On the normalization of scintillation autocovariance for generalized SCIDAR

The Generalized SCIDAR (Scintillation Detection and Ranging) technique consists in the computation of the mean autocorrelation of double-star scintillation images taken on a virtual plane located a few kilo-meters below the telescope pupil. This autocorrelation is normalized by the autocorrelation of the mean image. Johnston et al. in 2002 pointed out that this normalization leads to an inexact estimate of the optical-turbulence strength C(2)(N). Those authors restricted their analysis to turbulence at ground level. Here we generalize that study by calculating analytically the error induced by that normalization, for a turbulent layer at any altitude. An exact expression is given for any telescope-pupil shape and an approximate simple formula is provided for a full circular pupil. We show that the effect of the inexact normalization is to overestimate the C(2)(N) values. The error is larger for higher turbulent layers, smaller telescopes, longer distances of the analysis plane from the pupil, wider double-star separations, and larger differences of stellar magnitudes. Depending on the observational parameters and the turbulence altitude, the relative error can take values from zero up to a factor of 4, in which case the real C(2)(N) value is only 0.2 times the erroneous one. Our results can be applied to correct the C(2)(N) profiles that have been measured using the Generalized SCIDAR technique.