Atmospheric turbulence component of the innovation covariance

Abstract

ABSTRACT:The innovation covariance is derived for general atmospheric turbulence conditions and for climatologies of upper-level turbulence based on aircraft data. Error is defined in terms of the effective spatial filter of the forecast model to produce a consistent definition of measurement error and model error. Calculations of the atmospheric turbulence component (observation sampling-error covariance) are performed for rawinsonde data and various forecast model resolutions. Significant contributions from the atmospheric turbulence field are produced, especially for the older forecast models.

Measuring information content from observations for data assimilation: spectral formulations and their implications to observational data compression

“…This error can be caused by the limited spatial resolution in the calculation of the observation operator and it depends on the subgrid turbulent structures within the model effective resolution volume over which the true state is averaged. The model effective resolution, denoted by Δ x e , can be significantly coarser than the model grid resolution Δ x in the horizontal direction (Frehlich, 2008). This effective resolution Δ x e (>Δ x ) also limits the finest horizontal scale resolvable by the background error covariance, so the background error power spectrum S ( k i ) is essentially band‐limited and should become zero (or set to zero) for | k i | > k e , where k e =π/Δ x e is the effective Nyquist‐wavenumber.…”

The definition of ‘truth’ for Numerical Weather Prediction error statistics

“…However, a rigorous definition of error statistics is required to evaluate any results correctly. For example, the forecast error (innovation error) statistics depend on the observation sampling errors and therefore are a function of the local turbulence statistics, which must be included in the analysis (Frehlich, 2008).…”

“…However, this technique must include a rigorous definition of error statistics as well as the climatology of the atmospheric turbulence statistics (e.g. the structure function of Figure 1) and the effects of the spatial filter of the forecast model (Frehlich, 2008). This is especially important for data products that have a large spatial average, such as GPS occultation data (Kuo et al, 2004;Healy and Thépaut, 2006;Chen et al, 2009), where the observation sampling error can be large because of the large mismatch between the model filter and the spatial average of the observation.…”

Measuring information content from observations for data assimilations: utilities of spectral formulations demonstrated with radar observations

“…According to the analyses in sections 4.3 and 4.4 of X11, uniformly distributed observations with unbiased and spatially uncorrelated errors can be compressed into super-observations by local averaging with no loss of information until the superobservation resolution becomes coarser than the effective background resolution (Frehlich, 2008) or the background error covariance resolution (determined by π /k c where k c is the cutoff wavenumber of the background error power spectrum). In this case, the original observations are simply averaged over each x s , where x s is the super-observation resolution.…”

“…Purser et al, 2000;Liu et al, 2005;Alpert and Kumar 2007;Lu et al 2011, section 3.3). According to the theoretical analysis based on the spectral formulations in section 4.3 and remarks in section 4.4 of X11, super-Obbing by local averaging will cause no loss of information (measured globally by the dispersion part of relative entropy or the Shannon entropy difference) for uniformly distributed observations with unbiased and spatially uncorrelated errors unless the super-observation resolution becomes coarser than the effective background resolution (Frehlich, 2008) or the background error covariance resolution (determined by one half of the wavelength associated with the cut-off wavenumber of the background error power spectrum). This is an attractive global property for super-Obbing by local averaging, but it is not clear whether and to what extent this global property can be retained for non-uniformly distributed observations.…”