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Geophysical models in general, and atmospheric models more specifically, arealways limited in spatial resolutions. Due to this limitation, we face withtwo different needs. The first is a need for knowing (or "downscaling")more spatial details (e.g., precipitation distribution) than having modelsimulations for practical applications, such as hydrological modelling. Thesecond is a need for "parameterizing" the subgrid-scale physical processesin order to represent the feedbacks of these processes on to the resolvedscales (e.g., the convective heating rate).  The present article begins by remarking that it is essential to consider thedownscaling and parametrization as an "inverse" of each other: downscalingseeks a detail of the subgrid-scale processes, then the parameterizationseeks an integrated effect of the former into the resolved scales. Aconsideration on why those two closely-related operations are traditionallytreated separately, gives insights of the fundamental limitations of thecurrent downscalings and parameterizations.  The multiresolution analysis (such as those based on wavelet) provides animportant conceptual framework for developing a unified formulation for thedownscaling and parameterization. In the vocabulary of multiresolutionanalysis, these two operations may be considered as types of decompressionand compression. A new type of a subgrid-scale representation scheme,NAM-SCA (nonhydrostatic anelastic model with segmentally-constantapproximation), is introduced under this framework

Downscaling, parameterization, decomposition, compression: a perspective from the multiresolution analysis