  The removal of noise typically correlated in time and wavelength is one of
the main challenges for using the radial velocity method to detect Earth
analogues. We analyze radial velocity data of tau Ceti and find robust evidence
for wavelength dependent noise. We find this noise can be modeled by a
combination of moving average models and "differential radial velocities". We
apply this noise model to various radial velocity data sets for tau Ceti, and
find four periodic signals at 20.0, 49.3, 160 and 642 d which we interpret as
planets. We identify two new signals with orbital periods of 20.0 and 49.3 d
while the other two previously suspected signals around 160 and 600 d are
quantified to a higher precision. The 20.0 d candidate is independently
detected in KECK data. All planets detected in this work have minimum masses
less than 4$M_\oplus$ with the two long period ones located around the inner
and outer edges of the habitable zone, respectively. We find that the
instrumental noise gives rise to a precision limit of the HARPS around 0.2 m/s.
We also find correlation between the HARPS data and the central moments of the
spectral line profile at around 0.5 m/s level, although these central moments
may contain both noise and signals. The signals detected in this work have
semi-amplitudes as low as 0.3 m/s, demonstrating the ability of the radial
velocity technique to detect relatively weak signals.
