  We propose a solution to the problem of accretion disc sizes in active
galactic nuclei being larger when measured by reverberation mapping than
predicted by theory. Considering blackbody reprocessing on a disc with
thickness profile $H(r)$, our solution invokes a steep rim or rippled
structures irradiated by the central lamp-post. We model the continuum lags and
the faint and bright disc spectral energy distribution (SED) in the
best-studied case NGC 5548 (black hole mass $M = 7\times10^{7} M_\odot$, disc
inclination $i=45^\circ$). With the lamp-post off, the observed disc SED
requires a low accretion rate ($\dot{M} \sim 0.0014 M_\odot$/yr) and high
prograde black hole spin ($a \sim 0.93$). Reprocessing on the thin disc gives
time lags increasing with wavelength but 3 times smaller than observed.
Introducing a steep $H(r)$ rim, or multiple crests, near $r = 5$ light days,
reprocessing on their steep centre-facing slopes increases temperatures from
$\sim1500$ K to $\sim6000$ K and this increases optical lags to match the lag
data. Most of the disc surface maintains the cooler $T\propto r^{-3/4}$
temperature profile that matches the SED. The bright lamp-post may be powered
by magnetic links tapping the black hole spin. The steep rim occurs near the
sublimation radius for dust in the disc, as in the "failed disc wind model" for
broad-line clouds. Lens-Thirring torques aligning the disc and black hole spin
may also raise a warp and associated waves. In both scenarios, the small
density scale height implied by the inferred value of $H(r)$ suggests possible
marginal gravitational instability in the disc.
