Definition of error statistics for ensemble forecasts and OSSE's

Next generation data assimilation must include the state dependent observation errors, i.e., the spatial and temporal variations in observation error produced by the turbulent atmosphere. A rigorous analysis of optimal data assimilation algorithms and ensemble forecast systems requires a definition of model "truth" or perfect measurement which defines the total observation error and the forecast error. Truth is defined as the spatial average of the continuous atmospheric state variables centered on each model grid cell. To be consistent with the climatology of turbulence, the spatial average is chosen as the effective spatial filter of the numerical model. Then, the observation errors consist of two independent components: an instrument error and an observation sampling error which describes the mismatch of the spatial average of the observation and the spatial average of the perfect measurement or "truth". The observation sampling error is related to the "error of representativeness" but is defined only in terms of the local statistics of the atmosphere and the sampling pattern of the observation. Therefore, different measurement systems will have different total observation error which is a critical component of optimal data assimilation and observing system simulation experiments (OSSE's). For velocity measurements, the spatial variations in the observation error span more than a factor of 10 due to the typical spatial variations in the turbulence levels. These large variations in observation error are currently ignored for data assimilation and the evaluation of error statistics.

Optimal data assimilation requires an estimate of the local background error covariance (forecast error) as well as the local observation error covariance. Ensemble data assimilation methods can produce estimates for both of these error covariances. A rigorous evaluation of these optimal ensemble data assimilation techniques requires a definition of the ensemble members and the ensemble average that describes the error covariances. A new formulation is presented that is consistent with turbulence scaling laws and the climatology of atmospheric turbulence. The implications of this formulation for ensemble forecast systems and the evaluation of current and future measurements using OSSE's is also presented.