Feel free to skip forward over this subsection which is merely a mathematical proof.

When Fourier transformation is possible,
extrapolation operators are
complex numbers like *e*^{i kz z}.
With complex numbers *a* and *b* there
is never any question that .Then both **splitting** and **full separation** are always valid.

Suppose Fourier transformation has not been done,
or could not be done because of some spatial
variation of material properties.
Then extrapolation operators are built up
by combinations of
differential operators or their finite-difference representations.
Let and denote two such operators.
For example, could be a matrix containing
the second *x* differencing operator.
Seen as matrices,
the **boundary condition**s of a
differential operator are incorporated in the corners of the matrix.
The bottom line is whether ,so the question clearly involves the **boundary**
conditions as well as the differential operators.

Extrapolation downward a short distance can be done
with the operator .Let denote a vector where components of
the vector designate the wavefield at various locations on the *x*-axis.
Numerical analysis gives us a matrix operator, say , which
enables us to project forward, say,

(11) |

(12) |

(13) |

To march some distance down the *z*-axis we
apply the operator many times.
Take an interval *z _{1}*-

To prove the validity of **splitting**, we
take .Observe that the operator differs from the
operator by
something in proportion to or 1/*N ^{2}*.
So in the limit of a very large number of
subintervals, the error disappears.

It is much easier to establish the validity of the full-separation concept.

Commutativity is whether or not .Commutativity is always true for scalars.
With finite differencing the question is whether the two matrices commute.
Taking and to be differential operators,
commutativity is defined with the help of
the family of all possible wavefields *P*.
Then and are commutative if
.

The operator representing will
be taken to be .The simplest numerical
integration scheme using the **splitting** method is

(14) |

(15) |

(16) |

12/26/2000