  Extending the concept of multi-selfsimilar random field we study multi-scale
invariant (MSI) fields which have component-wise discrete scale invariant
property. Assuming scale parameters as $\lambda_i>1$, $i=1,\ldots,d$ and the
parameter space as $(1, \infty)^d$, the first scale rectangle is referred to
the rectangle $ (1, \lambda_1)\times \ldots \times (1, \lambda_d)$. Applying
certain component-wise geometric sampling of MSI field, the harmonic-like
representation and spectral density of the sampled MSI field are characterized.
Furthermore, the covariance function and spectral density of the sampled Markov
MSI field are presented by the variances and covariances of samples inside
first scale rectangle. As an example of MSI field, a two-dimensional simple
fractional Brownian sheet (sfBs) is demonstrated. Also real data of the
precipitation in some area of Brisbane in Australia for two days (25 and 26
January 2013) are examined. We show that precipitation on this area has MSI
property and estimate it as a simple MSI field with stationary increments
inside scale intervals.
  This structure enables us to predict the precipitation in surface and time.
We apply the mean absolute percentage error as a measure for the accuracy of
the predictions.
