Examination of various-order adjoint-based approximations of observation impact

The necessity, and potential implications, of considering higher-than-first-order approximations of the impact of observations on numerical weather forecasts are investigated. In particular, clarification is sought on issues raised by ERRICO (2007) concerning the interpretation of an adjoint-based measure of observation impact developed by LANGLAND and BAKER (2004). Various-order adjoint-based approximations of observation impact are examined in this context using adjoint versions of the NASA Goddard Earth Observing System (GEOS) atmospheric data assimilation system. Attention is given to examining the effects of nonlinearity on partial sums used to estimate the impact of particular subsets of observations, and especially whether contributions to these sums from observations outside a particular subset might cloud their interpretation. It is shown that higher-than-first-order accuracy (in terms of innovations) is required to capture adequately the impact of observations measured by the change in an energy-based metric of forecast error. When accounting for the impact of the entire set of observations, the error of the first-order approximation is roughly four times larger than that of the second- or third-order approximations. Just as importantly, no substantial evidence is found to suggest that nonlinear approximations of the impacts of selected subsets of observations, in this case corresponding to specific satellite and conventional observing systems assimilated in GEOS-5, are altered in any discernible way by cross terms with different observation types. An explanation for this is offered based on the relative contributions from linear and nonlinear terms in these approximations.