# Split-Band Interferometry-Assisted Phase Unwrapping for the Phase Ambiguities Correction

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## Abstract

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## 1. Introduction

_{f}) [1].

_{f}. In Section 3, the SBInSAR-assisted phase unwrapping is tested on spotlight images acquired over Copahue volcano. The test site is described, as well as the data set and the processing. An indirect validation procedure is also presented, along with an indicator to compare the precision of the results.

## 2. Methods

#### 2.1. Rationale of Split-Band Interferometry

^{th}partial interferogram is expressed by:

^{th}partial interferogram. This expression holds for independent data points, i.e., nonoverlapping subbands.

#### 2.2. Detection of Frequency-Persistent Scatterers

_{f}.

_{f}population and their efficiency will be compared to the one of the multifrequency phase error when applied to the test case.

#### 2.2.1. Multifrequency Phase Error

#### 2.2.2. Slope Standard Deviation

#### 2.2.3. Phase Variance Stability

_{f}. In practice, the spectral decomposition is always symmetrical with respect to the central carrier frequency. For the second assumption, we consider it as verified when the value of ${\sigma}_{\Delta \phi}$ varies of less than 5% when the squared sum is neglected.

_{f}may even be missed during the selection. However, the accuracy for the selected points is guaranteed, as we will demonstrate with the test case.

#### 2.3. SBInSAR-Assisted Phase Unwrapping

_{f}to derive this integer number of cycles. Let us specify that the phase-offset, or phase ambiguity, denotes the $2\pi n$-discrepancy. However, we will largely use these terms in the following sections to refer to the number of cycles n alone.

_{f}at pixel’s coordinates $(k,l)$ in the image, neglecting the noise and the phase unwrapping errors, the unknown number of cycles can be computed as:

_{f}of this region based on one of the criteria presented in the previous section. It will then estimate the phase ambiguities using Equation (12) for all the selected pixels and round each of these values to the nearest integer. The rounded value with the largest number of occurrences, i.e., the mode of the distribution, is assumed to be the phase ambiguity we are looking for. Finally, this value is multiplied by $2\pi $ and added to the unwrapped phase of all the pixels of the region in order to correct the phase ambiguity. This procedure is repeated for each region separately unwrapped. When the distribution of the rounded phase ambiguities has multiple modes, no correction is applied. Regions with a population below 10 PS

_{f}are not corrected either, since they frequently show multiple modes. The algorithm steps are presented in Figure 1. In the end of the process, an image of the leveled unwrapped phase is provided, showing only the shifted areas. Let us stress that, despite the loss of resolution in the subproducts, this final unwrapped phase image preserves the initial range resolution.

_{f}is enough to determine the phase-offset.

## 3. Copahue Test Case

#### 3.1. Test Site

^{TM}(Copahue, Neuquén, Argentina) optical view of the area is given in Figure 2b. The area of interest shows steep topography as well as moderate slopes, little vegetation and a snow cover that can vary over the year. Change in snow cover and frequent precipitations can cause local loss of coherence in InSAR products. Variety of topography, of slope orientations and of geometrical distortions, along with the presence of natural scatterers make Copahue volcano an interesting site to apply SBInSAR-assisted phase unwrapping.

#### 3.2. Data Set and Processing

#### 3.3. Validation Procedure

#### 3.4. Indicator of Quality

## 4. Results and Discussion

_{f}selection point of view: in the initial situation, we do not discriminate the PS

_{f}and keep all the pixels. In the other cases, the PS

_{f}are selected using either the multifrequency phase error ${\sigma}_{\nu}$, the threshold on standard deviation of the slope ${\sigma}_{s}$ of the linear regression or the stability of the phase variance ${\sigma}_{{\varphi}_{i}}^{2}$. We set a threshold of 0.5 on the multifrequency phase error. The number of selected pixels according to the region and the detection criterion is shown in Figure 7. Since the first region is noticeably larger than the three others, it shows therefore a larger population of selected pixels for any detection criterion. The number of detected pixels is much higher in the case of the multifrequency phase error than for the other two criteria. The phase variance stability classifies approximately 2–3% of the initial population as PS

_{f}while the proportion is about ten times higher for the standard deviation of the slope.

_{f}is applied, the distribution is spread over a large range of values with a low probability for the mode bin. This behavior is similar for histograms over all the other regions. The quality of the results without selection criterion applied is quantified by a $W/H$ ratio of about 45–50 for most regions, with the highest value of 85 for region 4 (see Table 3). The larger dispersion in region 4 is probably due to the large uncorrelated patch present in the split-band phase. Those targets are most probably unstable and they are not discriminated in this case. The $W/H$ aspect ratio is lowered when the multifrequency phase error criterion is applied to select stable pixels. In this case, the contrast between the dispersion in region 4 and the three others is significantly reduced.

_{f}selection criteria considered in this study, the phase variance stability appears to be the most efficient. However, when applied to a less favorable case, e.g., to images with a smaller bandwidth or small disconnected areas, this criterion can be too restrictive and the population of selected PS

_{f}will be too limited to estimate the phase ambiguity reliably. In such cases, the standard devation of the slope is a satisfactory alternative.

_{f}and if the phase-offset mode is unique. During the test on the artificially disconnected areas, we observed that the phase variance stability selected fewer targets than the standard deviation of the slope in a given region. We reach a similar conclusion for the naturally disconnected areas (Figure 9). It is interesting to note that the majority of the pixels identified by the phase variance stability are also identified by the standard deviation of the slope. Less than 1% of the PS

_{f}population is selected by the phase variance stability only.

_{f}population larger than 30 pixels, we observe in Figure 10 that the mode of the phase-offset distribution represents approximately 20–30% of the detected PS

_{f}for the standard deviation of the slope and a slightly higher value of 25–35% for the phase variance stability. Regions with smaller population of stable targets can exhibit even larger values. The regions with an occurrence of the mode below 20% of the PSf

_{f}population are an exception, whatever the detection criterion.

_{f}population. The stable nature of a target is probably related to its intrinsic characteristics and/or the geometry of observation, which causes a heterogeneous distribution of the targets across the scene. Since the population of stable targets is the key point to determine the phase ambiguity, and it cannot be related to the initial population of a given region, it is not possible to define the minimum size of an area where the SBInSAR-assisted phase unwrapping can be applied. However, we observe that when the standard deviation of the slope is considered, the largest region with no PS

_{f}at all is made of less than 400 pixels. For the phase variance stability criterion, the largest region has a size of 3665 pixels, but most of them contain less than 500 pixels.

## 5. Conclusions

_{f}in a standard case. Decreasing the bandwidth to 100–150 MHz, we expect a reduced population of PS

_{f}due to the loss of resolution but still reasonable results. In addition, the method has been validated on disconnected areas with an important initial population (>105 pixels) and consequently with a higher probability to include stable targets. The next step will be to apply and validate it for smaller regions. Finally, due to the dependency of the split-band phase accuracy on the frequency, we expect better results with C-band or L-band data. In the future, the SBInSAR-assisted phase unwrapping will be tested on Sentinel-1 data.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PS_{f} | Frequency-Persistent Scatterer(s) |

SAR | Synthetic Aperture Radar |

InSAR | Synthetic Aperture Radar Interferometry |

SBInSAR | Split-Band Interferometry |

DEM | Digital Elevation Model |

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**Figure 2.**(

**a**) Location of Copahue volcano (red triangle) on the border between Chile and Argentina. (

**b**) Google Earth™ image of Copahue volcano in 2017. The footprint of the InSAR pair is drawn in red.

**Figure 3.**(

**a**) coherence image of the test pair over the Copahue volcano. (

**b**) fully connected unwrapped phase. Color chart values are given in radians.

**Figure 4.**Phase measured with Split-Band Interferometry over Copahue volcano. Color chart values are given in radians.

**Figure 5.**Diagrams of the validation procedure for a one-dimensional simplified interferogram. (

**a**) If the connected InSAR phase $\Delta {\varphi}_{c}$ is subtracted from the split-band phase $\Delta \phi $, the difference gives an offset $2\pi n$. (

**b**) If a region of the one-dimensional interferogram is disconnected, an offset $2\pi m$ is introduced between the connected phase $\Delta {\varphi}_{c}$ and the disconnected InSAR phase $\Delta {\varphi}_{d}$. (

**c**) If the disconnected InSAR phase is subtracted from the split-band phase, the relative offset $2\pi m$ between the disconnected regions remains the same as in case (

**b**).

**Figure 6.**(

**a**) map of the disconnected regions. The main coherent region from which areas have been artificially cut is represented in blue. We refer to the blue region, the red rectangle, the green ellipse and the yellow area, respectively, as the regions 1, 2, 3 and 4. Regions in black are regions with coherence lower than the threshold applied during phase unwrapping. White areas are naturally disconnected parts of the unwrapped phase and they are not considered for the validation of the SBInSAR-assisted phase unwrapping. (

**b**) artificially disconnected version of the unwrapped phase over Copahue volcano. Color chart values are given in radians.

**Figure 7.**Number of selected PS

_{f}for regions 1 to 4 using the different detection criteria. The y-axis is on a logarithmic scale.

**Figure 8.**Normalized histograms of the estimated phase-offset values n for region 3. The gray line histogram represents the case where no selection of PS

_{f}is applied. For other cases, the PS

_{f}population is selected using three different criteria. The vertical dashed line indicates the expected phase ambiguity. Similar figures are obtained for the three other regions.

**Figure 9.**PS

_{f}population as a function of the size of the area, for both artificially and naturally disconnected areas. Regions with no PS

_{f}are not represented. Axes are in logarithmic scale.

**Figure 10.**Probability of the mode of the phase-offset distribution as a function of the amount of selected PS

_{f}in a given region, for both artificially and naturally disconnected areas. Regions with multiple modes are not represented. The x-axis is on the logarithmic scale.

Relative Phase-Offset | Cycles |
---|---|

${m}_{1}-{m}_{2}$ | -1 |

${m}_{1}-{m}_{3}$ | 0 |

${m}_{1}-{m}_{4}$ | -1 |

${m}_{2}-{m}_{3}$ | 1 |

${m}_{2}-{m}_{4}$ | 0 |

${m}_{3}-{m}_{4}$ | -1 |

Computed Phase-Offset | Cycles |
---|---|

${n}_{1}$ | -3 |

${n}_{2}$ | -2 |

${n}_{3}$ | -3 |

${n}_{4}$ | -2 |

PS_{f} Selector | Region 1 | Region 2 | Region 3 | Region 4 |
---|---|---|---|---|

$\mathit{W}/\mathit{H}$ | $\mathit{W}/\mathit{H}$ | $\mathit{W}/\mathit{H}$ | $\mathit{W}/\mathit{H}$ | |

None | 54 | 43 | 43 | 85 |

${\sigma}_{\nu}$ | 25 | 26 | 25 | 30 |

${\sigma}_{s}$ | 12 | 14 | 12 | 13 |

${\sigma}_{{\varphi}_{i}}^{2}$ | 8 | 9 | 8 | 9 |

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**MDPI and ACS Style**

Libert, L.; Derauw, D.; D’Oreye, N.; Barbier, C.; Orban, A.
Split-Band Interferometry-Assisted Phase Unwrapping for the Phase Ambiguities Correction. *Remote Sens.* **2017**, *9*, 879.
https://doi.org/10.3390/rs9090879

**AMA Style**

Libert L, Derauw D, D’Oreye N, Barbier C, Orban A.
Split-Band Interferometry-Assisted Phase Unwrapping for the Phase Ambiguities Correction. *Remote Sensing*. 2017; 9(9):879.
https://doi.org/10.3390/rs9090879

**Chicago/Turabian Style**

Libert, Ludivine, Dominique Derauw, Nicolas D’Oreye, Christian Barbier, and Anne Orban.
2017. "Split-Band Interferometry-Assisted Phase Unwrapping for the Phase Ambiguities Correction" *Remote Sensing* 9, no. 9: 879.
https://doi.org/10.3390/rs9090879