An ever-present feature in seismic data, noise affects outcomes of processing and imaging algorithms, causing uncertainty in the interpretation of results. Despite abundant evidence that noise is not white, stationary or Gaussian, these assumptions are commonly made when generating noise models and processing data. While synthetic seismic datasets have evolved to include geological complexities, a standardised approach to incorporating realistic noise does not yet exist. The aim of this work is to introduce a noise modelling methodology that avoids the above assumptions. A statistical analysis of three months of pre-injection noise from the vertical components of a 50 station, c.2.5km-wide, cross-shaped array at the Aquistore CO2 storage site, characterises noise sources originating from wellsite activity and passing traffic. A covariance modelling approach is then devised to generate realistic noise models that have close similarity to the recorded noise in both the time and frequency domain, with >65% noise realisations having >50% probability of arising from the same distribution as the recorded noise. The modelling procedure is finally applied to two cases: benchmarking and development of microseismic inversion algorithms on synthetic datasets; and noise suppression. In the former, the source location is correctly estimated at a signal-to-noise ratio of 0.1 with white, Gaussian noise (WGN) but 0.5 was required for realistic noise. Then, applying a microseismic source inversion algorithm, datasets with realistic noise identify pitfalls unobserved under WGN conditions. Thus, in both cases, a WGN assumption gives a misleadingly favourable assessment of efficiency. In the latter, a noise whitening technique that utilises the inverse of the covariance matrix reduces the total noise energy by a factor of 3.5, allowing both imaging of additional microseismic events and greater confidence in identified events. The proposed techniques are illustrated on passive surface data, but offer future applications in both active and passive seismic monitoring.