Dual formulation of four‐dimensional variational assimilation

Abstract A duality between two formulations of variational assimilation, three‐dimensionsal (3D‐Var) and the Physicalspace Statistical Analysis System (PSAS), is presented. It is shown that their conditioning is identical. the temporal extension of 3D‐Var leads to 4D‐Var. the temporal extension of PSAS, 4D‐PSAS, is achieved using an algorithm inspired by the representer technique but without the explicit computation of the representers. Assuming the model is perfect, both 4D‐Var and 4D‐PSAS are equivalent in terms of results produced and cost. Assuming the model is not perfect, the equivalence is preserved, but in 4D‐Var it is necessary to increase the size of the control variable while 4D‐PSAS remains almost unchanged. In duality, 4D‐Var remains almost unchanged by an increase in the number of observations used, whereas the size of the control variable in PSAS depends directly on this number.