Validity of the splitting and full-separation concepts

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 k_z z}. With complex numbers a and b there is never any question that $ab\ =\ ba$.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 ${\bf A}$ and ${\bf B}$ denote two such operators. For example, ${\bf A}$ could be a matrix containing the second x differencing operator. Seen as matrices, the boundary conditions of a differential operator are incorporated in the corners of the matrix. The bottom line is whether ${\bf A}{\bf B}= {\bf B}{\bf A}$,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 $( {\bf I}+ {\bf A}\, \Delta z )$.Let ${\bf p}$ 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 ${\bf A}$, which enables us to project forward, say,

$${\bf p} (z + \, \Delta z) \eq {\bf A}_1 \ {\bf p} (z)$$ (11)

The subscript on ${\bf A}$ denotes the fact that the operator may change with z. To get a step further the operator is applied again, say,

$${\bf p} (z + 2 \, \Delta z) \eq {\bf A}_2 \ [ {\bf A}_1 \ {\bf p} (z)]$$ (12)

From an operational point of view the matrix ${\bf A}$ is never squared, but from an analytical point of view, it really is squared.

$${\bf A}_2 \ [ {\bf A}_1 \ {\bf p} (z)] \eq ( {\bf A}_2 \ {\bf A}_1 ) \ {\bf p} (z)$$ (13)

To march some distance down the z-axis we apply the operator many times. Take an interval z₁-z₀, to be divided into N subintervals. Since there are N intervals, an error proportional to 1/N in each subinterval would accumulate to an unacceptable level by the time z₁ was reached. On the other hand, an error proportional to 1 / N² could only accumulate to a total error proportional to 1/N. Such an error would disappear as the number of subintervals increased.

To prove the validity of splitting, we take $\Delta z=( z_1 - z_0 ) / N$.Observe that the operator ${\bf I}+ ( {\bf A}+ {\bf B}) \Delta z$ differs from the operator $( {\bf I}+ {\bf A}\, \Delta z)( {\bf I}+ {\bf B}\, \Delta z)$ by something in proportion to $\Delta z^2$ or 1/N². 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 ${\bf A}\, {\bf B}\, = \, {\bf B}\, {\bf A}$.Commutativity is always true for scalars. With finite differencing the question is whether the two matrices commute. Taking ${\bf A}$ and ${\bf B}$ to be differential operators, commutativity is defined with the help of the family of all possible wavefields P. Then ${\bf A}$ and ${\bf B}$ are commutative if ${\bf A}\, {\bf B}\, P \, = \, {\bf B}\, {\bf A}\, P$.

The operator representing $\partial P/\partial z$ will be taken to be ${\bf A}+ {\bf B}$.The simplest numerical integration scheme using the splitting method is  

$$P( z_0\ +\ \Delta z)\eq ( {\bf I}\ +\ {\bf A}\, \Delta z )\ ( {\bf I}\ +\ {\bf B}\, \Delta z ) \ P ( z_0 )$$ (14)

Applying (14) in many stages gives a product of many operators. The operators ${\bf A}$ and ${\bf B}$ are subscripted with j to denote the possibility that they change with z.  

$$P ( z_1 ) \eq \prod_{j=1}^N\ [ ( {\bf I}\ +\ {\bf A}_j \, \Delta z ) ( {\bf I}\ +\ {\bf B}_j \, \Delta z ) ] \ P ( z_0 )$$ (15)

As soon as ${\bf A}$ and ${\bf B}$ are assumed to be commutative, the factors in (15) may be rearranged at will. For example, the ${\bf A}$ operator could be applied in its entirety before the ${\bf B}$ operator is applied:  

$$P ( z_1 ) \eq \left[ \prod_{j=1}^N\ ( {\bf I}\ +\ {\bf B}_j ... ...{j=1}^N\ ( {\bf I}\ +\ {\bf A}_j \, \Delta z) \right] P ( z_0 )$$ (16)

Thus the full-separation concept is seen to depend on the commutativity of operators.