### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Statistical description of the atmosphere and the definition of ‘truth’
- 3. Statistical description of error referenced to the model grid
- 4. Statistical description of error referenced to the observation coordinate
- 5. NWP model representation
- 6. Maximum likelihood data assimilation for error referenced to the model grid coordinates
- 7. Data assimilation for error referenced to the observation coordinates
- 8. Implementation of advanced data-assimilation algorithms
- 9. Simple example calculations
- 10. Operational issues
- 11. Summary and discussion
- Acknowledgements
- References

A consistent definition of ‘truth’ is presented to define the errors in a numerical weather prediction (NWP) forecast, analysis and observations resulting from the unresolved turbulent field. ‘Truth’ is defined as the convolution of the continuous atmospheric variables by the effective spatial filter of an NWP model. Direct measurements of atmospheric variables are represented as an instrument error and a convolution of the continuous atmospheric variables by the observation sampling function. This clearly separates the instrument error from the observation sampling error that describes the mismatch between the NWP model effective spatial filter and the observation sampling function. The ensemble average that defines error statistics is defined by an infinite number of atmospheric realizations with statistically similar random fluctuations in the unresolved model field. This results in large spatial variations in the observation sampling errors due to the atmospheric variations in turbulence statistics. Two approaches are discussed to describe these spatial variations: one that defines observation error referenced to each model coordinate and one that assigns observation error referenced to each observation coordinate. The observation-error statistics depend on the observation sampling function, the local spatial statistics of the turbulence field and the NWP model filter. The effects of imprecise knowledge of the shape of the model filter on observation sampling error are small for rawinsonde measurements and for observations that produce a linear average along a track. The modifications to data-assimilation algorithms (the maximum-likelihood (ML) method, minimum mean-square-error algorithms, Kalman filtering, variational data assimilation and ensemble data assimilation) to include the spatial variations in observation-error statistics are discussed. In addition, the generation of ensemble forecast members should be consistent with the spatial variations in total observation error. A rigorous definition of error statistics is essential for evaluating the many different types of current and future observing systems. Copyright © 2010 Royal Meteorological Society