# Reconciling Chord Length Distributions and Area Distributions for Fields of Fractal Cumulus Clouds

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## Abstract

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## 1. Introduction

- a higher chance of missing small clouds all together biases towards large chord lengths;
- a high likelihood of hitting clouds off-center biases towards small chord lengths; and
- irregularities in cloud shapes, such as gaps and the fractal dimension of the cloud edge [15], bias toward small chord lengths.

## 2. Methods

## 3. Results and Discussion

#### 3.1. Sensitivity and Convergence

#### 3.2. Chord Length Distribution Conditional on Linear Size

#### 3.3. Linear Cloud Size Distribution Conditional on Chord Length

#### 3.4. Robustness for Small Sample Sizes

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of potential biases in the chord length distribution. (

**a**) A linear observation through a cloud field. (

**b**) A naïve interpretation of the cloud field based on those chord lengths.

**Figure 2.**Linear cloud size vs. perimeter for different liquid water path thresholds. The slope of the best fit line signifies a fractal dimension of 1.32.

**Figure 3.**Box–whisker plot of the LES-retrieved fractional cloud cover and cloud size as a function of sample size: (

**a**,

**b**) cloud cover and (

**c**,

**d**) weighted average cloud size; (

**a**,

**c**) for area measurements and (

**b**,

**d**) based on chord length observations. Based on the 8th hour of RICO. Fifty percent of the measurements falls within the box, and 95% of the measurements within the whiskers. The line within the box gives the median value.

**Figure 4.**Chord length distribution through a single cloud from the 8th hour of the RICO simulation. (

**a**) The projected footprint of the cloud, with transects for illustrative purposes. (

**b**) The resulting chord length distribution.

**Figure 5.**(

**a**) Chord length distribution through all clouds with the same linear size in the 8th hour of the RICO simulation. (

**b**) Modal chord length as a function of linear cloud size.

**Figure 6.**Cloud size distribution conditional on the chord length for the 8th hour of the RICO simulation. (

**a**) $L=100$ m. (

**b**) $L=200$ m. (

**c**) $L=400$ m. (

**d**) $L=600$ m. (

**e**) $L=1000$ m. (

**f**) $L=1400$ m.

**Figure 7.**Fitting parameters of the gamma distribution, as a function of chord length. The vertical lines depict the standard deviation between different cases. (

**a**) The mean $\mu $. (

**b**) The shape parameter $\beta $.

**Figure 8.**Reconstructed linear cloud size distributions for different cases, with the linear cloud size (black), the chord length distribution (red), and the adjusted chord length distribution (blue). (

**a**) BOMEX 8th hour. (

**b**–

**d**) RICO hr 8, 24, and 48. (

**e**–

**f**) ARM 12LT and 16LT. (

**g**–

**j**) LASSO 11, 13, 14, and 16 LT.

**Figure 9.**Reconstructed linear cloud size distributions for different transect lengths, with the linear cloud size (black), the mean adjusted chord length distribution (blue), and the adjusted chord length distribution of individual transects (light blue) for transects of (

**a**) 6.25 km, (

**b**) 12.5 km, (

**c**) 25 km, (

**d**) 50 km, (

**e**) 100 km, and (

**f**) 200 km.

**Table 1.**The area-weighted average cloud size for different instances, calculated using the actual cloud area, the chord length, and the adjusted chord length, together with the relative error for the latter two methods. For RICO and BOMEX, the simulation time is shown. For ARM and LASSO, the local time is shown.

Linear | Chord Length | pct Change | Adjusted Chord Length | pct Change | |
---|---|---|---|---|---|

RICO h8 | 545 | 292 | −47 | 478 | −12 |

RICO h24 | 741 | 354 | −52 | 524 | −29 |

RICO h40 | 716 | 506 | −29 | 643 | −10 |

RICO h48 | 790 | 554 | −30 | 697 | −12 |

RICO h56 | 656 | 512 | −22 | 649 | −1 |

BOMEX h4 | 435 | 210 | −52 | 470 | 8 |

BOMEX h5 | 467 | 227 | −51 | 474 | 1 |

BOMEX h6 | 446 | 216 | −52 | 470 | 5 |

BOMEX h7 | 445 | 214 | −52 | 469 | 5 |

BOMEX h8 | 452 | 227 | −50 | 472 | 4 |

ARM h11 | 383 | 249 | −35 | 466 | 22 |

ARM h12 | 713 | 343 | −52 | 518 | −27 |

ARM h13 | 905 | 376 | −58 | 540 | −40 |

ARM h14 | 909 | 367 | −60 | 533 | −41 |

ARM h15 | 888 | 359 | −60 | 526 | −41 |

ARM h16 | 826 | 358 | −57 | 525 | −37 |

ARM h17 | 823 | 354 | −57 | 521 | −37 |

ARM h18 | 744 | 326 | −56 | 502 | −32 |

LASSO h12.0 | 392 | 219 | −44 | 511 | 30 |

LASSO h13.0 | 560 | 330 | −41 | 517 | −8 |

LASSO h14.0 | 743 | 345 | −54 | 515 | −31 |

LASSO h15.0 | 639 | 324 | −49 | 504 | −21 |

LASSO h16.0 | 504 | 306 | −39 | 494 | −2 |

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**MDPI and ACS Style**

Barron, N.R.; Ryan, S.D.; Heus, T.
Reconciling Chord Length Distributions and Area Distributions for Fields of Fractal Cumulus Clouds. *Atmosphere* **2020**, *11*, 824.
https://doi.org/10.3390/atmos11080824

**AMA Style**

Barron NR, Ryan SD, Heus T.
Reconciling Chord Length Distributions and Area Distributions for Fields of Fractal Cumulus Clouds. *Atmosphere*. 2020; 11(8):824.
https://doi.org/10.3390/atmos11080824

**Chicago/Turabian Style**

Barron, Nicholas R., Shawn D. Ryan, and Thijs Heus.
2020. "Reconciling Chord Length Distributions and Area Distributions for Fields of Fractal Cumulus Clouds" *Atmosphere* 11, no. 8: 824.
https://doi.org/10.3390/atmos11080824