After the introduction of the ionization-injection scheme in Laser Wake Field Acceleration and of related high-quality electron beam generation methods as two-color or the Resonant Multi Pulse Ionization injection (ReMPI), the theory of thermal emittance by C. Schroeder et al. , has been used to predict the beam normalised emittance obtainable with those schemes. We recast and extend such a theory, including both higher order terms in the polinomial laser field expansion and non polinomial corrections due to the onset of saturation effects on a single cycle. Also, a very accurate model for predicting the cycle-averaged distribution of the extracted electrons, including saturation and multi-process events, is proposed and tested. We show that our theory is very accurate for the selected processes of Kr 8 + →10 + and Ar 8 + →10 + , resulting in a a maximum error below 1% even in deep saturation regime. The accurate prediction of the beam phase-space can be implemented e.g. in laser-envelope or hybrid PIC/fluid codes, to correctly mimic the cycle-averaged momentum distribution without the need of resolving the intra-cycle dynamics. We introduce further spatial averaging, obtaining expressions for the whole-beam emittance fitting with simulations in a saturated regime, too. Finally, a PIC simulation for a LWFA injector in the ReMPI configuration is discussed.