For a synthetic aperture radar (SAR) onboard a geosynchronous-earthorbit (GEO) satellite, the track can be curvilinear. Thus, the validity of imaging algorithms based on the conventional hyperbolic range equation (CHRE) becomes questionable. A fourth-order range equation is adopted to improve the accuracy in the approximation of the range history for GEO SAR and a modified chirp scaling algorithm (CSA) is proposed. Simulation results show that the presented algorithm has better performance than the CHRE-based CSA in the GEO case, implying good application prospects.Introduction: With the higher orbit height, the integration time of a geosynchronous-earth-orbit synthetic aperture radar (GEO SAR) can be long. Thus, the linear track approximation cannot fit the actual GEO SAR track properly. The conventional hyperbolic range equation (CHRE) becomes inadequate. A range equation with higher-order terms and the corresponding imaging algorithms are needed for GEO SAR. Eldhuset [1] established a fourth-order range equation, and developed an extended exact transfer function (EETF) for high-resolution spaceborne SAR. Although Eldhuset achieved accurate range approximation, one drawback is that the EETF cannot correct the spatialvariant range cell migration (RCM). Liu et al.[2] studied a modified range migration algorithm (RMA) for GEO SAR; however, this algorithm requires interpolation, which is also based on CHRE. The standard CHRE-based chirp scaling algorithm (CSA) [3] is a very precise and efficient algorithm, which is widely used in the low-earth-orbit (LEO) cases. Therefore, a CSA based on the fourth-order range equation is derived in this Letter. It exceeds the CHRE-based CSA in terms of precision. Simulation results show the validity of the algorithm.

To improve further the accuracy in the approximation of the range history for a synthetic aperture radar (SAR) onboard a mediumearth-orbit (MEO) satellite a fourth power term has been added to the advanced hyperbolic range equation (AHRE), and the new one is called a modified AHRE (MAHRE). Then, a two-dimensional spectrum using the MAHRE was derived, and the accuracy of the spectrum analysed. Promising results were obtained. Introduction:The integration time of one synthetic aperture of a medium-earth-orbit (MEO) synthetic aperture radar (SAR) can be long or a fine spatial resolution can be achieved. However, the long integration time means that the approximation in range history with a straight flight path within one synthetic aperture could become invalid. Thus, the use of a typical hyperbolic range equation can be questionable. An accurate but potentially complicated equation is needed for the processing of the MEO SAR data. Eldhuset [1] approximated the relative earth and satellite motion using a fourth-order Taylor expansion of the equation in azimuth time. Although Eldhuset achieved accurate range approximation, one drawback was the complicated algebraic operations involved in the development of imaging algorithms for data processing with the approximation. Then, Huang et al. [2] studied an advanced hyperbolic range equation (AHRE) consisting of terms up to the third order of the range equation in [1] and a linear term. They simplified the range equation of [1]. Using the AHRE, Huang et al. derived the 2D spectrum and then developed an advanced nonlinear chirp scaling (ANLCS) algorithm [3]. Closely examining the AHRE, one notices that the AHRE only compensates for the range history up to the cubic term. Under some circumstances, the compensation might not be enough or the level of accuracy might be inadequate. To increase the accuracy level, we modify the AHRE with an addition of a fourthorder term or modified AHRE (MAHRE). This addition is not simply a roll-back to the original range equation in [1]. Instead, the addition is directly applied to the AHRE. Thus, the MAHRE is still concise in expression.