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The defining equation for a second
*difference*
operator is

| |
(30) |

The second
*derivative*
operator is defined by taking the limit
| |
(31) |

Many different definitions can all go to the same limit
as goes to zero.
The problem is to find an expression that is accurate when is
larger than zero and, on a practical level,
is not too complicated.
Our first objective is to see how the accuracy of
equation (30) can be evaluated quantitatively.
Second, we will look at an expression that is slightly more complicated
than (30) but much more accurate.
The basic method of analysis we will use is Fourier transformation.
Take the derivatives of
the complex exponential and
look at any errors as functions of the spatial frequency *k*.
For the second derivative,

| |
(32) |

Define by an expression analogous to the difference
operator:
| |
(33) |

Ideally would equal *k*.
Inserting
the complex exponential into (30),
we see that the definition (33)
gives an expression for in terms of *k*:
| |
(34) |

| |
(35) |

It is a straightforward matter
to make plots of versus from (35).
The half-angle trig formula allows us to take an
analytic square root of (35), which is
| |
(36) |

Series expansion shows that for low frequencies is
a good approximation to *k*.
At the Nyquist frequency,
defined by ,the approximation is a poor approximation to .

** Next:** The 1/6 trick
** Up:** FREQUENCY DISPERSION IN WAVE-MIGRATION
** Previous:** Spatial aliasing various space
Stanford Exploration Project

10/31/1997