Model fitting by least squares

Digression: curl grad as a measure of bad data

The relation (96) between the phases and the phase differences is

$$\left[ \begin{array}{rrrr} -1 & 1& 0& 0 \\ 0 & 0& -1& 1 ... ... \Delta \phi_{ac} \\ \Delta \phi_{bd} \end{array}\right]$$ (97)

Starting from the phase differences, equation (97) hope to 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:

$$\Delta \phi_{ac} + \Delta \phi_{cd} - \Delta \phi_{bd} - \Delta \phi_{ab} \eq 0$$ (98)

Rearranging to agree with the order in equation (97) yields

$$- \Delta \phi_{ab} + \Delta \phi_{cd} + \Delta \phi_{ac} - \Delta \phi_{bd} \eq 0$$ (99)

which says that the row vector $[-1,+1,+1,-1]$ premultiplies (97), yielding zero. Rearrange again

$$- \Delta \phi_{bd} + \Delta \phi_{ac} \eq \Delta \phi_{ab} - \Delta \phi_{cd}$$ (100)

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

$$(\Delta \phi_{db} - \Delta \phi_{ca}) \ -\ (\Delta \phi_{dc} - \Delta \phi_{ba}) \eq 0$$ (101)

This is the finite-difference equivalent of

$${\partial^2 \phi \over \partial x \partial y} \ -\ {\partial^2 \phi \over \partial y \partial x} \eq 0$$ (102)

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 (96), and then computed the sum with (98). 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 13 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 13 would be one-third covered with dots but as expected, we see considerably fewer.

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Figure 13. Values of curl at Vesuvius. The bad data locations at both coarse and fine resolution tend to occur in pairs of opposite polarity.

Model fitting by least squares