NUMERICAL MATRIZANTS

A differential equation relates field variables at a point to field variables at neighboring points. A matrizant relates field variables at one depth in a stratified material to variables at some other depth. A matrizant may also be regarded as the integral of the matrix differential equation (9-1-7). First we will show how to get the matrizant of (9-1-7) by numerical means. That is, we will solve the problem for arbitrary depth variations in density and in compressibility. Then we will come back and develop analytical solutions for the special case of constant material properties. We have

$$\frac{\partial \mbox{\bf Z}}{\partial z} \quad\approx\quad\f... ...d\approx\quad\mbox{\bf A}(z) \mbox{\bf X}(z) + \mbox{\bf S}(z)$$

or

$$\mbox{\bf X}(z + \triangle z) \eq (\mbox{\bf I} + \mbox{\bf A} \triangle z)\mbox{\bf X}(z) + \mbox{\bf S}(z)\triangle z$$ (1)

Given $\mbox{\bf X}$ for some particular z it is clear that (9-2-1) may be used recursively to get $\mbox{\bf X}$ for any z. For simplicity we may take $\triangle z = 1$ and use subscripts to indicate the z coordinate. Let $[\mbox{\bf I} + \mbox{\bf A}(z)\triangle z]$ be denoted by $\mbox{\bf Q}(z)$,then (9-2-1) becomes

$$\mbox{\bf X}_{k+1} \eq \mbox{\bf Q}_{k}\mbox{\bf X}_{k} + \mbox{\bf S}_{k}$$

or

$$\mbox{\bf X}_{1} \eq \mbox{\bf Q}_{0}\mbox{\bf X}_{0} + \mbox{\bf S}_{0}$$

hence hence likewise

$$\mbox{\bf X}_{4} \eq \mbox{\bf M}\mbox{\bf X}_{0} + \mbox{\bf T}$$ (2)

So we have in general a numerically determinable matrix $\mbox{\bf M}$(called the matrizant) and a vector $\mbox{\bf T}$ which relates the field variables at the top of the strata to those on the bottom by

$$\mbox{\bf X}_{z_{\rm top}} \eq \mbox{\bf M}\mbox{\bf X}_{z_{\rm bot}} + \mbox{\bf T}$$ (3)

The matrix $\mbox{\bf M}$ is also called an integral matrix. Physical problems present themselves in different ways with different boundary conditions. For the acoustic problem discussed earlier $\mbox{\bf X}$ is a two-component vector involving pressure and of the stratified medium. Thus (9-2-3) represents two equations for four unknowns. The solution to the problem comes only when two boundary conditions are introduced. If we are talking about sound waves in the ocean, (simplified) boundary conditions would be to prescribe zero pressure at the surface and zero vertical displacement at the sea floor. Then these boundary conditions with (9-2-3) would be two equations and two unknowns and consequently could be solved for surface displacement and bottom pressure. From these, pressure and displacement could be determined everywhere. Proper determination of boundary conditions is often the trickiest part of a problem; we will return to it for some other problems in a later section.

If portions of the material have constant material properties and contain no sources, then it is possible to find an analytical expression for the matrizant. A matrizant which takes one across such a layer of constant properties is called, appropriately enough, a layer matrix. It may be verified by substitution that

$$\mbox{\bf X}_{z} \eq e^{[\mbox{\bf A}(z-z_{0})]}\mbox{\bf X}_{z_{0}}$$ (4)

is the solution to $(\partial/\partial z)\mbox{\bf X} = \mbox{\bf A}\mbox{\bf X}$ where

$$e^{\mbox{\bf A}(z - z_{0})} \eq \mbox{\bf I} + \mbox{\bf A}(z - z_{0}) + \frac{\mbox{\bf A}^{2}(z - z_{0})^{2}}{2!} + \cdots$$ (5)

in a region of space where $\mbox{\bf A}$ is constant with z. Thus $e^{\mbox{\bf A}(z-z_{0})}$ is the required matrizant. The matrix exponential could be computed numerically either by the method of (9-2-2) or the method of (9-2-5) or the method of Sylvester's theorem described in Chap. 5. In the next section we will see how Sylvester's theorem leads directly to the ideas of up- and downgoing waves.

EXERCISES:

1. What is $\mbox{\bf Q}_{k}$ for the improved central difference approximation?

   $$\mbox{\bf X}(z + \triangle z) - \mbox{\bf X}(z) \eq \frac{\t... ...ox{\bf A}[\mbox{\bf X}(z + \triangle z) + \mbox{\bf X}(z)]}{2}$$