  CANARY is the multi-object adaptive optics (MOAO) on-sky pathfinder developed
in the perspective of Multi-Object Spectrograph on Extremely Large
Telescopes~(ELTs). In 2013, CANARY was operated on-sky at the William Herschel
telescope~(WHT), using three off-axis natural guide stars~(NGS) and four
off-axis Rayleigh laser guide stars~(LGS), in open-loop, with the on-axis
compensated turbulence observed with a H-band imaging camera and a Truth
wave-front sensor~(TS) for diagnostic purposes. Our purpose is to establish a
reliable and accurate wave-front error breakdown for LGS MOAO. This will enable
a comprehensive analysis of \cana on-sky results and provide tools for
validating simulations of MOAO systems for ELTs. To evaluate the MOAO
performance, we compared the CANARY on-sky results running in MOAO, in Single
Conjugated Adaptive Optics~(SCAO) and in Ground Layer Adaptive Optics~(GLAO)
modes, over a large set of data acquired in 2013. We provide a statistical
study of the seeing. We also evaluated the wave-front error breakdown from both
analytic computations, one based on a MOAO system modelling and the other on
the measurements from the CANARY TS. We have focussed especially on the
tomographic error and we detail its vertical error decomposition~(VED). We show
that CANARY obtained 30.1\%, 21.4\% and 17.1\% H-band Strehl ratios in SCAO,
MOAO and GLAO respectively, for median seeing conditions with 0.66" of total
seeing including 0.59" at the ground. Moreover, we get 99\% of correlation over
4,500 samples, for any AO modes, between two analytic computations of residual
phase variance. Based on these variances, we obtain a reasonable
Strehl-ratio~(SR) estimation when compared to the measured IR image SR. We
evaluate the gain in compensation for the altitude turbulence brought by MOAO
when compared to GLAO.
