Construction of a Hilbert curve on the sphere with an isometric parameterization of area

Of the various space-filling curves invented since their introduction by Peano the Hilbert curve is probably the simplest to define and to work with. Its value as a tool for optimizing both adaptive and static domain decompositions is now well established in the practice of massively parallel computing. Other important applications of interest to atmospheric scientists involve the thinning of data from remote sensing instruments in regions where these data are distributed with excessive density relative to the assimilation grid resolution, or the aggregation of clusters of such dense data into more manageable numbers of approximately equivalent surrogates known as `super-observations'. Hilbert curves also enable observational data to be partitioned into disjoint subsets suitable for applying cross-validation methods to the problem of estimating statistical parameters for an assimilation scheme. In these applications involving randomly distributed observations it is desirable that equal intervals in the parameter of the space-filling curve map to equal areas of the globe. This brief note shows how this can be most conveniently done for the sphere in a way that does not unduly distort the pattern traced by the curve. The extensions of these methods to curves filling spherical domains that include the vertical, and even the temporal dimensions, are also discussed briefly. R. James Purser, Manuel de Pondeca, Sei-Young Park. "June 29, 2009." "This is an unreviewed manuscript, primarily intended for informal exchange of information among the NCEP staff members." System requirements: Adobe Acrobat Reader. Includes bibliographical references (pages 17-18). 2009 NWS (National Weather Service) NCEP (National Centers for Environmental Prediction) Library Public Domain 2285