# An Accurate GEO SAR Range Model for Ultralong Integration Time Based on mth-Order Taylor Expansion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- As the azimuth resolution increases, the low-order Taylor expansion range model shows insufficient accuracy in ultralong integration time.
- (2)
- The iterative approximation range model has a high accuracy but cannot be expressed by an analytical expression.
- (3)
- The range models lack the flexibility of precision adjustment for different exposure times.

## 2. Characteristics of GEO SAR

#### 2.1. Invalidation of “Stop-and-Go” Assumption

#### 2.2. Ultralong Integration Time

- Set an initial search region [T
_{min}, T_{max}] according to the estimated integration time by (1), and this region is required to include the real integration time. - Define T
_{mid}= (T_{min}+ T_{max})/2. If T_{max}− T_{min}is smaller than the threshold, then return the T_{real}= T_{mid}and stop the iteration. - Calculate the synthetic aperture angle of different exposure time T
_{min}, T_{max}, and T_{mid}, denoted as θ_{min}, θ_{max}, and θ_{mid}. - Define the new search region: if θ
_{mid}> θ_{syn}, then T_{max}= T_{mid}, else let T_{min}= T_{mid}. Return to step 2.

#### 2.3. “Non-Stop-and-Go” Range Models

- (1)
- Fourth-order Taylor expansion model considering the invalidation of “stop-and-go” assumption. This model can be expressed in the form of a power series, which can obtain the 2-D spectrum to form an efficient frequency domain imaging algorithm. However, limited by the order of expansion, this model only has a satisfactory fitting effect in a short observation time. As the exposure time increases, the fitting error accumulates rapidly at both edges of the aperture.
- (2)
- Iterative approximation range model. The main idea of this range model is to use twice the transmitting delay to approximately calculate the satellite position in the orbit, then an iteration can be formed to calculate a relatively accurate result. The calculation of this model is simple, and only one iteration already has a high accuracy, which can be used in echo generation and time-domain imaging algorithms. However, this range model cannot be expressed by an analytical expression, so its usage for frequency domain imaging algorithms is limited.

## 3. Proposed Range Model and Imaging Algorithm

#### 3.1. Proposed Range Model

- (1)
- A general calculation method for mth-order expansion of pulse transmitting distance is given, and the expansion order can be adjusted according to the integration time and error accuracy requirement.
- (2)
- An accurate pulse receiving distance is obtained by using the thought of iterative approximation, and the analytical expression is obtained by Taylor expansion in the ECEF coordinate system.

#### 3.1.1. Pulse Transmitting Distance

**r**

_{1n}, we can first square the two sides of Equation (2),

**A**

_{sn}and

**A**

_{tn}are the acceleration of satellite and target, respectively.

_{m}represents the mth-order coefficient of azimuth time.

**R**

_{sn}and

**R**

_{tn}. Considering that the motion state of the target is known already, the key point to this problem lies in the derivative of the satellite position vector. During the calculation process, we found that the mth-order derivative of

**R**

_{sn}has a form that is easy to represent and iteratively calculate in the orbital coordinate system. Therefore, the derivatives of

**R**

_{sn}is first calculated in the orbital coordinate system, and are then rotated into the ECEF coordinate system. For convenience, the three coordinate systems used in the subsequent derivation are defined here—E

_{v}represents the orbital coordinate system, E

_{o}represents the ECI coordinate system, and E

_{g}represents the ECEF coordinate system.

_{v}can be expressed as:

^{14}m

^{3}/s

^{2}, P and e are the semi-latus rectum and the eccentricity of the orbit.

_{v}can be obtained. The first-order to fifth-order derivatives of f are given in Appendix A.

**R**

_{s_v}from E

_{v}to E

_{o}. Considering that the rotation matrix A

_{ov}is constant, it can be directly written as:

**R**

_{s_o}from E

_{o}to E

_{g}. However, the rotation matrix A

_{go}has the variable orbit time t

_{s}, so we also need to take the derivative of A

_{go}.

_{go}can be written as:

_{g}is the Greenwich hour angle, ω

_{e}is the angular velocity of the earth’s rotation, t

_{s}is the Greenwich mean time. The mth-order derivative of A

_{go}can be expressed as:

_{g}can be written as follows:

#### 3.1.2. Pulse Receiving Distance

_{n}is subsecond, its higher order terms have little effect on the fitting error. Therefore, this paper only performs a second-order Taylor expansion of the receiving distance on fast time, and the subsequent simulations in Section 4 also prove that the second-order expansion can achieve sufficiently high accuracy.

_{n}cannot be solved, the idea of one iteration approximation is employed here. The total propagation time is replaced by double the transmitting delay:

#### 3.1.3. Total Pulse Propagation Distance

_{1n}and ${\Delta}r$ represents the Taylor expansion of the pulse transmitting distance and the compensation term for the “stop-and-go” error, respectively. The fitting precision of this range model for the ultralong integration time is mainly determined by r

_{1n}, whose accuracy can be adjusted by the expansion order m. The calculation of ${\Delta}r$ mainly uses the ideas of iterative approximation and twice Taylor expansion, since the total propagation time is subsecond, there is often no need to expand ${\Delta}r$ to a very high order in actual use.

#### 3.2. Imaging Method

- (1)
- Calculating the true anomaly and its derivatives from 1st to (m − 1)-th order based on the satellite ephemeris data and orbit configuration;
- (2)
- Determining the Taylor expansion order m by the SAR system parameters and error accuracy, and then performing the mth-order Taylor expansion to obtain the transmitting distance for each grid point.
- (3)
- Calculating the compensation term of the “stop-and-go” assumption failure, and then bringing the complete pulse propagation distance into the azimuth compression.

## 4. Simulation Results and Discussion

#### 4.1. Fitting Error of the Pulse Transmitting Distance

#### 4.2. Fitting Error of Compensation Term

^{−5}and 10

^{−4}rad, which verified the effectiveness of the proposed compensation term for a long integration time.

#### 4.3. Focusing Results at Different Orbit Position

#### 4.4. Focusing Results with Ultralong Integration Time

#### 4.5. The Influence of Different Sources of Distortion on Imaging

- (1)
- Trajectory distortions caused by errors in measuring the parameters of the spacecraft motion.
- (2)
- Hardware distortions and instabilities during the formation and reception of probing signals (in particular, phase instability, instability of the microwave path parameters).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{s}is the separation distance between the satellite and earth center, f is the true anomaly. P, e, and a are the semi-latus rectum, eccentricity, and semi-major axis of the orbit, respectively.

_{s}and f can be expressed as:

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**Figure 1.**GEO SAR pulse propagation model in the ECEF coordinate system. P is a point target on the earth, S

_{0}is the satellite position when the pulse is transmitted, S

_{1}is the satellite position when the pulse reaches the target, S

_{2}is the satellite position when the pulse is received. r

_{1}and r

_{2}represent the propagation distance of pulse transmission and reception, respectively.

**Figure 3.**GEO SAR velocity. (

**a**) “8” shaped orbit in ECI; (

**b**) “8” shaped orbit in ECEF; (

**c**) Near-circular orbit in ECI; and (

**d**) Near-circular orbit in ECEF.

**Figure 5.**Fitting error comparison of several GEO SAR range models. (

**a**) “Stop-and-go” assumption; (

**b**) fourth-order Taylor expansion; and (

**c**) iterative approximation.

**Figure 6.**The propagation history of the transmitting pulse at the nth PRT. P is a point target on the earth.

**R**

_{sn},

**R**

_{tn},

**V**

_{sn}, and

**V**

_{tn}are the position vectors and velocity vectors of satellite and target, respectively.

**Figure 7.**The Flowchart of the mth-order Taylor expansion process of the pulse transmitting distance in GEO SAR.

**Figure 8.**The propagation history of transmitting and receiving pulse at the nth PRT. P is a point target on the earth.

**r**

_{1n},

**r**

_{2n}represent the transmitting distance vector and receiving distance vector, respectively. τ

_{n}is the total propagation time of the pulse.

**Figure 9.**Flowchart of the improved BP algorithm based on the mth-order Taylor expansion range model.

**Figure 10.**Pulse transmitting distance fitting error comparison of different order Taylor expansion. (

**a**) Fourth-order; (

**b**) fifth-order; and (

**c**) sixth-order.

**Figure 11.**The maximum phase error of the whole orbit for 6000 s observation time. (

**a**) “8” shaped orbit; and (

**b**) a near-circular orbit.

**Figure 12.**The fitting error of “stop-and-go” assumption compensation term through the iterative approximation for different integration times. (

**a**) 100 s; (

**b**) 1000 s; (

**c**) 2000 s.

**Figure 13.**The fitting error of the “stop-and-go” assumption compensation term by the proposed method in this paper for different integration times. (

**a**) 100 s; (

**b**) 1000 s; (

**c**) 2000 s.

**Figure 14.**The fitting errors of the fourth-order Taylor expansion with 1000 s observation time at different orbit positions.

**Figure 15.**The fitting error and focusing result of the fourth-order Taylor expansion at perigee. (

**a**) Fitting error; (

**b**) 2-D contour; and (

**c**) azimuth profile.

**Figure 16.**The fitting error and focusing result of the fourth-order expansion at 45° of true anomaly. (

**a**) Fitting error; (

**b**) 2-D contour; and (

**c**) azimuth profile.

**Figure 17.**The fitting error and focusing result of sixth-order expansion at 45° of true anomaly. (

**a**) Fitting error; (

**b**) 2-D contour; and (

**c**) azimuth profile.

**Figure 18.**The fitting errors of the sixth-order Taylor expansion with 2000 s observation time at different orbit positions.

**Figure 20.**Focusing results with the range model proposed in this paper for 2000 s integration time at perigee.

**Figure 22.**Focusing results with the range model proposed in this paper for 2000 s integration time at 55° of true anomaly.

**Figure 23.**The simulation results of measurement error. (

**a**) The constant measurement error in the x-axis direction. (

**b**) The quadratic polynomial measurement error in the x-axis direction. (

**c**) The sinusoidal measurement error in the x-axis direction. (

**d**) The imaging result of the constant error. (

**e**) The imaging result of quadratic polynomial error. (

**f**) The imaging result of sinusoidal error.

**Figure 24.**The simulation results of phase distortion. (

**a**) The constant phase error. (

**b**) The quadratic polynomial phase error. (

**c**) The sinusoidal phase error. (

**d**) The imaging result of constant phase error. (

**e**) The imaging result of quadratic polynomial phase error. (

**f**) The imaging result of sinusoidal phase error.

Parameter | Value | Parameter | Value |
---|---|---|---|

Semi-major Axis | 42,164 km | RAAN ^{1} | 0° |

Inclination | 53°/7.4° | Argument of Perigee | 270° |

Eccentricity | 0.07/0.1 | Down Angle | 4.65° |

PRF | 70 Hz/140 Hz | Wave Length | 0.24 m |

Bandwidth | 75 MHz /150 MHz | Pulsewidth | 20 μs |

^{1}RAAN = Right Ascension of Ascending Node.

Range Model | Mean Error (rad) | Max Error (rad) | Standard Deviation (rad) |
---|---|---|---|

“Stop-and-go” assumption | 47.29 | 153.72 | 12.79 |

Fourth-order Taylor expansion | 3.95 | 50.56 | 4.41 |

Iterative approximation | 1.84 × 10^{−6} | 1.21 × 10^{−5} | 1.16 × 10^{−6} |

Expansion Order | Mean Error (rad) | Max Error (rad) | Standard Deviation (rad) |
---|---|---|---|

Fourth-order | 1.97 | 25.28 | 2.20 |

Fifth-order | 0.05 | 0.66 | 0.05 |

Sixth-order | 1.16 × 10^{−3} | 0.02 | 1.55 × 10^{−3} |

Positions (km) | PSLR(dB) | ISLR(dB) | IRW(m) | |||
---|---|---|---|---|---|---|

Range | Azimuth | Range | Azimuth | Range | Azimuth | |

(−20, 20) | −13.29 | −13.29 | −9.97 | −10.54 | 0.88 | 1.13 |

(−10, −10) | −13.29 | −13.30 | −9.98 | −10.54 | 0.88 | 1.13 |

(0, 0) | −13.28 | −13.31 | −9.97 | −10.55 | 0.88 | 1.13 |

(10, 10) | −13.28 | −13.32 | −9.97 | −10.55 | 0.88 | 1.13 |

(20, −20) | −13.30 | −13.33 | −9.97 | −10.56 | 0.88 | 1.13 |

Positions (km) | PSLR(dB) | ISLR(dB) | IRW(m) | |||
---|---|---|---|---|---|---|

Range | Azimuth | Range | Azimuth | Range | Azimuth | |

(−20, 20) | −13.30 | −13.19 | −10.07 | −10.55 | 0.88 | 0.76 |

(−10, −10) | −13.29 | −13.19 | −10.07 | −10.57 | 0.88 | 0.76 |

(0, 0) | −13.32 | −13.21 | −10.07 | −10.55 | 0.88 | 0.76 |

(10, 10) | −13.31 | −13.20 | −10.07 | −10.57 | 0.88 | 0.76 |

(20, −20) | −13.31 | −13.19 | −10.07 | −10.56 | 0.88 | 0.76 |

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**MDPI and ACS Style**

Zhou, B.; Qi, X.; Zhang, H.
An Accurate GEO SAR Range Model for Ultralong Integration Time Based on *m*th-Order Taylor Expansion. *Remote Sens.* **2021**, *13*, 255.
https://doi.org/10.3390/rs13020255

**AMA Style**

Zhou B, Qi X, Zhang H.
An Accurate GEO SAR Range Model for Ultralong Integration Time Based on *m*th-Order Taylor Expansion. *Remote Sensing*. 2021; 13(2):255.
https://doi.org/10.3390/rs13020255

**Chicago/Turabian Style**

Zhou, Binbin, Xiangyang Qi, and Heng Zhang.
2021. "An Accurate GEO SAR Range Model for Ultralong Integration Time Based on *m*th-Order Taylor Expansion" *Remote Sensing* 13, no. 2: 255.
https://doi.org/10.3390/rs13020255