# Effect of Logarithmically Transformed IMERG Precipitation Observations in WRF 4D-Var Data Assimilation System

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The 4D-Var Method

#### 2.1.1. Cost Function with Nontransformed Observations

#### 2.1.2. Cost Function with Logarithmically Transformed Observations

#### 2.2. Precipitation Observations

#### 2.3. Model Configurations

#### 2.4. Experiments Design and Evaluation Metrics

## 3. Results

#### 3.1. Evaluation of Precipitation Estimates from Six-Hour-Window and Hourly-Window Experiments

#### 3.2. Evaluation of Precipitation Estimates from Nontransformed and Logarithmically Transformed Experiments

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The three precipitation events studied in this paper. (

**a**) The spatial distribution of total precipitation amount and (

**b**) the time series of hourly domain-mean precipitation during 0000–1800 UTC August 9, 2015. Figure sets (

**c**–

**f**) are as for (

**a**,

**b**), but for precipitation during 1800 UTC August 4–1800 UTC August 5, 2016 and 0600 UTC August 5–0000 UTC 7 August 2017, respectively.

**Figure 2.**The steps of running open loop (OPL) and data assimilation (DA) experiments with hourly and six-hour assimilation windows.

**Figure 3.**Scatter plot of hourly domain-mean observed precipitation and open-loop estimated precipitation (mm/h).

**Figure 4.**Scatter plot of the statistics of hourly domain-mean precipitation (mm/h) from WRFDA experiments against that from the open-loop experiment. The circles with green edges represent mean values.

**Figure 5.**Boxplots of statistical metrics of hourly precipitation estimates estimated from each assimilation experiment and the OPL experiment. The threshold for calculating the POD, FAR, and ETS is 0.001 mm/h, which is the threshold of distinguishing precipitation or no precipitation in the IMERG data. EXP2 and EXP3 represent experiments with nontransformed and logarithmically transformed precipitation with constant observation error, respectively; EXP4 and EXP5 are as for EXP2 and EXP3, but with increasing observation error with precipitation magnitude.

**Figure 6.**Detection scores of hourly precipitation estimated from assimilation experiments and the OPL experiment at different thresholds ranging from 1 mm/h to 30 mm/h. EXP2 and EXP3 represent experiments with nontransformed and logarithmically transformed precipitation with constant observation error, respectively; EXP4 and EXP5 are as for EXP2 and EXP3, but with increasing observation error with precipitation magnitude.

**Figure 7.**(

**a**) Fraction of observations used in each assimilation experiment to the total available observations at each hour. (

**b**) Number of observations used for all the hours at different magnitude classes with 1 mm/h intervals. EXP2 and EXP3 represent experiments with nontransformed and logarithmically transformed precipitation with constant observation error, respectively; EXP4 and EXP5 are as for EXP2 and EXP3, but with increasing observation error with precipitation magnitude.

Experiment Name | Assimilation Window | Transformation of Precipitation | Errors in Regular Space (mm/h) | Errors in Log Space (mm/h) |
---|---|---|---|---|

EXP1 | 6-hour | No transformation | 2 (mm/6 h) | -- |

EXP2 | Hourly | No transformation | 0.3 | 0.3/(y_{i} + 1) |

EXP3 | Hourly | Log transformation | 0.3 | 0.3/(y_{i} + 1) |

EXP4 | Hourly | No transformation | 0.3 × (y_{i} + 1) | 0.3 |

EXP5 | Hourly | Log transformation | 0.3 × (y_{i} + 1) | 0.3 |

Statistical Metrics | Equation | Perfect Value |
---|---|---|

Bias | Bias = X^{e} − X^{r} | 0 |

Mean Absolute Difference (MAD) | MAD = |X^{e} − X^{r}| | 0 |

Correlation Coefficient (CC) | CC = cov(X^{e}, X^{r})/ (std(X^{e})∙std(X^{r})) | 1 |

Probability of Detection (POD) | POD = a/(a + c) | 1 |

False Alarm Ratio (FAR) | FAR = b/(a + b) | 0 |

Equitable Threat Score (ETS) | ETS = (a − e)/(a + b + c − e), where e = (a + b)(a + c)/(a + b + c + d) | 1 |

^{1}X

^{e}and X

^{r}are estimates and reference, respectively. cov(X

^{e}, X

^{r}) is the covariance of X

^{e}and X

^{r}, and std(X) is the standard deviation of X. a represents the number of grids that X

^{r}≥ x

_{0}and X

^{e}≥ x

_{0}; b represents the number of grids that X

^{r}< x

_{0}but X

^{e}≥ x

_{0}; c represents the number of grids that X

^{r}≥ x

_{0}but X

^{e}< x

_{0}; d represents the number of grids that X

^{r}< x

_{0}and X

^{e}< x

_{0}; and x

_{0}is a threshold of precipitation magnitude.

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## Share and Cite

**MDPI and ACS Style**

Zhang, J.; Lin, L.-F.; Bras, R.L.
Effect of Logarithmically Transformed IMERG Precipitation Observations in WRF 4D-Var Data Assimilation System. *Water* **2020**, *12*, 1918.
https://doi.org/10.3390/w12071918

**AMA Style**

Zhang J, Lin L-F, Bras RL.
Effect of Logarithmically Transformed IMERG Precipitation Observations in WRF 4D-Var Data Assimilation System. *Water*. 2020; 12(7):1918.
https://doi.org/10.3390/w12071918

**Chicago/Turabian Style**

Zhang, Jiaying, Liao-Fan Lin, and Rafael L. Bras.
2020. "Effect of Logarithmically Transformed IMERG Precipitation Observations in WRF 4D-Var Data Assimilation System" *Water* 12, no. 7: 1918.
https://doi.org/10.3390/w12071918