In the context of interpolation, a distance metric is an underlying assumption of an interpolation method, as it is implicit in the calculation of the interpolating weights. Interpolating over a spherical surface (dimensionality 2) and for a generic space unit the weights are calculated in a

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[L]^2 |

unit, to which corresponds a distance

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[L] |

. Because the metric is implicit to the method implementation, the metric discussion is here in terms of the specific methods.

## Classical bi-linear method

The *classical bi-linear* method (EMOSLIB, MIR) interpolates using spherical coordinates, with curvilinear quadrilaterals as the supporting elements; calculating the interpolating weights (and distance) is fairly equivalent to (weights are de-normalized)

Mathinline w_{bi} \propto (\Delta lon)^2 + (\Delta lat)^2

Mathinline d_{bi} = \sqrt{ w_{bi} }

For input regular grids (*regular_ll* and *regular_ll*) this operates over a spherical surface tessellation — *ie.* weights are not overlapping for different grid elements. However, for reduced grids (*reduced_gg*, or quasi-regular) this as an approximation because the component

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\Delta^2 lon |

assumes the supporting elements are meridian-aligned. In practice, when the number of points per latitude changes (for a *reduced_gg* octahedral grid every parallel has a different number of points) calculating

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w_{bi} |

will not exactly match the supporting quadrilateral element, effectively blocking a possible local conservativity property.

Additionally, because the real distance (on the spherical surface) in longitude depends on the latitude parallel –the real distance at specific latitudes decreases approaching the poles– the weights are always distorted pole-wards away from the equator. This effect always present, however minimized due to the smaller latitude increments towards the poles of the Gaussian grids.

## Triangular linear method

The *triangular linear* interpolation method (MIR-only) uses the 3D Euclidean space, tessellating a spherical surface with planar triangles. Weights are calculated as (note similarity to Euclidean distance):

Mathinline w_{3D} \propto (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2

Mathinline d_{3D} = \sqrt{ w_{3D} }

The supporting elements form a spherical surface tessellation irrespective of the input grid format; Because the elements are described in 3D there is no special concern for the poles (the poles singularity is effectively avoided by operating in the dimension above). Using *planar* supporting elements does however introduce an approximation compared to the (real) spherical surface distance, an error minimized with resolution increase.

# Useful references

- Wikipedia article: Great-circle distance
- Wikipedia article: "Earth-centered, earth-fixed" geographic coordinate system (ECEF)