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