On the Smoothness Constraints for Four-Dimensional Data Assimilation

An algorithm for determination of the weights of smoothness penalty constraints for the four-dimensional variational data assimilation technique is proposed and evaluated. To study the nature of smoothness penalty constraints, a simple nonlinear harmonic oscillator problem is first considered. Penalizing smoothness constraints is found to make the modified Hessian matrix of the cost function more positive definite, akin to the idea behind the modified line search Newtonʹs methods. However, the use of the derivative smoothness constraints with a fixed coefficient does not warrant uniform imposition of these constraints at every iteration. A remedy is to control the ratio of the smoothness penalty function over the cost function, which can dramatically increase the positive definite area. On the other hand, the large smoothness coefficients obtained from this approach can deteriorate the convergence property of the minimization problem. Based on these observations, an algorithm for tuning the weights of smoothness constraints is proposed to overcome the aforementioned problems. The algorithm is first applied to a simple dynamic problem. It is then tested on the retrieval of microscale turbulent structures in a simulated convective boundary layer. This method is further evaluated on the retrieval of a strong meso-scale thunderstorm outflow from Doppler radar data. The results show that the algorithm yields efficient retrieval.