In this paper, a method for generating samples of a fully non-stationary zero-mean Gaussian process, having a target acceleration time-history as one of its own samples, is presented. The proposed method requires the following steps: i) divide the time axis of the target accelerogram in contiguous time intervals in which a uniformly modulated process is introduced as the product of a deterministic modulating function per a stationary zero-mean Gaussian sub-process, whose power spectral density (PSD) function is filtered by two Butterworth filters; ii) estimate, in the various time intervals, the parameters of modulating functions by least-square fitting the expected energy of the proposed model to the energy of the target accelerogram; iii) estimate the parameters of the PSD function of the stationary sub-process, once the occurrences of maxima and of zero-level up-crossings of the target accelerogram, in the various intervals, are counted; iv) obtain the evolutionary spectral representation of the fully non-stationary process by adding the various contribution evaluated in the various intervals. •The generation of artificial accelerograms consistent with real seismic records is addressed.•Main characteristics of target accelerograms are used to define fully non-stationary processes.•The target accelerogram is considered a sample of the fully non-stationary process.•The proposed method is suitable for selecting input motion for dynamic analyses of structural and geotechnical systems.•The proposed procedure is characterised by accuracy and computational efficiency.