Correlated observation errors.
In Numerical Weather Prediction there is a concerted move towards higher resolution models with the aim of forecasting small-scale yet destructive weather phenomena. In order to constrain these models, via data assimilation, there is a simultaneous need for high-resolution observations. However, the inabilty to account for signicant spatial correlations, due to representivity and forward model error, has historically restricted the use of observations that resolve all the scales modelled.
Recent advances in estimating and including observation error correlations in DA has meant that we are now moving towards being able to assimilate high-density observations in a more optimal way. This poses the following questions:
- How does including observation error correlations change the impact of the observations on the analysis?
- What density of observations that exhibit spatial error correlations is optimal?
- How sensitive is the use of dense observation to the accuracy of the estimated observation error correlations?
In general as the stregth of observation error correlations increase the entropy (a measure of uncertainty) associated with the observations reduces. This means that typically, measures of observation influence, such as degrees of freedom for signal and mutual information increase with increasing correlation strength. However, the impact of the observation error correlations on other metrics cannot be understood in isolation from the background-error statistics and the observation operator. When the prior and likelihood are accurate at different scales (i.e. the backgrond in observation space and observation error covariance matrices have very different structures), it can be shown that
- the analysis-error variances are smallest;
- there is the greatest spread in information (i.e. crosssensitivities are largest); and
- the observations have the greatest mutual information.
Observations with spatial error correlations are known to provide more information about small scales than observations with uncorrelated errors (e.g. Seaman 1977, Rainwater et al. 2015, Fowler et al. 2018). However, as the trace of the covariance matrix is conserved observations with correlated error have more uncertainty at large-scales. This implies that thinning or averaging observations with large error correlations removes a greater proportion of their information than observations with uncorrelated error. The importance of the small-scale information in the observations depends on what is already known a-priori (represented by the prior error covariance matrix). Using a metric such as the average information content per observation, it can be shown that there is the greatest benefit of having dense observations resolving the smallest scales modelled when the observations are more certain than the prior at those scales.
- Fowler, A. M.: Data compression in the presence of observational error correlations, Tellus A: Dynamic Meteorology and Oceanography, 71:1, 1-16, DOI: 10.1080/16000870.2019.1634937
- Fowler, A. M., Dance, S. and Waller, J.: On the interaction of observation and a-priori error correlations in data assimilation. Q.J.R.Met.Soc.144: 48–62. doi:10.1002/qj.3183