets or corresponding cross-term ambiguity points can be expressed as follows.

$$\{\begin{array}{l}{\xi }_e\left(k,m,n\right)=\arcsin \left\{\sin \gamma \left(n\right)\cos \left[{\theta }_{dt0}+{\theta }_{amb}\left(m\right)+{\theta }_{BFPA}\right]\right\}\\ {\zeta }_a\left(k,m,n\right)=\arctan \left\{\tan \gamma \left(n\right)\sin \left[{\theta }_{dt0}+{\theta }_{amb}\left(m\right)+{\theta }_{BFPA}\right]\right\}\end{array}$$

(44)

where γ(n) is the look angle of the azimuth distributed targets or the corresponding cross-term ambiguity points on different circles, as shown in Figure 11. Finally, according to (31), (32), (33), (36), (37), (41), and (44), the DT-CASR for the Geo-SAR staring mode could be advanced as follows.

$$CAS{R}_{ta}=\frac{{\displaystyle \sum _{k=1}^{k_p}{\displaystyle \sum _{m=\pm 1}^{\pm M}{\displaystyle \sum _{n=\pm 1}^{\pm N}{G}_{ea}^2\left[{\xi }_e\left(k,m,n\right),{\zeta }_a\left(k,m,n\right)\right]/\left(R_{st\_Abm}^4\sin {\theta }_{in\_Abm}\right)}}}}{{\displaystyle \sum _{\begin{array}{l}\begin{array}{cc}& k=1\end{array}\\ m=0,n=0\end{array}}^{k_p}{G}_{ea}^2\left[{\xi }_e\left(k,m,n\right),{\zeta }_a\left(k,m,n\right)\right]/\left(R_{st}^4\sin {\theta }_{in}\right)}}$$

(45)

4.5. Simulation Results

The Geo-SAR staring mode ambiguity characteristics are analyzed in this section through simulation. The whole orbital ergodic has to be analyzed considering the time-varying property. One typical orbital position, a latitude of 23.4 degrees, is selected for analysis due to the limited length of the article. Furthermore, the effects of different attitude steering methods on ASR are compared. The simulation parameters are listed in

Table 3. Simulation parameters.

Parameter Name	Value	Parameter Name	Value
Semi-major axis (km)	42,163.5	Wave length (m)	0.24
Argument of perigee (°)	90	Inclination (°)	28
Range swath width (km)	≥320	Bandwidth (MHz)	50
Incidence angle (°)	(18, 60)	Eccentricity	0.001
Ground squint angle (°)	0/60	Earth model	WSG84
Azimuth resolution (m)	25	Antenna size (m)	25
Ground range resolution (m)	25	Latitude argument (°)	23.4

.

4.5.1. RASR

The RASR and latitude argument are not strongly related. Figure 13 simulates the RASR of thirteen beam positions in the incidence angle scale of (18°,60°). The broadside and squint condition are, respectively, depicted in Figure 13a,b. The simulation results based on 2D-PRSC are represented by a blue solid line, and those based on 2D-YS are represented by a red dotted line.

The simulation results in Figure 13 show the following:

◼

In broadside condition, RASR in the 2D-PRSC condition is about 2–45 dB better than that in the 2D-YS condition. The smaller the incidence angle is, the greater the improvement. Due to BFPA, not all the antenna elevation pattern gain is in the range of the ambiguity region. So, the range ambiguity signal is weakened.

◼

In the squint condition, the differences of RASR between the different attitude steering methods are not obvious. This is because that BFPA, which is about −72 degrees at a latitude argument of 23.4 degrees, counteracts most of the antenna elevation pattern gain variety caused by GSA.

4.5.2. DT-AASR

There isa strong correlation between the AASR and latitude argument. Figure 14 simulates that the scene center’s AASR varies with PRF for thirteen beam positions in the 2D-PRSC condition. Different linear colors indicate different beam center incidence (BCI) angles. The small image embedded in each figure shows the DT-AASR in the azimuth range of 320 km when the incidence angle equals 45.5 degrees.

Furthermore, the influences of different attitude steering methods on AASR are compared. We choose an incidence angle of 45.5 degrees for simulation. The results are depicted in Figure 15.

To facilitate comparison, the analysis conclusions of Figure 14 and Figure 15 will be illustrated in the next section.

4.5.3. DT-CASR

Similar to AASR, CASR is analyzed here. Figure 16 simulates that the scene center’s CASR varies with PRF for thirteen beam positions in the 2D-PRSC condition. Different linear colors indicate different BCI angles. The small image embedded in each figure shows the DT-CASR in the azimuth range of 320 km when the incidence angle equals 45.5 degrees.

Furthermore, the influences of different attitude steering methods on CASR are compared. We choose an incidence angle of 45.5 degrees for simulation again. The results are depicted in Figure 17.

The simulation results in Figure 14, Figure 15, Figure 16 and Figure 17 show the following:

◼

CASR has a comparable magnitude as AASR in the Geo-SAR condition, so it cannot be ignored. The constraints of RASR, AASR, and CASR should be considered simultaneously in the process of system designing.

◼

Due to the strong time-varying property, different GSA and latitude argument both have obvious effects on AASR and CASR. Therefore, the GSA should be calculated according to the longitude and latitude of the expected imaging zone and satellite’s position. Furthermore, it is recommended to recalculate and update the system parameters for every five-degree change in latitude argument.

◼

The DT-AASR and DT-CASR of the distributed targets in the azimuth range change significantly for the staring mode. In order to ensure that the DT-AASR and DT-CASR of all targets in the azimuth range of 320 km meet the requirement of −20 dB, it is necessary to select an appropriate PRF in the scale of (160, 430) Hz.

◼

Different from RASR, the influence of different attitude steering methods on AASR and CASR is not obvious.

5. Conclusions

The staring mode properties and system parameters’ performance of Geo-SAR with reflector antenna are presented in this study. A new 2D-PRSC method for the fixed-pointing staring mode is firstly proposed. The satellite can continuously stare at any scene in the whole orbital period, and its maximum steering angle is about ±7.6 degrees. According to the simulation results of Beijing and Hong Kong, it can be inferred that about 10 h for any region of China could meet the Geo-SAR imaging requirement of 24 h. Furthermore, the staring mode model and Doppler history of Geo-SAR are analyzed and compared with those of Leo-SAR. For an L-band Geo-SAR satellite with 25 m × 25 m resolution, a circular antenna with a diameter of 25 m and peak radiation power no less than 20,000 W are required, and its worst NESZ will exceed −18.4 dB in the whole orbital period. Finally, the Geo-SAR’s ASR calculation formulas are proposed, and the differences between Geo-SAR and Leo-SAR are compared. Since CASR, AASR, and RASR of Geo-SAR are almost in comparable magnitude, all of them should be considered in the system designing process, and the whole orbit ergodicity analysis should be carried out. Furthermore, it is necessary to select an appropriate PRF in the scale of (160, 430) Hz for the staring mode to ensure that the DT-AASR and DT-CASR of all the targets in the azimuth range of 320 km meet the requirement of −20 dB.