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Diamond interpolation; a class of accurate compact-stencil grid interpolation methods
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    Class of accurate compact-stencil grid interpolation methods
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    Throughout the execution of a data assimilation algorithm, it is necessary to apply, with numerous repetitions, the operator of interpolation from the regular analysis grid to interior target points at the irregular observation locations, and the adjoints of these operators. In such a situation, and where a reasonably high level of numerical accuracy is required, it is sometimes worth the expenditure of a little extra effort in establishing interpolation stencils and their weights that achieve the formal level of required accuracy with significantly fewer stencil points than would be required in the more conventional case of an interpolation scheme achieving the same formal order of accuracy by employing a Cartesian product of basic one-dimensional interpolation operators. A family of so-called 'diamond interpolation' schemes provides a natural way to construct alternative compact stencils and weights for a quasi-uniform rectangular grid in two or three dimensions. These methods, which employ approximately 'diamond'-shaped stencils, allow both values and derivatives of the gridded quantity to be interpolated. If continuity and smoothness of the target value as a function of position is a requirement, these properties can also be accommodated by modifications of these methods. The techniques for generating and applying these interpolation schemes will be described here together with a brief discussion of some applications in numerical weather prediction.
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