Application to 45 degree wavefield extrapolation

The scalar wave equation for the extrapolation of a downgoing wavefield is  

$${d q\over dz } \eq i\, k_z q\eq - \, R\, q$$ (87)

where the R operator takes the usual form  

$$R\eq - \, i \, k_z \eq { - \, i \omega \over v }\ \sqrt { 1 \ -\ { v^2 \, k_x^2 \over \omega^2 } }$$ (88)

Our plan is to approximate the square root by the usual continued fraction expansion and then identify $i \, k_x$ with $\partial_x$ to obtain a space-domain equation. The main effort we must make stems from our refusal to make the usual assumption that v(x,z) is independent of x. Since $\partial_x v q$differs from $v \partial_x q$, the space representation does not seem to be unique, and we may wonder how the variable q relates to physical wave variables like pressure and displacement. Since (88) is purely imaginary, the depth-invariance of the quadratic $q^ { {\rm *} \, } q$can be interpreted as the downward energy flux across the datum at depth z. Our main effort will be to assure that $q^ { {\rm *} \, } q$does indeed remain depth-invariant when $v(x,z) \ \ != \ \ const$.The task of determining the relation between the energy flux variable q and the physical variables will be left to the reader.

First v² k_x² must be represented in the space domain. Thinking of the x-derivative operator $\partial / \partial x \ \ =\ \ \partial_x$ as a large bidiagonal matrix with $(1,-1)/ \Delta x$ along the diagonal and V(x) as a diagonal matrix, we are attracted to expressions like $( V\partial_x )^T( V\partial_x )$or $( V\partial_x ) ( V\partial_x )^T$because they are symmetric, positive, semidefinite matrices. In simplest form, such numerical representations are tridiagonal matrices that can be abbreviated as either  

$$T\ \eq \ { ( V\partial_x ) \, ( V\partial_x )^T}$$ (89)

or  

$$T\ \eq \ { ( V\partial_x )^T\, ( V\partial_x ) }$$ (90)

At a later time accuracy or some other consideration could determine the choice in (89) and (90). Even other expressions could be used, provided they are real, symmetric, and positive definite.

In the previous section the constant velocity, 45$^\circ$ expansion of (88) was shown to be  

$$R\eq {1 \over v }\ \ { v^2 k_x^2 \over - \,i \omega \,2 \ +\ { v^2 k_x^2 \over - \,i \omega \,2 }}$$ (91)

This scalar R always has a positive real part because $- \,i \omega$ is always represented in an impedance form, and the whole expression is built up satisfying Muir's rules for combining impedance functions. In going to the x-domain notice that $(i k_x )^2 \ =\ - \partial_{xx}$and $(\partial_x )^T\ =\ -\, \partial_x$.So the positive scalar v² k_x² corresponds to the positive eigenvalues of (89) and (90).

The expression of the bulletproof, square-root operator R in the space domain will now be given as

$$\begin{eqnarray} M\ \ \ &=&\ \ \ {T\over - \,i \omega\, 2 \,I\ +\ {T\over -\,i ... ...,2 } } \\ R\ \ \ &=&\ \ \ V^{ - {1\over 2}} \ M\ V^{ - {1\over 2}}\end{eqnarray}$$ (92) (93)

Use of the division sign in (92) and (93) is justifiable because the matrix T commutes with the identity matrix I. (A hazard in this work is that T does not commute with the diagonal matrix V). The matrix M has the properties required of R since a basic matrix theorem says that the eigenvalues of a polynomial of a real symmetric matrix are the polynomials of the eigenvalues. In other words, replacing T in (92) and (93) by one of its eigenvalues produces a complex M whose real part is positive, so that $M^ { {\rm *} \, } \ +\ M$ is positive as required. What is needed is to show that the following matrix is positive definite:  

$$R\ +\ R^ { {\rm *} \, } \eq V^{ - {1\over 2} } \ ( M\ +\ M^ { {\rm *} \, } ) \, V^{ - {1\over 2}}$$ (94)

A matrix A is positive definite if for arbitrary d, the scalar $d^ { {\rm *} \, } A\, d$ is positive. The diagonal matrix $V^{{-} {1\over 2} }$ can certainly be absorbed into d and d will still be arbitrary, so the proof is complete.

In programming it is a nuisance to put $V^{{-} {1\over 2} }$on each side of the matrix M. Actually you can put V^-1 on either side. In general, some other quadratic form such as $q^ { {\rm *} \, } U\, q$ where U is strictly positive definite will be decreasing if $R^ { {\rm *} \, } U\ +\ U\,R$ is positive definite.