The Power of the Duality in Spatial–Temporal Estimation

Space–time filtering has a long and often confusing history in the geosciences. It is called by different names in different areas of geoscience, where numerous applications have been developed. The variety of notations that have emerged adds to this confusion. A unified treatment of spatial–temporal estimation is presented, which highlights its duality and the associated trade-off in the construction of any optimal estimation algorithm. The duality in optimal estimation comes from the requirement that the representation of the spatial–temporal statistical structure of the increments between the true field and the system-operator model used by the filter be matched with the true ensemble structure of the increment field. The associated trade-off arises from the following dichotomy: the closer the system-operator model corresponds to the true system operator, the less ensemble structure remains in the increment field. Conversely, the simpler the model of the system operator, the more residual statistical structure remains to be represented. Several examples of estimation of spatial–temporal systems, in practice, are presented to illustrate the power of the duality. The rationale for determining the placement of effort in modeling the system operator vis-à-vis representing residual statistical structure is discussed.