The influence matrix is used in ordinary least-squares applications for monitoring statistical multipleregression analyses. Concepts related to the influence matrix provide diagnostics on the influence of individual data on the analysis, the analysis change that would occur by leaving one observation out, and the effective information content (degrees of freedom for signal) in any sub-set of the analysed data. In this paper, the corresponding concepts are derived in the context of linear statistical data assimilation in Numerical Weather Prediction. An approximate method to compute the diagonal elements of the influence matrix (the self-sensitivities) has been developed for a large-dimension variational data assimilation system (the 4D-Var system of the European Centre forMedium-Range Weather Forecasts). Results show that, in the ECMWF operational system, 18% of the global influence is due to the assimilated observations, and the complementary 82% is the influence of the prior (background) information, a short-range forecast containing information from earlier assimilated observations. About 20% of the observational information is currently provided by surface-based observing systems, and 80% by satellite systems.

A toy-model is developed to illustrate how the observation influence depends on the data assimilation covariance matrices. In particular, the role of high-correlated observation error and high-correlated background error with respect to uncorrelated ones is presented. Low-influence data points usually occur in data-rich areas, while high-influence data points are in data-sparse areas or in dynamically active regions. Background error correlations also play an important role: high correlation diminishes the observation influence and amplifies the importance of the surrounding real and pseudo observations (prior information in observation space). To increase the observation influence in the presence of high correlated background error, it is necessary to also take the observation error correlation into consideration. However, if the observation error variance is too large with respect to the background error variance the observation influence will not increase. Incorrect specifications of the background and observation error covariance matrices can be identified by the use of the influence matrix.

**KEYWORDS:** Observations Influence Data Assimilation Regression Methods

- Introduction
- Classical Statistical Definitions of Influence Matrix and Self-Sensitivity
- Observational Influence and Self-Sensitivity for a DA Scheme
- Results
- Conclusions
- Appendix:
- A. Influence Matrix Calculation in Weighted Regression Data Assim. Scheme
- B. Approximate calculation of self-sensitivity in a large variational analysis system

- References

10alecture notes10aNWP1 aCardinali, Carla uhttps://www.ecmwf.int/node/16938