The time and space derivatives must be discretized from a continuous
function to a discrete function. Methods used to
compute derivatives
for seismic modeling include Taylor Series, Chebechev, Fourier
Transforms, and Padé.
The Taylor Series (TS) methods will be
developed here; they are reasonably good approximations but
not optimal. The theories for optimal operators are
beyond the scope of this effort. The TS method
assumes that a function known at point **a** can be extended
to point **b** if sufficient number of derivatives exist and
are known at point **a**. The truncation error for the series
expansion has a maximum within the approximation interval.

The formulation preferred is a matrix representation for computing the coefficients of the Taylor Series. Given

then