  We investigate the problem of reconfigurable intelligent surface (RIS)-aided
near-field localization of a user equipment (UE) served by a base station (BS)
under phase-dependent amplitude variations at each RIS element. Through a
misspecified Cram\'{e}r-Rao bound (MCRB) analysis and a resulting lower bound
(LB) on localization, we show that when the UE is unaware of amplitude
variations (i.e., assumes unit-amplitude responses), severe performance
penalties can arise, especially at high signal-to-noise ratios (SNRs).
Leveraging Jacobi-Anger expansion to decouple range-azimuth-elevation
dimensions, we develop a low-complexity approximated mismatched maximum
likelihood (AMML) estimator, which is asymptotically tight to the LB. To
mitigate performance loss due to model mismatch, we propose to jointly estimate
the UE location and the RIS amplitude model parameters. The corresponding
Cram\'{e}r-Rao bound (CRB) is derived, as well as an iterative refinement
algorithm, which employs the AMML method as a subroutine and alternatingly
updates individual parameters of the RIS amplitude model. Simulation results
indicate fast convergence and performance close to the CRB. The proposed method
can successfully recover the performance loss of the AMML under a wide range of
RIS parameters and effectively calibrate the RIS amplitude model online with
the help of a user that has an a-priori unknown location.
