MEO satellite-borne bisar imaging method based on improved equivalent single base

Abstract

The application relates to a kind of MEO satellite machine BiSAR imaging methods based on improved equivalent single base, and relates to the field of microwave remote sensing imaging.The present application is to solve the problem of low precision and low efficiency of the existing satellite machine BiSAR imaging method.The present application comprises: establishing a non-stop MEO satellite machine BiSAR improved equivalent single base model based on the ground rectangular coordinate system;Using the non-stop MEO satellite machine BiSAR improved equivalent single base model to obtain the reflection echo of the MEO SAR transmitting linear frequency modulation signal, and then obtaining the closed expression of the echo signal two-dimensional spectrum;Using the compensation term in the closed expression of the echo signal two-dimensional spectrum to design a distance high-order compensation function, and correcting the distance error of the target;Using the compensation term in the closed expression of the echo signal two-dimensional spectrum to design a range-dependent compensation function, using the range-dependent compensation function and the echo corrected in step four to correct the phase error difference of the target at different azimuth positions, and obtaining a focused SAR image.The present application is used for satellite machine BiSAR imaging.

Description

Technical Field

[0001] This invention relates to the field of microwave remote sensing imaging, and in particular to a MEO satellite-to-aircraft BiSAR imaging method based on an improved equivalent single-base model. Background Technology

[0002] Bistatic Synthetic Aperture Radar (BiSAR), a novel radar system, employs a separate transmit and receive architecture on both satellite and airborne platforms. It boasts strong concealment and excellent anti-jamming capabilities, making it a cutting-edge technology in Earth observation. Medium Earth orbit (MEO) BiSAR systems combine the wide-area coverage of MEO SAR with the flexibility of airborne platforms, demonstrating significant application potential. Firstly, MEO SAR, as the illumination source, combines the high resolution of low-Earth orbit SAR with the wide coverage of high-Earth orbit SAR, enabling large-scale, rapid revisit Earth observation while maintaining good imaging performance. Secondly, the airborne platform only needs to passively receive signals, eliminating the need for a transmitter, significantly reducing system complexity and hardware costs. Furthermore, the system's passive receiving mode effectively enhances concealment and anti-jamming capabilities, making it invaluable.

[0003] However, MEO-based BiSAR imaging faces severe technical challenges. First, MEO orbits exhibit significant curvature and long signal propagation delays, rendering traditional slant range models based on the assumption of uniform linear motion ineffective. Furthermore, existing BiSAR imaging algorithms suffer from both insufficient modeling accuracy and low computational efficiency when dealing with MEO orbit curvature, long propagation delays, and dynamic geometric configurations, resulting in low accuracy and efficiency in BiSAR imaging. Summary of the Invention

[0004] The purpose of this invention is to address the problems of low accuracy and low efficiency in existing spacecraft BiSAR imaging methods, and to propose a MEO spacecraft BiSAR imaging method based on an improved equivalent single-base method.

[0005] A MEO satellite-based BiSAR imaging method based on an improved equivalent monobase includes the following steps:

[0006] Step 1: Establish a ground-based rectangular coordinate system and convert the satellite's velocity and acceleration in the geocentric rectangular coordinate system into the satellite's velocity and acceleration in the ground-based rectangular coordinate system;

[0007] Step 2: Establish an improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on the ground rectangular coordinate system;

[0008] Step 3: Use the improved equivalent single-base model of MEO satellite-based BiSAR to obtain the reflected echo of the MEO SAR transmitted linear frequency modulated signal, and use the reflected echo of the MEO SAR transmitted linear frequency modulated signal to obtain the closed expression of the two-dimensional spectrum of the echo signal.

[0009] Step 4: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal to correct the range error of the target and obtain the corrected echo signal.

[0010] Step 5: Design an azimuth spatial variation compensation function using the compensation term in the closed expression of the two-dimensional spectrum of the echo signal. Use the azimuth spatial variation compensation function and the echo signal corrected in Step 4 to perform phase error differential correction on targets at different azimuth positions to obtain a focused SAR image.

[0011] Furthermore, in step one, a ground-based rectangular coordinate system is established, and the satellite's velocity and acceleration in the geocentric rectangular coordinate system are converted into the satellite's velocity and acceleration in the ground-based rectangular coordinate system. Specifically:

[0012] Step 11: Obtain the target's coordinates in the geocentric rectangular coordinate system O-XYZ:

[0013]

[0014] Among them, R t (X es ,Y es Z es H1(t), H2, and H3(t) are the coordinates of the target in the geocentric rectangular coordinate system, and H1(t), H2, and H3(t) are intermediate variables. st It is the slant range vector from the satellite to the target; r(t) is the distance between the satellite and the Earth's center, t is the satellite time, and ω is the slant range vector from the satellite to the target. e ω is the Earth's rotational angular velocity, ω is the argument of perigee, and θ is the angle of rotation. f (t) is the true anterior angle of the medium-orbit synthetic aperture radar at time t, Ω is the right ascension of the ascending node, and i is the orbital inclination.

[0015] Steps 1 and 2: Establish a ground rectangular coordinate system PX g Y g Z g Obtain PX g Y g Z g The three-dimensional coordinate vector of the midpoint includes the following steps:

[0016] First, establish a ground rectangular coordinate system PX. g Y g Z g :

[0017] Take the center point of the beam transmitted from the transmitter to the receiver as the origin P, and Direction as Z g The axis direction, with point P along the surface direction as the Y-axis. g In the axial direction, according to Z g Axial direction and Y g The axis direction is obtained by following the right-hand rule. g direction;

[0018] Where O is the Earth's center of mass;

[0019] Then, obtain PX g Y g Z g The three-dimensional coordinate vector of the midpoint:

[0020]

[0021] Where (x,y,z) is PX g Y g Z g The three-dimensional coordinate vector of the midpoint, where M is the satellite position;

[0022] Finally, the three-dimensional coordinate vectors of each point in the geocentric rectangular coordinate system O-XYZ are plotted along PX. g Y g Z g Perform a projection transformation to obtain the satellite's velocity and acceleration in the ground rectangular coordinate system, specifically:

[0023]

[0024] Where M0 and M1 are intermediate matrices, V s It is the satellite's velocity in the geocentric rectangular coordinate system, A s It is the acceleration of the satellite in the geocentric rectangular coordinate system, V. t =[v tx ,v ty ,v tz ] is the satellite's velocity in a Cartesian coordinate system on the ground, A t =[a tx ,a ty ,a tz θ represents the satellite's acceleration in a Cartesian coordinate system on the ground. lo θ is the latitude of point P. la It is the longitude of point P, v tx ,v ty ,v tz They are V t In X g ,Y g Z g Components in direction, a tx ,aty ,a tz A respectively t It is in X g ,Y g Z g Components in direction.

[0025] Furthermore, the establishment of the improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on the ground rectangular coordinate system in step two is specifically as follows:

[0026] Step 2: 1. Calculate the sum of instantaneous distances from the transceiver platform to the target point using the satellite's velocity and acceleration in the ground Cartesian coordinate system;

[0027] Step 22: Utilize the sum of instantaneous distances R from the transceiver platform to the target point. sag (t a Obtain the propagation delay τ based on stop-and-go movement. sag Based on τ sag Obtain the actual bistatic instantaneous slant range of MEO satellite-mounted BiSAR;

[0028] Steps 2 and 3: Using the actual bistatic instantaneous slant range of MEO satellite-based BiSAR, the actual slant range of MEO satellite-based BiSAR is equivalent to monostatic SAR, thereby obtaining the parameters in the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation, and then constructing the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation.

[0029] Furthermore, the step 21, which involves obtaining the sum of instantaneous distances from the transceiver platform to the target point using the satellite's velocity and acceleration in a Cartesian coordinate system on the ground, specifically involves:

[0030] Step 2.11: Obtain the slant range equation R from the transmitter to any point target N. T (t a ):

[0031]

[0032] Among them, R t0 It is an intermediate variable, μ t1 μ t2 μ t3 It is the coefficient of time in each direction, k t1 k t2 k t3 It is an intermediate variable, t a For location and time, (x t ,y t ,z t (x) is the initial position of the transmitter. n ,y n ,z n() represents the coordinates of any target point;

[0033] Step 2.12: Obtain the instantaneous slant range R of the carrier aircraft. R (t a )as follows:

[0034]

[0035] Where, θ r It is the receiver's oblique angle, R r0 t is the slant range from the receiver's beam center to the target. a It is location and time, v r It is the receiver speed, (x) r ,y r ,z r () is the receiver position;

[0036] Steps 2-3: Utilize the instantaneous slant range R of the carrier aircraft R (t a ) and the slant range R from the transmitter to any point target N T (t a Obtain the sum of instantaneous distances from the transceiver platform to the target point:

[0037] R sag (t a ) = R T (t a )+R R (t a (17)

[0038] Among them, R sag (t a () is the sum of the instantaneous distances from the transceiver platform to the target point.

[0039] Furthermore, in step two, the sum of the instantaneous distances R from the transceiver platform to the target point is used. sag (t a Obtain the propagation delay τ based on stop-and-go movement. sag Based on τ sag The actual bistatic instantaneous slant range of the MEO satellite-mounted BiSAR is obtained as follows:

[0040] Step 221: Utilize the sum of instantaneous distances R from the transceiver platform to the target point. sag (t a Obtain the propagation delay τ based on stop-and-go movement. sag Specifically:

[0041] τ sag = R sag (t a ) / c (18)

[0042] Where c is the speed of light;

[0043] Step 2: Approximate the pulse propagation delay τ of the receiver slant range function as τ sag The approximate instantaneous slant range of the carrier aircraft is obtained as follows:

[0044]

[0045] Steps 2-3: Obtain the actual bistatic instantaneous slant range of the MEO satellite-to-aircraft BiSAR using the approximated instantaneous slant range, as follows:

[0046]

[0047] Furthermore, in steps two and three, the actual bistatic instantaneous slant range of the MEO satellite-to-aircraft BiSAR is used to convert the actual slant range of the MEO satellite-to-aircraft BiSAR into a monostatic SAR, thereby obtaining the parameters in the improved equivalent monostatic model of the MEO satellite-to-aircraft BiSAR based on non-stop operation, and then constructing the improved equivalent monostatic model of the MEO satellite-to-aircraft BiSAR based on non-stop operation, specifically as follows:

[0048] Step 231: Construct the equivalent slope distance expression and perform a third-order Taylor expansion on the equivalent slope distance expression:

[0049] First, obtain the equivalent slope expression:

[0050]

[0051] Among them, R M0 It is the equivalent slant range at the beam center time, v M It is the equivalent velocity, θ M It is the equivalent oblique angle, and β is the coefficient of the track curvature compensation term;

[0052] Then, a third-order Taylor expansion of the equivalent slant distance expression yields:

[0053]

[0054] Step 232: Perform a third-order Taylor expansion on the actual bistatic instantaneous slant range of the MEO satellite-based BiSAR, as follows:

[0055]

[0056] Where i is the index, K i It is an intermediate variable, n takes the values ​​0, 1, 2, ..., d n It is the sign of differentiation with respect to a variable. It is for t a Find the nth derivative;

[0057] Step 233: Equivalent the actual slant range of MEO satellite-based BiSAR to monostatic SAR, and formulate the following equations:

[0058] R(t a ) = 2R M (t a (25)

[0059] Steps 2, 3, and 4: Obtain the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation according to formula (25), and substitute the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation into formula (21) to obtain the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation.

[0060] The parameters in the improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation are obtained according to formula (25), specifically as follows:

[0061]

[0062] Furthermore, in step three, the reflected echo of the MEO SAR transmitted linear frequency modulated signal is obtained using the improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on non-stop MEO operation, and the closed-form expression of the two-dimensional spectrum of the echo signal is obtained using the reflected echo of the MEO SAR transmitted linear frequency modulated signal. Specifically:

[0063] Step 31: Using the improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on non-stop operation, obtain the reflected echo of the linear frequency modulated signal transmitted by MEO SAR, and perform range FFT on the reflected echo, specifically:

[0064] First, using an improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation, the reflected echo of the linear frequency modulated signal transmitted by MEO SAR is obtained:

[0065]

[0066] in, It is fast time, γ is the range-direction frequency modulation slope, ω r and ω a It is the distance and orientation window, λ is the wavelength of the signal, and j is the imaginary unit;

[0067] Then, a range-direction FFT is performed on the reflected echo of the linear frequency modulated signal transmitted by the MEO SAR, as follows:

[0068]

[0069] Among them, f r It is the distance frequency, W r It is the distance window, and f0 is the carrier frequency;

[0070] Step 3.2: Construct a compression function. Using the compression function and the reflected echo after range-direction FFT, obtain the expression of the echo signal in the two-dimensional frequency domain, specifically:

[0071] Step 321: Construct the compression function:

[0072]

[0073] Among them, H rc It is a compression function;

[0074] Step 322: Perform range compression on the SAR signal using a compression function to obtain the expression of the echo signal in the two-dimensional frequency domain:

[0075] First, perform phase multiplication of (28) and (29) to compress the SAR signal, and rewrite the echo signal as:

[0076]

[0077] Then, perform an azimuth FFT transform on equation (30) to obtain the expression of the echo signal in the two-dimensional frequency domain:

[0078] S(f r ,f a ) = W r (f r )·W a (f a )·exp[jφ(f r ,f a (31)

[0079] Where φ is the phase, W a It is a directional window, f a It is the azimuth frequency;

[0080] (31) can be rearranged into the following form:

[0081]

[0082] Among them, S1(f r ,t a () is an echo signal without the introduction of a first phase;

[0083] Based on the properties of the Fourier transform, formula (32) can be rearranged into the following form:

[0084]

[0085] Where ρ is an intermediate variable;

[0086] Step 3: Obtain the echo signal S1(f) without introducing a first-order phase. r ,t a The two-dimensional spectral phase φ1(f) r ,f a ):

[0087]

[0088] Steps three and four: Using the expression of the echo signal in the two-dimensional frequency domain and φ1(f) r ,f a Obtain the closed-form expression of the two-dimensional spectrum of the echo signal, and perform a third-order Taylor expansion on the closed-form expression of the two-dimensional spectrum.

[0089] Furthermore, in steps three and four, the expression for the echo signal in the two-dimensional frequency domain and φ1(f) are used. r ,f a Obtain the closed-form expression of the two-dimensional spectrum of the echo signal, and perform a third-order Taylor expansion on the closed-form expression of the two-dimensional spectrum, specifically:

[0090] First, using the expression for the echo signal in the two-dimensional frequency domain and φ1(f r ,f a The closed-form expression for obtaining the two-dimensional spectrum of the echo signal is:

[0091]

[0092] Then, in f r The third-order Taylor expansion of the closed-form expression for the two-dimensional spectrum of the echo signal at the point = 0 is as follows:

[0093] φ(f r ,f a )≈φ0(f a )+φ1(f r ,f a )+φ2(f r ,f a )+φ3(f r ,f a (36)

[0094]

[0095] Among them, Y(f a ), ζ, υ(f a ) is an intermediate variable, φ0(f a ,R M0 ) is the azimuth modulation term, φ0(f a ,R M0 ), φ2(f r ,f a ,RM0 ), φ3(f r ,f a ,R M0 These are the first, second, and third compensation terms for the distance dimension, respectively.

[0096] Furthermore, in step four, a higher-order range compensation function is designed using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal to correct the range error of the target and obtain the corrected echo signal. Specifically:

[0097] Step 41: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal.

[0098] H COM (f a ,f r ,R Mc0 )=exp(-jφ1(f a ,f r ,R Mc0 )-jφ2(f a ,f r ,R Mc0 )-jφ3(f a ,f r ,R Mc0 ))(38)

[0099] Among them, R Mc0 It is the nearest slant distance to the beam center point P;

[0100] Step 42: Using the range-compressed SAR signal and the higher-order range compensation function, perform range migration and higher-order range error correction on the target's range error, and obtain the expression for the corrected signal:

[0101] S(f r ,f a ) = W r (f r )·W a (f a )·exp(jφ0(f a ,R M0 ))(39).

[0102] Furthermore, in step five, an azimuth spatial variation compensation function is designed using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal. This azimuth spatial variation compensation function, along with the echo corrected in step four, is used to perform phase error differential correction on targets at different azimuth positions, thereby obtaining a focused SAR image. Specifically:

[0103] Step 51: Design the azimuth compression reference function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal.

[0104]

[0105] Δr=R M0 -R Mc0 (41)

[0106] Where Δr is the difference in slant range between any target and the target at the center of the scene at the moment of beam crossing;

[0107] Step 52: Perform range-direction fast inverse Fourier transform on the echo obtained in Step 42 after target range error correction. Then multiply the echo after target range error correction processed by fast inverse Fourier transform with the azimuth compression reference function obtained in Step 51, and perform azimuth-direction IFFT on the obtained product to obtain a focused SAR image.

[0108] The beneficial effects of this invention are as follows:

[0109] This invention considers the orbital characteristics of MEO SAR and the influence of Earth's rotation, as well as the transformation relationship from the geocentric rectangular coordinate system to the ground rectangular coordinate system. It analyzes the high orbital characteristics of MEO SAR and the high maneuverability of the airborne receiver, introduces a curvature compensation factor, and establishes an improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on non-stop-and-go operation. This overcomes the modeling defects of traditional equivalent single-base methods for curved trajectories, significantly improves the adaptability of MEO orbit curvature characteristics and long propagation delays, and solves the geometric distortion problem caused by the traditional assumption of uniform linear motion. Furthermore, this invention utilizes a series inversion method to perform a high-order expansion of the phase history of the echo signal, solves the closed-form expression of the signal's two-dimensional spectrum, achieves accurate phase error compensation in the frequency domain, and effectively suppresses phase errors through an azimuth spatial variation effect compensation mechanism. This invention addresses the degradation in imaging focusing performance caused by the complex geometry of MEO SAR (Medium Orbital Spacecraft BiSAR). It quantitatively analyzes the adverse effects of MEO SAR trajectory curvature and high-orbit characteristics on imaging quality. By introducing an orbit curvature compensation factor, an improved non-stop-and-go equivalent slant range model is constructed, overcoming the limitations of traditional equivalent single-base methods in curved trajectory modeling and improving modeling accuracy and computational efficiency. This invention analyzes the precise two-dimensional spectral expression of MEO SAR, analyzes each component, and effectively compensates for phase errors, thereby improving the accuracy and efficiency of spacecraft BiSAR imaging. The proposed MEO BiSAR imaging algorithm based on improved equivalent single-base significantly improves performance compared to traditional equivalent single-base algorithms, reducing image entropy by 32.71% and increasing contrast by 22.28% in extended target imaging, effectively improving image clarity and detail resolution, and meeting the high-quality imaging requirements of wide-area monitoring. Attached Figure Description

[0110] Figure 1This is a flowchart of the present invention;

[0111] Figure 2 This is a geometric model diagram of the MEO satellite-based BiSAR system.

[0112] Figure 3 This is a schematic diagram of the improved equivalent single-base slant distance model of the present invention;

[0113] Figure 4 A distribution map of point targets;

[0114] Figure 5(a) shows the focusing results of the traditional equivalent single-base algorithm;

[0115] Figure 5(b) shows the focusing results of this invention;

[0116] Figure 6(a) shows the contour map of point P1 in the traditional equivalent single-base algorithm;

[0117] Figure 6(b) shows the contour map of point P2 in the traditional equivalent single-base algorithm;

[0118] Figure 6(c) shows the contour map of point P3 in the traditional equivalent single-base algorithm;

[0119] Figure 6(d) shows the contour plot of point P4 in the proposed algorithm;

[0120] Figure 6(e) shows the contour plot of point P5 in the proposed algorithm;

[0121] Figure 6(f) shows the contour plot of point P6 using the proposed algorithm;

[0122] Figure 7(a) shows the contour plot of point P1 in the proposed algorithm;

[0123] Figure 7(b) shows the contour plot of point P2 in the proposed algorithm;

[0124] Figure 7(c) shows the contour plot of point P3 in the proposed algorithm;

[0125] Figure 7(d) shows the contour plot of point P4 using the proposed algorithm;

[0126] Figure 7(e) shows the contour plot of point P5 in the proposed algorithm;

[0127] Figure 7(f) shows the contour plot of point P6 using the proposed algorithm. Detailed Implementation

[0128] Specific implementation method one: as follows Figure 1 As shown, the specific process of a MEO satellite-to-aircraft BiSAR imaging method based on an improved equivalent monobase in this embodiment is as follows:

[0129] Step 1: Establish a ground-based rectangular coordinate system, obtain the transformation relationship from geocentric rectangular coordinates to the ground-based rectangular coordinate system, and convert the satellite's velocity and acceleration in the geocentric rectangular coordinate system to the satellite's velocity and acceleration in the ground-based rectangular coordinate system:

[0130] Step 11: Obtain the target's coordinates in the geocentric rectangular coordinate system O-XYZ:

[0131]

[0132]

[0133] Among them, R t (X es ,Y es Z es H1(t), H2, and H3(t) are the coordinates of the target in the geocentric rectangular coordinate system, and H1(t), H2, and H3(t) are intermediate variables. st It is the slant range vector from the satellite to the target; r(t) is the distance between the satellite and the Earth's center, t is the satellite time, and ω is the slant range vector from the satellite to the target. e =7.292×10 -5 rad / s is the Earth's angular velocity of rotation, ω is the argument of perigee, and θ is the angle of attack. f (t) is the true anterior angle of the medium-orbit synthetic aperture radar at time t, Ω is the right ascension of the ascending node, and i is the orbital inclination.

[0134] Steps 1 and 2: Establish a ground rectangular coordinate system PX g Y g Z g Obtain PX g Y g Z g The three-dimensional coordinate vector of the midpoint includes the following steps:

[0135] First, establish a ground rectangular coordinate system PX. g Y g Z g :

[0136] The origin P is the center point of the beam transmitted from the transmitter to the receiver;

[0137] Z g The axial direction is: along direction;

[0138] Y g The axis direction is: point P along the direction of the Earth's surface;

[0139] X g The direction is according to Z g Axial direction and Y g The axis direction is obtained following the right-hand rule;

[0140] Then, obtain PX g Y g Z g The three-dimensional coordinate vector of the midpoint:

[0141]

[0142] Where (x,y,z) is PX g Y g Z g The three-dimensional coordinate vector of the midpoint, where M is the satellite position and O is the Earth's center of mass;

[0143] Finally, by tracing the three-dimensional coordinate vectors of each point in the geocentric rectangular coordinate system O-XYZ along PX g Y g Z g A projection transformation converts the image to a ground-based rectangular coordinate system. This transformation allows the MEO SAR imaging model to be modeled and analyzed using heading, range, and elevation parameters with intuitive physical meaning. The resulting coordinate system is defined as a ground-based rectangular coordinate system. The position of the target in the ground-based rectangular coordinate system, and the velocity and acceleration of the satellite in the ground-based rectangular coordinate system after the transformation are as follows:

[0144]

[0145] in, This represents the target's position in a Cartesian coordinate system on the ground. M0 and M1 are intermediate matrices, and V... s It is the satellite's velocity in the geocentric rectangular coordinate system, A s It is the acceleration of the satellite in the geocentric rectangular coordinate system, V. t =[v tx ,v ty ,v tz ] is the satellite's velocity in a Cartesian coordinate system on the ground, A t =[a tx ,a ty ,a tz θ represents the satellite's acceleration in a Cartesian coordinate system on the ground. lo θ is the latitude of point P. la It is the longitude of point P, R e It is the Earth's radius, v tx ,v ty ,v tz They are V t In X g ,Y g Z g Components in direction, a tx ,a ty ,a tz A respectively t It is in Xg ,Y g Z g Components in direction.

[0146] In this step, in order to establish MEO SAR and the airborne receiver in the same coordinate system, a coordinate transformation method is used to map the geocentric rectangular coordinate system to the ground rectangular coordinate system.

[0147] Step 2: Establish an improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on a ground-based rectangular coordinate system, specifically as follows:

[0148] Step 21: Obtain the sum of the instantaneous distances from the transceiver platform to the target point using the satellite's velocity and acceleration in the ground Cartesian coordinate system.

[0149] Step 2.11: Obtain the slant range equation R from the transmitter to any point target N. T (t a ):

[0150] First, due to the high orbital characteristics of MEO SAR and the high mobility of its airborne receiver, its signal propagation time is significantly longer than that of low-Earth orbit BiSAR platforms. Therefore, traditional echo models based on the "stop-and-go" assumption cannot be directly applied to MEO satellite-to-air BiSAR systems. This is because the motion of the receiver must be considered during signal propagation. Figure 2 This demonstrates the difference between the actual propagation path of the MEO satellite-based BiSAR signal and the propagation path established based on the "walk-stop" approximation. Considering the orbital curvature of the MEO SAR transmitter, the slant range equation R from the transmitter to any point target N is also presented. T (t a The expression for ) is:

[0151]

[0152] Among them, (x t ,y t ,z t ) is the initial position of the transmitter, t a For location and time, (x n ,y n ,z n () represents the coordinates of any target point;

[0153] Then, the slant range equation R from the transmitter to any point target N is... T (t a The expression can be rewritten as follows:

[0154]

[0155] Among them, R t0 It is an intermediate variable, μt1 μ t2 μ t3 These are coefficients for time in each direction;

[0156] Then, after performing a third-order Taylor expansion of formula (10), retaining the first four terms, and simplifying, we obtain:

[0157]

[0158] Where, k t1 k t2 k t3 It is an intermediate variable;

[0159] Step 2.12: Obtain the instantaneous slant range R of the carrier aircraft. R (t a )as follows:

[0160]

[0161] Where, θ r It is the receiver's oblique angle, R r0 t is the slant range of the receiver from the beam center to the target. a It is location and time, v r It is the receiver speed, (x) r ,y r ,z r () is the receiver position;

[0162] Steps 2-3: Utilize the instantaneous slant range R of the carrier aircraft R (t a ) and the slant range R from the transmitter to any point target N T (t a Obtain the bistatic slant range history of traditional satellite-based BiSAR, which is the sum of the instantaneous distances from the transceiver platform to the target point:

[0163] R sag (t a ) = R T (t a )+R R (t a (17)

[0164] Step 22: Utilize the sum of instantaneous distances R from the transceiver platform to the target point. sag (t a Obtain the propagation delay τ based on stop-and-go movement. sag Based on τ sag Obtain the actual bistatic instantaneous slant range of MEO satellite-to-air BiSAR:

[0165] Step 221: Utilize the sum of instantaneous distances R from the transceiver platform to the target point. sag (ta Obtain the propagation delay τ based on stop-and-go movement. sag Specifically:

[0166] τ sag =R sag (t a ) / c(18)

[0167] Where c is the speed of light;

[0168] Step 2.2. Considering that the velocity of the airborne platform in MEO satellite-based BiSAR remains constant during the pulse propagation period, the pulse propagation delay τ of the receiver slant range function can be approximated as τ0. sag The approximate process is as follows:

[0169]

[0170] Step 223: Use formula (19) to obtain the actual bistatic instantaneous slant range of the MEO satellite-to-bistatic SAR, as follows:

[0171]

[0172] Steps 2 and 3: Based on the concept of equivalent single-base SAR, using the actual bistatic instantaneous slant range of MEO satellite-to-aircraft BiSAR, the actual slant range of MEO satellite-to-aircraft BiSAR is equivalent to single-base SAR, thereby obtaining the parameters in the improved equivalent single-base SAR model based on non-stop MEO satellite-to-aircraft BiSAR, and constructing the improved equivalent single-base SAR model based on non-stop MEO satellite-to-aircraft BiSAR:

[0173] Step 231: Design the equivalent slope distance expression and perform a third-order Taylor expansion on the equivalent slope distance expression:

[0174] First, design the equivalent slant distance expression:

[0175]

[0176] Among them, R M0 It is the equivalent slant range at the beam center time, v M It is the equivalent velocity, θ M It is the equivalent oblique angle, and β is the coefficient of the track curvature compensation term;

[0177] Then, performing a third-order Taylor expansion on equation (21) yields:

[0178]

[0179] Step 232: Perform a third-order Taylor expansion on the actual bistatic instantaneous slant range of the MEO satellite-based BiSAR, as follows:

[0180]

[0181] Where i is the index, K i It is an intermediate variable, n takes the values ​​0, 1, 2, ..., d n It is the sign for differentiation with respect to a variable. It is for t a Find the nth derivative;

[0182] Step 233: Based on the concept of equivalent monostatics, we treat the actual slant range of MEO satellite-to-air BiSAR as equivalent to monostatic SAR, and the equations are as follows:

[0183] R(t a ) = 2R M (t a (25)

[0184] Steps 2, 3, and 4: Obtain the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation according to formula (25), and substitute the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation into formula (21) to obtain the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation:

[0185]

[0186] This step designs an improved equivalent single-basis algorithm, enhancing compensation accuracy by adding a compensation factor. Compared to the traditional hyperbolic approximation, which only considers the second-order term of the Taylor expansion, this invention achieves higher-precision error compensation by introducing additional variables. The improved equivalent single-basis model proposed in this invention is as follows: Figure 3 As shown, the improved algorithm of this invention overcomes the approximation error caused by the traditional method ignoring higher-order terms.

[0187] Step 3: Obtain the reflected echo of the MEO SAR transmitted linear frequency modulated signal using the improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on non-stop operation, and obtain the closed-form expression of the two-dimensional spectrum of the echo signal using the reflected echo, specifically:

[0188] Step 31: Using the improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation, obtain the mathematical model of the reflected echo of the linear frequency modulated signal transmitted by MEO SAR, and perform range FFT (Fast Fourier Transform) on the reflected echo:

[0189] First, using an improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation, a mathematical model of the reflected echo of the linear frequency modulated signal transmitted by MEO SAR is obtained:

[0190]

[0191] in, It is fast time, γ is the range-direction frequency modulation slope, ω r and ω a It is the distance and orientation window, λ is the wavelength of the signal, and j is the imaginary unit;

[0192] Then, perform a distance-oriented FFT on equation (27) as follows:

[0193]

[0194] Among them, f r It is the distance frequency, W r It is the distance window, and f0 is the carrier frequency;

[0195] Step 3.2: Design a compression function. Using the compression function and the reflected echo after range-direction FFT, obtain the expression of the echo signal in the two-dimensional frequency domain, specifically:

[0196] Step 321: The first step in echo signal processing is to perform pulse compression in the distance dimension. Therefore, the compression function is designed as follows:

[0197]

[0198] Among them, H rc It is a compression function;

[0199] Step 322: Perform range compression on the SAR signal using a compression function to obtain the expression of the echo signal in the two-dimensional frequency domain:

[0200] First, multiply (28) and (29) by phase to compress the SAR signal, and the echo signal becomes:

[0201]

[0202] Then, an azimuth FFT transform is performed on formula (30), and the expression of the echo signal in the two-dimensional frequency domain is as follows:

[0203] S(f r ,f a ) = W r (f r )·W a (f a )·exp[jφ(f r ,f a )](31)

[0204] Where φ is the phase, W a It is a directional window, f a It is the azimuth frequency;

[0205] After introducing a phase, directly solving the above equation becomes quite complex. We assume that the echo signal without introducing a phase is represented as S1(fr ,t a ), rearranging (31) gives:

[0206]

[0207] Among them, S1(f r ,t a () is an echo signal without the introduction of a first phase;

[0208] Based on the properties of the Fourier transform, formula (32) can be rearranged into the following form:

[0209]

[0210] Where ρ is an intermediate variable;

[0211] Step 3: Obtain the echo signal S1(f) without introducing a first-order phase. r ,t a The two-dimensional spectral phase φ1(f) r ,f a ):

[0212]

[0213] Steps three and four: Using the expression of the echo signal in the two-dimensional frequency domain and φ1(f) r ,f a Obtain the closed-form expression of the two-dimensional spectrum of the echo signal, and perform a third-order Taylor expansion on the closed-form expression of the two-dimensional spectrum:

[0214] First, the closed-form expression for the two-dimensional spectrum of the echo signal is obtained using formulas (33) and (34):

[0215]

[0216] Then, in f r The third-order Taylor expansion of formula (35) at the point = 0 is as follows:

[0217] φ(f r ,f a )≈φ0(f a )+φ1(f r ,f a )+φ2(f r ,f a )+φ3(f r ,f a (36)

[0218]

[0219] Among them, Y(f a ), ζ, υ(fa ) is an intermediate variable, φ0(f a ,R M0 ) is the azimuth modulation term, φ0(f a ,R M0 ), φ2(f r ,f a ,R M0 ), φ3(f r ,f a ,R M0 These are the first, second, and third compensation terms for the distance dimension, respectively.

[0220] Step 4: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal to correct the range error of the target, eliminate range defocus caused by orbit curvature and bistatic configuration, and obtain the corrected echo signal. Specifically:

[0221] Step 41: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal.

[0222] H COM (f a ,f r ,R Mc0 )=exp(-jφ1(f a ,f r ,R Mc0 )-jφ2(f a ,f r ,R Mc0 )-jφ3(f a ,f r ,R Mc0 ))(38)

[0223] Among them, R Mc0 It is the nearest slant distance to the beam center point P;

[0224] This step designs a higher-order range compensation function to correct the range error of the target and eliminate range defocus caused by orbit curvature and bistatic configuration; it also corrects range migration and higher-order range errors in the echo.

[0225] Step 4.2: Multiply formulas (30) and (38) to correct the target's range error for range migration and higher-order range error. The expression for the corrected signal is then:

[0226] S(f r ,f a ) = W r (f r )·W a (f a )·exp(jφ0(fa ,R M0 (39)

[0227] Step 5: Design an azimuth spatial variation compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal. Use the azimuth spatial variation compensation function and the echo after target range error correction to perform phase error differential correction on targets at different azimuth positions, complete high-precision imaging, and obtain a precisely focused SAR image.

[0228] Step 51: Design the azimuth compression reference function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal.

[0229]

[0230] Δr=R M0 -R Mc0 (41)

[0231] Where Δr is the difference in slant range between any target and the target at the center of the scene at the moment of beam crossing;

[0232] This step takes into account the characteristic of the azimuth modulation term changing with distance and designs an adaptive reference function to implement azimuth matched filtering;

[0233] Step 5.2. Perform a fast inverse Fourier transform on the range direction of the echo after the target range error correction (Formula (39)), then multiply the processed Formula (39) with Formula (40), and then perform an azimuth IFFT on the product to obtain a precisely focused SAR image.

[0234] Example: To verify the beneficial effects of the present invention, the following experiments were conducted:

[0235] This embodiment uses point target simulation experiments to verify the invention and compares it with the traditional single-basis equivalent method. To test the correction performance of the proposed improved equivalent single-basis algorithm for errors caused by track curvature, the traditional equivalent single-basis model is applied with a non-"stop-and-go" approach considered, and the target distribution is as follows: Figure 4 As shown.

[0236] The focusing results for points P1-P6 using the traditional equivalent single-base algorithm and the proposed method are shown in Figures 5(a) and 5(b). Comparing the two figures, it can be seen that the proposed algorithm achieves better focusing results for point targets compared to the traditional equivalent single-base algorithm. This is because the proposed algorithm eliminates the phase errors introduced by MEO SAR orbit curvature and scene spatial variations.

[0237] Furthermore, to demonstrate the superiority of the improved equivalent single-base algorithm of this invention, this embodiment selects... Figures 7(a)-7(f) The marked points are used for a detailed evaluation of image quality, which is then used in comparative experiments. For example... Figures 6(a)-6(f) As shown, the contour map generated by the traditional equivalent single-base algorithm exhibits defocusing in the azimuth direction, indicating that the existence of equivalent slant range error leads to deviations in azimuth compensation. Experimental results confirm that in MEO satellite-machine bistatic SAR systems, the satellite orbit curvature effect has a significant impact on imaging quality. The marker point contour map obtained by the improved equivalent single-base algorithm of this invention is shown below. Figures 7(a)-7(f) As shown, the focusing effect of all six marker points is ideal, and their coupling phase residual error has been completely corrected. Comparison shows that, through this invention, the azimuth defocusing caused by the complex motion of MEO SAR has been completely corrected. After implementation of this invention, the impulse response curves of the point targets are all standard sinc functions, thus proving that the point targets achieved good focusing using this invention.

Claims

1. A MEO (Metal-Operated Oscillator) BiSAR imaging method based on an improved equivalent monobase, characterized in that... The specific process of the method is as follows: Step 1: Establish a ground-based rectangular coordinate system and convert the satellite's velocity and acceleration in the geocentric rectangular coordinate system into the satellite's velocity and acceleration in the ground-based rectangular coordinate system; Step 2: Establish an improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on the ground rectangular coordinate system; Step 3: Use the improved equivalent single-base model of MEO satellite-based BiSAR to obtain the reflected echo of the MEO SAR transmitted linear frequency modulated signal, and use the reflected echo of the MEO SAR transmitted linear frequency modulated signal to obtain the closed expression of the two-dimensional spectrum of the echo signal. Step 4: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal to correct the range error of the target and obtain the corrected echo signal. Step 5: Design an azimuth spatial variation compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal. Use this azimuth spatial variation compensation function and the echo signal corrected in Step 4 to perform phase error differential correction on targets at different azimuth positions, obtaining a focused SAR image. Specifically: Step 51: Design the azimuth compression reference function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal. （40） （41） in, It is the difference in slant range between any target and the target at the center of the scene at the moment of beam crossing. It is the azimuth frequency. It is the imaginary unit. It is an equivalent oblique angle. It is the equivalent speed. It is an intermediate variable. It is the wavelength of the signal. It is the equivalent slant range at the beam center time. For the track curvature compensation term coefficient, It is the nearest slant distance to the beam center point P; Step 52: Perform a range-direction fast inverse Fourier transform on the echo obtained in Step 4 after target range error correction. Then multiply the echo after target range error correction processed by the fast inverse Fourier transform with the azimuth compression reference function obtained in Step 51, and perform an azimuth-direction IFFT on the obtained product to obtain a focused SAR image.

2. The MEO satellite-based BiSAR imaging method according to claim 1, characterized in that: In step one, a ground-based rectangular coordinate system is established, and the satellite's velocity and acceleration in the geocentric rectangular coordinate system are converted to the satellite's velocity and acceleration in the ground-based rectangular coordinate system. Specifically: Step 11: Obtain the target in the geocentric rectangular coordinate system Coordinates in: （1） （2） （3） （4） in, These are the coordinates of the target in a geocentric rectangular coordinate system. , , It is an intermediate variable. It is the slant range vector from the satellite to the target; The distance between the satellite and the Earth's center. It's satellite time. It is the Earth's rotational angular velocity. It is the perigee argument. yes True perimeter angle of mid-orbit synthetic aperture radar at any given time. It is the right ascension of the ascending node. It is the track inclination angle; Steps 1 and 2: Establish a ground rectangular coordinate system , obtain The three-dimensional coordinate vector of the midpoint includes the following steps: First, establish a ground rectangular coordinate system. : Take the center point of the beam transmitted from the transmitter to the receiver as the origin P, and Direction as Axial direction, Point along the surface direction as axial direction, according to Axial direction and The axial direction is obtained by following the right-hand rule. direction; Where O is the Earth's center of mass; Then, obtain The three-dimensional coordinate vector of the midpoint: （5） in, yes The three-dimensional coordinate vector of the midpoint, where M is the satellite position; Finally, the geocentric rectangular coordinate system The three-dimensional coordinate vectors of each point along Perform a projection transformation to obtain the satellite's velocity and acceleration in the ground rectangular coordinate system, specifically: （6） （7） （8） in, , It is the intermediate matrix. It is the satellite's velocity in the geocentric rectangular coordinate system. It is the acceleration of the satellite in the geocentric rectangular coordinate system. It is the satellite's velocity in the Cartesian coordinate system on the ground. It is the acceleration of the satellite in the Cartesian coordinate system on the ground. It is the latitude of point P. It is the longitude of point P. They are exist Components in direction, respectively Is Components in direction.

3. The MEO satellite-based BiSAR imaging method according to claim 2, characterized in that: The second step, establishing an improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on a ground-based rectangular coordinate system, specifically involves: Step 2:

1. Calculate the sum of instantaneous distances from the transceiver platform to the target point using the satellite's velocity and acceleration in the ground Cartesian coordinate system; Step 22: Utilize the sum of instantaneous distances from the transceiver platform to the target point. Obtain the propagation delay based on stop-and-go traffic ,based on Obtain the actual bistatic instantaneous slant range of MEO satellite-mounted BiSAR; Steps 2 and 3: Using the actual bistatic instantaneous slant range of MEO satellite-based BiSAR, the actual slant range of MEO satellite-based BiSAR is equivalent to monostatic SAR, thereby obtaining the parameters in the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation, and then constructing the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation.

4. The MEO satellite-based BiSAR imaging method according to claim 3, characterized in that: Step 2.1, which involves using the satellite's velocity and acceleration in the ground-based Cartesian coordinate system to obtain the sum of the instantaneous distances from the transceiver platform to the target point, specifically involves: Step 2.11: Obtain the slant range equation from the transmitter to any target point N. : （13） （14） （11） （12） in, It is an intermediate variable. , , These are coefficients for time in each direction. , , It is an intermediate variable. For location and time, This is the transmitter's initial position. Let the target coordinates be any point; Step 2.12: Obtain the instantaneous slant range of the carrier aircraft. as follows: （15） （16） in, It is the receiver's angled view. This is the slant range from the receiver's beam center to the target. It refers to location and time. It's the receiver speed. It is the receiver location; Steps 2-3: Utilizing the instantaneous slant range of the carrier aircraft and the slant distance from the transmitter to any point target N Obtain the sum of instantaneous distances from the transceiver platform to the target point: （17） in, It is the sum of the instantaneous distances from the sending and receiving platform to the target point.

5. The MEO satellite BiSAR imaging method based on an improved equivalent monobase as described in claim 4, characterized in that: In step two, the sum of the instantaneous distances from the transceiver platform to the target point is used. Obtain the propagation delay based on stop-and-go traffic ,based on The actual bistatic instantaneous slant range of the MEO satellite-mounted BiSAR is obtained as follows: Step 221: Utilize the sum of instantaneous distances from the transceiver platform to the target point. Obtain the propagation delay based on stop-and-go traffic Specifically: （18） in, It's the speed of light; Step 222: Delay the pulse propagation of the receiver's slant range function. Approximately The approximate instantaneous slant range of the carrier aircraft is obtained as follows: （19） Steps 2-3: Obtain the actual bistatic instantaneous slant range of the MEO satellite-to-aircraft BiSAR using the approximated instantaneous slant range, as follows: （20）。 6. The MEO satellite BiSAR imaging method based on an improved equivalent monobase as described in claim 5, characterized in that: In steps two and three, the actual bistatic instantaneous slant range of MEO satellite-based BiSAR is used to convert the actual slant range of MEO satellite-based BiSAR into a monostatic SAR, thereby obtaining the parameters in the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation. This leads to the construction of the improved equivalent monostatic model of MEO satellite-based BiSAR based on non-stop operation. Specifically: Step 231: Construct the equivalent slope distance expression and perform a third-order Taylor expansion on the equivalent slope distance expression: First, obtain the equivalent slope expression: （21） Then, a third-order Taylor expansion of the equivalent slant distance expression yields: （22） Step 232: Perform a third-order Taylor expansion on the actual bistatic instantaneous slant range of the MEO satellite-based BiSAR, as follows: （23） （24） in, It is an index. It is an intermediate variable. Take 0, 1, 2, ... It is the sign of differentiation with respect to a variable. Yes Find the nth derivative; Step 233: Equivalent the actual slant range of MEO satellite-based BiSAR to monostatic SAR, and formulate the following equations: （25） Steps 2, 3, and 4: Obtain the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation according to formula (25), and substitute the parameters in the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation into formula (21) to obtain the improved equivalent single-base model of MEO satellite BiSAR based on non-stop operation. The parameters in the improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation are obtained according to formula (25), specifically as follows: （26）。 7. The MEO satellite BiSAR imaging method based on an improved equivalent monopole as described in claim 6, characterized in that: Step three involves obtaining the reflected echo of the MEO SAR transmitted linear frequency modulated signal using an improved equivalent single-base model based on non-stop MEO satellite BiSAR, and then using the reflected echo of the MEO SAR transmitted linear frequency modulated signal to obtain a closed-form expression for the two-dimensional spectrum of the echo signal. Specifically: Step 31: Using the improved equivalent single-base model of MEO satellite-to-aircraft BiSAR based on non-stop operation, obtain the reflected echo of the linear frequency modulated signal transmitted by MEO SAR, and perform range FFT on the reflected echo, specifically: First, using an improved equivalent single-base model of MEO satellite-based BiSAR based on non-stop operation, the reflected echo of the linear frequency modulated signal transmitted by MEO SAR is obtained: （27） in, It's a fast time. It is the range-direction frequency modulation slope. and It is a distance and orientation window; Then, a range-direction FFT is performed on the reflected echo of the linear frequency modulated signal transmitted by the MEO SAR, as follows: （28） in, It is distance frequency. It's a distance window. It is the carrier frequency; Step 3.2: Construct a compression function. Using the compression function and the reflected echo after range-direction FFT, obtain the expression of the echo signal in the two-dimensional frequency domain, specifically: Step 321: Construct the compression function: （29） in, It is a compression function; Step 322: Perform range compression on the SAR signal using a compression function to obtain the expression of the echo signal in the two-dimensional frequency domain: First, perform phase multiplication of (28) and (29) to compress the SAR signal, and rewrite the echo signal as: （30） Then, perform an azimuth FFT transform on equation (30) to obtain the expression of the echo signal in the two-dimensional frequency domain: （31） in, It is phase. It is a directional window; (31) can be rearranged into the following form: （32） in, It is an echo signal without introducing a first phase; Based on the properties of the Fourier transform, formula (32) can be rearranged into the following form: （33） in, It is an intermediate variable; Step 33: Obtain the echo signal without introducing a first phase. Two-dimensional spectral phase : （34） Steps three and four: Using the expression of the echo signal in the two-dimensional frequency domain and Obtain the closed-form expression of the two-dimensional spectrum of the echo signal, and perform a third-order Taylor expansion on the closed-form expression of the two-dimensional spectrum.

8. The MEO satellite BiSAR imaging method based on an improved equivalent monobase as described in claim 7, characterized in that: The expression for the echo signal in the two-dimensional frequency domain in steps three and four is as follows: Obtain the closed-form expression for the two-dimensional spectrum of the echo signal, and perform a third-order Taylor expansion on the closed-form expression for the two-dimensional spectrum, as follows: First, using the expression of the echo signal in the two-dimensional frequency domain and The closed-form expression for obtaining the two-dimensional spectrum of the echo signal: （35） Then, in The closed-form expression for the two-dimensional spectrum of the echo signal is expanded using a third-order Taylor series, as follows: （36） （37） in, , , It is an intermediate variable. It is the azimuth modulation term. , , These are the first, second, and third compensation terms for the distance dimension, respectively.

9. The MEO satellite-based BiSAR imaging method according to claim 8, characterized in that: In step four, a higher-order range compensation function is designed using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal to correct the range error of the target and obtain the corrected echo signal. Specifically: Step 41: Design a higher-order range compensation function using the compensation term in the closed-form expression of the two-dimensional spectrum of the echo signal. （38） Step 42: Using the range-compressed SAR signal and the higher-order range compensation function, perform range migration and higher-order range error correction on the target's range error, and obtain the expression for the corrected signal: （39）。

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