  In this article, using kernel convolution of order based dependent Dirichlet
process (Griffin and Steel (2006)) we construct a nonstationary, nonseparable,
nonparametric space-time process, which, as we show, satisfies desirable
properties, and includes the stationary, separable, parametric processes as
special cases. We also investigate the smoothness properties of our proposed
model. Since our model entails an infinite random series, for Bayesian model
fitting purpose we must either truncate the series or more appropriately
consider a random number of summands, which renders the model dimension a
random variable. We attack the variable dimensionality problem using
Transdimensional Transformation based Markov Chain Monte Carlo introduced by
Das and Bhattacharya (2019b), which can update all the variables and also
change dimensions in a single block using essentially a single random variable
drawn from some arbitrary density defined on a relevant support. For the sake
of completeness we also address the problem of truncating the infinite series
by providing a uniform bound on the error incurred by truncating the infinite
series.
  We illustrate the effectiveness of our model and methodologies on a simulated
data set and demonstrate that our approach significantly outperforms that of
Fuentes and Reich (2013) which is based on principles somewhat similar to ours.
We also fit two real, spatial and spatio-temporal datasets with our approach
and obtain quite encouraging results in both the cases.
