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curl grad
The relation (76) between the phases and the phase differences is
| |
(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
.The phase increments that sum to zero are:
| |
(78) |

Rearranging to agree with the order in equation (77) yields
| |
(79) |

which says that the row vector [-1,+1,+1,-1]
premultiplies (77),
yielding zero.
Rearrange again
| |
(80) |

and finally interchange signs and directions
(i.e., )
| |
(81) |

This is the finite-difference equivalent of
| |
(82) |

and is also
the *z*-component of the theorem that the curl of a gradient
vanishes for any .
The four summed around the 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 '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 .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 to have a neighbor with .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.

**screw90
**

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.

** Next:** Estimating the inverse gradient
** Up:** VESUVIUS PHASE UNWRAPPING
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Stanford Exploration Project

4/27/2004