pypfda.localization

Spatial localization for ensemble data assimilation.

Localization mitigates the spurious long-range correlations that arise when ensemble size is small relative to the state-space dimension. The canonical choice in geoscience DA is the fifth-order piecewise rational function of Gaspari & Cohn (1999), which is compactly supported, smooth, and a valid covariance kernel.

References

Gaspari, G. & Cohn, S. E. (1999). Construction of correlation functions in two and three dimensions. Quarterly Journal of the Royal Meteorological Society, 125, 723-757.

Module Attributes

EARTH_RADIUS_KM

Mean radius of Earth used by haversine_distance(), in kilometres.

Functions

gaspari_cohn(distance, localization_radius)	Gaspari-Cohn fifth-order piecewise rational localization function.
haversine_distance(lat1, lon1, lat2, lon2)	Great-circle distance in kilometres between points on the sphere.
pairwise_distance_matrix(lats, lons)	Symmetric pairwise great-circle distance matrix.

pypfda.localization.EARTH_RADIUS_KM: float = 6371.0

Mean radius of Earth used by haversine_distance(), in kilometres.

pypfda.localization.haversine_distance(lat1, lon1, lat2, lon2)

Great-circle distance in kilometres between points on the sphere.

Inputs are in degrees and may be scalars or broadcastable arrays. Result is in the same shape as the broadcast of the inputs.

Parameters:

• lat1 (array_like) – Latitudes and longitudes in degrees.

• lon1 (array_like) – Latitudes and longitudes in degrees.

• lat2 (array_like) – Latitudes and longitudes in degrees.

• lon2 (array_like) – Latitudes and longitudes in degrees.

Returns:

distance – Great-circle distance in kilometres.

Return type:

ndarray

pypfda.localization.gaspari_cohn(distance, localization_radius)

Gaspari-Cohn fifth-order piecewise rational localization function.

The function is \(1\) at zero distance, smoothly decays to \(0\) at twice the localization radius, and is exactly zero beyond that. The half-width at half-maximum is approximately equal to the localization radius.

Parameters:

• distance (array_like) – Distance from the observation. Same units as localization_radius.

• localization_radius (float) – Localization radius. Must be positive.

Returns:

weight – Localization weight in \([0, 1]\), same shape as distance.

Return type:

ndarray

pypfda.localization.pairwise_distance_matrix(lats, lons)

Symmetric pairwise great-circle distance matrix.

Parameters:

• lats (array_like, shape (n,)) – Latitudes and longitudes in degrees.

• lons (array_like, shape (n,)) – Latitudes and longitudes in degrees.

Returns:

D – D[i, j] is the distance in kilometres between point i and point j. Diagonal is zero.

Return type:

ndarray, shape (n, n)