  High-resolution Doppler spectroscopy is a powerful tool for identifying
molecular species in the atmospheres of both transiting and non-transiting
exoplanets. Currently, such data is analysed using cross-correlation techniques
to detect the Doppler shifting signal from the orbiting planet. In this paper
we demonstrate that, compared to cross-correlation methods currently used, the
technique of Doppler tomography has improved sensitivity in detecting the
subtle signatures expected from exoplanet atmospheres. This is partly due to
the use of a regularizing statistic, which acts to suppress noise, coupled to
the fact that all the data is fit simultaneously. In addition, we show that the
technique can also effectively suppress contanimating spectral features that
may arise due to overlapping lines, repeating line patterns, or the use of
incorrect linelists. These issues can confuse conventional cross-correlation
approaches, primarily due to aliasing issues inherent in such techniques,
whereas Doppler tomography is less susceptible to such effects. In particular,
Doppler tomography shows exceptional promise for simultaneously detecting
multiple line species (e.g. isotopologues), even when there are high contrasts
between such species -- and far outperforms current CCF analyses in this
respect. Finally, we demonstrate that Doppler tomography is capable of
recovering molecular signals from exoplanets using real data, by confirming the
strong detection of CO in the atmosphere of Tau Boo b. We recover a signal with
a planetary radial velocity semi-amplitude Kp = 109.6 +/- 2.2 km/s, in
excellent agreement with the previously reported value of 110.0 +/- 3.2 km/s.
