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.
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