Resolution analysis for geostationary spaceborne-airborne bistatic forward-looking SAR

: Owing to the spatial separation of transmitter and receiver, bistatic SAR is able to image the area in front of the moving direction of transmitter or receiver, but the geometry model is more complex than that of monostatic SAR, which brings great difficulties to system design and imaging process. Besides, the special geometry configuration of geostationary spaceborne-airborne bistatic forward-looking SAR results in non-orthogonal and non-uniform ground resolution, which is not considered in the traditional range and azimuth resolution. To solve this problem, the resolution ellipse based on the generalised ambiguity function (GAF) is used to maintain the best and worst ground resolution in the resolution cell. The receiver's flight direction is designed to obtain the optimal uniform ground resolution distribution, which has great value for system design and the receiver's path planning. Finally, the accuracy of the resolution ellipse and correctness of the deduced formulas are verified by simulation.


Introduction
Geostationary spaceborne-airborne bistatic forward-looking SAR is a kind of bistatic SAR that place the transmitter on the geostationary satellite and the receiver on the airborne platform [1]. The transmitter located at the geostationary satellite is stationary relative to the ground and has the advantages of large coverage and strong security [2][3][4]. The receiver placed on the airborne platform has the advantage of flexibility. At the same time, bistatic SAR has the capability of forward-looking imaging because the transmitter and receiver are located at different platforms [5].
Although the spatial separation of transmitter and receiver brings a lot of advantages, it also makes it difficult to calculate the resolution of bistatic SAR. There are two main methods for bistatic SAR resolution calculation: the gradient method and the generalised ambiguity function. The gradient method proposed in [6] can be used to determine the range resolution, Doppler resolution, and the cross resolution. The generalised ambiguity function derived in [7] can be used to calculate the spatial resolution of arbitrary bistatic configuration, but the resolution in any other direction except range direction and azimuth direction need to be calculated by numerical method.
Based on these existing resolution calculation methods, the ground resolution for GEO-UAV bistatic SAR has been analysed and the path for UAV is also planned, and the unevenness of the earth's surface has also been considered [8,9]. However, the current analysis for bistatic SAR resolution are based on the range resolution and Doppler resolution, the resolutions in other direction are not considered. The path planning of the receiver is to achieve the orthogonality of the range resolution and azimuth resolution on ground, while it's impossible to achieve in geostationary spaceborne-airborne bistatic forward-looking SAR because the incident angle of receiver is often greater than transmitter in actual conditions.
To represent the distribution spatial resolution of GEO SAR, the resolution ellipse is adapted to approximate the resolution area in [10], and the semi-major axis and semi-minor axis of the ellipse are used to present the worst and best resolutions in the resolution area.
However, GEO SAR is a monostatic SAR, the geometry model and ambiguity function are quite different from that of bistatic SAR.
Thus, this article will derive the resolution ellipse for geostationary spaceborne-airborne bistatic forward-looking SAR and calculate the receiver's flight direction for optimal uniformity of ground resolution distribution. First, the generalised ambiguity function is used to determine the spatial resolution expression of geostationary spaceborne-airborne bistatic forward-looking SAR. Then, the resolution ellipse for geostationary spaceborne-airborne bistatic forward-looking SAR is obtained, and the uniformity of ground resolution distribution is characterised by the ratio of major and minor axes of the ellipse. The flight direction is found through maximising the ratio. Finally, the calculation formulas for spatial resolution, the flight direction, and the resolution ellipse are verified by simulation.

Geometry model
To calculate the ground resolution of geostationary spaceborneairborne bistatic forward-looking SAR, the geometry model needs to be established first. O is the origin of the scene coordinate system in Fig. 1, P is the target location, R is the location of the receiver, and T is the location of the transmitter. φ r and φ t are the incident angles of the receiver and transmitter, and φ r is assumed to be >φ t to comply with the actual situation. V r is the speed of the receiver and is supposed to have no component along Z-axis. β is the bistatic angle, and θ is the projection of β on the XOY plane. The angle θ is introduced to describe the flight direction of receiver and the relative position of transmitter and receiver on XOY plane. While the receiver platform is moving towards, β is always changing, but θ is constant.
According to the geometry model, the slant range of receiver and transmitter can be expressed as Eng where R t is the length of vector TP, R r is the length of vector RP.
The velocity vector of the receiver can be expressed as

Ground resolution
The generalised ambiguity function is given by where Φ TP and Φ RP are the unit vector of TP and RP, respectively.
Q is another point in a vicinity of P. p( ⋅ ) is the IFT of the signal power spectrum, and m A ( ⋅ ) is the IFT of the normalised received signal magnitude pattern and responsible for the Doppler resolution V r⊥ is the component of V r that is perpendicular to RP and can be expressed as and the vector P − Q can be written as Thus, combining (1)-(5), we can obtain where the coefficient k1 − k3 can be written as k 1 = cos 2 φ r k 2 = sin φ t sin θ k 3 = sin φ t cos θ + sin φ r ρ a and ρ r are the −4 dB resolution on the slant range plane where B is the signal bandwidth and T a is the integration time. The results shown in (8) are two times that of monostatic SAR, because the range resolution here refers to the resolution of total slant range, which equals to the sum of R r and R t , and the Doppler bandwidth is contributed by receiver only. From (6), the ground resolution in range direction and azimuth direction can be acquired The range resolution on ground is along the direction bisector of bistatic angle, the azimuth resolution on ground is along the direction of Y-axis. The angle between the range resolution and azimuth resolution on ground is given by It is obvious that the range of η depends on the relative size of φ r and φ t . However, in the case of geostationary spaceborne-airborne bistatic forward-looking SAR, the incident angle of receiver is usually greater than that of transmitter. That means tan η is finite and η cannot reach π/2, which is expected in the currently resolution design. Fig. 2 shows the relationship between η and θ, while φ t = 45°a nd φ r = 60°, the maximum value of η is obtained while θ = 145°. From (9) and Fig. 3, we can know that the greater θ is, the worse the range resolution on ground is. So how to choose between η and ρ gr is a difficult problem.

Resolution ellipse
To describe the non-uniformity of ground resolution area rather than range resolution of an azimuth resolution, the resolution ellipse is introduced. According to (6), the resolution area can be expressed as k 2 Δx + k 3 Δy ρ r /2 2 + k 1 Δy ρ a /2 2 = 1 (11) (11) is the equation of oblique elliptic equation, the semi-major axis and semi-minor axis of the ellipse is solved as The semi-major axis and semi-minor axis of the ellipse are the directions that the resolution area obtains the worst and best resolution. Thus, the ratio of B and A is used to describe the uniformity of resolution area, and the ratio is defined as where q is expressed as 0 < q = 2 k 1 k 2 / ρ a ρ r (k 2 2 + k 3 2 )/ ρ r 2 + k 1 2 / ρ a 2 ≤ 1 (14) K d describes the difference between different direction, The bigger K d is, the smaller the difference is, and when K d = 1, that means the ground resolution is same in any direction. However, it is possible for K d to get 1 only if k 3 = 0, but this is not satisfied under the assumption that φ r is >φ t . When q is maximum, K d gets the maximum value. Combining (7) and (8) (14) can be written as q = ρ a ρ r sin φ t cos 2 φ r sin θ ρ a 2 (sin 2 φ t + sin 2 φ r + 2sin φ t sin φ r cos θ) + ρ r 2 cos 4 φ r (15) and (15) maintain the maximum value when θ satisfies θ = cos −1 − 2ρ a 2 sin φ t sin φ r ρ a 2 (sin 2 φ t + sin 2 φ r ) + ρ r 2 cos 4 φ r At this time, the difference between different direction of the resolution area is smallest, and the ratio of resolution of any two directions is not smaller than K d . Equation (16) is the direction that obtains the best uniformity.

Simulation results
In order to verify the correctness of (16), the computer simulations were performed, simulation parameters are shown in Table 1. According to (16), the maximum value of K d is taken when θ is equal to 132.5° and the maximum value of K d is 0.4937. Fig. 4 shows the simulation results of BP imaging algorithm using the parameter in Table 1, and the spatial resolution in different direction are listed in Table 2. In Fig. 4, the angle between range resolution and azimuth resolution on ground is 53.46°, which is quite close to the 53.32° calculated by (10). The results in Table 2 show that the resolution calculated by the formulas above are close to the simulation results, and the errors may be introduced during the measurement of image resolution. The results verified the correctness of the above formulas. Fig. 5 shows the value of K d at different θ calculated by (13) and the solution of ambiguity function obtained by the numerical method. The maximum value of K d calculated by numerical method is 0.4929, when θ is 133 degrees. The results calculated by (13) are quite close to the theoretical value calculated by numerical method in most cases. When θ is close to zero, the resolution ellipse is no longer suitable for describing resolution distribution.

Conclusion
Due to the characteristics of non-orthogonal and non-uniform ground resolution, traditional range resolution and azimuth resolution are not suitable to describe the resolution distribution of the geostationary spaceborne-airborne bistatic forward-looking SAR, but the resolution ellipse can represent the non-uniformity of ground resolution in different direction. The optimal flight direction calculated based on optimal uniformity can always find, and the results are verified by computer simulation. This work has great value for bistatic forward-looking SAR system design and the receiver's path planning.