Spatio-temporal phase unwrapping using integer least-squares
Freek van Leijen(1) , Petar Marinkovic(1)
, Bert Kampes(2)
, and Ramon Hanssen(1)
Delft University of Technology,
2629 HS Delft,
(2) German Aerospace Center (DLR) - Oberpfaffenhofen , Münchner Strasse 20, 82234 Wessling, Germany
Conventional phase unwrapping algorithms utilize the spatial correlation
between neighboring points to unwrap data using heuristic assumptions
on phase gradients between adjacent points. Decorrelation, for instance
induced by geometry or physical change of the earth's surface, often
results in poor performance of these algorithms. Moreover, a quantitative quality description of the results cannot be derived.
With the introduction of Persistent Scatterer techniques, the unwrapping
problem shifted from the spatial to the temporal domain. Typically,
time series of interferometric phases at a single location or the
differential phases between two of these locations (arcs) are used
to solve the unwrapping problem. Various techniques to unwrap the time
series are proposed, such as the search for maximal temporal coherence or integer least-squares. Usually, however, the spatial correlation between neighboring arcs is ignored in this process. Since the time series of two nearby points may be very correlated, it is evident that this should be used to improve or validate the results of the individual point time series. Hence, by introducing this spatial correlation into the temporal unwrapping problem, an improved or more reliable solution is feasible.
In this study the mathematical framework for a spatio-temporal unwrapping algorithm based on integer least-squares will be derived. This includes the functional model, which describes the functional relation between the parameters and the observations, and the stochastic model, which represents the correlation between the points in the 3D data cube formed by a stack of interferograms. Covariance functions describing spatial and temporal phase behavior related to, e.g., point coherence, deformation, topography, and atmosphere are designed and included inthe model. Apart from unwrapped data, a quality description of the results will be obtained. Using the integrated 3D approach, a more reliable and consistent solution is obtained compared to sequentially applying a temporal and a spatial unwrapping algorithm.
The method is tested on data covering Flevoland in the Netherlands, which contains urban areas as well rural areas. Furthermore, comparisons with conventional methods are shown and evaluated.