Continuous versus discrete advection adjoints in chemical data assimilation with CMAQ

Data assimilation obtains improved estimates of the state of a physical system by combining imperfect model results with sparse and noisy observations of reality. In the four dimensional variational (4D-Var) framework data assimilation is formulated as an optimization problem, which is solved using gradient based optimization methods. The 4D-Var gradient is obtained by forcing the adjoint model with observation increments. The construction of the adjoint model requires considerable development effort. In the continuous approach the adjoint differential equations are discretized. In the discrete approach the numerical solution of the forward equations is differentiated. The two routes lead to different gradients. s paper we investigate numerically the effect of using discrete and continuous adjoints of the advection equation in chemical transport modeling. Continuous advection adjoints are easily implemented by calling the same advection subroutines as the forward model, with a sign change for the winds, and with rescaling the solution. Discrete advection adjoints involve the differentiation of a nonlinear, monotonic advection scheme like the piecewise parabolic method. The numerical experiments are carried out with CMAQ-ADJ. The results show that, while discrete advection adjoints are more accurate in point-to-point comparisons against finite differences, the continuous adjoints of advection perform better as gradients for optimization in 4D-Var data assimilation.