Phase unwrapping of angle-domain common image gathers

$L^0$ solution and weighted iterations

The solution of the above system of equations (2) can be found by minimizing a chosen indicator. Given the particular nature of the unwrapping problem, the $L^0$ measure is considered a good choice. The point is that we are not looking for a smooth solution that tries to accomodate all equations (like the $L^2$ norm does); we instead want the algorithm to make hard choices between alternatives and to produce a solution that satisfies, with no approximation, the highest possible number of equations. Ghiglia and Romero (1996) describe a way to minimize the $L^0$ measure via successive steps that are computed solving weighted least squares problems. Ghiglia and Romero's algorithm is more general and provides a way to minimize any $L^p$ measure, with $p$ in $[0,2]$. An application of the $L^1$-norm is found in Lomask (2006).

The following is the outline of the suggested algorithm, (setting $p=0$ for our specific case):

• Set up the initial weights, $\bf W_0$.
• Set $i=0$.
• Until $i$ has reached the maximum number of iterations, repeat the following steps:
  1. Solve (to convergence) the Weighted Least Square (WLS) system:
     

     $$\bf G^T W_i G \Phi_i = G^T W_i d.$$ (3)

     
     

  2. Compute new weights according to the formula
     

     $${W_{i+1}(n)} = \frac{\epsilon_0}{\epsilon_0 + \vert{\bf g}(n){\bf\Phi_i}-d(n)\vert^{2-p}}{W_{i}(n)}.$$ (4)

     
     

  3. Increase $i$ by 1.
• End.

$\bf W_i$ is a diagonal matrix with elements $W_i(n)$, the weights for each equation. The vector ${\bf g}(n)$ is the $n^{th}$ row of $\bf G$, so that ${\bf g}(n){\bf\Phi_i}$ is a scalar and $\epsilon_0$ an adequately small value. For efficiency reasons the WLS step is implemented by preconditioned conjugate gradient.

With this iterative mechanism and this particular choice of weights, each equation which is not satisfied at a given iteration is almost ignored for the next iteration, provided that more trusted equations exist that involve the same points.

Thus the choice of the initial weights is critical to yielding good results. We preliminarily used the amplitude information as a measure for the phase reliability: each equation was given a weight proportional to the harmonic average between the amplitudes of the two points involved.

velocity
Figure 1. The velocity used for modeling the seismic data.

Phase unwrapping of angle-domain common image gathers