i4DVar: An Integral Correcting Four‐Dimensional Variational Data Assimilation Method

Four‐dimensional variational data assimilation (4DVar) has become an increasingly important tool in data science with wide applications in many engineering and scientific fields. The current state‐of‐the‐art 4DVar offers only two choices in incorporating the forecast model which lead to the strongly and weakly constrained 4DVar approaches. The former ignores the model error and only corrects the initial condition error at the expense of reduced accuracy; while the latter accounts for both the initial and model errors but corrects them separately, which increases computational costs and uncertainty. To overcome these limitations, in this study we develop an integral correcting 4DVar (i4DVar) approach by treating all errors together as a whole and correcting them simultaneously and indiscriminately. Our main idea is to introduce an averaged penalization term in the cost functional to correct the error evolution at selected time steps with same interval, which is amount to dividing the assimilation window into several sub‐windows. As a result, i4DVar greatly enhances the capability of the strongly constrained 4DVar for correcting the model error while also overcomes the limitation of the weakly constrained 4DVar for being prohibitively expensive with added uncertainty. The new i4DVar approach has the potential to be applicable to various scientific and engineering fields as well as industrial sectors which involve big observation data because of its ease of implementation and superior performance.