Four-dimensional variational data assimilation (4DVar) has become an increasingly important tool in data science with wide applications in many engineering and scientific fields such as geoscience1-12, biology13 and the financial industry14. The 4DVar seeks a solution that minimizes the departure from the background field and the mismatch between the forecast trajectory and the observations within an assimilation window. The current state-of-the-art 4DVar offers only two choices by using different forms of the forecast model: the strong- and weak-constrained 4DVar approaches15-16. 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 and corrects them separately, which increases computational costs and uncertainty. To overcome these limitations, here we develop an integral correcting 4DVar (i4DVar) approach by treating all errors as a whole and correcting them simultaneously and indiscriminately. To achieve that, a novel exponentially decaying function is proposed to characterize the error evolution and correct it at each time step in the i4DVar. As a result, the i4DVar greatly enhances the capability of the strong-constrained 4DVar for correcting the model error while also overcomes the limitation of the weak-constrained 4DVar for being prohibitively expensive with added uncertainty. Numerical experiments with the Lorenz model show that the i4DVar significantly outperforms the existing 4DVar approaches. It has the potential to be applied in many scientific and engineering fields and industrial sectors in the big data era because of its ease of implementation and superior performance.
Four-dimensional variational data assimilation (4DVar) has become an increasingly important tool in data science with wide applications in many engineering and scientific fields such as geoscience [1][2][3][4][5][6][7][8][9][10][11][12] , biology 13 and the financial industry 14 . The 4DVar seeks a solution that minimizes the departure from the background field and the mismatch between the forecast trajectory and the observations within an assimilation window. The current state-of-the-art 4DVar offers only two choices by using different forms of the forecast model: the strong-and weak-constrained 4DVar approaches [15][16] . 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 and corrects them separately, which increases computational costs and uncertainty. To overcome these limitations, here we develop an integral correcting 4DVar (i4DVar) approach by treating all errors as a whole and correcting them simultaneously and indiscriminately. To achieve that, a novel exponentially decaying function is proposed to characterize the error evolution and correct it at each time step in the i4DVar. As a result, the i4DVar greatly enhances the capability of the strong-constrained 4DVar for correcting the model error while also overcomes the limitation of the weak-constrained 4DVar for being prohibitively expensive with added uncertainty. Numerical experiments with the Lorenz model show that the i4DVar significantly outperforms the existing 4DVar
approaches. It has the potential to be applied in many scientific and engineering fields and industrial sectors in the big data era because of its ease of implementation and superior performance.
Data assimilation (DA), whose latest development is represented by the four-dimensional variational (4DVar) approach [17][18] , has experienced explosive growth and development, especially in the context of the big data era [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . This is because DA can effectively incorporate the time series of observational data into model simulations and predictions to improve estimates of all the current and future states of a natural (e.g., the atmosphere) [1][2][3][4][5][6][7][8][9][10][11][12][13] or social (e.g., the financial markets) system 14 . For example, most numerical weather prediction (NWP) centers around the world have adopted the 4DVar approach [8][9][10][11][12] to assimilate asynchronous observations simultaneously, which has greatly improved the accuracy of weather prediction.
Currently, the 4DVar only has two approaches: the strong-and weak-constrained methods depending on whether the 4DVar solution is required to satisfy the forecast model exactly [15][16] .
The strong-constrained 4DVar (s4DVar) assumes the forecast model is perfect and all errors in the prediction originate from the initial conditions. This is clearly unrealistic in many cases, and recent studies [15][16][19][20][21] show that incorporating the model error into the 4DVar improves its performance. The weak-constrained 4DVar (w4DVar) allows for both initial and model errors and it corrects them separately [15][16] , which adds computational costs significantly and increases uncertainty. To reduce the computational costs, various simplifications are used to specify the model error in the w4DVar [20][21] . However, these simplifications still have many issues; for example, it is very difficult or expensive to determine the parameters used to represent the model error covariance 21 .
To overcome the limitations of both the s4DVar and w4DVar, we develop a new 4DVar approach in which all errors are treated indistinctly and corrected simultaneously. Our approach is based on the observation that the influence of all errors would decay gradually in time if the correction-term is incorporated into the model integration sequentially.
Here, we first show a less recognized fact that the s4DVar actually has a hidden mechanism that can correct the model error at the initial (i.e., analysis) time. The s4DVar seeks an analysis increment '
x m x of the initial background condition 0
x , such that ’ x minimizes the following cost function (so that *’ 00 x x x represents the corrected estimate of the state variable
x at the initial time 0 t , see Fig. 1) 20 :
under the constraint 1 ()
, where '
x is assumed to be Gaussian with the covariance 1.
The traditional s4DVar assumes that the forecast model 1 ()
describes the underlying system exactly 15 , thus the model error is negligible. However, the model error is often non-negligible due to errors from discretization of continuous fields, approximation of certain physical processes, parameter uncertainties, boundary conditions, and round-off errors.
Moreover, various studies show that the model error prevails over the initial error in many circumstances 22 . The use of the 4DVar at several major
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