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Implicit equations

Recall that the inflation-of-money equation
 (73)
is a simple explicit finite differencing of the differential equation .And recall that a better approximation to the differential equation is given by the Crank-Nicolson form
 (74)
that may be rearranged to
 (75)
or
 (76)
The amplification factor (76) has magnitude less than unity for all negative r values, even r equal to minus infinity. Recall that the heat-flow equation corresponds to
 (77)
where k is the spatial frequency. Since (76) is good for all negative r, the heat-flow equation, implicitly time-differenced, is good for all spatial frequencies k. The heat-flow equation is stable whether or not the space axis is discretized (then ) and regardless of the sizes of and .Furthermore, the 15 wave-extrapolation equation is also unconditionally stable. This follows from letting r in (76) be purely imaginary: the amplification factor (76) then takes the form of some complex number 1 + r/2 divided by its complex conjugate. Expressing the complex number in polar form, it becomes clear that such a number has a magnitude exactly equal to unity. Again there is unconditional stability.

At this point it seems right to add a historical footnote. When finite-difference migration was first introduced many objections were raised on the basis that the theoretical assumptions were unfamiliar. Despite these objections finite-difference migration quickly became popular. I think the reason for its popularity was that, compared to other methods of the time, it was a gentle operation on the data. More specifically, since (76) is of exactly unit magnitude, the output has the same -spectrum as the input. There may be a wider lesson to be learned from this experience: any process acting on data should do as little to the data as possible.

Next: Leapfrog equations Up: INTRODUCTION TO STABILITY Previous: Explicit 15 degree migration
Stanford Exploration Project
10/31/1997