Estimation of key analysis errors using the adjoint technique

An iteration procedure minimizing the short-range forecast error leads, after some iterations, to so-called key analysis errors. These are estimates of the part of analysis errors that is largely responsible for the short-range forecast errors. The first step of the minimization procedure provides a scaled gradient of the two-day forecast errors for which the ‘energy’ inner-product provides an efficient way of identifying the analysis errors at scales that are relevant for forecast error growth. By using an ‘enstrophy’ like inner-product as an alternative to ‘energy’ the sensitivity gradient obtains an unrealistically large scale. Performing a few more steps in the minimization provides better estimates of the analysis error in the directions spanned by the leading singular vectors of the tangent-linear model. On a case study it is shown that three steps provide key analysis increments which, when added to the analysis, both significantly improve the fit to the available data, and substantially improve the subsequent model integration. It does not appear to be beneficial to do more steps of the minimization because of the uncertainty in the definition of the short-range forecast error, and of approximations in the tangent-linear model. Key analysis errors represent an improved estimate of analysis errors compared to the scaled gradient of day-2 forecast errors. In particular the geographical distribution shows the stability dependence of the scaled gradient. The projection of the gradient on the fastest growing errors limits maximum sensitivity to the major baroclinic zones. The close correspondence of evolved key analysis errors and forecast errors shows that key analysis errors are more realistically projecting on to full analysis errors. The close link between the stability of the flow and the gradient of the forecast errors implies an unreasonably strong seasonal variation of analysis errors estimates. In contrast, key analysis errors are nearly seasonally independent, which means that their detrimental effect on forecast errors in absolute terms in summer and winter is comparable.