We quantify the effects of morphological k-correction at $1<z<2$ by comparing morphologies measured in the K and I-bands in the COSMOS area. Ks-band data have indeed the advantage of probing old stellar populations for $z<2$, enabling a determination of galaxy morphological types unaffected by recent star formation. In paper I we presented a new non-parametric method to quantify morphologies of galaxies on seeing limited images based on support vector machines. Here we use this method to classify $∼$$50 000$ $Ks$ selected galaxies in the COSMOS area observed with WIRCam at CFHT. The obtained classification is used to investigate the redshift distributions and number counts per morphological type up to $z∼2$ and to compare to the results obtained with HST/ACS in the I-band on the same objects from other works. We associate to every galaxy with $Ks<21.5$ and $z<2$ a probability between 0 and 1 of being late-type or early-type. The classification is found to be reliable up to $z∼2$. The mean probability is $p∼0.8$. It decreases with redshift and with size, especially for the early-type population but remains above $p∼0.7$. The classification is globally in good agreement with the one obtained using HST/ACS for $z<1$. Above $z∼1$, the I-band classification tends to find less early-type galaxies than the Ks-band one by a factor $∼$1.5 which might be a consequence of morphological k-correction effects. We argue therefore that studies based on I-band HST/ACS classifications at $z>1$ could be underestimating the elliptical population. Using our method, we observe an increase of the early-type population from $z∼2$ ($21.9%±8%$) to the present ($32.0%±5%$) probably reflecting a progressive building up of the red sequence from late-type objects.
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