In this paper, we show that an atom interferometer inertial sensor, when associated with the auxiliary measurement of external vibrations, can be operated beyond its linear range and still keep a high acceleration sensitivity. We propose and compare two measurement procedures (fringe fitting and non-linear lock) that can be used to extract, without adding any bias, the mean phase of the interferometer when the interferometer phase fluctuations exceed 2π. Despite operating in the urban environment of inner Paris without any vibration isolation, the use of a low noise seismometer for the measurement of ground vibrations allows our atom gravimeter to reach at night a sensitivity as good as 5.5 × 10−8g at 1 s. Robustness of the measurement to large vibration noise is also demonstrated by the ability of our gravimeter to operate during an earthquake with excellent sensitivity. For such low vibration frequency though, high pass filtering of the seismometer degrades its correlation with the interferometer signal, so that low frequency seismic vibrations appear on the gravity measurement. Nevertheless, our high repetition rate allows for efficient sampling of these perturbations, ensuring proper averaging. Such techniques open new perspectives for applications in other fields, such as navigation and geophysics.