A review on the use of the adjoint method in four-dimensional atmospheric-chemistry data assimilation

In this paper we review a theoretical formulation of the adjoint method to be used in four-dimensional (4D) chemistry data assimilation. The goal of t he chemistry data assimilation is to combine an atmospheric-chemistry model and actual observations to produce the best estimate of the chemistry of t he atmosphere. The observational dataset collected during the past decades is an unprecedented expansion of our knowledge of the atmosphere. The explo itation of these data is the best way to advance our understanding of atmos pheric chemistry, and to develop chemistry models for chemistry-climate pre diction. The assimilation focuses on estimating the state of the chemistry in a chemically and dynamically consistent manner (if the model allows onli ne interactions between chemistry and dynamics). In so doing, we can: produ ce simultaneous and chemically consistent estimates of all species (includi ng model parameters), observed and unobserved; fill in data voids; test the photochemical theories used in the chemistry models. In this paper, the Hi lbert space is first formulated from the geometric structure of the Banach space, followed by the development of the adjoint operator in Hilbert space . The principle of the adjoint method is described, followed by two example s which show the relationship of the gradient of the cost function with res pect to the Output vector and the gradient of the cost function with respec t to the input vector. Applications to chemistry data assimilation are pres ented for both continuous and discrete cases. The 4D data variational adjoi nt method is then tested in the assimilation of stratospheric chemistry usi ng a simple catalytic ozone-destruction mechanism, and the test results ind icate that the performance of the assimilation method is good.