Phase unwrapping of angle-domain common image gathers

Phase Unwrapping

When a signal is delayed, the phases of its Fourier spectral components are rotated proportionally. However, due to the periodic nature of Fourier components, the observable phases are always limited to the interval $[-\pi,\pi]$; i.e. there is no record of the number of entire cycles that may have intervened. This phenomenon is usually referred to as phase ambiguity, because different delays can correspond to the same observed phase shift. Phase unwrapping is the problem of recovering the number of $2\pi$ cycles that unambiguously reconstructs the original delay.

Phase unwrapping can be approached in various ways. In this work we follow the recipe presented in Ghiglia and Romero (1996) and Ghiglia and Romero (1994), where the unwrapped phase is found as the solution of a linear system.

The general principle is that even though the unwrapped phases are usually outside the interval $[-\pi,\pi]$, differences in unwrapped phases of ``neighboring'' points are often included in that interval, so that they can be recovered also from the wrapped values, which are available. Thus we write a number of equations that describe differences in the unwrapped phases and rely on the solution of the system to integrate those differences.

Of course some of the original equations are wrong (they assume the phase difference to be within the interval $[-\pi,\pi]$, when in fact it is not) and conflict with others. The algorithm that solves the system will eventually have to make a decision and discard some equations, favoring others.

First we have to define a graph that represents the equations we will use. Then we describe the algorithm for the solution of the system.

Phase unwrapping of angle-domain common image gathers