at different scales? What can and cannot be inferred about the continuous process within which sampling is done? These concerns also involve different types of estimates such as second- and higher-order spectral estimates, probability density estimates, and regression estimates. Such questions should be considered under the assumptions of both stationary and nonstationary processes. These problems are connected with those involving non-Gaussian observations (see Chapter 8). Suitably selected and designed multiscale wavelets may be helpful in this situation.
There are statistical research opportunities in modeling a random field given:
observational data representing averages over regions (pixels) of a given size (as determined, e.g., by a satellite footprint), and
observational data obtained by irregular sampling (spatial and temporal data gaps, etc.) of a random field.
An analysis of extrema of non-Gaussian fields is needed. It will depend partly on what one can say in the stationary case about the tails of the instantaneous distributions. Such an analysis will have both a probabilistic and a statistical aspect; i.e., given a nice probabilistic characterization, can some aspect of it be effectively estimated from data? Progress on these questions may also carry over to notions of intermittency. Specific issues for focus include:
analysis of asymptotics of extrema of a non-Gaussian field,
analysis of behavior of outlying observations in a case of non-Gaussian data, and
modeling of a random field with given statistics of extrema.
Additional issues and problems concerning non-Gaussian random fields and processes are listed at the end of Chapter 8.