Increased absolute accuracy may always be purchased by reducing .Increased accuracy relative to the Nyquist frequency may be purchased at a cost of computer time and analytical clumsiness by adding higher-order terms, say,

(37) |

(38) |

(39) |

Figure 21

Taking *b* in (38) and (39) to be 1/12, then
(38), (39) and (37) would agree
to second order in .The 1/12 comes from series expansion, but the 1/6 fits over a
wider range and is a value in common use.
Francis Muir has pointed out
that the value 1/6.726 gives an
*exact*
fit at the Nyquist frequency and an accurate fit over all lower frequencies!
Few explorationists consider the remaining
accuracy deficiency of (38) and (39)
to be sufficient to warrant interpolation
of field-recorded values.
Figure 22 compares hyperbolas for various values of *b*.
Observe in Figure 22 that the longest wavelengths
travel at the same speed regardless of *b*.
The time axis in Figure 22 is only 256 points long,
whereas in practice it would be a thousand or more.
So Figure 22 exaggerates the frequency dispersion
attributable in practice to finite differencing the *x*-axis.

Figure 22

Let us be sure it is clear how (38) and (39) are put
into use.
Take *b* = 1/6.
The simplest prototype equation is the heat-flow equation:

(40) |

(41) |

10/31/1997