Spectral Analytical Filter

This block performs multiplication in spectral space by an analytical function depending only on the total wavenumber. Typical use cases include horizontal localization and frequency filters.

Example yaml

saber outer blocks:
- (...)
- saber block name: spectral analytical filter
  normalize filter variance: false  # Optional. Default is true (for localization)
  function:
    shape: gaussian                 # Optional. Default is gaussian.
    horizontal daley length: 2000e3
- (...)

The parameter normalize filter variance decides whether the analytical function should be normalized to act as a localization function. The only function shape currently implemented is gaussian.

Usage for horizontal localization

This block can be used for horizontal localization, relying on the fact that convolution on the physical space is equivalent to multiplication on the spectral space.

To model convolution by a Gaussian localization function \(f(d)\) in physical space (where \(d\) is the great circle separation distance), we multiply by a Gaussian localization function \(\widehat{f}(n)\) in spectral space (where \(n\) is the total wavenumber). If the physical space Gaussian is given by

\[\begin{split}f(d) &= \exp\biggl(-\frac{d^2}{2 L^2}\biggr)\\ \text{Or equivalently }f(\lambda) &= \exp\biggl(-\frac{\lambda^2}{2 \sigma_{\lambda}^2}\biggr)\end{split}\]

where \(L\) is the horizontal localization length, \(d = \lambda R\), \(R\) is the Earth radius and \(\sigma_\lambda = L / R\), then the associated Gaussian in spectral space is given by:

\[\widehat{f}(n) = C\exp\biggl(-\frac{n^2}{2\sigma_n}\biggr)\]

where \(\sigma_n = 1/ \sigma_\lambda = R / L\) and \(C\) is some normalization constant.

The normalization constant \(C\) is computed by ensuring the total variance of the localization function in spectral space is 1.

- saber block name: spectral analytical filter
  function:
    horizontal daley length: 2000e3

Here, the localization is modeled as a Gaussian with Daley length specified from the yaml, in meters. For a Gaussian function, the Daley length \(\sqrt{-f(0) / f''(0)}\) is parameter \(L\) in the equations above (see for instance Pannekoucke et al., 2008).

References

Errera, Q., & Ménard, R. (2012). Technical Note: Spectral representation of spatial correlations in variational assimilation with grid point models and application to the Belgian Assimilation System for Chemical Observations (BASCOE). Atmospheric Chemistry and Physics, 12(21), 10015–10031. doi:10.5194/acp-12-10015-2012

Pannekoucke, O., Berre, L., & Desroziers, G. (2008). Background-error correlation length-scale estimates and their sampling statistics. Quarterly Journal of the Royal Meteorological Society, 134(631), 497–508. doi:10.1002/qj.212