Context: Below 1 mHz, the power spectrum of helioseismic velocity measurements is dominated by the spectrum of convective motions (granulation and supergranulation) and it is difficult to detect the low-order acoustic modes and gravity modes. Aims: We attempt to understand more clearly the behavior of solar granulation as a function of observing height in the solar atmosphere and with magnetic activity during solar cycle 23. Methods: We analyze the Power Spectral Density (PSD) of eleven years of GOLF/SOHO velocity time series data using a Harvey-type model to characterize the properties of the convective motions in the solar oscillation power spectrum. We study the evolution of the granulation with both altitude in the solar atmosphere and solar activity. Results: We first demonstrate that the traditional use of a Lorentzian profile to describe the envelope of the p modes is unsuitable for GOLF data. To model properly the solar spectrum, we must instead adopt a second Lorentzian profile. Secondly, we show that the granulation clearly evolves with height in the photosphere but does not present any significant variation with the activity cycle.