  The space-based gravitational-wave observatory LISA, a NASA-ESA mission to be
launched after 2012, will achieve its optimal sensitivity using Time Delay
Interferometry (TDI), a LISA-specific technique needed to cancel the otherwise
overwhelming laser noise in the inter-spacecraft phase measurements. The TDI
observables of the Michelson and Sagnac types have been interpreted physically
as the virtual measurements of a synthesized interferometer. In this paper, I
present Geometric TDI, a new and intuitive approach to extend this
interpretation to all TDI observables. Unlike the standard algebraic formalism,
Geometric TDI provides a combinatorial algorithm to explore exhaustively the
space of second-generation TDI observables (i.e., those that cancel laser noise
in LISA-like interferometers with time-dependent armlengths). Using this
algorithm, I survey the space of second-generation TDI observables of length
(i.e., number of component phase measurements) up to 24, and I identify
alternative, improved forms of the standard second-generation TDI observables.
The alternative forms have improved high-frequency gravitational-wave
sensitivity in realistic noise conditions (because they have fewer nulls in the
gravitational-wave and noise response functions), and are less susceptible to
instrumental gaps and glitches (because their component phase measurements span
shorter time periods).
