# Evaluation of a Cubature Kalman Filtering-Based Phase Unwrapping Method for Differential Interferograms with High Noise in Coal Mining Areas

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## Abstract

**:**

## 1. Introduction

## 2. Phase Unwrapping Theory and Local Phase Slope Estimation

#### 2.1. Phase Unwrapping Theory

_{ϕ}(k) refers to the true phase error and ${\tilde{\epsilon}}_{\phi}(k)$ is the mapped phase error.

#### 2.2. Phase Slope Estimation

_{m}× B

_{n}) can be generally expressed as [21,27,28]:

_{m}, n = 1,2,…,B

_{n}; m and n refer to the row and column indices respectively; B

_{m}and B

_{n}are the row and column length of the local window respectively; a(m,n) is the random amplitude of the complex signal; f

_{x}and f

_{y}refer to the true local frequency in the row and column direction respectively; and w(m,n) is the corresponding noise.

## 3. Cubature Kalman Filtering Based Phase Unwrapping

#### 3.1. System Model for Phase Unwrapping

#### 3.1.1. State-Space Equation

#### 3.1.2. Observation Equation

_{1}(k) and v

_{2}(k) are the errors in the real and imaginary parts of the complex measurements, respectively. In a strict sense, the right-hand side of the second equal sign is actually more of a definition or a model than an equality [16], but it is written as shown for simplicity:

#### 3.2. CKF Phase Unwrapping (CKFPU) Algorithms

#### 3.2.1. One-Dimensional CKFPU Algorithm

_{x}and n

_{x}is the dimension of the state vector.

_{xx}can be obtained by Cholesky decomposition or singular value decomposition.

_{k}is the measurement noise, which has a Gaussian distribution.

#### 3.2.2. Two-Dimensional CKFPU Algorithm

_{(a,s)}is the estimated error variance matrix for phase slope at pixel (a,s); B denotes eight adjacent pixels for pixel (m,n) and L denotes the whole image. The optimal weight d(a,s) can be calculated as follows [21]:

#### 3.3. The Main Factors Affecting CKFPU Performance

#### 3.3.1. Number of Multi-Looks

#### 3.3.2. Quality Indexes for Guiding Path Tracking

#### Maximum Coherence

#### Phase Derivative Variance

#### Fisher Distance

## 4. Experiments

#### 4.1. Dataset Introduction and DInSAR Processing Strategy

#### 4.1.1. Dataset Introduction

Parameters | Values |
---|---|

Frequency | 9.6 GHZ |

Wavelength | 3.1 cm |

Polarisation | HH |

Swath Width | 50 km |

Incidence Angle | ~26° |

Range Pixel Spacing | 0.9 m |

Azimuth Pixel Spacing | 1.9 m |

Orbit Repeat Cycle | 11 days |

Precise Orbit Accuracy | ~10 cm |

Datasets | Target Working Face | Master | Slave | Perpendicular Baseline B_{⊥}/m | Temporal Baseline B_{T}/day |
---|---|---|---|---|---|

A | 18a203 | 4 January 2013 | 15 January 2013 | −103.3 | 11 |

B | 52304 | 2 April 2013 | 24 April 2013 | −107.2 | 22 |

C | 18a203 | 16 September 2012 | 27 September 2012 | −3 | 11 |

27 September 2012 | 8 October 2012 | −7.2 | 11 | ||

8 October 2012 | 19 October 2012 | −70.2 | 11 | ||

19 October 2012 | 10 November 2012 | −30.5 | 22 | ||

10 November 2012 | 21 November 2012 | 62.2 | 11 | ||

21 November 2012 | 2 December 2012 | 160.7 | 11 | ||

2 December 2012 | 13 December 2012 | −36.3 | 11 | ||

13 December 2012 | 24 December 2012 | −99.8 | 11 |

**Figure 2.**Interferograms (left) and coherence maps (right) under different numbers of multi-looks in the working face 18a203: (

**a**) interferogram of 1 × 1 look; (

**b**) interferogram of 2 × 2 looks; (

**c**) interferogram of 4 × 4 looks; (

**d**) coherence map of 1 × 1 look; (

**e**) coherence map of 2 × 2 looks; (

**f**) coherence map of 4 × 4 looks.

#### 4.1.2. DInSAR Processing Strategy

#### 4.2. Results and Analysis

#### 4.2.1. Case 1

**Figure 3.**Unwrapped maps (left) and rewrapped maps (right) under different numbers of multi-looks: (

**a**) unwrapped map of 1 × 1 look; (

**b**) unwrapped map of 2 × 2 looks; (

**c**) unwrapped map of 4 × 4 looks; (

**d**) rewrapped map of 1 × 1 look; (

**e**) rewrapped map of 2 × 2 looks; (

**f**) rewrapped map of 4 × 4 looks.

**Figure 4.**Three phase quality maps with FD, MC and PDV: (

**a**) FD quality map; (

**b**) MC quality map; (

**c**) PDV quality map.

**Figure 5.**Unwrapped maps (left) and rewrapped maps (right) based on MC and PDV: (

**a**) unwrapped map of MC; (

**b**) unwrapped map of PDV; (

**c**) rewrapped map of MC; (

**d**) rewrapped map of PDV.

#### 4.2.2. Case 2

**Figure 6.**Results based on MCF method: (

**a**) unwrapped map with pre-filtering filter threshold set at 0.1; (

**b**) rewrapped map of (

**a**); (

**c**) unwrapped map with adaptive filter threshold set at 0.25; (

**d**) rewrapped map of (

**c**).

**Figure 7.**Experimental results for Dataset 2: (

**a**) original differential interferogram; (

**b**) corresponding coherence map; (

**c**) unwrapped map of CKFPU; (

**d**) rewrapped map of CKFPU; (

**e**) unwrapped map of MCF when the pre-filtering threshold is set at 0.1; (

**f**) rewrapped map of (e); (

**g**) unwrapped map of MCF based on MCF when the pre-filtering threshold is set at 0.25 and (

**h**) its rewrapped map.

#### 4.2.3. Case 3

ID | SAR Data | Pre-Filtering + MCF | CKFPU | GPS Result | |||
---|---|---|---|---|---|---|---|

Master | Slave | SSV/m | ASV/m | SSV/m | ASV/m | ASV/m | |

1 | 16 September 2012 | 27 September 2012 | −0.024 | −0.024 | −0.036 | −0.036 | —— |

2 | 27 September 2012 | 8 October 2012 | −0.013 | −0.037 | −0.016 | −0.052 | −0.057 |

3 | 8 October 2012 | 19 October 2012 | −0.023 | −0.060 | −0.042 | −0.094 | —— |

4 | 19 October 2012 | 10 November 2012 | −0.051 | −0.111 | −0.044 | −0.138 | −0.176 |

5 | 10 November 2012 | 21 November 2012 | −0.032 | −0.143 | −0.024 | −0.162 | —— |

6 | 21 November 2012 | 2 December 2012 | −0.020 | −0.164 | −0.021 | −0.183 | —— |

7 | 2 December 2012 | 13 December 2012 | −0.011 | −0.174 | −0.013 | −0.196 | —— |

8 | 13 December 2012 | 24 December 2012 | 0.002 | −0.173 | −0.005 | −0.201 | −0.227 |

**Figure 8.**Time series displacement of a Corner Reflector over 88 days, derived from MCF, CKFPU and GPS.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wegmüller, U.; Werner, C.; Strozzi, T.; Wiesmann, A. Monitoring mining induced surface deformation. In Proceedings of the 2004 IEEE Geoscience and Remote Sensing Symposium, IGARSS’04, Anchorage, AK, USA, 20–24 September 2004; Volume 3, pp. 1933–1935.
- Liu, D.; Shao, Y.; Liu, Z.; Riedel, B.; Sowter, A.; Niemeier, W.; Bian, Z. Evaluation of InSAR and TomoSAR for monitoring deformations caused by mining in a mountainous area with high resolution satellite-based SAR. Remote Sens.
**2014**, 6, 1476–1495. [Google Scholar] [CrossRef] - Liu, Z.; Bian, Z.; Lei, S.; Liu, D.; Sowter, A. Evaluation of PS-DInSAR technology for subsidence monitoring caused by repeated mining in mountainous area. Trans. Nonferrous Met. Soc. China
**2014**, 24, 3309–3315. [Google Scholar] [CrossRef] - Ghiglia, D.C.; Pritt, M.D. Two-dimentional Phase Unwrapping. In Theory, Algorithms and Software; Wiley: New York, NY, USA, 1998. [Google Scholar]
- Gens, R. Two-dimensional phase unwrapping for radar interferometry: Developments and new challenges. Int. J. Remote Sens.
**2003**, 24, 703–710. [Google Scholar] [CrossRef] - Goldstein, R.M.; Zebker, H.A.; Werner, C.L. Satellite radar interferometry: Two-dimensional phase unwrapping. Radio Sci.
**1988**, 23, 713–720. [Google Scholar] [CrossRef] - Xu, W.; Cumming, I. A region-growing algorithm for InSAR phase unwrapping. IEEE Trans. Geosci. Remote Sens.
**1999**, 37, 124–133. [Google Scholar] [CrossRef] - Flynn, T.J. Two dimensional phase unwrapping with minimum weighted discontinuity. J. Opt. Soc. Am.
**1997**, 14, 2692–2710. [Google Scholar] [CrossRef] - Costantini, M. A novel phase unwrapping method based on network programming. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 813–821. [Google Scholar] [CrossRef] - Ghiglia, D.; Romero, L. Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J. Opt. Soc. Am. A
**1994**, 11, 107–117. [Google Scholar] [CrossRef] - Lee, J.-S.; Papathanassiou, K.P.; Ainsworth, T.L.; Grunes, M.R.; Reigbe, A. A new technique for noise filtering of SAR interferometric phase images. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 1456–1465. [Google Scholar] - Wu, N.; Feng, D.-Z.; Li, J. A locally adaptive filter of interferometric phase images. IEEE Trans. Geosci. Remote Sens. Lett.
**2006**, 3, 73–77. [Google Scholar] [CrossRef] - Yu, Q.; Yang, X.; Fu, S.; Liu, X.; Sun, X. An adaptive contoured window filter for interferometric synthetic aperture radar. IEEE Trans. Geosci. Remote Sens. Lett.
**2007**, 4, 23–26. [Google Scholar] [CrossRef] - Suksmono, A.B.; Hirose, A. Adaptive noise reduction of InSAR images based on a complex-valued MRF model and its applications to phase unwrapping problem. IEEE Trans. Geosci. Remote Sens.
**2002**, 40, 699–709. [Google Scholar] [CrossRef] - Kim, M.; Griffiths, H. Phase unwrapping of multibaseline interferometry using Kalman filtering. In Proceedings of the Seventh International Conference on Image Processing and Its Applications, Manchester, UK, 13–15 July 1999; pp. 813–817.
- Loffeld, O.; Nies, H.; Knedlik, S.; Wang, Y. Phase unwrapping for SAR interferometry—A data fusion approach by kalman filtering. IEEE Trans. Geosci. Remote Sens.
**2008**, 46, 47–58. [Google Scholar] [CrossRef] - Nies, H.; Loffeld, O.; Wang, R. Phase unwrapping using 2d-kalman filter—Potential and limitations. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium, Boston, MA, USA, 7–11 July 2008; pp. 1213–1216.
- Osmanoǧlu, B.; Wdowinski, S.; Dixon, T.H.; Biggs, J. InSAR phase unwrapping based on extended Kalman filtering. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 1197–1202. [Google Scholar] - Martinez-Espla, J.; Martinez-Marin, T.; Lopez-Sanchez, J. A Particle Filter Approach for InSAR Phase Filtering and Unwrapping. IEEE Trans. Geosci. Remote Sens.
**2009**, 47, 1197–1211. [Google Scholar] [CrossRef] - Osmanoǧlu, B.; Dixon, T.H.; Wdowinski, S. Three-Dimensional Phase Unwrapping for Satellite radar interferometry, I DEM generation. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 1059–1075. [Google Scholar] [CrossRef] - Xie, X.; Pi, Y. Phase noise filtering and phase unwrapping method based on unscented Kalman filter. J. Syst. Eng. Electron.
**2011**, 22, 365–372. [Google Scholar] - Xie, X.; Pi, Y.; Peng, B. Phase unwrapping: An unscented particle filtering approach. Acta Electron. Sin.
**2011**, 39, 705–709. (In Chinese) [Google Scholar] - Liu, W.; Zhang, Q. New Strong Tracking Filter Based on Cubature Kalman Filter. J. Syst. Simul.
**2014**, 26, 1102–1107. (In Chinese) [Google Scholar] - Ienkaran, A.; Simon, H. Cubature kalman filters. IEEE Trans. Autom. Control
**2009**, 54, 1254–1269. [Google Scholar] - Ienkaran, A. Cubature Kalman Filtering: Theory & Applications. Ph.D. Thesis, McMaster University, Hamilton, ON, Canada, 2009. [Google Scholar]
- Jia, B.; Xin, M.; Cheng, Y. High-degree Cubature Kalman filter. Automatica
**2013**, 49, 510–518. [Google Scholar] [CrossRef] - Spagnolini, U. 2-D phase unwrapping and instantaneous frequency estimation. IEEE Trans. Geosci. Remote Sens.
**1995**, 33, 579–589. [Google Scholar] [CrossRef] - Trouve, E.; Nicolas, J.M.; Maitre, H. Improving phase unwrapping techniques by the use of local frequency estimates. IEEE Trans. Geosci. Remote Sens.
**1998**, 36, 1963–1965. [Google Scholar] [CrossRef] - Jiang, M.; Li, Z.; Ding, X.; Zhu, J.; Feng, G. Modeling minimum and maximum detectable deformation gradients of interferometric SAR measurements. Int. J. Appl. Earth Obs. Geoinf.
**2011**, 13, 766–777. [Google Scholar] [CrossRef] - Osmanoǧlu, B.; Dixon, T.H.; Wdowinski, S.; Cabral-Cano, E. On the importance of Path for Phase Unwrapping in Synthetic Aperture Radar Interferometry. Appl. Opt.
**2011**, 50, 3205–3220. [Google Scholar] [CrossRef] [PubMed] - Just, D.; Bamler, R. Phase statistics of interferograms with application to synthetic aperture radar. Appl. Opt.
**1994**, 33, 4361–4368. [Google Scholar] [CrossRef] [PubMed]

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**MDPI and ACS Style**

Liu, W.; Bian, Z.; Liu, Z.; Zhang, Q.
Evaluation of a Cubature Kalman Filtering-Based Phase Unwrapping Method for Differential Interferograms with High Noise in Coal Mining Areas. *Sensors* **2015**, *15*, 16336-16357.
https://doi.org/10.3390/s150716336

**AMA Style**

Liu W, Bian Z, Liu Z, Zhang Q.
Evaluation of a Cubature Kalman Filtering-Based Phase Unwrapping Method for Differential Interferograms with High Noise in Coal Mining Areas. *Sensors*. 2015; 15(7):16336-16357.
https://doi.org/10.3390/s150716336

**Chicago/Turabian Style**

Liu, Wanli, Zhengfu Bian, Zhenguo Liu, and Qiuzhao Zhang.
2015. "Evaluation of a Cubature Kalman Filtering-Based Phase Unwrapping Method for Differential Interferograms with High Noise in Coal Mining Areas" *Sensors* 15, no. 7: 16336-16357.
https://doi.org/10.3390/s150716336