Phase unwrapping of angle-domain common image gathers

Graph and linear system

In our domain, the signal is a function of angle ($\alpha$), vertical wavenumber ($kz$) and midpoint inline position ($x$). Each equation we include in our system connects two points, so that each equation corresponds to a link and the entire system to a graph. A simple cartesian grid was used in the angle-kz plane. Each point is connected to its four neighbors, so for example the point $A(\alpha,kz,x)$ is connected to $A(\alpha,kz\pm 1,x)$ and to $A(\alpha\pm 1,kz,x)$. Points at the boundary of the domain have fewer connections.

To increase the robustness of the unwrapping procedure we do not consider each gather independently but connect several gathers in the inline direction, presuming continuity along that axis too. So $A(\alpha,kz,x)$ is also connected to $A(\alpha,kz,x\pm 1)$, raising to six the number of equations in which a given point typically appears.

An example of the basic equation is the following:

$$\phi(\alpha,kz,z) - \phi(\alpha-1,kz,z) = [\varphi(\alpha,kz,z) - \varphi(\alpha-1,kz,z)]_{2\pi}$$ (1)

where the other cases are straightforward. The expression $[\cdot]_{2\pi}$ represents the wrapping operator, or the remainder after integer division by $2\pi$; $\phi$ are the unwrapped values and $\varphi$ their wrapped, observed counterparts.

The system is not complete without some boundary equations that serve as a phase reference. We set to zero the zero-angle phases of a reference gather for all the considered wavenumbers.

The whole system can be written in matrix form:

$${\bf G \Phi= d},$$ (2)

where $\bf G$ is the graph incidence matrix plus border equations, $\bf\Phi$ is the unknown vector of unwrapped phases and $\bf d$ is a function of the observed phases (the wrapped differences). $\bf G$ is a very sparse matrix with typically two non-zero entries per row.

Phase unwrapping of angle-domain common image gathers