Digression: curl grad as a measure of bad data

curl grad The relation (76) between the phases and the phase differences is  

$$\left[ \begin{array} {rrrr} -1 & 1& 0& 0 \\ 0 & 0& -1& 1 ... ... \\ \Delta \phi_{ac} \\ \Delta \phi_{bd} \end{array}\right]$$ (77)

Starting from the phase differences, equation (77) cannot find all the phases themselves because an additive constant cannot be found. In other words, the column vector [1,1,1,1]' is in the null space. Likewise, if we add phase increments while we move around a loop, the sum should be zero. Let the loop be $a \rightarrow c \rightarrow d \rightarrow b \rightarrow a$.The phase increments that sum to zero are:  

$$\Delta \phi_{ac} + \Delta \phi_{cd} - \Delta \phi_{bd} - \Delta \phi_{ab} \eq 0$$ (78)

Rearranging to agree with the order in equation (77) yields

$$- \Delta \phi_{ab} + \Delta \phi_{cd} + \Delta \phi_{ac} - \Delta \phi_{bd} \eq 0$$ (79)

which says that the row vector [-1,+1,+1,-1] premultiplies (77), yielding zero. Rearrange again

$$- \Delta \phi_{bd} + \Delta \phi_{ac} \eq \Delta \phi_{ab} - \Delta \phi_{cd}$$ (80)

and finally interchange signs and directions (i.e., $\Delta \phi_{db} = -\Delta \phi_{bd}$)

$$(\Delta \phi_{db} - \Delta \phi_{ca}) \ -\ (\Delta \phi_{dc} - \Delta \phi_{ba}) \eq 0$$ (81)

This is the finite-difference equivalent of

$${\partial^2 \phi \over \partial x \partial y} \ -\ {\partial^2 \phi \over \partial y \partial x} \eq 0$$ (82)

and is also the z-component of the theorem that the curl of a gradient $\nabla\times\nabla\phi$ vanishes for any $\phi$.

The four $\Delta\phi$ summed around the $2\times 2$ mesh should add to zero. I wondered what would happen if random complex numbers were used for a, b, c, and d, so I computed the four $\Delta\phi$'s with equation (76), and then computed the sum with (78). They did sum to zero for 2/3 of my random numbers. Otherwise, with probability 1/6 each, they summed to $\pm2\pi$.The nonvanishing curl represents a phase that is changing too rapidly between the mesh points. Figure shows the locations at Vesuvius where bad data occurs. This is shown at two different resolutions. The figure shows a tendency for bad points with curl $2\pi$ to have a neighbor with $-2\pi$.If Vesuvius were random noise instead of good data, the planes in Figure would be one-third covered with dots but as expected, we see considerably fewer.

 

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Figure 4 Values of curl at Vesuvius. The bad data locations at both coarse and fine resolution tend to occur in pairs of opposite polarity.