  Developing statistical models for seismic noise is an exercise of high value
in seismic data analysis since these models play a critical role in detecting
the onset of seismic events. A majority of these models are usually built on
certain critical assumptions, namely, stationarity, linearity, and Gaussianity.
Despite their criticality, very little reported literature exists on validating
these assumptions on real seismic data. The objectives of this work are (i) to
critically study these long-held assumptions and (ii) to propose a systematic
procedure for developing appropriate time-series models. A rigorous statistical
analysis reveals that these standard assumptions do not hold for most of the
data sets under study; rather they exhibit additional special features such as
heteroskedasticity and integrating effects. Resting on these novel discoveries,
ARIMA-GARCH models are developed for seismic noise. Studies are carried out on
$185$ real-time data sets over different time intervals to study the daily and
seasonal variations in noise characteristics and model structure. Nearly all
datasets tested positive for first-order non-stationarity, heteroskedasticity,
and Gaussianity, while $19\%$ tested negative for linearity. Analysis of the
structural uniformity of the developed models with respect to daily and
seasonal variations is also presented.
