The current projects of Extremely Large Telescopes rely on adaptive optics systems using several sodium laser guide stars (LGSs). Because of the thickness of the sodium layer in the mesosphere, the subapertures of a Shack-Hartmann wavefront sensor will see the LGS all the more elongated as its position is distant from the launching point of the laser. This effect is significant and prompts the lasers to be launched from behind the secondary instead of from around the telescope. The elongations increase the centroiding errors and new smarter algorithms have been designed to mitigate this effect, but the loss of accuracy is still significant. Further, the measurement uncertainties are no more uniform across the pupil and correlations are introduced between the two coordinates of the gradients. From numerical simulations, we analyze the benefit of taking into account this structured correlations in wavefront reconstruction algorithms and compare the reconstruction accuracy when using least squares, weighted least squares, or minimum variance using von Karman turbulence priors. For a single LGS launched behind the secondary, numerical simulations show effective improvements when using both noise correlations and priors in wavefront reconstruction. When combining the measurements from several LGSs in a Ground Layer adaptive optics system, we show that taking into account the noise covariances yields better reconstructions when LGSs are launched from around the telescope than from behind the secondary. Further, results indicate that we could discard the measurements along the elongated direction where this elongation is greater than a given threshold.© (2008) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.