# A Neural-Network Based MPAS—Shallow Water Model and Its 4D-Var Data Assimilation System

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. MPAS-SW Dynamics

#### 2.2. NN Emulator of MPAS-SW

#### 2.3. The Tangent Linear and Adjoint Models

#### 2.4. A Continuous 4D-Var DA System

## 3. Experiment Design

#### 3.1. A Single Observation Experiment

#### 3.2. Full Vector Observation Experiment

#### 3.3. Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**(

**a**) Spatial distribution of the Spherical Centroidal Voronoi Tessellation (SCVT) mesh at 1000 km with 611 cells globally. (

**b**) The neural network diagram showing the structure of the NN-based MPAS-SW model. The actual number of the neurons for the input and output layers is N = 1833, and N = 3666 for the hidden layer.

**Figure 3.**(

**a**) The spatial distribution of the fields of height (shaded) and wind (vectors) at 00 UTC 1 January 2021. (

**b**,

**c**) The 12-h forecasts made by (

**b**) the NN-based SW model and (

**c**) MPAS-SW model.

**Figure 4.**The differences in height (shaded) and wind (vectors) between the 12-h forecasts by (

**a**) NN-based SW and (

**b**) MPAS-SW with respect to the initial conditions.

**Figure 5.**Variations in the gradient-check results $log\left(\right|\Phi \left(\alpha \right)-1\left|\right)$ as a function of the log of the scaling factor $\alpha $.

**Figure 6.**(

**a**) Variation of the cost function (solid curve) and the norm of the gradient (dashed curve) with respect to the number of iterations when assimilating a single point observation. (

**b**) The analysis increment in height (shaded) and wind (vectors) after assimilating only the height at the location marked with a white cross.

**Figure 7.**(

**a**) Variation of the cost function (solid curve) and the norm of the gradient (dashed curve) with respect to the number of iterations when assimilating the observations of the entire model state. (

**b**) The analysis increment in height (shaded) and wind (vectors).

**Figure 8.**The differences in root mean squared errors (RMSE) between the control and DA experiments with respect to the forecast lead time.

**Figure 9.**Differences in 2-day forecasts (

**a**) between the control experiment and referenced high resolution simulations and (

**b**) between the DA experiment and the referenced high-resolution simulations.

**Figure 10.**Differences in 5-day forecasts (

**a**) between the control experiment and referenced high resolution simulations and (

**b**) between the DA experiment and the referenced high-resolution simulations.

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**MDPI and ACS Style**

Tian, X.; Conibear, L.; Steward, J.
A Neural-Network Based MPAS—Shallow Water Model and Its 4D-Var Data Assimilation System. *Atmosphere* **2023**, *14*, 157.
https://doi.org/10.3390/atmos14010157

**AMA Style**

Tian X, Conibear L, Steward J.
A Neural-Network Based MPAS—Shallow Water Model and Its 4D-Var Data Assimilation System. *Atmosphere*. 2023; 14(1):157.
https://doi.org/10.3390/atmos14010157

**Chicago/Turabian Style**

Tian, Xiaoxu, Luke Conibear, and Jeffrey Steward.
2023. "A Neural-Network Based MPAS—Shallow Water Model and Its 4D-Var Data Assimilation System" *Atmosphere* 14, no. 1: 157.
https://doi.org/10.3390/atmos14010157