TY - JOUR
AU - Lasserre,, Cécile
AB - SUMMARY Interferometric Synthetic Aperture Radar (InSAR) is commonly used in Earth Sciences to study surface displacements or construct high resolution topographic maps. Recent satellites such as those of the Sentinel-1 constellation allow to derive dense deformation maps with millimetric precision with high revisit frequency. However, InSAR is still limited by interferometric coherence. Interferometric phase noise resulting from a loss of coherence, due to changes in scattering properties between repeated SAR acquisitions, may lead to unwrapping errors, which then in turn lead to centimetric errors in time-series reconstruction. We present an algorithm based on interferometric phase closure to automatically correct unwrapping errors. We describe the algorithm and highlight its performances with two case studies, in Lebanon with Envisat satellite data and in Central Turkey with Sentinel-1 data. The first data set is particularly affected by unwrapping errors because of long spatial (500 m) and temporal baseline interferograms (6 yr) and decorrelation due, in particular, to vegetation. The second data set contains unwrapping errors because of temporal changes in the scattering properties of the ground. For these two examples, the algorithm allows the correction of almost all detectable unwrapping errors, without requiring visual inspection or manual deletions. Our algorithm is efficient especially on large data sets, such as with Sentinel-1 constellation, where interferometric phase is redundant and improves eventually the reconstruction of time-series. Creep and deformation, Radar interferometry, Image processing 1 INTRODUCTION Interferometric Synthetic Aperture Radar (InSAR) is a geodetic technique developped in the 1970s for geophysical applications and, originally, to construct topographic maps of the Earth (Graham 1974; Zebker & Goldstein 1986), Venus (Rogers & Ingalls 1970) and the Moon (Zisk 1972a,b; Margot et al.2000). In the 1990s, InSAR was then used for the study of surface displacements related to earthquakes (Massonnet et al.1993; Zebker et al.1994), inflation of volcanoes (Massonnet et al.1995) or ice sheet motion (Goldstein et al.1993). InSAR is based on the acquisition of successive SAR images over the same area and from close positions by a side looking radar onboard a plane or a satellite. The complex conjugate product of two SAR images is called an interferogram. The phase of an interferogram, hereafter called the interferometric phase, corresponds to the relative traveltime difference of the electromagnetic wave between two SAR acquisitions. The interferometric phase depends on satellite orbits, topography, spatio-temporal variations in the refractive index of the atmosphere between two acquisitions, ground deformation along the satellite line-of-sight (LOS) and various sources of noise, including digital elevation model, orbits errors and instrumental noise. With two simultaneous acquisitions from two points of view, InSAR is a measurement of topography used to build DEMs, while with successive acquisitions it can be used to measure ground deformation. Measurements of deformation and ground velocity using InSAR have now reached a millimetre accuracy (Simons & Rosen 2015; Elliott et al.2016). Reconstructing continuous signals, including deformation, involves phase unwrapping, which consists in adding the appropriate multiple of 2π to the interferometric phase and multiple methods have been developed to do so. Branch-cut algorithms consist in identifying consistent and inconsistent paths to integrate the phase signal (Goldstein et al.1988; Prati et al.1990; Lin et al.1994; Herszterg et al.2018). Least-squares techniques, weighted or unweighted, minimize the mean deviation between the estimated (wrapped) and unknown (unwrapped) discrete derivatives of the phase (Ghiglia & Romero 1994; Flynn 1997; Costantini 1998; Chen & Zebker 2001), sometimes using external data such as GPS to constrain the unwrapping process (Agram & Zebker 2010). Ultimately, Permanent or Persistent Scatterers InSAR (PS-InSAR) methods, based on the identification of pixels with stable backscattering properties in time, use the temporal information of multiple interferograms to unwrap the phase in time and space (Pepe & Lanari 2006; Hooper & Zebker 2007; Hussain et al.2016). However, phase unwrapping may fail, especially within low coherence regions (Rosen et al.1996). In an interferogram, each pixel phase value corresponds to the phase of the coherent sum of backscattered electromagnetic wave from scatterers on the ground within the pixel. If scattering properties change over time or if the geometry of acquisition is too different between each pass of the satellite, the phase change between two neighbouring pixels may exceed one phase cycle (i.e. 2π). Coherence is a measure of the spatial correlation of phase (Lee et al.1994). A coherence of 1 indicates the phase is constant within a cell surrounding the pixel. Over low coherence regions, higher phase noise may lead to phase differences between neighbouring pixels higher than π. In addition, large deformation gradients may lead to similar situation, for instance close to a fault that ruptured in a large earthquake (e.g. Simons et al.2002). Phase unwrapping is based on the hypothesis that the phase of two neighbouring pixels of an interferogram only differs by a fraction of π. This hypothesis is only valid in high coherence regions with a moderate fringe rate. When this assumption breaks down, unwrapping methods may fail, creating erroneous offsets of multiples of 2π in the unwrapped phase. The size of the affected region may vary from a few pixels to a significant fraction of the image. In Earth science applications, almost all interferograms have large regions where phase decorrelates due to changes in scattering properties (e.g. vegetation, humidity, anthropic changes), high topographic gradients or high deformation areas and unwrapping becomes challenging (Simons et al.2002; Zebker et al.2007). Unwrapping errors bias estimations of surface deformation by introducing inconsistencies in the interferometric network in case of time-series analysis. Unwrapping errors are sometimes manually detected and masked (e.g. Jolivet et al.2012) and methods based on interferometric network misclosure analysis (e.g. López-Quiroz et al.2009) and time-series analysis have been proposed (e.g. Hussain et al.2016). We propose an efficient algorithm, named CorPhU (CORrection of Phase Unwrapping errors), for the correction of unwrapping errors after phase unwrapping, based on the phase closure of interferogram triplets within an interferometric network. A proof of concept of this algorithm has been presented by Pinel-Puysségur et al. (2018) and we describe in details the formulation, implementation and performances of the algorithm in this paper. Phase unwrapping errors detected by the algorithm are automatically and iteratively corrected. In the following sections, we describe the algorithm and present qualitative results focusing on two case studies where decorrelation is high and could be a limiting factor, including data from the Envisat satellite over Lebanon and data from the Sentinel-1 constellation over Turkey. We then perform a quantitative assessment of the algorithm. Finally, we discuss limitations and possible improvements of our approach. 2 METHOD By construction, the sum of the phase of three unwrapped interferograms forming a closed loop equals 0 (Jennison 1958). For a triplet T of three SAR acquisitions k, l and m, the triplet phase closure ΦT is: $$\begin{eqnarray*} \Phi _{T} = \phi _{kl} + \phi _{lm} - \phi _{km}, \end{eqnarray*}$$(1) where ϕkl, ϕlm and ϕkm are the unwrapped phase of interferograms Ikl, Ilm and Ikm computed from acquisitions k, l and m. By construction, phase closure ΦT should be equal to 0, up to phase inconsistencies due to variations of the backscattering properties of the ground, including for instance soil moisture (De Zan et al.2015). Once this later contribution is removed, the remaining phase closure inconsistencies correspond to phase unwrapping errors exactly equal to a multiple of 2π. Our algorithm detects and corrects such unwrapping errors within a stack of coregistred interferograms formed from SAR images (Fig. 1). First, we identify all triplets in the interferogram network. Secondly, we compute the phase closure for each triplet following eq. (1) and identify unwrapping errors. Thirdly, for each of these incorrectly unwrapped regions, we identify the interferogram incorrectly unwrapped among the three possible ones using the so-called ‘flux’ or ‘mean closure’ methods, described in Sections 2.2 and 2.3, respectively. Once we have identified the interferogram incorrectly unwrapped, we correct the unwrapping error. We proceed iteratively through the network of triplets. Figure 1. Open in new tabDownload slide Algorithm implementation. First, we build the network of triplets. We process then each triplet. We identify unwrapping errors and reference regions using triplet phase closure. We correct each unwrapping error using a two-steps detection, with the flux method or with the mean closure method, in case the flux method cannot determine which interferogram to correct. Figure 1. Open in new tabDownload slide Algorithm implementation. First, we build the network of triplets. We process then each triplet. We identify unwrapping errors and reference regions using triplet phase closure. We correct each unwrapping error using a two-steps detection, with the flux method or with the mean closure method, in case the flux method cannot determine which interferogram to correct. 2.1 Automatic identification of unwrapping errors For all available triplets, we start by building masks mkl, mlm and mkm associated to interferograms Ikl, Ilm and Ikm, based on the coherence map. Pixels with a coherence lower than a given threshold (0.8 by default) are masked out. If none of the three individual masks is empty, we construct the total mask of the triplet |$m_{T}^{\rm {tot}}$| as the intersection of masks mkl, mlm and mkm. We then compute triplet closure on unwrapped interferograms using eq. (1). We distinguish two sources of misclosure in unwrapped interferograms. The first one is unwrapping errors and is specific to unwrapped interferograms. The second one arises from interferogram multilooking prior to unwrapping. Indeed, multilooking is a non-coherent summation of different pixels, leading to small phase inconsistencies in the wrapped interferograms and thus to non-zero closure (De Zan et al.2015). We therefore calculate the closure of wrapped interferograms, defined as: $$\begin{eqnarray*} \Phi _{T}^{w} = (\phi _{I_{kl}}^{w} + \phi _{I_{lm}}^{w} - \phi _{I_{km}}^{w}) [2\pi ], \end{eqnarray*}$$(2) where |$\phi _{kl}^{w}$|⁠, |$\phi _{lm}^{w}$| and |$\phi _{km}^{w}$| are the phase of wrapped interferograms computed from acquisitions k, l and m and modulo [2π] indicates phase signals are within the interval [0; 2π]. We substract closure of wrapped interferograms |$\Phi _{T}^{w}$| from closure ΦT computed on unwrapped interferograms in order to remove misclosures related to phase consistency loss in multilooking (eq. 3, Fig. 2). The total triplet closure |$\Phi _{T}^{tot}$| hence writes: $$\begin{eqnarray*} \Phi _{T}^{tot} = (\Phi _{T} - \Phi _{T}^{w}) m_{T}^{tot}. \end{eqnarray*}$$(3) We then round the total triplet closure |$\Phi _{T}^{tot}$| modulo 2π to estimate how many multiples of 2π should be corrected. We consider non-zero values as unwrapping errors and group them into regions using structuring elements (Fig. 3b; Verveer 2003). Remaining zero-values (i.e. pixel has been correctly unwrapped) are grouped into regions, considered as reference for the flux method described in the next section. Phase unwrapping errors generally arise in noisy or high fringe rate areas on interferograms. The error spreads from this area, forming a connected region on which phase has been locally correctly unwrapped but is inconsistent with neighbouring regions. We then associate each unwrapping error region to the largest reference region in the vicinity. Only regions containing more than 200 (minSize) pixels are considered. Now that we know where each unwrapping error is, we need to determine which interferogram of the triplet has been incorrectly unwrapped using a two-steps detection approach. Figure 2. Open in new tabDownload slide Closure maps (left-hand panel) and profiles across an unwrapping error (right-hand panel). Top panel: closure from unwrapped interferograms. Centre panel: closure from wrapped interferograms. Non-zero closure is due to multilooking. Bottom panel: total closure computed by removing misclosures due to multilooking effects. Figure 2. Open in new tabDownload slide Closure maps (left-hand panel) and profiles across an unwrapping error (right-hand panel). Top panel: closure from unwrapped interferograms. Centre panel: closure from wrapped interferograms. Non-zero closure is due to multilooking. Bottom panel: total closure computed by removing misclosures due to multilooking effects. Figure 3. Open in new tabDownload slide Steps to identify and correct an unwrapping error by the so-called flux method. (a) Total phase closure of the triplet. (b–c) Masked pixels within the unwrapping error zone are filled by erosion and dilation tool. (d) Erosion and dilation of the unwrapping error zone to identify inner (red) and outer (blue) border. (e) Computation of flux vectors between outer and inner pixels of the unwrapping error. Figure 3. Open in new tabDownload slide Steps to identify and correct an unwrapping error by the so-called flux method. (a) Total phase closure of the triplet. (b–c) Masked pixels within the unwrapping error zone are filled by erosion and dilation tool. (d) Erosion and dilation of the unwrapping error zone to identify inner (red) and outer (blue) border. (e) Computation of flux vectors between outer and inner pixels of the unwrapping error. 2.2 Step 1: flux method selection We first try to identify which interferogram of a triplet shows an abnormal phase offset, called ‘flux’, between an unwrapping error and its associated reference region. To compute this flux, we need to both erode and dilate the incorrectly unwrapped region to, respectively, isolate pixels within the unwrapping error from outside adjacent pixels, which are within a reference region (i.e. phase closure equals to zero). As explained in details in Pinel-Puysségur et al. (2018), we first fill up masked pixels within the error region (Figs 3b and c; Verveer 2003) and we erode and dilate using a structuring element, here chosen as a square of 2×2 pixels (Matheron 1967). The difference between the dilated and the original regions determines the outer border of the unwrapping error. Similarly, the difference between the eroded and the original regions determines the inner border of the unwrapping error (Fig. 3d). The size of the structuring element used for erosion and dilatation has been empirically chosen for the borders to be thin enough to compute the flux between neighbouring pixels but wide enough to ensure a sufficient number of flux measurements. We then discard pixels of the inner border that do not have any neighbour in the outer border, for example when they are on the image border, close to a masked region or far from the reference region. We calculate flux vectors along this boundary by differencing the phase of an inner pixel with the phase of the neighbouring outer pixel (Fig. 3e). We define pflux as the minimum proportion of flux vectors to correct an interferogram. For each interferogram of a triplet, we estimate the proportion of flux vectors equal to a multiple of 2π. If only one interferogram has more than pflux of its flux vectors equal to a multiple of 2π, this interferogram is marked as incorrectly unwrapped and the error is corrected by adding the appropriate multiple of 2π. If two or three interferograms have a proportion greater than pflux, we cannot discriminate which interferogram is to be corrected. In our case, pflux is set to 30 per cent. This choice is empirical and based on a user decision. In our two case studies, if pflux is too small (i.e. we have very few flux vectors), the algorithm misses most interferograms to correct. 2.3 Step 2: mean closure method selection If the flux method cannot determine which interferogram has to be corrected, we try to identify the interferogram incorrectly unwrapped by computing the mean closure of the three interferograms for all their triplets. We consider interferogram Ikl that belongs to |$N_{I_{kl}}$| triplets. The mean closure of interferogram Ikl, noted |$\Phi _{I_{kl}}^{\rm {mean}}$|⁠, is defined as the sum of the phase closure Φn on its |$N_{I_{kl}}$| triplets, normalized by the effective number of triplets : $$\begin{eqnarray*} \Phi _{I_{kl}}^{mean} = \frac{\sum \limits _{n=1}^{N_{I_{kl}}} \Phi _{n}}{m_{I_{kl}}} M_{I_{kl}}, \end{eqnarray*}$$(4) where |$M_{I_{kl}}$| is the intersection of all masks associated to each triplet and |$m_{I_{kl}}$| is an image containing for each pixel the number of defined triplets (without mask) for interferogram Ikl. We define pmc as the minimum proportion of pixels equal to a multiple of 2π to correct an interferogram. For each interferogram of the triplet, we compute the proportion of pixels in the unwrapping error zone such that the mean closure is equal to a multiple of 2π. If only one interferogram has more than pmc pixels equal to a multiple of 2π in the unwrapping error region, this one is marked as incorrectly unwrapped and the error is corrected. If two interferograms fulfill this condition and if the ratio between the two proportions, noted rmc, is greater than 2 (by default), the interferogram of highest proportion is corrected. Otherwise, we cannot discriminate which interferogram to correct. As the error may be corrected in another triplet, the algorithm then processes the following triplet. In our case studies, pmc is set to 50 per cent, a number that has been found empirically to increase the reliability of corrections. This threshold should be chosen on a case-by-case basis and is region dependent. In general, the algorithm should be iteratively run multiple times until no unwrapping corrections are needed. Several parameters such as the size of the structuring element used for dilation and erosion and threshold values may influence the performances of the algorithm. Users have to determine which set of parameters provides adequate unwrapping error corrections for each data set. We propose in the following to evaluate the performance of our method and provide guidelines on how to choose various parameters. 3 A QUALITATIVE EXAMINATION ON TWO CASE STUDIES We experiment our algorithm on two sets of SAR acquisitions and present the effects of unwrapping errors on time-series analysis. First, we process the archive of SAR acquisitions from the Envisat C-Band satellite over Lebanon. There, unwrapping errors arise because of low phase coherence due to interferograms with long perpendicular baselines (max. perpendicular baseline: 500 m; max. temporal baseline: 6 yr) and to the presence of vegetation. Secondly, we process SAR acquisitions from the constellation of Sentinel-1 C-Band satellites over Central Turkey (max. perpendicular baseline: 250 m; max. temporal baseline: 1 yr). This constellation offers a much shorter revisit time and a larger coverage compared to products from Envisat (revisit time of 6 days, 300 km wide). Manual corrections of unwrapping errors cannot be performed because of the untractable size of the resulting data set. The two case studies below differ by satellites and processing approach, however, both aim at retrieving slow tectonic deformations. As we are attempting to measure slow deformation rates at a regional scale and because of strong decorrelation effects, due for example to residual atmospheric artefacts, vegetation and snow in the winter, strong multilooking is required to enhance coherence in both cases, otherwise unwrapping of the phase is almost impossible. 3.1 Application to Envisat data set in Lebanon The Levant fault system is a complex active fault system of 1200 km long, where large earthquakes of magnitude up to 7.5 occurred in the past (e.g. Elias et al.2007). This major continental fault bounds the Arabian and African plates. We process data from Envisat ASAR track 78 with NSBAS (Doin et al.2011), a processing chain based on the Repeat Orbit Interferometry PACkage (ROI_PAC, Rosen et al.2004). We coregister SLCs to a master image taking into account local topography (Guillaso et al.2008). We use DORIS orbits from the European Space Agency (ESA) and SRTM Digital Elevation Model (DEM) Version 3.0 (Farr et al.2007) to compute the orbital and topographic phase contributions. We multilook wrapped interferograms by a factor of 4 in range and 20 in azimuth. We use MuLSAR (Multi-Link Interferograms) in order to increase the signal-to-noise ratio of interferograms (Pinel-Puysségur et al.2012). We then correct wrapped interferograms from stratified tropospheric delays estimated from the ERA-Interim global atmospheric reanalysis from ECMWF (Doin et al.2009; Jolivet et al.2011). We evaluate and compensate DEM errors by estimating the bias induced by perpendicular baselines (Ducret et al.2014). Finally, we filter interferograms using a Goldstein filter (Goldstein & Werner 1998), multilook by an additional factor of 4 (16 looks in range, 80 looks in azimuth) and unwrap them using the branch-cut method (Goldstein et al.1988). Our final data set is made of 165 unwrapped interferograms. Our algorithm identifies 282 triplets, among which 186 are corrected. We illustrate automatic corrections with a long temporal baseline interferogram, spanning 4 yr, where three corrections are performed (Fig. 4). The first error (number 1 in Fig. 4) is clearly well corrected. The two other errors (number 2 and 3 in Fig. 4) are more challenging due to the effect of filtering on high fringe rate areas. In both cases, a sharp fringe, partially visible on the interferogram before filtering (arrows in Figs 4d and f), disappears through filtering (arrows in Figs 4e and g) hence leading to discontinuities in the unwrapped interferograms (red circles in Fig. 4b). After correction, the algorithm restores continuity where a 2π phase offset was inconsistently introduced by the unwrapping procedure (red circles in Fig. 4c) and discontinuity in high fringe rate areas (arrows in Fig. 4c). Figure 4. Open in new tabDownload slide Results for Envisat data set in Lebanon. (a) Example of an interferogram spanning 2004/08/01–2008/07/06 which contains unwrapping errors (red circles). (b) and (c) Zooms of not corrected and corrected unwrapped interferograms. Error 1 is well corrected by the algorithm. Errors 2 and 3 are challenging areas, where the high fringe gradient, visible on wrapped interferograms, disappears by filtering before unwrapping [arrows in (b)]. The algorithm restores the correct positions of offsets [arrows in (c). (d), (e), (f) and (g)] Zooms of unwrapping errors 2 and 3 on wrapped interferograms, before and after filtering. Filtering erases fringes in high fringe rate regions (arrows). Figure 4. Open in new tabDownload slide Results for Envisat data set in Lebanon. (a) Example of an interferogram spanning 2004/08/01–2008/07/06 which contains unwrapping errors (red circles). (b) and (c) Zooms of not corrected and corrected unwrapped interferograms. Error 1 is well corrected by the algorithm. Errors 2 and 3 are challenging areas, where the high fringe gradient, visible on wrapped interferograms, disappears by filtering before unwrapping [arrows in (b)]. The algorithm restores the correct positions of offsets [arrows in (c). (d), (e), (f) and (g)] Zooms of unwrapping errors 2 and 3 on wrapped interferograms, before and after filtering. Filtering erases fringes in high fringe rate regions (arrows). 3.2 Application to Sentinel-1 data set in Central Turkey The North Anatolian Fault is an active right-lateral strike-slip fault that accommodates the rotation of Anatolia with respect to Eurasia. During the 19th century, seismic activity was characterized by a westward propagation of large earthquakes (∼Mw 7.0) along this 1200-km-long fault (Stein et al.1997). The last earthquake to date is the Izmit event Mw 7.5 in 1999, east of the Sea of Marmara (e.g. Reilinger et al.2000). We process data from Sentinel-1 track 87 with the InSAR Scientific Computing Environment (ISCE) software (Gurrola et al.2010). We define the acquisition of July 9th, 2017 as the master Single Look Complex (SLC) and coregister all SLCs to this master image. Coregistration is enhanced using the spectral diversity of burst overlaps refined within the network of interferograms (Fattahi et al.2017). We generate interferograms, accounting for digital elevation model (SRTM Version 3.0; Farr et al.2007) and orbital contributions, and merge tiles for each of them using bursts and swaths overlaps. We multilook merged interferograms with factors of, respectively, 81 and 27 in azimuth and range directions for a final pixel size of 540 × 420 m, in range and azimuth, respectively. We correct the phase from tropospheric signals using ERA-5, the latest global atmospheric reanalysis from ECMWF (Hersbach & Dee 2016). Finally, we filter (Goldstein & Werner 1998) and unwrap interferograms using the branch-cut method (Goldstein et al.1988). Before building triplets, we discard low coherence interferograms which cannot be sufficiently unwrapped (less than 20 per cent of the area). Our final data set is made of 686 coregistered and unwrapped interferograms. Our algorithm identifies 2880 triplets, among which 986 triplets are corrected eventually (Fig. 5a). We calculate the percentage of corrected pixels per interferogram by summing the number of pixels detected as unwrapping errors and corrected by the algorithm in all triplets of the interferogram. We see that most of the interferograms are totally corrected from unwrapping errors during a first pass of the algorithm (Fig. 5b). We illustrate automatic corrections with two examples of corrected interferograms, one with a large unwrapping error of 10 388 pixels (4 per cent of the interferogram, Fig. 5c) and another with two unwrapping errors localized in different places (Fig. 5d). In both cases, 99 per cent of the unwrapping error is automatically detected and corrected by the algorithm. Uncorrected pixels are located in the masked region of the triplet. The second example shows that the algorithm can perform multiple corrections in a single interferogram (Fig. 5d). In this case, it detects two unwrapping errors in the same interferogram and corrects them in the same triplet. Figure 5. Open in new tabDownload slide Results for Sentinel-1 data set in Turkey. (a) Perpendicular baseline plot with corrected triplets in black. Dots are SAR acquisitions and lines are interferograms. (b) Histogram of the number of interferograms corrected as a function of percentage of corrected pixels. (c) and (d) Examples of corrections spanning 2017/04/22–2017/11/12 and 2017/03/05–2017/11/12, respectively. Figure 5. Open in new tabDownload slide Results for Sentinel-1 data set in Turkey. (a) Perpendicular baseline plot with corrected triplets in black. Dots are SAR acquisitions and lines are interferograms. (b) Histogram of the number of interferograms corrected as a function of percentage of corrected pixels. (c) and (d) Examples of corrections spanning 2017/04/22–2017/11/12 and 2017/03/05–2017/11/12, respectively. 4 DISCUSSION AND QUANTITATIVE TESTS 4.1 Unwrapping errors and time-series analysis One potential application of SAR interferometry is to perform time-series analysis and estimate ground velocity over a given region from a stack of interferograms. We illustrate the effect of automatic corrections of unwrapping errors on the estimation of ground velocity and the associated decrease in errors on ground surface deformation measurement. We perform two time-series analysis on the Sentinel-1 data set (Section 3.2): the first one is applied to the original stack of interferograms not corrected from unwrapping errors, the second one is applied to the interferograms corrected by the proposed approach. We invert the temporal evolution of the phase for both data sets identically using the small baseline NSBAS approach (Doin et al.2011) implemented in the Generic InSAR Analysis Toolbox (GIAnT, Agram et al.2012). In this method, we consider each pixel independently to recover the phase change with time (López-Quiroz et al.2009; Doin et al.2011; Jolivet et al.2012). In addition to phase reconstruction, NSBAS includes a time dependent model of the phase to predict the phase evolution with time when interferometric links are missing between two disconnected subsets of interferograms. For each time-series analysis, we first remove interferograms that have less than 35 per cent unwrapped pixels, hence reducing the data set to 627 interferograms. We then multilook interferograms by a factor of 2 in order to reduce noise on the interferograms (due to the presence of vegetation and snow) and spatially reference them by choosing a region where the phase is set to be equal in all interferograms. We correct orbital biases in interferograms by estimating a linear ramp. Terms of the ramp are refined accounting for the interferometric network (Lin et al.2010; Jolivet et al.2012). We then perform a least squares inversion of phase delays of each pixel to solve for the total phase delay of each date relative to the first date and for a parametric evolution of phase change across the whole acquisition period. The parametric evolution of surface deformation is a combination of a linear term and a seasonal-annual function. We obtain two velocity maps over Central Turkey (Figs 6a and b). If we do not correct interferograms from unwrapping errors before the inversion, surface velocity is strongly affected by unwrapping errors (Figs 6a, b, d and e). In particular, several suspicious discontinuities visible on the first velocity map (Figs 6a and d) are not detected on the second one (Figs 6b and e). The difference in velocity between the two fields reaches up to 4 mm yr–1 in large regions (Fig. 6c), corresponding in our case to about 20 per cent of the expected tectonic displacement in the area. We can also identify small differences of about 1 mm yr–1 (Figs 6c and f), due to a difference in referencing between the two velocity maps. If we choose a reference region within an unwrapping error, the inversion will differ hence the resulting velocity maps will be different. Figure 6. Open in new tabDownload slide Influence of unwrapping error corrections on time-series analysis. (a) Velocity map calculated from a stack of interferograms not corrected from unwrapping errors and (b) from a stack of interferograms corrected from unwrapping errors. (c) Differences between a and b. (d) (e) and (f) Profiles across a, b and c, respectively. Figure 6. Open in new tabDownload slide Influence of unwrapping error corrections on time-series analysis. (a) Velocity map calculated from a stack of interferograms not corrected from unwrapping errors and (b) from a stack of interferograms corrected from unwrapping errors. (c) Differences between a and b. (d) (e) and (f) Profiles across a, b and c, respectively. The effect of unwrapping errors can be evaluated quantitatively by computing a root mean square (RMS) map, defined as: $$\begin{eqnarray*} \Phi _{RMS} = \frac{1}{N} \left[ \sum \limits _{N} \left( \phi _{ij} - \sum \limits _{k=i}^{j-1} m_k \right)^2 \right]^{1/2}, \end{eqnarray*}$$(5) where ϕij is the measured phase between acquisitions i and j and |$\sum \limits _{k=i}^{j-1} m_k$| is the reconstructed phase between the same acquisitions (fig. 7; Cavalié et al.2007). This RMS evaluates the quality of the time-series reconstruction and should then reflect interferometric misclosure. If we do not correct interferograms from unwrapping errors before the inversion, RMS reaches 12 mm (Fig. 7a), compared to few millimeters if unwrapping errors are corrected with the proposed approach (Fig. 7c). Average RMS is of 1.61 mm without corrections and 0.98 mm with corrections. In the case where unwrapping errors are not corrected, deviation in RMS is much larger than when errors are corrected, with extreme values of 8–14 mm (Fig. 7b). Since pixels with a large RMS after time-series analysis cannot be trusted for further interpretation, our approach allows to extend the area over which we can interpret the LOS displacement signal. Therefore, correcting unwrapping errors allows to expand the zone over which we confidently measure ground velocity, in the present case by 20 per cent with a RMS threshold of 3 mm. Figure 7. Open in new tabDownload slide Influence of unwrapping errors on root mean square (RMS) maps. (a) and (c) RMS maps where unwrapping errors are not corrected and corrected, respectively. Unwrapping errors have a large contribution on the estimation of RMS. (b) and (d) Histograms of RMS maps a and c, respectively. Figure 7. Open in new tabDownload slide Influence of unwrapping errors on root mean square (RMS) maps. (a) and (c) RMS maps where unwrapping errors are not corrected and corrected, respectively. Unwrapping errors have a large contribution on the estimation of RMS. (b) and (d) Histograms of RMS maps a and c, respectively. 4.2 Parametric tests on the Lebanon data set The validation of an automatic correction algorithm for phase unwrapping errors is a difficult task. The ideal way to assess the performance of such an algorithm would be to have a comprehensive ground truth where every pixel of every interferogram of the database is labelled as correctly or incorrectly unwrapped. Then, for each pixel incorrectly unwrapped, the number of cycles of the error and its associated sign should be known. In addition, such ground truth would allow identifying false alarms, that is pixels correctly unwrapped but detected as incorrectly unwrapped. Practically, on a real database, there is no simple and efficient way to determine which pixels have been incorrectly unwrapped. It should be stressed that pixels incorrectly unwrapped may be detected easily on a triplet thanks to its closure but that determining them directly on interferograms is a tedious task. Indeed, even a thorough visual examination of interferograms does not always allow determining if a region is incorrectly unwrapped and if so to which extent. Furthermore, no independent data set provides comparable measurement at the resolution allowed by InSAR. There are two strategies to quantitatively validate the results of the proposed algorithm. The first one is to establish an experimental ground truth on a part of a real database by visual inspection. The second one would be to create a synthetic database on which unwrapping errors would be known. This last one will be explored in future work. In this paper, we chose to derive quantitative performances attempting to compare with our experimental ground truth. Because we need to identify manually unwrapping errors, we will apply our validation on the Lebanon data set. We select 22 interferograms with easily identifiable unwrapping errors. For each interferogram, we manually detect and label the regions of unwrapping errors. For each region, we also identify the signed number of phase cycles of the error. We thoroughly perform this manual task to ensure that the experimental ground truth does not contain any error. The ground truth contains 26 error regions with a total of 120 235 pixels. To check if the algorithm detects false alarms, we select 15 interferograms and manually cut out and store regions that do not contain any unwrapping errors, with a total of 983 310 pixels. CorPhU depends on four parameters (see Table 1). Instead of a full parametric study which is beyond the scope of this paper, we focus our parametric tests on two out of the four parameters identified as the most important ones, including the thresholds pflux and pmc used for the identification of the incorrectly unwrapped interferogram of a triplet. pflux and pmc, two proportions of flux or pixels, are defined between 0 and 100 per cent. For the parametric tests, we test values ranging from 10 to 90 per cent with a step of 20 per cent. We also test the algorithm with only one of the two steps. To do so, we set pflux (resp. pmc) to a value strictly greater than 100 per cent. As a proportion would never attain such a value, we only use the 2nd step (resp. 1st step) during the run. In practice, the case with both pflux and pmc strictly greater than 100 per cent prohibits any correction and is not applicable. We run CorPhU with each of these combinations on the whole Lebanon data set, with two iterations. We then compute the true detection (TD) and false alarm (FA) rates on the ground truth as follows. For each region labelled as incorrectly unwrapped in the ground truth, we compute the difference between the interferogram obtained after CorPhU’s processing and the original one. We compare this difference to the expected correction, which is known from our estimates of ground truth. If these quantities are equal, then the error region has been well detected. Then, we compute the ratio of well detected regions as the TD rate. For the FA rate, we compare all regions of the ground truth labelled as correctly unwrapped to the difference of the interferograms before and after CorPhU’s processing. This difference should be zero. If not, we count the region as a FA. The results do not show any sufficient variation as a function of the parameters. For almost every test, the TD rate is quasi constant between 46 and 50 per cent. The only cases where the TD rate decreases significantly (under 40 per cent) are for pmc or pflux strictly greater than 100 per cent (only one of the steps is used). Similarly, the FA rate is almost always equal to 0 except for some cases where it reaches 2 per cent. These results do not sufficiently vary to draw any conclusion on the influence of the parameters. However, we can draw two conclusions. First, only half of unwrapping errors of this particular ground truth are detected. This poor performance can be explained as follows. In the Lebanon test case, there are only 282 triplets for 165 interferograms or an average of 1.71 triplets per interferogram. Moreover, many interferograms are partially masked. As triplet closure is only defined on the intersection of the definition domain of the three interferograms, the effective number of triplets per interferogram is even strictly smaller than 1.71. CorPhU should work better when the number of triplets increases for several reasons: first, the errors can only be detected on triplets closures; second, the first step needs a neighboring reference region on which the triplet closure is defined; third, the second step needs at least two triplets per interferogram under examination to determine which one is badly unwrapped. The poor TD rate on this database could thus be explained by the lack of triplets and the large masked areas, especially compared to the Turkey data set where an interferogram belongs to 8.2 triplets on average. Second, we think our estimate of ground truth is not sufficient and not precise enough to assess the parameters influence on CorPhU’s performances. It highlights the need of a synthetic database where the number of interferograms and triplets, the masked areas surface or other variables could be changed and their effect on CorPhU’s performances properly assessed. Consequently, we choose to assess the influence of tunable parameters on the performance by counting the number of pixels incorrectly unwrapped before and after the run. We automatically compute the total number of pixels on all triplets with non-zero triplet closure, W. We then derive the difference ΔW between W before (Winit) and after (Wfinal) the run to derive the percentage of corrected pixels (here, Winit is equal to 1847 173 pixels, see Table 2). Table 2 shows that the percentage of corrected pixels highly depends on the parameter value. We highlight in green in Table 2 the range of best parameters. We highlight acceptable sets of parameters in blue while red values correspond to settings that should not be used. Settings highlighted in green allow more than 90 per cent of the detected pixels to be corrected. The algorithm performs the best when pflux = 50 per cent, whatever the value of pmc. However, for pflux < = 30 per cent, the performances slightly deteriorate for decreasing pflux. Although still very acceptable for pflux = 30 per cent, the performances are less good for pflux = 10 per cent, especially if pmc is small. Surprisingly, for pflux between 10 and 50 per cent, CorPhU still performs well even without the second step (see last line of Table 2). Table 1. Default values for the algorithm thresholds. Name . Value (default) . minSize 200 pflux 30 per cent pmc 50 per cent rmc 2 Name . Value (default) . minSize 200 pflux 30 per cent pmc 50 per cent rmc 2 Open in new tab Table 1. Default values for the algorithm thresholds. Name . Value (default) . minSize 200 pflux 30 per cent pmc 50 per cent rmc 2 Name . Value (default) . minSize 200 pflux 30 per cent pmc 50 per cent rmc 2 Open in new tab Table 2. Percentage of corrected pixels for the 35 parametric tests performed on the Lebanon data set. Column pflux >100 per cent (respectively pmc >100 per cent) corresponds to step 2 (resp. step 1) only. NA: not applicable. pmc/pflux . 10 per cent . 30 per cent . 50 per cent . 70 per cent . 90 per cent . >100 per cent . 10 per cent 51.24 87.57 91.88 83.30 70.52 34.28 30 per cent 87.83 87.49 90.83 83.16 40.22 7.89 50 per cent 87.72 87.27 90.66 83.12 39.76 8.07 70 per cent 87.36 86.99 90.52 83.29 45.66 13.30 90 per cent 87.25 86.71 90.30 82.90 44.20 11.97 >100 per cent 91.25 90.49 81.92 44.45 8.85 NA pmc/pflux . 10 per cent . 30 per cent . 50 per cent . 70 per cent . 90 per cent . >100 per cent . 10 per cent 51.24 87.57 91.88 83.30 70.52 34.28 30 per cent 87.83 87.49 90.83 83.16 40.22 7.89 50 per cent 87.72 87.27 90.66 83.12 39.76 8.07 70 per cent 87.36 86.99 90.52 83.29 45.66 13.30 90 per cent 87.25 86.71 90.30 82.90 44.20 11.97 >100 per cent 91.25 90.49 81.92 44.45 8.85 NA Open in new tab Table 2. Percentage of corrected pixels for the 35 parametric tests performed on the Lebanon data set. Column pflux >100 per cent (respectively pmc >100 per cent) corresponds to step 2 (resp. step 1) only. NA: not applicable. pmc/pflux . 10 per cent . 30 per cent . 50 per cent . 70 per cent . 90 per cent . >100 per cent . 10 per cent 51.24 87.57 91.88 83.30 70.52 34.28 30 per cent 87.83 87.49 90.83 83.16 40.22 7.89 50 per cent 87.72 87.27 90.66 83.12 39.76 8.07 70 per cent 87.36 86.99 90.52 83.29 45.66 13.30 90 per cent 87.25 86.71 90.30 82.90 44.20 11.97 >100 per cent 91.25 90.49 81.92 44.45 8.85 NA pmc/pflux . 10 per cent . 30 per cent . 50 per cent . 70 per cent . 90 per cent . >100 per cent . 10 per cent 51.24 87.57 91.88 83.30 70.52 34.28 30 per cent 87.83 87.49 90.83 83.16 40.22 7.89 50 per cent 87.72 87.27 90.66 83.12 39.76 8.07 70 per cent 87.36 86.99 90.52 83.29 45.66 13.30 90 per cent 87.25 86.71 90.30 82.90 44.20 11.97 >100 per cent 91.25 90.49 81.92 44.45 8.85 NA Open in new tab For pflux > = 70 per cent, performances degrade with increasing pflux. In particular, for pflux >1 (only step 2), the percentage of corrected pixels drops under 35 per cent. When pflux increases, less corrections are possible with step 1 so CorPhU moves to step 2 to determine the wrong interferogram. The loss of performance suggests that step 2 is less effective than step 1 and that relying only on step 2 degrades the performances of CorPhU in our case. This suggests that many efficient corrections due to step 1 disappear when pflux is too high. It should be noted however that the poor performance of the second step in the Lebanon case may be due to the relatively small number of triplets as a greater number of triplets should enhance the robustness of the second step. In general, if pflux is set between 10 and 50 per cent and pmc between 30 and 90 per cent, at least 85 per cent of the pixels are corrected. Nonetheless, analysis of this difference might not be a perfect assessment method for several reasons. First, some unwrapping errors cannot be detected by triplet misclosure, because the interferogram does not belong to any triplet or because of the masked areas. Secondly, errors cannot be detected either when they compensate each other on two interferograms of the same triplet, thus leading to a zero closure. This is the case when one acquisition of the triplet contains a sharp atmospheric delay incorrectly unwrapped in two interferograms. Thirdly, the true parameter of interest would be the total number of incorrectly unwrapped pixels on all interferograms. Instead, W counts several times the same error of an interferogram as soon as it belongs to several triplets. Thus there is no simple relationship between these two variables. Fourthly, if the misclosure of a triplet disappears after the run, it is hopefully due to a right correction but it also may be due to an incorrect correction. Although such a wrong correction may then increase the number of pixels with non-zero closure on other triplets, it is not always the case. Nonetheless, W seems to vary for a given range of parameters and tendances arises, hence it is a proxy of the performance of CorPhU. Even if this parametric analysis is partial, some conclusions can be drawn: the most determining factor is pflux which should be set ideally around 50 per cent whereas pmc should lie above 30 per cent. However, these results are relative to a specific data set, hence they should be taken with caution if applied to another data set. Among others, the number of interferograms and triplets, the multilooking factors, the shape and size of decorrelating areas should influence the optimal choice of parameters. A solution to determine the best set of parameters for a given data set would be to first run CorPhU for different sets of parameters in order to make an automatic diagnosis. The algorithm should then be run again with appropriate values of parameters. Another solution would be to vary the parameters settings during the iterations. The thresholds could be set to high values during the first iterations to allow only few corrections with a high confidence level and progressively decrease during the following iterations. The aim is to avoid any false corrections at the beginning of the process that may later induce other false corrections. 4.3 Effectiveness of the algorithm in correcting unwrapping errors in high redundancy data sets We also assess the effectiveness of the algorithm in correcting unwrapping errors in large data sets using the proxy W, corresponding to the total number of pixels on all triplets with non-zero closure (unwrapping errors) automatically computed before and after two successive runs. We run the algorithm four times, iteratively, on the 686 interferograms of the Turkey data set. Before performing any corrections, the algorithm detects around 2 millions of pixels incorrectly unwrapped on the 2880 triplets built (Fig. 8). At the end of the first run, the number of pixels corresponding to unwrapping errors is about 500 000, indicating that the algorithm corrects 75 per cent of the initial pixels in a single iteration. At the end of the second run, the number of pixels decreases to 100 000, illustrating that the algorithm corrects 95 per cent of the initial incorrectly unwrapped pixels in only two iterations. Next runs show that the algorithm converge in two iterations, as the number of pixels incorrectly unwrapped does not decrease anymore after the second run. This might be due to the size of residual errors, too small to be corrected considering our threshold, for this particular case, of a minimum 200 pixels for a single unwrapping error, or to errors which are not or no longer connected to references regions (no misclosure), and therefore where the flux method cannot be performed. As a conclusion, the strength of this algorithm is that it automatically corrects almost all of the unwrapping errors from a large data set in only two iterations. Figure 8. Open in new tabDownload slide Number of pixels incorrectly unwrapped by a branch-cut method (unwrapping errors) detected by the algorithm during 4 successive runs. The algorithm converge in two iterations, leading to correct 95 per cent of the 2 millions pixels incorrectly unwrapped at the beginning. Figure 8. Open in new tabDownload slide Number of pixels incorrectly unwrapped by a branch-cut method (unwrapping errors) detected by the algorithm during 4 successive runs. The algorithm converge in two iterations, leading to correct 95 per cent of the 2 millions pixels incorrectly unwrapped at the beginning. 4.4 Effectiveness of the algorithm in correcting unwrapping errors using a least-square unwrapping method We test our algorithm not only on interferograms unwrapped using a branch-cut method but also on those unwrapped using a least-square approach. We unwrap the 686 interferograms of the Turkey data set using the Snaphu algorithm and run the algorithm four times as described above. Before performing any corrections, the algorithm detects around 20 millions of pixels incorrectly unwrapped on the 2880 triplets built, compared to the 2 millions of pixels for the branch-cut method (Fig. 9). At the end of the first run, the number of pixels corresponding to unwrapping errors does not decrease significantly, as only 20 000 pixels have been corrected during the iteration. Other iterations do not lead to perform a high number of corrections. Figure 9. Open in new tabDownload slide Comparison of the number of pixels incorrectly unwrapped by a branch-cut (blue diamonds) and a least-square (red squares) approach detected by the algorithm during 3 successive runs. Figure 9. Open in new tabDownload slide Comparison of the number of pixels incorrectly unwrapped by a branch-cut (blue diamonds) and a least-square (red squares) approach detected by the algorithm during 3 successive runs. Results suggest that the algorithm is not adapted for interferograms unwrapped using a least-square approach, such as Snaphu. First, as Snaphu is a global minimization procedure, resulting triplets phase closure maps are not obvious to interpret, compared to those computed for interferograms unwrapped using a branch-cut method. Closure maps are equal to 0 plus or minus a residual that might not necessarily equal a multiple of 2π. Therefore, the algorithm detects, in this case, 10 times more pixels incorrectly unwrapped with Snaphu than with the branch-cut method. Secondly, global minimization leads to a less marked spatial signature of unwrapping errors (lower phase gradients between outer and inner pixels of an unwrapping error). Consequently, the so-called flux method of our algorithm, designed for the detection of unwrapping errors using phase gradients, fails most of the time. The mean closure method also fails most of the time in correcting unwrapping errors because errors are not necessarily a multiple of 2π. Further work is required to adapt our method to global minimization unwrapping methods. 5 CONCLUSIONS, LIMITS AND FUTURE WORK We developed an algorithm called CorPhU, using phase closure of triplets of interferograms to correct unwrapping errors left after phase unwrapping. We assess its efficiency on two data sets in Lebanon and Turkey, respectively with Envisat and Sentinel-1 satellites. Our algorithm helps the interpretation of the interferometric phase in low coherence regions, polluted by unwrapping errors, without requiring visual interferogram inspection or manual deletions of unwrapping errors. As the contribution of unwrapping errors to velocity maps may reach up to 1 cm yr–1 and as they lead to RMS errors up to 1 cm, it is critical to correct these errors for interseismic strain measurements in active tectonic environments, where deformation rates are typically on the order of millimeters per year. As the algorithm is based on triplet information, the more interferograms are constructed, the largest the network of triplets is built, hence the higher the probability to correct recurrent unwrapping errors. The algorithm is particularly powerful for large data sets such as from Sentinel-1, where the revisit time is 6 days hence allowing to construct large networks. However, there are some limitations. Processing time is one of the main constraints and depends on the size of the data set. For example, the algorithm takes about six hours to process the Turkey data set, which corresponds to 2880 triplets, using 24 threads on a classic desktop machine. One way to increase the speed of processing is to take more benefits from triplets information considering the first iteration. The goal is to determine which interferograms to correct first so that it helps for the correction of other interferograms, hence reducing processing time. For instance, triplets with small-baseline interferograms should be corrected in priority as they are supposed to be less affected by decorrelation and therefore less affected by unwrapping errors. Long-baseline interferograms should be corrected afterwards, using triplets where small-baseline interferograms have been corrected. Another improvement would be to parallelize some of the steps of the algorithm, for instance to deal with independent triplets in parallel. Our automatic method, designed for dense networks of interferograms, requires technical improvements but, in overall, fits well into existing lines of research, where we increasingly face ‘big data’ related challenges, which must be converted from a highway to hell to a stairway to heaven. ACKNOWLEDGEMENTS This project received fundings from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Geo-4D project, grant agreement 758210). We use the Sentinel-1 and Envisat products, respectively provided by the Plateforme d’Exploitation des Produits Sentinel (PEPS) for Turkey and ESA through Cat1 proposal and EOLi-SA platform for Lebanon. Data analysis in Lebanon was supported by the CNES through the TOSCA program. It is based on observations with ASAR embarked on Envisat. 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TI - CorPhU: an algorithm based on phase closure for the correction of unwrapping errors in SAR interferometry
JO - Geophysical Journal International
DO - 10.1093/gji/ggaa120
DA - 2020-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/corphu-an-algorithm-based-on-phase-closure-for-the-correction-of-RmYValcHY0
SP - 1959
VL - 221
IS - 3
DP - DeepDyve
ER -