Interpretations of an adjoint-derived observational impact measure

The measure developed by Langland and Baker to estimate and compare the utility of arbitrary sets of observations for reducing forecast error is re-derived here. The formula uses adjoints of both a forecast model and a data assimilation system. Although the final expression here is nearly identical to that developed by Langland and Baker, the third-order character of their measure is now explicitly presented. The appropriateness of going beyond first order is explained and a warning of the measure's nonlinearity possibly clouding its interpretation is offered.