  The WS time delay matrix relates a lossless and reciprocal system's
scattering matrix to its frequency derivative, and enables the synthesis of
modes that experience well-defined group delays when interacting with the
system. The elements of the WS time delay matrix for surface scatterers and
antennas comprise renormalized energy-like volume integrals involving electric
and magnetic fields that arise when exciting the system via its ports. Here,
direct and indirect methods for computing the WS time delay matrix are
presented. The direct method evaluates the energy-like volume integrals using
surface integral operators that act on the incident electric fields and current
densities for all excitations characterizing the scattering matrix. The
indirect method accomplishes the same task by computing scattering parameters
and their frequency derivatives. Both methods are computationally efficient and
readily integrated into existing surface integral equation codes. The proposed
techniques facilitate the evaluation of frequency derivatives of antenna
impedances, antenna patterns, and scatterer radar cross sections in terms of
renormalized field energies derived from a single frequency characterization of
the system.
