The differences between the observations and the forecast background used for the analysis (the innovations or O-F for short) and those between the observations and the final analysis (O-A) are by-products of any assimilation system and provide information about the quality of the analysis and the impact of the observations. Innovations have been traditionally used to diagnose observation, ... background and analysis errors at observation locations (Hollingsworth and Lonnberg 1989; Dee and da Silva 1999). At the most simplistic level, innovation variances can be used as an upper bound on background errors, which are, in turn, an upper bound on the analysis errors. With more processing (and the assumption of optimality), the O-F and O-A statistics can be used to estimate observation, background and analysis errors (Desroziers et al. 2005). They can also be used to estimate the systematic and random errors in the analysis fields. Unfortunately, such data are usually not readily available with reanalysis products. With MERRA, however, a gridded version of the observations and innovations used in the assimilation process is being made available. The dataset allows the user to conveniently perform investigations related to the observing system and to calculate error estimates. Da Silva (2011) provides an overview and analysis of these datasets for MERRA.
The innovations may be thought of as the correction to the background required by a given instrument, while the analysis increment (A-F) is the consolidated correction once all instruments, observation errors, and background errors have been taken into consideration. The extent to which the O-F statistics for the various instruments are similar to the A-F statistics reflects the degree of homogeneity of the observing system as a whole. Using the joint probability density function (PDF) of innovations and analysis increments, da Silva (2011) introduces the concepts of the effective gain (by analogy with the Kalman gain) and the contextual bias. In brief, the effective gain for an observation is a measure of how much the assimilation system has drawn to that type of observation, while the contextual bias is a measure of the degree of agreement between a given observation type and all other observations assimilated.
With MERRAs gridded observation and innovation data sets, a wealth of information is available for examination of the quality of the analyses and how the different observations impact the analyses and interact with each other. Such examinations can be conducted regionally or globally and should provide useful information for the next generation of reanalyses.