Implementation of a Block Krylov Algorithm in Variational Data Assimilation

Abstract: Numerical weather prediction relies on two major components: sophisticated atmospheric forecast models and equally important data assimilation algorithms. Data assimilation (DA) is the process used to produce the best estimate of the state of a model, using an Earth system numerical model and observations. This “best estimate” will then be an accurate initial condition to the model. More and more applications of weather or weather-related forecasts (such as air quality) require estimates of the uncertainty in the forecasts, or even full probability distribution, to inform decision making. An estimate of the probability distribution of the analysis can be obtained by generating an ensemble of analysis. For each member of the ensemble, all inputs to the assimilation process are perturbed randomly according to their respective error statistics. Each data assimilation instance is an optimization problem and in the context of Ensemble Data Assimilation, many almost identical optimization problems are solved. So far, these problems have been solved independently of each other, using for example a Lanczos algorithm. However, it is possible to use information from all the members to construct a better approximation of the eigen-structure of the matrix at the heart of the optimization problem, and accelerate the convergence. The block-Lanczos algorithm is one of the block methods that exist to solve an Ensemble of Data Assimilation. This project consists in implementing the block Lanczos algorithm in the Joint Effort for Data assimilation Integration (JEDI) framework and demonstrate the relevance of the technique with the use of the Quasi-Geostrophic model (QG). Results show that when comparing a Lanczos and a block Lanczos, two effects compete against each other: the block requires less iterations to converge but each iteration takes more time. The fastest time to converge is reached when using around 16 members. Though, some issues are still encountered and requires to look more into the kind of operators we use in the block algorithm or it is subject to divergence.