Evaluation of Trajectory Error Effects in BP Based Space Target ISAR Imaging

, , ling

Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China

Abstract: The space target imaging is important in the development of space technology. Due to the availability of trajectory information of the space targets and the arising of rapid parallel processing hardware, the back projection (BP) method has been applied to synthetic aperture radar (SAR) imaging and shows a number of advantages as compared with conventional Fourier-domain imaging algorithms. However, the practical processing shows that the insufficient accuracy of the trajectory information results in the degrading of the imaging results. On the other hand, the autofocusing algorithms for BP imaging are not well developed, which is a bottleneck for the application of BP imaging. Here, an analysis of the effect of trajectory errors on the space target imaging using microlocal technology is presented. Our analysis provides an explicit quantitative relationship between the trajectory errors of the space target and the positioning errors in the reconstructed images. The explicit form of the position errors for some typical trajectory errors is also presented. Numerical simulations demonstrate our theoretical findings. The measured position errors obtained from the reconstructed images are consistent with the analytic errors calculated by using the derived formulas. Our results will be used in the development of effective autofocusing methods for BP imaging.

Key words: inverse synthetic aperture radar (ISAR); imaging; space targets; trajectory error; back projection (BP)

0 Introduction

Conventional inverse synthetic aperture radar(ISAR) imaging uses range-Doppler (RD) method, in which a simple fast Fourier transform (FFT) is applied in the cross-range dimension to achieve the cross-range resolution. The RD method works under the assumptions of plane wave-front and small rotation-angle in the coherent processing interval (CPI)[1]. Motion compensation including range alignment and phase compensation is necessary before the FFT in cross-range to remove the translational motion component between the target and the radar. This FFT-based imaging method becomes invalid if the target undergoes a larger rotation relative to the radar accompanied by migration through resolution cell (MTRC)[1-3].

Many literatures for ISAR imaging of space targets have appeared[4-9]. In space target imaging, the trajectory information of space targets is always available. The targets mostly undergo stationary motion. This in fact enables the applications of the polar format algorithm (PFA)[10,11]and back-projection (BP)[12-14]imaging method to integrate longer data than conventional RD method, and hence obtain higher image resolution. As compared to RD and PFA, the BP imaging method has several advantages: (1) it does not need motion compensation required by RD algorithm; (2) it does not need the assumption of plane wave-front; (3) it is applicable to arbitrary trajectories and imaging geometries; (4) it can be implemented efficiently exploiting parallel processing and high speed hardware, e.g. graphic processing unit (GPU). In this paper, we are interested in performing the image formation of space targets using BP method with the known trajectory information.

Due to the limited accuracy of the trajectory data, the errors in the trajectory may lead to mispositioning and smearing of the space target in the reconstructed image. We provide an analysis of the effect of trajectory errors on the image reconstruction and use the microlocal analysis to get explicit expressions of the positioning errors. The results show the relationship between the positioning errors and the trajectory errors, and the dependency of the positioning errors on the space target trajectory, the flying velocity of the space target, etc. We specify the positioning errors due to trajectory errors for some typical types of trajectory errors. Numerical simulations are performed to demonstrate our analysis. Our results can be easily extended to the bistatic ISAR imaging case. The approach for analyzing the positioning errors due to space target trajectory errors is initially presented in our previous conference paper[15].

1 ISAR Imaging Principles

The ISAR imaging from the perspective of BP theory is briefly described in this section.

The received signal of ISAR is given as

(1)

wheretdenotes the fast time,sthe slow-time,c0the electromagnetic wave speed,ωthe angular frequency of the transmitted waveform,X(s) the coordinates of the scatterers on the target, andT(X(s)) the target reflectivity. Thesdependency of the scatterer position accounts for the position change as the space target flies along the trajectory. Note that the stop-go assumption is used in Eq. (1), which is valid for the cases of pulsed transmitted waveform and little radial motion existing between the space target and the radar.A(X(s),ω) is a complex amplitude function including the transmitted waveform and the antenna beam patterns.R(X(s)) denotes the total range of the two-way propagation of the electromagnetic wave. For monostatic ISAR,R(X(s))is given by

R(X(s))=2|X(s)-p|

(2)

wherepdenotes the position of radar. Without the loss of generality, we focus on monostatic ISAR imaging in the rest of the discussion.

LetX(s)=γ(s)+X′

(3)

whereγ(s) denotes the trajectory of space target andX′the coordinates of the target scatterers in the local coordinate system embedded on the target, as shown in Fig.1.

Fig.1 Illustration of imaging geometry for space target imaging where the radar is stationary on the ground

Substituting Eqs. (2),(3) into Eq. (1), and making the change of variablesX→X′, we have

(4)

In the rest of our discussion,X′ will be replaced withXfor notational simplicity.

A three-dimensional (3D) image can be reconstructed as long as the data is collected in 3D space, i.e.,γ(s)undergoes sufficient variation inR3. We focus on the two-dimensional (2D) image formation. Letz∈R2denote the position in the reconstructed 2D image. By using the filtered BP (FBP) method[12], the image can be formed as

(5)

Here

R(s,Z)=2|γ(s)+Z-p|

(6)

whereZ∈R3,Z=(z,0). The image is assumed to be reconstructed on a zero-height plane, i.e. ,z3=0.

In Eq. (5), the phase term accounts for the matched filtering. The filterQcan be chosen according to a variety of criteria[16, 17]to compensate the amplitude term in Eq. (5) or to enhance the edges in the images, etc.

Substituting Eq. (4) into Eq. (5), we have

(7)

The H?rmander-Sato lemma[18-22]tells us that the imaging operator in fact reconstructs a scatterer at locationxon the target at positionzin the image satisfying the following conditions[12, 19, 20].

?ωΦ(Z,X,s,ω)=0?R(s,Z)=R(s,X)

(8)

?sΦ(Z,X,s,ω)=0?fd(s,Z)=fd(s,X)

(9)

Herefddenotes the Doppler frequency which is given by

(10)

2 Analysis of Positioning Errors Due to Trajectory Errors

In this section, we analyze reconstruction errors in ISAR images due to errors in the trajectory of the space target using microlocal analysis[18, 24,25].

Letγ(s) denote the ideal or assumed trajectory andγε(s)=γ(s)+εΔγ(s) the true trajectory of the space target whereεis a small constant. In an ideal case where the target is tracked with sufficient accuracy, i.e.Δγ(s)≈0, the scattererXon the target is mapped toz0in the image. We refer toz0as the correct position of the scatterer in the reconstructed image. Note that the mapping fromXtoz0is associated with the projection of 3D coordinates to 2D coordinates.

Letz=z0+Δzbe the erroneous reconstructed position due to trajectory error Δγ(s). Using Eqs. (8) and (9), we have[23]

By expanding the terms on the left side of Eqs. (11) and (12) in Taylor series aroundε=0 and keeping the first-order terms, Eqs. (11) and (12) will respectively become

By simplifying Eqs. (13) and (14), we will obtain

(15)

(16)

where

(17)

(18)

and

(19)

(20)

From Eqs. (15) and (16), we have the following findings:

(1) The radial positioning error Δzris determined by the radial component of the trajectory errors. The transverse positioning error Δztdepends on the radial component of the velocity error, the transverse component of the trajectory error, the velocity of the space target and the range between the radar and target. Note that we refer to the radial component as the projection of a vector onto the radar look-direction and the transverse component as the projection of a vector onto the plane perpendicular to the radar look-direction.

(2) The first term in the square bracket of Eq. (16) makes more contribution to the transverse positioning error than the second term due to the large range term in the denominator of the second term. This implies that the value of Δztis more affected and determined by the radial velocity error of the space target.

(3) The dependency of Δzrand Δztonsgives the explanation of smearing in the reconstructed image. In the case of wide-aperture imaging where the change of the radar look-direction becomes significant during the CPI and the case where the trajectory errors are time-varying, the positioning errors accordingly change with time leading to the defocusing of the targets.

The positioning errors induced by typical trajectory errors including constant, linear, quadratic and sinusoidal trajectory errors are analyzed. The results are listed in Table 1. The angles are defined as follows.

Table1Positioningerrorsduetodifferenttypesoftrajectoryerrors

Δγ(s)ΔzrΔztΔγ0-Δγr0cosψ-Δγt0 cosθsinφcosψ⊥ ves-vrecosψs-vreγ(s)+Z-γ⊥(s)cosψ⊥-vtecosθsinφcosψ⊥s12aes2-are2cosψs2-areγ(s)+Z-γ⊥(s)cosψs-atecosθ2sinφcosψ⊥s2Aesinωs-(Aesinωs)rcosψ-(Aeωcosωs)rγ(s)+Z-γ⊥(s)cosψ-(Aesinωs)tcosθsinφcosψ⊥

Fig.2 Illustration of angles and vectors used in trajectory error analysis

Nextly the positioning errors for each case in Table 1 are analyzed in detail.

(1)ConstanttrajectoryerrorΔγ0The second row of Table 1 gives positioning errors induced by the constant trajectory error. The radial error Δzris dependent on the radial component of Δγ0and the transverse error Δztis dependent on the transverse component of Δγ0. For a relatively short synthetic aperture, the radar look-direction does not change much. Δzrand Δztare approximately constant with only position shift in the reconstructed image rather than defocusing. However, for a large synthetic aperture, the radar look-direction undergoes relatively large change leading to large changes of anglesψ,ψ⊥andθ. Thus, Δzrand Δztmay vary significantly with slow time, which leads to the smearing of the reconstructed target even if the trajectory error is constant.

(4)SinusoidaltrajectoryerrorAesinωsAs show in the fifth row of Table 1, Δzrand Δztoscillate with time in the case of sinusoidal trajectory errors. Similar to the cases (2) and (3), Δztis much larger than radial positioning error and mainly depends on the radial component of the sinusoidal trajectory error. Thus, the target gets smeared in the transverse radar look-direction if there is a radial component of the sinusoidal trajectory error (Aeωcosωs)r.

3 Demonstrations

3.1 Configuration and parameters

In this section, simulations are conducted to demonstrate our analysis. The satellite tool kit (STK) is used to obtain the real trajectory data of in-orbit space targets, which is required by the raw data generation. The transmitted waveform is a linear frequency modulated signal. The parameters of the orbit, waveform and system configuration are listed in Table 2.

The total passing time of the orbit used in the simulations is 860.43 s. We select a part of the trajectory in the simulations, which corresponds a CPI of 3.75 s having 1 500 pulses. The range between the target and the radar at the starting time is 680.85 km.

Table 2 System parameters used in simulations

In the current configuration, the synthetic aperture length is much smaller than the radar-to-target range. Thus, the radar look-direction and transverse look-direction do not change significantly during the coherent integration time.

The position errors in the reconstructed images are measured and compared with the analytic ones calculated using Eqs. (15),(16). The sub-aperture processing is used in measuring the position errors. The data is divided into several patches with or without overlapping. The images are reconstructed using each patch of data, which refer to as the sub-images. The position errors in each sub-image are measured and then interpolated to obtain the positioning errors over the aperture.

3.2 Results

Fig.3 Image of a point target reconstructed using correct trajectory information

3.2.1 Constant trajectory errors

We assumed a constant trajectory error, Δγ0=[2 m,2 m,0 m] in the image formation. Fig.4 shows the reconstructed result using the erroneous trajectory. The yellow circle denotes the true position of the target.

Fig.4 Reconstructed ISAR image for the case of a constant trajectory error

It can be seen that the target is well reconstructed, however at an erroneous position of [-2.01 m,-2.01 m]. The positioning error is a constant position shift, which is constant with our analysis in Section 2.

Fig.5 presents the analytic and measured radial and transverse positioning error. It can be seen that the measured and analytic values match very well. Note that the discretization of the image and the imaging resolution determined by the configuration and waveform parameters affect the accuracy of the measured position errors, which leads to a discrepancy between the measured and analytic values.

Fig.5 The analytic and measured radial and transverse positioning errors in the case of a constant trajectory error

By comparing the radial and transverse components of the positioning errors, we can see that Δztis smaller than Δzr. This is also indicated in the image shown in Figure 4, in which the position shift in the radial direction is dominant. The curves in Fig. 5 show that Δzris nearly constant during the integration time, while Δztundergoes a larger change with the slow time than Δzr. This may be due to the larger change of anglesθandφ, as shown by the term in the second row and the third column. However, since Δztvaries in a small range, there is no visible smearing in the reconstructed image, as shown in Fig.4.

3.2.2 Liner trajectory error

Fig.6 The reconstructed ISAR image in the case of a linear trajectory error ve=[0.001 m/s,0.001 m/s,0 m/s]

We considered a linear trajectory error, Δγ(s)=ves, whereve=[0.001 m/s,0.001 m/s,0 m/s] in the image formation. Fig. 6 shows the reconstructed image under the assumption of linear trajectory error. It can be seen that the target is reconstructed at a position shifting mainly in the transverse look-direction as compared to the target true position. This is consistent with our analysis in Section 2. The radial component of the velocity error induces a much larger transverse positioning error than the radial positioning error.

Fig.7 shows the analytic and measured radial and transverse positioning error. It can be seen that the measured values match with the analytic ones well. Both the radial and transverse error change linearly with time. Δztis larger than Δzr, as indicated by the red and black lines in Fig. 7. This is consistent with the image shown in Fig. 6 and the theoretical result in Section 2.

Fig.7 The analytic and measured radial and transverse positioning errors in the case of a linear trajectory error, ve=[0.001 m/s,0.001 m/s,0 m/s]

Fig.8 The reconstructed image in the case of a quadratic trajectory error, ae=[0.000 2 m/s2,0.000 2 m/s2,0 m/s2]

3.2.3 Quadratic trajectory errors

It is seen that the target is defocused with a streaking artifact. As compared to the correct reconstruction in Fig. 3, the smearing occurs mainly in the transverse look-direction.

Fig.9 The analytic and measured radial and transverse positioning errors in the case of a quadratic trajectory error, ae=[0.000 2 m/s2,0.000 2 m/s2,0 m/s2]

Fig.9 presents the analytic and measured radial and transverse positioning errors in the case of a quadratic trajectory error. The measured positioning errors indicated by the dashed lines match with the analytic ones indicated by the solid lines. The radial positioning error approaches zero, while the transverse positioning error is much larger than the radial component and undergoes a large change during the integration time. This in fact leads to the smeared target signature, as shown in Fig.8. It can be also seen from Fig. 8 that the transverse positioning error changes quadratically with the time. This is consistent with our theoretical analysis in Section 2.

3.2.4 Sinusoidal trajectory errors

We assume a sinusoidal error, Δγ(s)=[0.01sin(πs/16) m,0.01cos(πs/16) m,0 m] in the trajectory of the target. Fig.10 shows the reconstructed image under the assumption of this sinusoidal trajectory error.

From Fig.10, it is seen that the target is not focused at its true position. The target energy spreads severely in the transverse look-direction. Comparing the target signature with that of in Fig.8, i.e. the image reconstructed in the case of a quadratic trajectory error, we see the smearing occurs mainly in the transverse look-direction similarly, but in the case of a sinusoidal trajectory error, the smearing is centered around the true position of the target rather than the one-side smearing in Fig. 8 in the case of a quadratic trajectory error.

Fig.10 The reconstructed image in the case of a sinusoidal trajectory error Δγ(s)=[0.01sin(πs/16 m, 0.01cos(πs/16) m,0 m]

Fig.11 shows the analytic and measured radial and transverse components of the positioning error. The consistency of the measured and analytic values verifies the correctness of our formulas of the positioning errors.

Fig.11 The analytic and measured radial and transverse positioning errors in the case of a sinusoidal trajectory error, Δγ(s)=[0.01sin(πs/16) m,0.01cos(πs/16) m,0 m]

It can be seen from Fig. 11 that the radial positioning error is much close to zero, but the transverse positioning error is large and varies in a sinusoidal law. This explains the smeared target signature in Fig. 10, which is due to the oscillation of the transverse positioning error around zero.

3.3 Remarks

From the demonstration above, we see that our analysis provides a qualitative relationship between the smeared target signatures in the reconstructed images and the trajectory errors. The deterministic trajectory errors in the demonstration including constant, linear, quadratic and sinusoidal errors are mainly considered. The random trajectory errors are wideband noise-like errors. Given a specified data set, the realization of a random trajectory error can be regarded as a combination of a number of sinusoidal signals, which leads to a superposition of the defocusing effects as shown in Fig. 10.

Our results given by Eqs. (15),(16) indicate that a straightforward inversion from 2D positioning errors (a common 2D imaging is assumed.) to six independent unknowns (three components of the transmitter trajectory errors and three for the receiver) is not solvable. However, under certain scenarios, the relationship between the positioning errors and the trajectory errors may be simplified, e.g., in the monostatic configuration. In the monostatic case, the trajectory error can be expressed in terms of a radial component and a transverse component. Thus, the inversion may be solvable. Since the true position of the target is not known a prior, the position errors cannot be determined with only the measurements of the erroneous positions. We may divide the data into several sub-apertures and measure the change of the erroneous positions through the sub-apertures. The differential position errors in horizontal and vertical directions are then used to retrieve the radial and transverse trajectory errors. We are developing autofocusing methods on this track.

4 Conclusions

We presented an approach to analyze the reconstruction errors in ISAR imaging of space targets due to errors in the trajectory information. We derived the explicit formulas of the positioning errors in the radar look-direction and transverse look-direction using microlocal analysis and obtained a quantitative relationship between the trajectory errors of the space targets and positioning errors in the reconstructed images. The constant trajectory error leads to constant position shifts in the reconstructed images for a relatively short imaging time. The linear trajectory error leads to a much large position shift in the transverse look-direction for a relatively short imaging time. For wide-aperture imaging or long imaging time, the dependency of the positioning errors on slow time cannot be ignored and may lead to defocusing of the target rather than only a constant position shift. The quadratic trajectory error leads to obvious smearing in radar traverse look-direction and small radial positioning error. The sinusoidal trajectory error leads to smearing of the target in radar transverse look-direction centered at the target true position. Numerical simulation results demonstrated our theoretical analysis. The autofocusing research may benefit from our results. To develop the autofocusing method exploiting the obtained quantitative relationship between the trajectory error and reconstructed position error is one of our undergoing works.

Acknowledgements

This work is supported by the National Natural Science Foundation of China(No.61871217) and the Foundation of Graduate Innovation Center in Nanjing University of Aeronautics and Astronautics (No. kfjj20170404), China.