  Models of X-ray reverberation from extended coronae are developed from
general relativistic ray tracing simulations. Reverberation lags between
correlated variability in the directly observed continuum emission and that
reflected from the accretion disc arise due to the additional light travel time
between the corona and reflecting disc. X-ray reverberation is detected from an
increasing sample of Seyfert galaxies and a number of common properties are
observed, including a transition from the characteristic reverberation
signature at high frequencies to a hard lag within the continuum component at
low frequencies, as well a pronounced dip in the reverberation lag at 3keV.
These features are not trivially explained by the reverberation of X-rays
originating from simple point sources. We therefore model reverberation from
coronae extended both over the surface of the disc and vertically. Causal
propagation through its extent for both the simple case of constant velocity
propagation and propagation linked to the viscous timescale in the underlying
accretion disc is included as well as stochastic variability arising due to
turbulence locally on the disc. We find that the observed features of X-ray
reverberation in Seyfert galaxies can be explained if the long timescale
variability is dominated by the viscous propagation of fluctuations through the
corona. The corona extends radially at low height over the surface of the disc
but with a bright central region in which fluctuations propagate up the black
hole rotation axis driven by more rapid variability arising from the innermost
regions of the accretion flow.
