# Assessing the Accuracy of 3D-VAR in Supercell Thunderstorm Forecasting: A Regional Background Error Covariance Study

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Data Assimilation and Background Error Theory

#### 2.1. The 3D-VARDA Approach

^{7}degrees of freedom is not possible because the requirement increases the computational cost significantly. To reduce the computational cost, ${\mathrm{J}}_{\mathrm{b}}$ is calculated in terms of control variable vectors, defined via the relation $\mathrm{x}{}^{\prime}=\mathrm{U}\mathrm{v}$, where $\mathrm{x}{}^{\prime}$ denotes the analysis increment, ${\mathrm{x}}^{\prime}=\mathrm{x}-{\mathrm{x}}^{\mathrm{b}}$. Using the incremental formulation [32,34] and the control variable transform, Equation (1) can be rewritten as:

#### 2.2. Background Error Disciplines

_{u}), unbalanced temperature (t

_{u}), pseudo-relative humidity (rh

_{s}), and unbalanced surface pressure (ps

_{u}). CV6 option is similar to CV5, but it has six extra correlation coefficients in the definition of the balanced part of analysis control variables, as well as the moisture control variables is the unbalanced portion of the pseudo-relative humidity (rh

_{s,u}). CV7 option uses a different set of control variables, which are u, v, temperature, pseudo-relative humidity (rh

_{s}), and surface pressure (ps). Table 1 outlines the specific atmospheric CV associated with each configuration option.

## 3. Case Study Overview

## 4. Experimental Setup and Data Sources

#### 4.1. WRF Model Setup

#### 4.2. Geographic Description and Area Details

#### 4.3. Parametrization Schemes in WRF

#### 4.4. Sources and Types of Observations

#### 4.5. Initial Conditions for Model Runs

## 5. Results

#### 5.1. Initial Run

#### 5.2. Background Error Covariance Analysis

_{u}) pattern suggests a growth in the lower levels, while the rh pattern suggests an interaction up to the intermediate levels. However, studying the rest of the eigenvectors, the relation grows more intricate, emphasizing the dynamic influence of temperature on the atmospheric structure.

_{u}highlight a dynamic equilibrium between temperature and other constituents, pivotal for the determination of moisture levels and circulation patterns.

#### 5.3. Background Error Length Scales Analysis

#### 5.4. Data Assimilation Using CV5, CV6, and CV7

_{o}) and background (J

_{b}) components. CV7 stands out with the lowest overall value of J, whereas CV6 stands out with the lowest value of J

_{b}.

_{b,}as well as the calculation of the observation term J

_{o}, based on $\mathrm{J}\left(\mathrm{v}\right)={\mathrm{J}}_{\mathrm{b}}+{\mathrm{J}}_{0}$ (Equation (2)), shows that CV6 has the lowest value of J

_{o}, followed by CV5 and then CV7. Having as criteria the J

_{b}value and not the J value, CV6 exhibited the best performance based on the results. The contribution of Jo in the final J value of CV6 is the biggest compared to CV5 and CV7, and this serves as observations that are well fitted to the analysis field much more efficiently than the other two CVs.

#### 5.5. Runs and Analysis Post Data Assimilation

^{2}, slope, and RMSE values were calculated. In particular, the verification was between the observational data and forecasts for each ORIG, RCV5, RCV6, and RCV7. Table 3 illustrates the results for specific humidity parameters in periods with sufficient data available. The dates presented in Table 3, Table 4 and Table 5 were chosen in order to have sufficient data samples for verification, as there were collection issues of archives where intermediate hours were faulty. The linear regression between the observations and the forecasts over the entire set of observations for the whole domain indicated that CV6 configuration forecasts (RCV6 run) had the best fit, as the higher values of the slope. Table 4 further supports this outcome, showing the RCV6 runs to have the lowest RMSE among most of the runs.

## 6. Conclusions

_{b}, and the produced run, namely RCV6 forecasts, were enriched with the most rain gauge values while notably influencing the weather evolution, having the biggest amount of precipitation over the Chalkidiki region, and the highest score in the forecasts when validated to surface observations of humidity among RCV5, RCV7, and ORIG runs. Such an alignment of CV6 with localized mesoscale processes suggests that the representation of BE, especially those closely adhering to ground observations, can considerably affect the accuracy of NWP and DA mechanisms. This indicates that when our BE aligns closely with observations, our weather forecasting accuracy improves significantly, as shown by the minimization of the J

_{b}term of Equation (2). Hence, CV6 seems to perform better in resolving the necessary scales associated with the extreme and complex convective event, producing more accurate precipitation forecasts, although only METAR observations were taken into account in the DA.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Surface pressure chart analysis of Europe for 09.07.2019 12:00 UTC, (

**b**) Chalkidiki area and Thessaloniki radar maximum dBz for 10.07.2019 at 18:47 UTC.

**Figure 4.**Five eigenvectors for (

**a**) CV5 (ψ, x

_{u}, t

_{u,}and rh variables), (

**b**) CV6 (ψ, x

_{u}, t

_{u,}and rh

_{u}variables), and (

**c**) CV7 (u, v, t

_{u,}and rh variables).

**Figure 5.**Length scale analysis for (

**a**) CV5 (ψ, x

_{u}, t

_{u}and rh variables), (

**b**) CV6 (ψ, x

_{u}, t

_{u}and rh

_{u}variables), and (

**c**) CV7 (t

_{u}variable). Blue lines: ψ, red: x

_{u}, magenta: rh and green: t

_{u}.

**Figure 6.**Cost function convergence analysis. A comparative plot showcasing the cost function J values across different iterations for each of the CVs. The figure illustrates the efficiency of the minimization process within the DA for CV5, CV6, and CV7, highlighting the relative convergence patterns and the optimization dynamics of each CV.

**Figure 8.**

**The**24 h accumulated precipitation during 10.07.2019 06:00 UTC to 11.07.2019 06:00 UTC covering Greece (

**a**) as estimated by the IMERG dataset, (

**b**) forecasted by the original model (ORIG) without DA, (

**c**) forecast post DA in the CV5 BE DA scenario, (

**d**) forecast post DA in the CV6 BE DA scenario, and (

**e**) forecast post DA in the CV7 BE DA setup.

CV Option | Control Variables |
---|---|

CV3 | ψ, χ_{u}, t_{u}, q, ps_{u} |

CV5 | ψ, χ_{u}, t_{u}, rh_{s}, ps_{u} |

CV6 | ψ, χ_{u}, t_{u}, rh_{s,u}, ps_{u} |

CV7 | u, v, t, rh_{s}, ps |

CV Option | Final Value of J | Final Value of J_{o} | Final Value of J_{b} |
---|---|---|---|

CV5 | 2379.51 | 2146.56 | 232.95 |

CV6 | 2462.67 | 2279.39 | 183.29 |

CV7 | 1841.01 | 1418.01 | 423.01 |

Slope | ||||
---|---|---|---|---|

ORIG | RCV5 | RCV6 | RCV7 | |

10.07.2019 09:00 UTC | 0.574596407 | 0.579820502 | 0.591004545 | 0.580909283 |

10.07.2019 12:00 UTC | 0.528259375 | 0.530443518 | 0.536777983 | 0.530303622 |

10.07.2019 15:00 UTC | 0.541017141 | 0.545911233 | 0.547654013 | 0.547349451 |

11.07.2019 00:00 UTC | 0.763604456 | 0.764116932 | 0.770858861 | 0.765469658 |

11.07.2019 18:00 UTC | 0.617027986 | 0.618871979 | 0.61865578 | 0.61854543 |

11.07.2019 21:00 UTC | 0.722005614 | 0.72469613 | 0.72496216 | 0.723679649 |

12.07.2019 00:00 UTC | 0.755929304 | 0.757677061 | 0.756957918 | 0.758342565 |

$\mathbf{RMSE}({10}^{-3}\mathbf{g}\times \mathbf{k}{\mathbf{g}}^{-1})$ | ||||
---|---|---|---|---|

ORIG | RCV5 | RCV6 | RCV7 | |

10.07.2019 09:00 UTC | 2.939403097 | 2.906229779 | 2.790401193 | 2.901062785 |

10.07.2019 12:00 UTC | 2.99862999 | 2.978911749 | 2.931584172 | 2.976548541 |

10.07.2019 15:00 UTC | 2.415220807 | 2.371456968 | 2.389589067 | 2.374596434 |

11.07.2019 00:00 UTC | 2.396224849 | 2.38678574 | 2.358999465 | 2.389230651 |

11.07.2019 18:00 UTC | 3.151521169 | 3.144182149 | 3.142446493 | 3.13495034 |

11.07.2019 21:00 UTC | 2.623882552 | 2.612348846 | 2.619757037 | 2.613839418 |

12.07.2019 00:00 UTC | 2.34083961 | 2.348286227 | 2.341523816 | 2.339594189 |

24 h Total Precipitation | ||||
---|---|---|---|---|

CV | ORIG | RCV5 | RCV6 | RCV7 |

Slope | 0.2326 | 0.2316 | 0.2488 | 0.2337 |

RMSE (mm) | 9.0845 | 9.0700 | 9.1989 | 9.4180 |

R-square | 0.3093 | 0.3044 | 0.3469 | 0.2917 |

24 h Total Precipitation | ||||
---|---|---|---|---|

CV | ORIG | RCV5 | RCV6 | RCV7 |

Slope | 0.2822 | 0.2629 | 0.2988 | 0.277 |

RMSE (mm) | 15.5242 | 16.0893 | 15.3334 | 16.5668 |

R-square | 0.1100 | 0.0088 | 0.2337 | 0.0598 |

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**MDPI and ACS Style**

Samos, I.; Louka, P.; Flocas, H.
Assessing the Accuracy of 3D-VAR in Supercell Thunderstorm Forecasting: A Regional Background Error Covariance Study. *Atmosphere* **2023**, *14*, 1611.
https://doi.org/10.3390/atmos14111611

**AMA Style**

Samos I, Louka P, Flocas H.
Assessing the Accuracy of 3D-VAR in Supercell Thunderstorm Forecasting: A Regional Background Error Covariance Study. *Atmosphere*. 2023; 14(11):1611.
https://doi.org/10.3390/atmos14111611

**Chicago/Turabian Style**

Samos, Ioannis, Petroula Louka, and Helena Flocas.
2023. "Assessing the Accuracy of 3D-VAR in Supercell Thunderstorm Forecasting: A Regional Background Error Covariance Study" *Atmosphere* 14, no. 11: 1611.
https://doi.org/10.3390/atmos14111611