# Effect of 6-DOF Oscillation of Ship Target on SAR Imaging

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The state-of-the-art researches concerning ship oscillation and SAR imaging are relatively dispersed, and there is no systematic description and derivation of the phase distortion introduced by ship 6-DOF oscillation;
- (2)
- The ship oscillation in most of the existing analyses are obtained through numerical simulation, and part of the conclusions lack the verification by measured attitude data.

## 2. Model of Oscillatory Ship Target

#### 2.1. Range Model of a Ship Target Based on 6-DOF Motion

_{x}, N

_{y}, N

_{z}indicate the numbers of frequency components associated with surge, sway, and heave.

_{x}, M

_{y}, M

_{z}indicate the numbers of frequency components associated with roll, pitch, and yaw.

_{p}, Y

_{p}, Z

_{p}). Thus, the position of point P in the o′-x′y′z′ coordinate system at time t can be derived by the following coordinate transformation:

_{0}, y

_{0}, z

_{0}).

#### 2.2. Ship Linear Oscillation

#### 2.2.1. Single-Frequency Linear Oscillation

_{0}(x

_{0}, y

_{0}, z

_{0}), the displacement of point P can be decomposed into three axes:

_{0}represents the cosine value of the angle between the oscillating axis and RLOS at the center moment.

_{t}is the reciprocal of a quadratic polynomial, and it can be ignored when observation time is short or grazing angle is small.

_{t}and the first-order term in the bracket. In the derivation of the new range approximate equation, the main modification was replacing the constant $\mathrm{cos}\langle \overrightarrow{P{P}_{0}},\overrightarrow{{R}_{0}{P}_{0}}\rangle $ with variable $\mathrm{cos}\langle \overrightarrow{P{P}_{0}},\overrightarrow{R{P}_{0}}\rangle $, which can better represent the projection direction of target oscillation. In the second item of Equation (14), the coefficient of the linear term is much smaller than the constant C

_{0}in most scenarios, which does not have a great impact on the range. However, when C

_{0}≈ 0, such as the radar works in the side-looking mode and the target oscillates along the azimuth direction, the linear term would become dominant. However, in this case, the range model in [7] shows that the range distortion introduced by target linear oscillation is zero, which is unreasonable. In order to compare the fitting effects of these two approximate methods, an experiment was conducted with a set of typical parameters: ${A}_{l}=1$ m, ${\omega}_{l}=2\pi /3$ rad/s, ${\phi}_{0}=0{}^{\xb0}$, ${v}_{a}=140$ m/s, $H=6$ km, ${f}_{c}=5.4$ GHz, ${\alpha}_{0}=90{}^{\xb0}$, ${\beta}_{0}=40{}^{\xb0}$. The simulation results are shown in Figure 4.

_{t}in Equation (14). In Figure 4b, the model in [7] showed a quite large fitting error when the target oscillates along the azimuth direction, and the distortion envelope is linearly modulated. This error is mainly derived from ignoring the change of projection direction, and it was also improved by adding the first-order term to the constant C

_{0}.

_{0}is close to zero. Figure 5 shows the micro-Doppler history caused by target linear oscillation.

#### 2.2.2. Multi-Frequency Linear Oscillation

_{l}indicates the number of frequency components associated with linear oscillation.

#### 2.3. Ship Angular Oscillation

_{0}(x

_{0}, y

_{0}, z

_{0}). The geometric model is shown in Figure 6.

#### 2.3.1. Single-Frequency Angular Oscillation

_{p}, Y

_{p}, Z

_{p}). Based on Equation (5), the coordinate of P in space coordinate system o-xyz can be expressed as

_{a}/H is usually close to zero, the first component will be dominant in most scenarios.

_{t}which is close to 1, the range distortion in Equation (28) can be rewritten as

_{x}is small, J

_{n}(B

_{x}) will decrease rapidly with the increase of n, so they can be approximated by several sinusoidal functions. The results are similar when there is only pitch motion, but are a little different for yaw motion, the specific results are shown in Appendix B.

- (1)
- Under the given coordinate (10 m, 10 m, 10 m), the magnitude of the micro-Doppler introduced by the three angular oscillations is ${f}_{d\_roll}>{f}_{d\_pitch}>{f}_{d\_yaw}$. The roll motion seems to be dominant when the ship is small, and this dominance will gradually weaken as the ship size increases.
- (2)
- When all the three angular oscillations exist at the same time, the micro-Doppler will become a relatively complex form, and the its periodicity will also be weakened, which may require more sinusoidal terms to better fit it.
- (3)
- With different heading angles, the micro-Doppler caused by angular oscillations has significant differences. According to Equation (33), the mean micro-Doppler introduced by roll motion is determined by $2\mathrm{sin}{\alpha}_{v}\mathrm{sin}{\beta}_{0}{X}_{p}{v}_{a}/\left(\lambda H\right)-2{L}_{x\_2}\mathrm{cos}{\theta}_{x0\_2}{J}_{0}\left({B}_{x}\right)/\lambda $. Bring the simulation parameters into this formula, when the heading angle is 0° and 90°, the mean micro-Doppler of point P introduced by carrier rolling is –5.45 Hz and 5.45 Hz, respectively. The calculation results agree well with simulation results in Figure 9e, which proves the validity of the proposed model.

#### 2.3.2. Multi-Frequency Angular Oscillation

_{t}which is close to 1, the range distortion introduced by multi-frequency roll motion can be expressed as

_{x}frequency components, we took the dual-frequency roll motion as an example to make a tentative analysis. When M

_{x}= 2, based on Jacobi–Anger expansion, the composite cosine in Equation (34) can be expanded as

## 3. The Effect of Oscillation on Imaging

#### 3.1. CPI Less Than the Oscillation Period

#### 3.1.1. Ship Linear Oscillation

_{t}is ignored, the range distortion introduced by linear oscillation can be concluded to the following form:

_{0}, the maximum variation of each order phase error component can be expressed as follows:

_{l}is equal to A

_{l}and k

_{l}is approaching zero. When the target oscillates along the azimuth direction, b

_{l}is close to zero and k

_{l}is far less than A

_{l}. Therefore, the maximum variation of phase error components will present sinusoidal forms, as shown in Figure 10. To explicitly illustrate the impact of linear oscillation on imaging and verify the validity of the presented analysis, we generated the echoes of a point target with linear oscillation. The algorithm used for imaging is the classic Range Doppler (RD) algorithm, and the results are shown in Figure 11.

#### 3.1.2. Ship Angular Oscillation

_{0}, the maximum variation of each order phase error component can be expressed as follows:

#### 3.2. CPI Greater Than the Oscillation Period

#### 3.2.1. Ship Linear Oscillation

_{l}is proportional to v

_{a}/H, so its value will be far less than 1 and its impact can be ignored in most scenarios except for b

_{l}= 0. In order to use the Bessel function to analyze this linear modulated sinusoidal phase error, we rewrite this term as follows:

_{l}is small, the linear modulated sinusoidal phase error will produce two symmetrical echoes on both sides of $n\Delta t$, and the echoes will decrease rapidly with the increase of t, as shown in Figure 16b.

#### 3.2.2. Ship Angular Oscillation

## 4. Measured Data and Experimental Results

#### 4.1. Measured Data of Ship Attitude

#### 4.1.1. Experimental Condition

#### 4.1.2. Measured Data

- (1)
- Set off from the harbor to experimental region;
- (2)
- Turned off the power and allowed the ship to drift along the current, then recorded the attitude data of the ship, this process lasted about 1.5 h;
- (3)
- Anchored the ship to the center of the anchorage, then recorded the attitude data of the ship, this process continued for 2 h;
- (4)
- Weighed anchor and drove the ship back-and-forth along a specific route, recorded the attitude data of the ship during this process, this step took about 40 min;
- (5)
- Back to the harbor.

#### 4.2. Experiments Based on the Measured Attitude Data

#### 4.2.1. Focusing Results of the Oscillatory Ship

#### 4.2.2. Phase Compensation Based on the Proposed Range Model

- (1)
- verifying the accuracy of the proposed range model for ship angular oscillation;
- (2)
- exploring the feasibility of phase compensation by fitting ship attitude angles with multi-frequency oscillation model.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Specific Derivation Result of Range Distortion When Roll, Pitch, and Yaw Motion All Exist

## Appendix B. The Range Distortion and micro-Doppler Introduced by Pitch and Yaw

## References

- Cumming, I.; Wong, F. Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation; Artech House: Boston, MA, USA, 2005; pp. 113–168. [Google Scholar]
- Vessel Motion Calculator. Available online: https://calqlata.com/productpages/00059-help.html (accessed on 16 February 2021).
- Yao, G.; Xie, J.; Huang, W. Ocean Surface Cross Section for Bistatic HF Radar Incorporating a Six DOF Oscillation Motion Model. Remote Sens.
**2019**, 11, 2738. [Google Scholar] [CrossRef] [Green Version] - Yao, G.; Xie, J.; Huang, W. HF Radar Ocean Surface Cross Section for the Case of Floating Platform Incorporating a Six-DOF Oscillation Motion Model. IEEE J. Ocean. Eng.
**2021**, 46, 156–171. [Google Scholar] [CrossRef] - Ouchi, K.; Iehara, M.; Morimura, K.; Kumano, S.; Takami, I. Nonuniform azimuth image shift observed in the Radarsat images of ships in motion. IEEE Trans. Geosci. Remote Sens.
**2002**, 40, 2188–2195. [Google Scholar] [CrossRef] - Doerry, A. Ship Dynamics for Maritime ISAR Imaging; No. SAND2008-1020; Sandia National Laboratories: Albuquerque, NM, USA, 2008. [Google Scholar]
- Li, X.; Deng, B.; Qin, Y.; Wang, H.; Li, Y. The influence of target micromotion on SAR and GMTI. IEEE Trans. Geosci. Remote Sens.
**2011**, 49, 2738–2751. [Google Scholar] [CrossRef] - Liu, P.; Jin, Y. A study of ship rotation effects on SAR image. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 3132–3144. [Google Scholar] [CrossRef] - Liu, W.; Sun, G.; Xia, X.; Fu, J.; Xing, M.; Bao, Z. Focusing Challenges of Ships with Oscillatory Motions and Long Coherent Processing Interval. IEEE Trans. Geosci. Remote Sens.
**2020**. [Google Scholar] [CrossRef] - Biondi, F. COSMO-SkyMed Staring Spotlight SAR Data for Micro-Motion and Inclination Angle Estimation of Ships by Pixel Tracking and Convex Optimization. Remote Sens.
**2019**, 11, 766. [Google Scholar] - Biondi, F.; Addabbo, P.; Orlando, D.; Clemente, C. Micro-Motion Estimation of Maritime Targets Using Pixel Tracking in Cosmo-Skymed Synthetic Aperture Radar Data—An Operative Assessment. Remote Sens.
**2019**, 11, 1637. [Google Scholar] [CrossRef] [Green Version] - Xu, G.; Xing, M.; Xia, X.; Zhang, L.; Chen, Q.; Bao, Z. 3D Geometry and Motion Estimations of Maneuvering Targets for Interferometric ISAR With Sparse Aperture. IEEE Trans. Image Process.
**2016**, 25, 2005–2020. [Google Scholar] [CrossRef] - Wang, F.; Xu, F.; Jin, Y. 3-D information of a space target retrieved from a sequence of high-resolution 2-D ISAR images. In Proceedings of the 2016 IEEE International Geoscience and Remote Sensing Symposium, Beijing, China, 10–15 July 2016; pp. 5000–5002. [Google Scholar]
- Ruegg, M.; Meier, E.; Nuesch, D. Vibration and rotation in millimeter-wave SAR. IEEE Trans. Geosci. Remote Sens.
**2007**, 45, 293–304. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Sun, J.; Lei, P.; Hong, W. SAR-based paired echo focusing and suppression of vibrating targets. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 7593–7605. [Google Scholar] [CrossRef] - Liu, Y.; Zhu, D.; Li, X.; Zhuang, Z. Micromotion characteristic acquisition based on wideband radar phase. IEEE Trans. Geosci. Remote Sens.
**2013**, 52, 3650–3657. [Google Scholar] [CrossRef] - Zhu, M.; Zhou, X.; Zang, B.; Yang, B.; Xing, M. Micro-Doppler Feature Extraction of Inverse Synthetic Aperture Imaging Laser Radar Using Singular-Spectrum Analysis. Sensors
**2018**, 18, 3303. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ubeyli, E.D.; Güler, I. Spectral analysis of internal carotid arterial Doppler signals using FFT, AR, MA, and ARMA methods. Comput. Biol. Med.
**2003**, 34, 293–306. [Google Scholar] [CrossRef] - Margarit, G.; Mallorqui, J.; Rius, J.; Sanz-Marcos, J. On the usage of GRECOSAR, an orbital polarimetric SAR simulator of complex targets, to vessel classification studies. IEEE Trans. Geosci. Remote Sens.
**2006**, 44, 3517–3526. [Google Scholar] [CrossRef] - Margarit, G.; Mallorqui, J.; Fortuny-Guasch, J.; Lopez-Martinez, C. Phenomenological vessel scattering study based on simulated inverse SAR imagery. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 1212–1223. [Google Scholar] [CrossRef] - Zhao, Y.; Zhang, M.; Zhao, Y.; Geng, X. A bistatic SAR image intensity model for the composite ship–ocean scene. IEEE Trans. Geosci. Remote Sens.
**2015**, 53, 4250–4258. [Google Scholar] [CrossRef] - Huo, W.; Huang, Y.; Pei, J.; Zhang, Y.; Yang, J. A New SAR Image Simulation Method for Sea-Ship Scene. In Proceedings of the IEEE 2018 International Geoscience and Remote Sensing Symposium, Valencia, Spain, 22–27 July 2018; pp. 721–724. [Google Scholar]
- Cochin, C.; Pouliguen, P.; Delahaye, B.; Le Hellard, D.; Gosselin, P.; Aubineau, F. MOCEM-An ‘all in one’ tool to simulate SAR image. In Proceedings of the 7th European Conference on Synthetic Aperture Radar, Friedrichshafen, Germany, 2–5 June 2008; pp. 1–4. [Google Scholar]
- Cochin, C.; Louvigne, J.C.; Fabbri, R.; Le Barbu, C.; Knapskog, A.O.; Ødegaard, N. Radar simulation of ship at sea using MOCEM V4 and comparison to acquisitions. In Proceedings of the 2014 International Radar Conference, Lille, France, 13–17 October 2014; pp. 1–6. [Google Scholar]
- Das, S.; Shiraishi, S.; Das, S. Mathematical modeling of sway, roll and yaw motions: Order-wise analysis to determine coupled characteristics and numerical simulation for restoring moment’s sensitivity analysis. Acta Mech.
**2010**, 213, 305–322. [Google Scholar] [CrossRef] - Neves, M.; Rodríguez, C. A coupled non-linear mathematical model of parametric resonance of ships in head seas. Appl. Math. Model.
**2009**, 33, 2630–2645. [Google Scholar] [CrossRef] - Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions (Applied Mathematics Series 55); The National Bureau of Standards: Washington, DC, USA, 1964. [Google Scholar]
- Donald, R. High-Resolution Radar, 2nd ed.; Artech House: Boston, MA, USA, 1995; pp. 408–411. [Google Scholar]
- Wang, L. Study on Key Technologies of Inverse Synthetic Aperture Radar Imaging. Ph.D. Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, China, 2006. [Google Scholar]
- Han, X.; Liu, Q.; Yang, W.; Guo, J.; Song, Y. The Effect of Amplitude and Phase Distortion on the Quality of One-dimensional High Resolution Range Profile. In Proceedings of the 2019 IEEE International Conference on Signal, Information and Data Processing, Chongqing, China, 11–13 December 2019; pp. 1–5. [Google Scholar]
- Barber, B. Some effects of target vibration on SAR images. In Proceedings of the 7th European Conference on Synthetic Aperture Radar, Friedrichshafen, Germany, 2–5 June 2008; pp. 1–4. [Google Scholar]
- Zhang, C. Synthetic Aperture Radar: Principle, System Analysis and Application; Science Press: Beijing, China, 1989; pp. 163–178. [Google Scholar]
- Hu, G.; Xiang, J.; Wang, F. Limitations of cubic phase error on SAR azimuth resolution. Acta Electron. Sin.
**2005**, 33, 2366–2369. [Google Scholar] - Yan, X.; Chen, J.; Nies, H.; Loffeld, O. Analytical Approximation Model for Quadratic Phase Error Introduced by Orbit Determination Errors in Real-Time Spaceborne SAR Imaging. Remote Sens.
**2019**, 11, 1663. [Google Scholar] [CrossRef] [Green Version]

**Figure 4.**The range distortions introduced by target linear oscillation. (

**a**) Oscillating along the range direction; (

**b**) oscillating along the azimuth direction. Red solid lines represent the true values of the range distortions obtained by numerical calculation. Blue dotted lines are the results obtained by proposed model, which are basically coincide with the red lines. Yellow dot-dash lines are the results obtained by the model in reference [7].

**Figure 5.**The micro-Doppler introduced by target linear oscillation. (

**a**) Oscillating along the range direction; (

**b**) oscillating along the azimuth direction.

**Figure 7.**The trajectories of point P in the fixed space coordinate system during 30 s. (

**a**) Point oscillates with the destroyer; (

**b**) point oscillates with the carrier.

**Figure 8.**The range distortions of point P caused by ship angular oscillations. (

**a**) Destroyer, roll motion; (

**b**) destroyer, pitch motion; (

**c**) destroyer, yaw motion; (

**d**) destroyer, all the three angular oscillations; (

**e**) carrier, roll motion; (

**f**) destroyer, pitch motion; (

**g**) destroyer, yaw motion; (

**h**) carrier, all the three angular oscillations. Red solid lines represent the true values of the range distortions obtained by numerical calculation. Blue dotted lines are the results obtained by proposed model, which are basically coincide with the red lines. Yellow dot-dash lines are the results obtained by the model in reference [7].

**Figure 9.**The micro-Doppler of point P introduced by ship oscillations. (

**a**) Destroyer, roll motion; (

**b**) destroyer, pitch motion; (

**c**) destroyer, yaw motion; (

**d**) destroyer, all the three angular oscillations; (

**e**) carrier, roll motion; (

**f**) destroyer, pitch motion; (

**g**) destroyer, yaw motion; (

**h**) carrier, all the three angular oscillations.

**Figure 10.**The maximum variation of phase error components from 1st-order to 4th-order, the oscillation period and amplitude is set as 30 s and 1 m, respectively. (

**a**) Target oscillates along the range direction; (

**b**) target oscillates along the azimuth direction.

**Figure 11.**Focusing results of point target with linear oscillations. (

**a**) Target oscillates along the range direction, φ

_{l}= 0; (

**b**) target oscillates along the range direction, φ

_{l}= π/2; (

**c**) target oscillates along the azimuth direction, φ

_{l}= 0; (

**d**) target oscillates along the azimuth direction, φ

_{l}= π/2. The red dot represents the true position of the target.

**Figure 12.**The maximum variation of each order phase error components introduced by destroyer angular oscillation. (

**a**) Roll motion; (

**b**) pitch motion; (

**c**) yaw motion; (

**d**) all the three angular oscillations.

**Figure 13.**Focusing results of the point target with angular oscillations of the destroyer on sea-state 5. (

**a**) Roll motion, ${\Psi}_{x}=0$; (

**b**) pitch motion, ${\Psi}_{y}=0$; (

**c**) yaw motion, ${\Psi}_{z}=0$; (

**d**) all the three angular oscillations, ${\Psi}_{x}={\Psi}_{y}={\Psi}_{z}=0$; (

**e**) roll motion, ${\Psi}_{x}=\pi /2$; (

**f**) pitch motion, ${\Psi}_{y}=\pi /2$; (

**g**) yaw motion, ${\Psi}_{z}=\pi /2$; (

**h**) all the three angular oscillations, ${\Psi}_{x}={\Psi}_{y}={\Psi}_{z}=\pi /2$.

**Figure 14.**The maximum variation of phase error components introduced by carrier angular oscillation. (

**a**) Roll motion; (

**b**) pitch motion; (

**c**) yaw motion; (

**d**) all the three-axis oscillations.

**Figure 15.**Focusing results of the point target with angular oscillations of the carrier on sea-state 5. (

**a**) Roll motion, ${\Psi}_{x}=0$; (

**b**) pitch motion, ${\Psi}_{y}=0$; (

**c**) yaw motion, ${\Psi}_{z}=0$; (

**d**) all the three angular oscillations exist, ${\Psi}_{x}={\Psi}_{y}={\Psi}_{z}=0$; (

**e**) roll motion, ${\Psi}_{x}=\pi /2$; (

**f**) pitch motion, ${\Psi}_{y}=\pi /2$; (

**g**) yaw motion, ${\Psi}_{z}=\pi /2$; (

**h**) all the three angular oscillations exist, ${\Psi}_{x}={\Psi}_{y}={\Psi}_{z}=\pi /2$.

**Figure 16.**The effect of sinusoidal phase error. (

**a**) Sinusoidal phase error; (

**b**) linearly modulated sinusoidal phase error. The blue line represents the pulse compression result of the original LFM signal, the red line represents the pulse compression result of the signal added phase error.

**Figure 17.**Focusing results of point target with linear oscillation. (

**a**) Target oscillates along the range direction, A

_{l}= 1 m; (

**b**) target oscillates along the range direction, A

_{l}= 0.05 m; (

**c**) target oscillates along the azimuth direction, A

_{l}= 1 m; (

**d**) target oscillates along the azimuth direction, A

_{l}= 0.05 m.

**Figure 18.**Focusing results of the point target with angular oscillations of two ships on sea-state 5. (

**a**) Destroyer, roll motion; (

**b**) destroyer, pitch motion; (

**c**) destroyer, yaw motion; (

**d**) destroyer, all the three angular oscillations; (

**e**) carrier, roll motion; (

**f**) carrier, pitch motion; (

**g**) carrier, yaw motion; (

**h**) carrier, all the three angular oscillations.

**Figure 19.**Experimental condition of this test: (

**a**) experimental vessel (20 m × 3 m); (

**b**) fixed position and appearance of the IMU; (

**c**) experimental region (an anchorage near Yantai Port).

**Figure 20.**Part of the measured ship motion data during 200 s. (

**a**–

**c**) Attitude angles of the experimental ship in the unanchored state, anchored state, and navigating state; (

**d**–

**f**) velocity of the experimental ship in the unanchored state, anchored state, and navigating state.

**Figure 22.**The ship attitude angles used for imaging. (

**a**) In the unanchored state; (

**b**) in the anchored state; (

**c**) in the navigating state.

**Figure 23.**The focusing results of the oscillatory ship in the different states. (

**a**–

**e**) In the unanchored state; (

**f**–

**j**) in the anchored state; (

**k**–

**o**) in the navigating state. The motion types from left to right are roll, pitch, yaw, all the three angular oscillations, and stationary.

**Figure 24.**The measured ship attitude angles with a duration of 20 s. (

**a**) A drifting ship in the unanchored state; (

**b**) a drifting ship in the anchored state; (

**c**) a moving ship in the navigating state.

**Figure 25.**The fitting results of the measured ship attitude angles by using 1~8 terms of Fourier series. (

**a**–

**c**) Roll, pitch, heading angles in the unanchored state; (

**d**–

**f**) roll, pitch, heading angles in the navigating state. The abscissa of the graph is the number of the terms used for fitting, and the ordinate is the R-square (coefficient of determination), which indicates the fitting effect.

**Figure 26.**The residual phase errors of Point A, B, and C after phase compensation. (

**a**–

**d**) In the unanchored state; (

**e**–

**h**) in the navigating state. The number of sinusoidal terms used for fitting attitude angles from left to right is 1, 2, 3, 4.

**Figure 27.**The refocusing results of the oscillatory ship after phase compensation. (

**a**–

**d**) In the unanchored state; (

**e**–

**h**) in the navigating state. The number of terms used for fitting attitude angles from left to right is 1, 2, 3, 4.

Symbols | The Meaning of Symbol |
---|---|

O-XYZ | The ship-fixed coordinate system |

o′-x′y′z′ | The interim space coordinate system |

o-xyz | The fixed space coordinate system |

$\Delta X,\Delta Y,\Delta Z$ | The coordinate changes caused by the ship’s surge, sway, and heave |

${A}_{x},{A}_{y},{A}_{z}$ | The amplitude of the ship’s surge, sway, and heave |

${\omega}_{x},{\omega}_{y},{\omega}_{z}$ | The angular frequency of the ship’s surge, sway, and heave |

${\phi}_{x},{\phi}_{y},{\phi}_{z}$ | The initial phase of the ship’s surge, sway, and heave |

${A}_{l},{\omega}_{l},{\phi}_{l}$ | The amplitude, angular frequency, and initial phase of a linearly oscillating target along an axis of the space |

${\theta}_{x},{\theta}_{y},{\theta}_{z}$ | The roll angle, pitch angle, and yaw angle of the ship |

${B}_{x},{B}_{y},{B}_{z}$ | The amplitude of the ship’s roll, pitch, and yaw |

${\Omega}_{x},{\Omega}_{y},{\Omega}_{z}$ | The angular frequency of ship’s roll, pitch, and yaw |

${\Psi}_{x},{\Psi}_{y},{\Psi}_{z}$ | The initial phase of the ship’s roll, pitch, and yaw |

$H$ | The height of the radar platform (airplane) |

${v}_{a}$ | The velocity of the radar platform |

${v}_{s}$ | The velocity of the ship |

${\alpha}_{v}$ | Heading angle, the angle between ship’s sailing direction and x-axis |

${\alpha}_{0}$ | Radar observation angle, the angle between the RLOS ^{1} projection direction and the platform moving direction |

${\alpha}_{1}$ | The angle between the projection of linear oscillation axis and y-axis |

${\beta}_{0}$ | Grazing angle, the angle between the RLOS direction and the sea level |

${\beta}_{1}$ | The angle between the linear oscillation axis and the sea level |

^{1}RLOS = radar line of sight.

Serial Number | Motion Name | Description |
---|---|---|

1 | Surge | The linear oscillation of a ship along its longitudinal axis. |

2 | Sway | The linear oscillation of a ship along its transverse axis. |

3 | Heave | The linear oscillation of a ship along its vertical axis. |

4 | Roll | The angular oscillation of a ship around its longitudinal axis. |

5 | Pitch | The angular oscillation of a ship around its transverse axis. |

6 | Yaw | The angular oscillation of a ship around its vertical axis. |

Ship Type | Motion Type | Double Amplitude (deg) | Average Period (sec) |
---|---|---|---|

Destroyer | Roll | 38.4 | 12.2 |

Pitch | 3.4 | 6.7 | |

Yaw | 3.8 | 14.2 | |

Carrier | Roll | 5.0 | 26.4 |

Pitch | 0.9 | 11.2 | |

Yaw | 1.33 | 33.0 |

Symbol | Parameter | Values |
---|---|---|

$f$ | Center Frequency | 5.4 GHz |

${f}_{a}$ | Pulse Repetition Frequency (PRF) | 420 Hz |

${B}_{r}$ | Signal Bandwidth | 300 MHz |

$H$ | Platform Height | 6 km |

${\beta}_{0}$ | Grazing Angle | 40° |

${\theta}_{rc}$ | Squint Angle | 0° |

${v}_{a}$ | Platform Velocity | 140 m/s |

${L}_{D}$ | Antenna Length | 1 m |

${T}_{0}$ | CPI | 3.73 s |

Phase Error Order | Effects on SAR Azimuth Images |
---|---|

1 | Peak displacement |

2 | Defocus of impulse response, decrease of peak amplitude |

3 | Unbalanced sidelobes, peak displacement, and amplitude decrease |

4 | Symmetrical increase of sidelobe, decrease of peak amplitude |

Higher-order | Paired echoes, ghost images |

Data | Center Position | Parameter | Value |
---|---|---|---|

28-12-2019 08:14 (local time) | 37°39.815N 121°29.812E | Wind speed | 4.8 m/s |

Wind direction | 211.2° | ||

Wave height | 0.7 m | ||

28-12-2019 14:14 (local time) | 37°40.395N 121°31.293E | Wind speed | 4.0 m/s |

Wind direction | 209.0° | ||

Wave height | 0.5 m |

Ship State | Variation Range (deg) | Standard Deviation (deg) | ||||
---|---|---|---|---|---|---|

Roll | Pitch | Yaw | Roll | Pitch | Yaw | |

Unanchored | 7.14 | 3.66 | 14.59 | 1.23 | 0.49 | 4.84 |

Anchored | 6.30 | 2.63 | 5.71 | 1.22 | 0.45 | 1.64 |

Navigating | 6.53 | 2.71 | 33.39 | 0.99 | 0.48 | 9.36 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, B.; Qi, X.; Zhang, J.; Zhang, H.
Effect of 6-DOF Oscillation of Ship Target on SAR Imaging. *Remote Sens.* **2021**, *13*, 1821.
https://doi.org/10.3390/rs13091821

**AMA Style**

Zhou B, Qi X, Zhang J, Zhang H.
Effect of 6-DOF Oscillation of Ship Target on SAR Imaging. *Remote Sensing*. 2021; 13(9):1821.
https://doi.org/10.3390/rs13091821

**Chicago/Turabian Style**

Zhou, Binbin, Xiangyang Qi, Jiahuan Zhang, and Heng Zhang.
2021. "Effect of 6-DOF Oscillation of Ship Target on SAR Imaging" *Remote Sensing* 13, no. 9: 1821.
https://doi.org/10.3390/rs13091821