  Reverberation mapping uses time-delayed variations in photoionised emission
lines to map the geometry and kinematics of emission-line gas in active
galactic nuclei. In previous work, the light travel time delay
tau=R(1+cos(theta))/c and Doppler shift v give a 2-d map Psi(tau,v) for each
emission line. Here we combine the velocity-delay information with
photoionisation physics in a maximum entropy fit to the full reverberating
spectrum F_lam(lam,t) to recover a 5-d map of the differential covering
fraction f(R,theta,n,N,v), with n and N the density and column density of the
gas clouds. We test the method for a variety of geometries (shells, rings,
disks, clouds, jets) by recovering a 3-d map f(R,theta,n) from reverberations
in 7 uv emission lines. The best test recovers a hollow shell geometry,
defining R to 0.15 dex, n to 0.3 dex, and ionisation parameter U ~ 1/(n R^2) to
0.1 dex. The results are sensitive to the adopted distance and luminosity,
suggesting that these parameters may be measurable as well.
