# A Frequency-Dependent Assimilation Algorithm: Ensemble Optimal Smoothing

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## Abstract

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## 1. Introduction

## 2. Assimilation Theory

#### 2.1. Overview of Sequential Ensemble-Based Assimilation

#### 2.2. Ensemble Optimal Smoother (EnOS)

## 3. Idealized Linear Ocean Model

## 4. Sensitivity Analysis of Time-Dependent Term

#### 4.1. Data

#### 4.2. Single-Point Observation Experiments

## 5. Validation Using the Reanalysis Data

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Hovmöller diagrams of stream function predicted by the forced Rossby wave model. In each panel, the horizontal axis shows the zonal position $\left(x\right)$ scaled by the width of the ocean $\left(L\right)$ and the vertical axis shows the time from 0 to 1000 days.(

**a**): stream function resulting from white noise forcing; (

**b**): analyzed stream function based on assimilating perfect observations $x/L=0.2$ when using EnOI and no forcing; (

**c**): As for the central figure observations, but assimilated using EnOS with lags of 0 and 1 time steps of the model.

**Figure 2.**Standard deviations of stream function predicted by the forced Rossby wave model. The horizontal axis shows the zonal position scaled by the width of the ocean $\left(x/L\right)$ and the vertical axis shows the standard deviation of the time-varying stream function for each grid point. Blue line: standard deviation of stream function resulting from white noise forcing. Green line: standard deviation of analysis errors resulting from assimilation of perfect observations at $x/L=0.2$ using EnOI and no forcing. Red line: as for the green line, but for observations assimilated using EnOS with lags of 0 and 1 time step.

**Figure 3.**The locations of the group A, B, and C single-point experiments on 1 January 2015. The black box represents the study area within plus or minus 2° of the black dot, which was the observation point.

**Figure 4.**Results of the group A experiment (the Kuroshio Current in the East China Sea) on 1 January 2015, the black dot in the figure is the observation point; (

**a**) background field of the single-point observation experiment; (

**b**) target field of the single-point observation experiment; (

**c**) analysis field of the single-point observation experiment (EnOI); (

**d**) analysis field of the single-point observation experiment (EnOS).

**Figure 5.**Distribution of the gain for group A on 1 January 2015, and the black dot in the figure is the observation point: (

**a**) Kalman gain $K$ at moment $t$ in the EnOI scheme; (

**b**) Kalman ${K}_{0}$ at moment $t$ in the EnOS scheme; (

**c**) gain ${K}_{1}$ acting at moment $t$ in the EnOS scheme at historical moment $t-1$.

**Figure 6.**Distribution of the gain for group B on 1 January 2015, and the black dot in the figure is the observation point: (

**a**) Kalman gain $K$ at moment $t$ in the EnOI scheme; (

**b**) Kalman ${K}_{0}$ at moment $t$ in the EnOS scheme; (

**c**) gain ${K}_{1}$ acting at moment t in the EnOS scheme at historical moment $t-1$.

**Figure 7.**Distribution of the gain for group C on 1 January 2015, and the black dot in the figure is the observation point. (

**a**) Kalman gain $K$ at moment $t$ in the EnOI scheme; (

**b**) Kalman gain ${K}_{0}$ at moment $t$ in the EnOS scheme; (

**c**) gain ${K}_{1}$ acting at moment t in the EnOS scheme at historical moment $t-1$.

**Figure 8.**(

**a**): RMSE distribution of the analysis and target fields of EnOI for 365 days in 2015; (

**b**): RMSE distribution of the analysis and target fields of EnOS for 365 days in 2015; (

**c**): RMSE distribution of the background and target fields for 365 days in 2015; (

**d**): distribution of observations over the difference of RMSE of EnOI minus RMSE of EnOS, and the black dot in the figure is the observation point.

**Figure 9.**The different distributions of RMSE of EnOI minus RMSE of EnOS in the winter (12, 1, 2) and autumn (9, 10, 11) of 2015.

**Figure 10.**Hovmöller diagrams of fluctuations over time on the longitude line at 21° N. In each panel, the horizontal axis shows the latitudinal range of the ocean and the vertical axis shows the time range of 1 to 365 days. The corresponding results of the three figures are from the target field, EnOI, and EnOS.

**Figure 11.**RMSE of the target and analysis fields for the entire year obtained via spatial averaging of the twin trials for 365 days in 2015. Black line: results of EnOI. Red line: results of EnOS.

Evaluation Criteria | Method | Winter | Spring | Summer | Autumn |
---|---|---|---|---|---|

Root-Mean-Square error (RMSE) | EnOI | 0.54 | 0.58 | 0.41 | 0.40 |

EnOS | 0.48 | 0.54 | 0.39 | 0.38 | |

Performance gain | 11.02% | 7.15% | 3.03% | 2.35% |

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**MDPI and ACS Style**

He, Z.; Zhao, Y.; Fu, X.; Sheng, X.; Xu, S.
A Frequency-Dependent Assimilation Algorithm: Ensemble Optimal Smoothing. *J. Mar. Sci. Eng.* **2022**, *10*, 1324.
https://doi.org/10.3390/jmse10091324

**AMA Style**

He Z, Zhao Y, Fu X, Sheng X, Xu S.
A Frequency-Dependent Assimilation Algorithm: Ensemble Optimal Smoothing. *Journal of Marine Science and Engineering*. 2022; 10(9):1324.
https://doi.org/10.3390/jmse10091324

**Chicago/Turabian Style**

He, Zhongjie, Yueqi Zhao, Xiachuan Fu, Xin Sheng, and Siwen Xu.
2022. "A Frequency-Dependent Assimilation Algorithm: Ensemble Optimal Smoothing" *Journal of Marine Science and Engineering* 10, no. 9: 1324.
https://doi.org/10.3390/jmse10091324