**Phase History Retrieval in Interferogram Stacks Using Integer Least Squares Estimation**
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Samiei Esfahany, Sami; Hanssen, Ramon
Delft University of Technology, NETHERLANDS
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In the recent years, algorithms have been proposed with focus on distributed scatterers in order to retrieve maximum information from interferometric stacks of SAR acquisitions , extending the applicability of time series InSAR methodologies to rural areas which are affected by geometrical and temporal deccorrelation but still preserve some degree of coherence in some interferograms. These methods are based on two fundamental steps. First, a stack of multi-looked wrapped interferograms are exploited in order to optimally retrieve the phase time series for each pixel. This step - called phase triangulation, phase linking or phase multi-linking - is the key part of these new algorithms. The goal of phase triangulation is to optimally retrieve wrapped interferometric phases from all possible interferometric combinations preserving useful information (i.e. wrapped phase time series corresponding to the optical path differences between the targets and the sensor) and filtering thermal and decorrelation noise. The advantage compared to conventional approaches, such as small baseline subsets (SBAS), is that the phase retrieval step can be applied before phase unwrapping. In the second step, the retrieved phase time series can be processed using the traditional time series InSAR methodologies for unwrapping and estimation of parameters of interest (e.g. deformation time series, velocity, and height). In this contribution, an alternative approach is presented for phase triangulation based on the integer least squares (ILS) estimation. We model the phase triangulation problem as a system of linear observation equations with some integer and real unknowns and use ILS to estimate them. ILS is a geodetic estimation method, originally designed for GPS applications, and later also applied for InSAR temporal phase unwrapping. The main advantages of this new formulation is that the ILS method considers all the mutual correlations between interferometric phases and allows the formal error propagation from observations to the final estimates, and so additionally provides as an output the precision of the retrieved phase time series. We have performed extensive simulation based on different coherence behaviors and acquisition scenarios using characteristics of different satellite missions. The results of the simulations demonstrate the capability of our method for each scenario with respect to different parameters such as coherence variability, revisit time, radar wavelength, and satellite baselines. The same simulation was performed using the characteristics of the Sentinel mission. The results show the expected improvement by using Sentinel data, in extracting information from distributed scatterers (e.g. in rural areas) for InSAR applications. Finally we show how the application of our method can significantly improve the characterization of deformation mechanism in a case study area of land subsidence in the Netherlands.