# Parameter Estimation Based on a Local Ensemble Transform Kalman Filter Applied to El Niño–Southern Oscillation Ensemble Prediction

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Zebiak–Cane Model

#### 2.2. LETKF-Based Parameter Estimation

#### 2.3. Covariance Inflation Scheme

#### 2.4. Model Parameters

#### 2.5. Experimental Designs

#### 2.5.1. Experimental Designs in OSSE Framework

#### 2.5.2. Experimental Designs in the Real World

## 3. Results

#### 3.1. Results of the OSSE Framework

#### 3.2. Results in Real-World Experiments

#### 3.3. Validation of Optimized Parameters

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Integral domain of the LDEO5 model, including the atmospheric model contoured by solid black lines, oceanic model by blue lines, and SSTA model by red lines. The Niño3.4 region is also labeled by dotted grey lines.

**Figure 2.**Temporal evolution of (

**a**) estimated gam2, and (

**b**) its RMSE relative to the “observations” and ensemble spread with assimilation length in SPE: bold solid line indicates the ensemble mean of gam2 and the dashed line is the truth in panel (

**a**).

**Figure 3.**Temporal evolution of six estimated parameters with assimilation length in MPE: (

**a**–

**f**) the six subfigures representing gam1, gam2, tda1, tda2, tdb1, and tdb2, respectively; bold solid lines indicate the ensemble mean of the parameters, and the dashed lines are their truths.

**Figure 4.**Time series of RMSE between the simulated state variables and the corresponding “observed” values during the last 100 years in three experiments: From (

**a**) to (

**d**), the four subfigures represent RMSEs of SSTA, ZWA, ULDA, and TeA, respectively.

**Figure 5.**Variation in ACCs averaged over the last 100 years in the three forecasting experiments: (

**a**) onlySE, (

**b**) SPE, and (

**c**) MPE.

**Figure 6.**Temporal evolution of the six estimated parameters with assimilation length in MPE: (

**a**–

**f**) the six subfigures represent gam1, gam2, tda1, tda2, tdb1, and tdb2, respectively.

**Figure 7.**The (

**a**) ACCs and (

**b**) RMSEs of the predicted SST anomalies against the true counterparts in the Niño3.4 region during the period 1981–2000.

**Figure 8.**Retrospective predictions of El Niños and La Niñas during 1981−2000: the red curve is the observed values and the blue curves are the predicted values at a 3-month and 6-month lead in (

**a**) and (

**b**) subfigures, respectively.

**Figure 9.**Composites of El Niños and La Niñas: (

**a**,

**b**) represent composites of observations, (

**c**,

**d**) represent predictions in onlySE, and (

**e**,

**f**) represent predictions in MPE.

**Figure 10.**The (

**a**) ACCs and (

**b**) RMSEs of the predicted SST anomalies against the true counterparts in the Niño3.4 region during the period 2001–2020.

**Table 1.**Convergence times, the estimated parameter values, and relative errors in parameter estimation with six inflation schemes.

Schemes | FI | CCI | EPES | RTPS | RTPP | New |
---|---|---|---|---|---|---|

Convergence times | 296 | 323 | 27 | 312 | 365 | 288 |

Estimated values | 0.7631 | 0.7793 | 0.8006 | 0.7318 | 0.7249 | 0.7492 |

Relative errors(%) | 1.75 | 3.91 | 6.75 | 2.42 | 3.35 | 0.11 |

**Table 2.**List of parameters (Pars.) in the Z-C model and their truths and initial guesses. Here, h represents the upper layer depth.

Pars. | Physical Meanings | Units | Truths | Initial Guesses |
---|---|---|---|---|

gam1 | Strength of mean upwelling advection term | - | 0.75 | 0.6 |

gam2 | Strength of anomalous upwelling advection term | - | 0.75 | 0.6 |

tda1 | Amplitude of subsurface temperature anomaly +h perturbations | °C | 28 | 22.4 |

tda2 | Amplitude of subsurface temperature anomaly for −h perturbations | °C | −40 | −48 |

tdb1 | Affect the nonlinearity of subsurface temperature anomaly for +h perturbations | m^{−1} | 1.25 | 1.0 |

tdb2 | Affect the nonlinearity of subsurface temperature anomaly for −h perturbations | m^{−1} | 3.0 | 2.4 |

**Table 3.**Experimental designs in the OSSE framework. Note that six parameters are estimated in MPE, so there are six pairs of values of a and b in MPE.

Experiments | Assimilated | Estimated | (a,b) |
---|---|---|---|

onlySE | SSTA | SSTA | (-,-) |

SPE | SSTA | gam2\SSTA | (0.15,0.08) |

MPE | SSTA | gam1\gam2\tda1\tda2\ tdb1\tdb2\SSTA | (0.31,0.016)\(0.30,0.015)\(4.12,1.69)\ (6.58,3.44)\(0.09,0.05)\(0.45,0.20) |

**Table 4.**Experimental designs in the real world. Note that six parameters are estimated in MPE, so there are six pairs of values of a and b in MPE.

Experiments | Assimilated | Estimated | (a,b) |
---|---|---|---|

OnlySE | SSTA | SSTA | (-,-) |

MPE | SSTA | gam1\gam2\tda1\ tda2\tdb1\tdb2\SSTA | (0.02,0.01)\(0.02,0.01)\ (0.80,0.38)\(1.14,0.54)\ (0.04,0.02)\(0.08,0.04) |

Relative Errors | gam1 | gam2 | tda1 | tda2 | tdb1 | tdb2 |
---|---|---|---|---|---|---|

Initial | −20% | −20% | 20% | 20% | 20% | 20% |

Estimated | 0.03% | −6% | −2.7% | 10.1% | 0.05% | −6.2% |

**Table 6.**Averaged correlation coefficients and RMSEs of the simulated SSTA, ZWA, ULDA, and TeA in two experiments against the corresponding counterparts in the SODA reanalysis dataset.

State Variables | SSTA | ZWA | ULDA | TeA | ||||
---|---|---|---|---|---|---|---|---|

Exps. | onlySE | MPE | onlySE | MPE | onlySE | MPE | onlySE | MPE |

Correlation coefficients | 0.9256 | 0.9594 | 0.8248 | 0.8356 | 0.7172 | 0.8205 | 0.7605 | 0.7893 |

RMSEs | 0.3071 | 0.2818 | 0.2318 | 0.2271 | 6.6496 | 6.5208 | 0.9197 | 0.9092 |

Experiments | Parameter Values (Truncated to Two Decimal Places) |
---|---|

Optimized Pars | gam1 = 0.73, gam2 = 0.70, tda1 = 26.24, tda2 = −37.48, tdb1 = 1.17, tdb2 = 2.81 |

Default Pars | gam1 = 0.75, gam2 = 0.75, tda1 = 28.00, tda2 = −40.00, tdb1 = 1.25, tdb2 = 3.00 |

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## Share and Cite

**MDPI and ACS Style**

Gao, Y.; Tang, Y.; Song, X.; Shen, Z.
Parameter Estimation Based on a Local Ensemble Transform Kalman Filter Applied to El Niño–Southern Oscillation Ensemble Prediction. *Remote Sens.* **2021**, *13*, 3923.
https://doi.org/10.3390/rs13193923

**AMA Style**

Gao Y, Tang Y, Song X, Shen Z.
Parameter Estimation Based on a Local Ensemble Transform Kalman Filter Applied to El Niño–Southern Oscillation Ensemble Prediction. *Remote Sensing*. 2021; 13(19):3923.
https://doi.org/10.3390/rs13193923

**Chicago/Turabian Style**

Gao, Yanqiu, Youmin Tang, Xunshu Song, and Zheqi Shen.
2021. "Parameter Estimation Based on a Local Ensemble Transform Kalman Filter Applied to El Niño–Southern Oscillation Ensemble Prediction" *Remote Sensing* 13, no. 19: 3923.
https://doi.org/10.3390/rs13193923