The Split-Band Interferometry Approach to Determine the Phase Unwrapping Offset

The Split-Band Interferometry Approach to Determine the Phase

Unwrapping Offset

Ludivine Libert1_{, Dominique Derauw}1_{, Nicolas d’Oreye}2,3_{, François Kervyn}4_{, Anne Orban}1 _{and Christian Barbier}1

1_{Centre Spatial de Liège (CSL), Université de Liège, Avenue du Pré-Aily, 4031 Angleur, Belgium}

2_{European Center for Geodynamics and Seismology, Rue Josy Welter 19, L-7256 Walferdange, Grand-Duchy of Luxembourg} 3_{National Museum of Natural History, Rue de Munster 25, L-2160 Luxembourg, Grand-Duchy of Luxembourg}

4_{Royal Museum of Central Africa, Leuvensesteenweg 13, 3080 Tervuren, Belgium}

Contact email : llibert@ulg.ac.be

Split-Band Interferometry

Figure 1. Sp lit -Ba nd Ph ase

Split-Band Interferometry (SBInSAR) takes advantage of the large range bandwidth of recent sensors to split it into subbands and produce several subrange images of a same scene, with lower resolution and shifted frequencies. In a first time, the same spectral decomposition is applied to both master and slave images of a given pair. Then interferometry is applied to the subrange images of the pair, yielding a stack of subrange interferograms. Finally, the linear phase trend is explored through the stack of interferograms in order to provide absolute phase

measurements. The issued phase is called the split-band phase (Figure 1).

Phase Ambigui:es

Conventional phase unwrapping algorithms for InSAR processing perform a relative measurement of the phase, i.e. the phase is reconstructed for each pixel with respect its neighbours. In practice, this causes distinct phase unwrapping of regions separated by noncoherent patches and it introduces unknown phase-offsets that prevent from comparing the phase from one region to another (Figure 2). Since SBInSAR is supposed to provide absolute phase measurements, it can potentially solve the phase

ambiguities and reconnect the phase of separately unwrapped regions.

In this study, we propose to use SBInSAR to assist the phase unwrapping and reconnect separately unwrapped areas. We consider the general

case where the phase ambiguity is an unknown number of cycles 2πn,

with n being an integer. This phase ambiguity has the same value for all

the points through a given region of continuously unwrapped phase.

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Disconnected Unwrapped Phase 2017/12/15-26 [radians]

-40 -30 -20 -10 0 10 20 30 40 Figure 2. U nw ra pp ed Ph ase 0 200 400 600 800 1000 0 500 1000 1500 2000 Azimuth [pixels] Range [pixels]

Disconnected Zone Map 2017/12/15-26

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Figure 3. Ma p of th e Ma nu al ly D isco nn ect ed R eg io ns

Frequency-Persistent Sca>erers

Targets with a stable response across the spectral domain are called

frequency-persistent scatterers (PS_f). The spectral stability of a PS_f

ensures the linear behaviour of the phase with the frequency and therefore the accuracy of the split-band phase. Consequently, absolute and spatially independent measurements are possible using SBInSAR at

these points only. We propose two criteria to select PS_f assuming a

one-cycle accuracy of the split-band phase: first, the standard deviation of

the slope of the linear regression; second, the stability of the phase variance throughout the subrange interferograms.

In a first time, the SBInSAR-assisted phase unwrapping selects the PS_f

of a given region based on one criterion or the other. Then it computes the difference in cycles between the split-band phase and the unwrapped

phase for each selected PS_f. The value of the phase offset with the

highest number of occurrences, i.e. the mode of the distribution, is assumed to be the correction that must be applied to the unwrapped area.

SBInSAR-Assisted Phase Unwrapping

0 0.05 0.1 0.15 0.2 0.25 0.3 -15 -10 -5 0 5 10 15 P(n) n

Phase variance stability Standard deviation on the slope Multifrequency phase error No selection Figure 4. N orma lize d D ist rib ut io ns of the D iff ere nce in C ycl es be tw ee n th e Sp lit -Ba nd Ph ase a nd th e U nw ra pp ed Ph ase a t t he PS f Se le ct ed fo r Region 3

SBInSAR-assisted phase unwrapping was tested on a pair of Spotlight images acquired by TerraSAR-X over Copahue Volcano on December 15 and 26, 2014. A spectral decomposition into 5 nonoverlapping subbands of 60 MHz was applied. In order to validate our results, we artificially disconnected regions of the unwrapped phase (Figure 3). This approach enables a validation based on the relative values of the phase-offsets.

Test Case

Results

When applied to the test case, the method provides the expected phase ambiguities for the artificially disconnected areas, whatever the selection criterion. However, the distribution shows less dispersion when the phase variance stability is used (Figure 4). Moreover, the procedure has been tested on

both artificially and naturally disconnected areas and we observed that the percentage of PS_f

contributing to the mode of the distribution stabilizes around 25-35% when the region contains more

than 30 PS_f. This amount is considered to be the minimum number of PS_f to have a statisticallly

significant population (Figure 5).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 101 102 103 104 105 106

Occurence of the mode / Number of PS

f

PS_f population

Probability of the mode as a function of the number of PS_f

Slope standard deviation Phase variance stability

Figure 5. Pe rce nt ag e of Mo de O ccu rre nce s fo r al l D ist in ct Regions

This work was carried out in the framework of the MUZUBI project financed by Belgian Science Policy Contract NR SR/00/324. This work was carried out in the framework of the MUZUBI project financed by Belgian Science Policy Contract NR SR/00/324.

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