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 the chemistry data assimilation is to combine an atmospheric-chemistry model and actual observations to produce the best estimate of the chemistry of the atmosphere. The observational dataset collected during the past decades is an unprecedented expansion of our knowledge of the atmosphere. The exploitation of these data is the best way to advance our understanding of atmospheric chemistry, and to develop chemistry models for chemistry-climate prediction. The assimilation focuses on estimating the state of the chemistry in a chemically and dynamically consistent manner (if the model allows online interactions between chemistry and dynamics). In so doing, we can: produce simultaneous and chemically consistent estimates of all species (including model parameters), observed and unobserved; fill in data voids; test the photochemical theories used in the chemistry models. In this paper, the Hilbert 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 examples which show the relationship of the gradient of the cost function with respect to the output vector and the gradient of the cost function with respect to the input vector. Applications to chemistry data assimilation are presented for both continuous and discrete cases. The 4D data variational adjoint method is then tested in the assimilation of stratospheric chemistry using a simple catalytic ozone-destruction mechanism, and the test results indicate that the performance of the assimilation method is good.

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