# Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology

^{*}

## Abstract

**:**

## Highlights:

- The statistical scaling of annual discharge maxima with the drainage area A of catchments is revisited in the light of rainfall climatology.
- Neglecting the dependence of annual discharge maxima on rainfall characteristics may lead to diverse findings regarding the observed type of scaling.
- The multiplicative fluctuations of discharge maxima relative to their corresponding means closely follow the scaling properties of actual rainfields.
- For spatial scales below approximately 100 km
^{2}, where rainfall deviates from multiflactal scale invariance, a break of scaling is also observed for the multiplicative fluctuations.

## 1. Introduction

_{i}, and ${Q}_{max}^{\left(j\right)}\left({A}_{j}\right)$ to be the annual maximum discharge at an ungauged location j upstream of i that drains area ${A}_{j}{A}_{i}$. In the most general case of statistical scale invariance, referred to as stochastic self-similarity, multiscaling, or multifractality see e.g., [3,4,7,8,9,10,11,14,17,19,20,28,30,32,33] one can obtain the statistical properties of ${Q}_{max}^{\left(j\right)}\left({A}_{j}\right)$ as a function of the statistical properties of ${Q}_{max}^{\left(i\right)}\left({A}_{i}\right)$ through (see e.g., the review in Veneziano and Langousis [32] and references therein):

## 2. Data

^{2}with a mean value of 359 km

^{2}. For the total of 805 stations, the average SAAR value (i.e., SAAR = (SAAR1 + SAAR2)/2), corresponding to the mean annual precipitation in mm for the 60-year period 1941–1990, varies from 558.5 to 2910.5 mm with a mean value of 1059.5 mm. Figure 1 shows the spatial distribution of the average SAAR for the 805 considered catchments across the United Kingdom. SAAR values tend to decrease when moving from the West to the East coast, with lowest values located in the southeastern region of the United Kingdom. The observed rainfall gradient is directly linked to Gulf Stream and the local topography, as moist air masses from the sea move upslope and cool causing precipitation to form see e.g., [44,46].

## 3. Analysis and Results

^{2}, with much smaller slopes, which concurs with the observed break of scaling in spatial rainfall (see above). In addition, for spatial scales larger than 100 km

^{2}, the empirical moment scaling function $K\left(q\right)$ in Equation (4) (see stars in Figure 5), estimated as the negative log-log slopes of the LS fitted lines in Figure 6, remains non-linear and close to that of the standardized discharge maxima ${Q}_{max}^{\prime}$.

^{2}and, also, showed that although significantly influenced by soil infiltration and abstraction mechanisms, the dependence of the CV of flood annual maxima on the area A of basins is theoretically linked to rainfall scaling. In any case, a definite argument on the origins of the observed scaling cannot be reached in the absence of concurrent rainfall and discharge data at high temporal resolution (i.e., at least hourly).

## 4. Conclusions

^{2}where significant deviations of rainfall from multiflactal scale invariance have been observed see e.g., [32,50,61,62,73,85,87], a break of scaling is also observed for the amplification factor ${\gamma}_{max}$ (see Figure 6). This break is not observable when studying the initial moments of ${Q}_{max}$ (or the ratio ${Q}_{max}\u2215A$) with A (see Figure 4), as it is smeared out by the considerable variability of the rainfall characteristics of basins.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Spatial distribution of the average SAAR values for the 805 considered catchments across the United Kingdom.

**Figure 2.**Spatial distribution of the mean value of the standardized discharge maxima ${Q}_{max}^{\prime}={Q}_{max}\u2215A$.

**Figure 3.**(

**a**) Scatterplot (points) and linear least-squares (LS) fit (solid line) of the mean values $E\left[{Q}^{\prime}\right]$ of the standardized discharges ${Q}^{\prime}=Q\u2215A$ for the 805 considered catchments, with respect to their corresponding SAAR values. (

**b**) Same as (a) but for the means $E\left[{Q}_{max}^{\prime}\right]$ of the standardized annual discharge maxima ${Q}_{max}^{\prime}={Q}_{max}/A.$

**Figure 4.**Log-log plots of the empirical moments $E\left[{\left({Q}_{max}^{\prime}\right)}^{q}\right]$ of standardized annual discharge maxima ${Q}_{max}^{\prime}={Q}_{max}\u2215A$, as a function of the catchment area A, for different moment orders $q=0.5:0.5:3$. Black solid lines correspond to least-squares (LS) fits.

**Figure 5.**Empirical moment scaling functions $K\left(q\right)$ for the standardized discharge maxima ${Q}_{max}^{\prime}={Q}_{max}\u2215A$ (circles), and the amplification factor ${\gamma}_{max}={Q}_{max}/E\left[Q\right]$ (stars).

**Figure 6.**Log-log plots of the empirical moments $E\left[{\left({\gamma}_{max}\right)}^{q}\right]$ of the amplification factor ${\gamma}_{max}={Q}_{max}/E\left[Q\right]$, as a function of the catchment area A, for different moment orders $q=0.5:0.5:3$. Black solid lines correspond to least-squares (LS) fits.

**Table 1.**Classification of the selected catchments, according to their size, into 10 approximately equally sized groups.

Group | Catchment Area A (km^{2}) | Number of Catchments | Mean Value (km^{2}) | Median Value (km^{2}) |
---|---|---|---|---|

1 | A < 31.6 | 80 | 18.09 | 19 |

2 | 31.6 ≤ A < 52.3 | 82 | 41.62 | 41.75 |

3 | 52.3 ≤ A < 74.4 | 83 | 63.24 | 62.8 |

4 | 74.4 ≤ A < 109.2 | 81 | 89.99 | 89.9 |

5 | 109.2 ≤ A < 148.1 | 81 | 128.87 | 128.9 |

6 | 148.1 ≤ A < 195.4 | 82 | 170.72 | 170.95 |

7 | 195.4 ≤ A < 272.1 | 79 | 228.73 | 229 |

8 | 272.1 ≤ A < 407.3 | 81 | 335.38 | 334.6 |

9 | 407.3 ≤ A < 900 | 80 | 602.97 | 570.35 |

10 | A ≥ 900 | 76 | 2240.1 | 1490 |

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Perdios, A.; Langousis, A.
Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology. *Water* **2020**, *12*, 610.
https://doi.org/10.3390/w12020610

**AMA Style**

Perdios A, Langousis A.
Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology. *Water*. 2020; 12(2):610.
https://doi.org/10.3390/w12020610

**Chicago/Turabian Style**

Perdios, Anastasios, and Andreas Langousis.
2020. "Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology" *Water* 12, no. 2: 610.
https://doi.org/10.3390/w12020610