## 5aUW8. Construction of exact left symbols of the vertical slowness operator.

### Session: Friday Morning, December 6

### Time: 9:28

**Author: Maarten V. de Hoop**

**Location: Ctr. for Wave Phenomena, Colorado School of Mines, Golden, CO 80401-1887**

**Author: Mattheus J. N. van Stralen**

**Location: Delft Univ. of Technol., 2600 GA Delft, The Netherlands**

**Author: Hans Blok**

**Location: Delft Univ. of Technol., 2600 GA Delft, The Netherlands**

**Abstract:**

The Bremmer series solution of the acoustical wave equation in generally
inhomogeneous media, requires the introduction of pseudodifferential operators.
Such operators describe the (de)composition, propagation and interaction of the
up- and downgoing waves. The exact solutions of the corresponding symbols can be
derived for several medium profiles. Of particular interest is the symbol
associated with the vertical slowness or square root operator, involved in the
propagation. This symbol constitutes the generalized slowness surface. These
symbols serve as benchmarks for numerical one-way wave schemes involving
waveguiding structures. Starting from the kernel representation of the
transverse Helmholtz operator's resolvent, a normal mode expansion is set up.
The Schwartz kernels of negative fractional powers of the transverse Helmholtz
operator and their left symbols can then be expressed in these normal modes as
well. Using the composition relation for the original operator and its inverse
square, a recursive relation for the left symbols is obtained. Results of the
quadratic waveguiding (focusing) profile and a localized profile will be
presented. The connection with the eigenfunctions and eigenvalues (normal mode
expansion) will be highlighted.

ASA 132nd meeting - Hawaii, December 1996