# A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes

^{*}

## Abstract

**:**

## 1. Introduction

^{10}data values from over 2 × 10

^{5}hydroclimatic global-scale stations on a daily, hourly, and subhourly resolution, with some dating even back to 1800. Specifically, we transform to hourly resolution the subhourly timeseries of air temperature, dew point, humidity, wind-speed, and sea level pressure (with around 15,000 active stations each). Similarly, we transform to hourly resolution over 10,000 timeseries of streamflow and 5,000 timeseries of precipitation located in the USA, and we merge them with over 600 USA streamflow and over 10

^{5}worldwide precipitation timeseries of daily resolution. It is noted that after quality control, only the 10% of records are finally selected for the analysis (Figure 1 and Table 1).

## 2. Methodology

#### 2.1. Dependence Structure Metrics

#### 2.2. Marginal Structure Metrics

#### 2.3. Global-Scale Data Extraction and Processing

^{6}records of longitudinal wind velocity along the flow direction, all measured by X-wire probes placed downstream of the grid and in different positions, and with a sampling temporal resolution of 25 μs [136]. To shift from the spatial to the temporal domain [137], we apply a standardization to all time series by subtracting their mean and dividing by their standard deviation (see more information and results in [76]). In this manner, we may directly estimate the expected marginal and temporal dependence structure by combining the estimations from all the time series, approximately as if the same experiment was performed multiple times at the same position. For the buoyancy behavior, we discuss the results from several studies of Papanicolaou and List [138,139] and Dimitriadis et al. [18,140,141], where more than 10 time series of horizontal and vertical positively buoyant thermal jets of temperature concentration, recorded with the laser-induced-fluorescence technique, and with a 30 ms resolution, various nozzle diameters, discharges, initial and ambient temperature, and of more than 10

^{4}sample length each, were analyzed.

## 3. Results

## 4. Discussion

## 5. Conclusions

- (1)
- A hierarchy related to the hydrological cycle was identified with the dew point, temperature, relative humidity, solar radiation, and sea level pressure all exhibiting a lower skewness over kurtosis absolute ratio than the turbulent processes, wind speed, and ocean waves, and with a stronger long-term persistence (LTP) behavior in the dependence structure (H > 0.75), followed by streamflow and precipitation, both of which exhibit a smaller skewness–kurtosis absolute ratio and a weaker LTP behavior (H ≤ 0.75).
- (2)
- All the examined processes can be adequately simulated by the truncated mixed-PBF distribution, adjusting for probability dry and lower (or upper) truncation, in terms of the first four moments, and ranging from (truncated) nearly Gaussian to Pareto-type tails.
- (3)
- As the sample size increases, different records of the same process from several locations converge to a smaller area of the nondimensionalized statistics (skewness–kurtosis), indicating a common marginal behavior.
- (4)
- All the examined hydrological-cycle processes exhibit a similar dependence structure that extends from the fractal behavior with roughness (M < 0.5) located at the small-intermittent scales to the LTP behavior at large scales (H > 0.5), while both indicate large uncertainty and high climatic variability.
- (5)
- Finally, since the above empirical findings are consistent with previous studies and can be justified by the principle of maximum entropy, they allow for a uniting stochastic view of the hydrological-cycle processes under the Hurst–Kolmogorov (HK) dynamics in terms of both the marginal and dependence structures.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Locations of the selected stations for each hydrological-cycle process: (

**a**) near-surface air-temperature; (

**b**) relative humidity and dew-point; (

**c**) precipitation; (

**d**) streamflow; (

**e**) wind; and (

**f**) sea-level pressure.

**Figure 2.**Near-surface air temperature: (

**a**) hourly sample (Boston, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 3.**Near-surface dew point: (

**a**) hourly sample (Boston, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 4.**Near-surface relative humidity: (

**a**) hourly sample (Boston, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 5.**Sea level pressure: (

**a**) hourly sample (Boston, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 6.**Wind speed: (

**a**) hourly sample (Boston, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 7.**Streamflow: (

**a**) hourly sample (Potomac River, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 8.**Precipitation: (

**a**) hourly sample (Potomac River, USA); (

**b**) climacogram and climacospectrum (mean, and 5% and 95% quantiles); (

**c**) mean vs. standard deviation; (

**d**) C-skewness vs. C-kurtosis.

**Figure 9.**L-skewness vs. L-kurtosis, and K-skewness vs. K-kurtosis estimated through the hyper-central K-moments, for the key hydrological-cycle and the grid-turbulence processes.

**Figure 10.**L-skewness vs. L-kurtosis (modified, i.e., 4/5 × λ

_{4}/λ

_{2}+ 6/5), estimated through the hyper-central K-moments, for the key hydrological-cycle and the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-Burr-Feller distribution for different probabilities of zero values.

**Figure 11.**K-skewness vs. K-kurtosis, estimated through the hyper-central K-moments, for the key hydrological-cycle and the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-Burr-Feller distribution for probabilities of zero values at 25% and 75%. The mean values of the K-skewness and K-kurtosis for each process are depicted by the square markers with the x-symbol inside.

**Figure 12.**The mean climacogram for the key hydrological-cycle processes and the grid-turbulence process accustomed for illustration to a 25 μs-scale rather than to the hourly-scale. Dashed and continuous lines at streamflow and precipitation correspond to the hourly and daily stations.

**Figure 13.**The mean climacospectrum for the key hydrological-cycle processes and the grid-turbulence process accustomed for illustration to a 25 μs-scale rather than to the hourly scale. Dashed and continuous lines in streamflow and precipitation correspond to the hourly and daily stations.

**Table 1.**Mean Time Resolution of Each Source, Total Number of Selected Stations and Total Records for Each Hydrological-Cycle Process, and from the Massive Set of Databases. The Humidity Time Series Are Extracted by the Combining the Temperature and Dew-Point Datasets.

Near-Surface Temperature | Dew Point | Humidity | Sea Level Pressure | Wind Speed | Precipitation | Streamflow | |
---|---|---|---|---|---|---|---|

Temporal resolution | Hourly | hourly | hourly | hourly | hourly | hourly/daily | hourly/daily |

Total number of stations/time series | 6613 | 5978 | 4025 | 4245 | 6503 | 93,904 | 1815 |

Total number of data values (×10^{6}) | 907.1 | 730.0 | 540.2 | 364.9 | 781.7 | 938.7 | 13.5 |

Time period | 1938–today | 1938–today | 1940–today | 1939–today | 1939–today | 1778–today | 1900–today |

**Table 2.**Summary Statistics of the Mean Values of the C-, L-, and K-Moments (and Their Standard Deviation in Parentheses) for Time Series Lengths of N ≥ 60 Years (1st Row), 30 < N < 60 (2nd Row) and N ≤ 30 (3rd Row).

Near-Surface Temperature | Relative Humidity | Dew Point | Sea Level Pressure | Wind Speed | Streamflow | Precipitation | |
---|---|---|---|---|---|---|---|

Mean | 14.6 (9.3) | 0.68 (0.1) | 8.3 (8.1) | 1013.9 (3.3) | 3.7 (1.2) | 1498.7 * | 2.3 (1.5) |

12.6 (3.6) | 0.72 (0.2) | 9.0 (2.3) | 1013.9 (187) | 3.51 (0.9) | 9.5 (1.5) | 2.5 (1.9) | |

15.3 (3.1) | 0.71 (0.1) | 6.3 (1.9) | 1014.1 (158) | 3.53 (0.8) | 7.6 (0.2) | 2.8 (2.0) | |

Standard deviation | 8.2 (3.2) | 0.2 (0.04) | 8.0 (3.2) | 7.1 (2.9) | 2.4 (0.5) | 1007.0 * | 7.2 (4.0) |

7.3 (2.1) | 0.2 (0.04) | 6.6 (1.8) | 7.4 (1.6) | 2.5 (0.6) | 16.3 (2.2) | 7.4 (4.8) | |

8.8 (1.9) | 0.2 (0.04) | 8.0 (1.6) | 8.0 (1.3) | 2.4 (0.5) | 17.9 (0.5) | 7.8 (5.0) | |

C-skewness | −0.2 (0.3) | −0.3 (0.5) | −0.6 (0.4) | −0.1 (0.3) | 0.9 (0.5) | 2.3 * | 7.7 (3.8) |

−0.2 (0.2) | −0.4 (0.2) | −0.8 (0.2) | −0.4 (0.2) | 2.1 (0.5) | 8.5 (0.7) | 6.6 (3.7) | |

−0.2 (0.1) | −0.4 (0.1) | −0.5 (0.1) | −0.2 (0.2) | 1.1 (0.4) | 9.0 (0.2) | 5.5 (3.1) | |

C-kurtosis | 3.3 (0.6) | 3.3 (0.7) | 4.0 (1.5) | 4.0 (2.7) | 5.9 (3.3) | 7.5 * | 136 (218) |

6.7 (3.4) | 3.9 (0.9) | 10.2 (3.4) | 20.8 (8.0) | 30.2 (10.8) | 160.7 (17.6) | 93 (115) | |

5.2 (3.0) | 3.6 (0.7) | 4.6 (1.5) | 6.5 (6.1) | 9.6 (7.6) | 160.8 (3.2) | 53 (50) | |

L-skewness | −0.04 (0.05) | −0.05 (0.09) | −0.1 (0.06) | −0.02 (0.04) | 0.1 (0.07) | 0.4 * | 0.7 (0.1) |

−0.03 (0.01) | −0.07 (0.03) | −0.09 (0.02) | −0.04 (0.01) | 0.2 (0.04) | 0.05 (0.04) | 0.7 (0.3) | |

−0.04 (0.01) | −0.08 (0.02) | −0.09 (0.02) | −0.03 (0.01) | 0.1 (0.03) | 0.6 (0.01) | 0.7 (0.3) | |

L-kurtosis (modified) | 1.3 (0.01) | 1.3 (0.02) | 1.3 (0.02) | 1.3 (0.01) | 1.3 (0.02) | 1.4 * | 1.6 (0.1) |

1.3 (0.3) | 1.3 (0.3) | 1.3 (0.3) | 1.3 (0.2) | 1.3 (0.3) | 1.5 (0.1) | 1.6 (0.7) | |

1.3 (0.3) | 1.3 (0.2) | 1.3 (0.2) | 1.3 (0.2) | 1.3 (0.3) | 1.6 (0.02) | 1.6 (0.8) | |

K-skewness | −0.1 (0.2) | −0.1 (0.3) | −0.3 (0.2) | −0.07 (0.1) | 0.4 (0.2) | 1.5 * | 1.7 (0.1) |

−0.1 (0.05) | −0.2 (0.1) | −0.3 (0.07) | −0.1 (0.04) | 0.6 (0.1) | 1.6 (0.1) | 1.7 (0.7) | |

−0.1 (0.04) | −0.2 (0.1) | −0.3 (0.06) | −0.1 (0.04) | 0.5 (0.1) | 1.6 (0.02) | 1.7 (0.8) | |

K-kurtosis | 2.1 (0.05) | 2.1 (0.07) | 2.1 (0.07) | 2.1 (0.02) | 2.1 (0.05) | 2.1 * | 2.7 (0.1) |

2.1 (0.5) | 2.1 (0.5) | 2.1 (0.5) | 2.2 (0.4) | 2.2 (0.5) | 2.7 (0.2) | 2.6 (1.1) | |

2.1 (0.4) | 2.1 (0.4) | 2.1 (0.4) | 2.1 (0.3) | 2.1 (0.4) | 2.7 (0.04) | 2.6 (1.2) |

**Table 3.**Summary Statistics of the Scale, Fractal and Hurst Parameters of the Second-Order Dependence Structure Adjusted for Bias Based on the Climacogram Estimation, with the 5% and 95% Quantiles in Parentheses, and for Each Key Hydrological-Cycle Process of Hourly Resolution.

q (h) | Fractal Parameter (M) | LTP Parameter (H) | |
---|---|---|---|

Near-surface temperature | 135.1 (9.2–323.1) | 0.16 (0.01–0.22) | 0.81 (0.61–0.82) |

Relative humidity | 17.4 (5.6–57.3) | 0.23 (0.2–0.27) | 0.83 (0.62–0.85) |

Dew point | 120.3 (16.4–213.2) | 0.23 (0.15–0.46) | 0.77 (0.58–0.79) |

Sea level pressure | 36.5 (10.0–67.2) | 0.36 (0.25–0.55) | 0.7 (0.53–0.77) |

Wind speed | 9.1 (0.1–25.9) | 0.15 (0.07–0.3) | 0.85 (0.69–0.86) |

Streamflow | 96.5 (16.8–533.1) | 0.43 (0.2–0.46) | 0.78 (0.67–0.86) |

Precipitation | 2.1 (0.1–3.0) | 0.25 (0.18–0.67) | 0.61 (0.52–0.69) |

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**MDPI and ACS Style**

Dimitriadis, P.; Koutsoyiannis, D.; Iliopoulou, T.; Papanicolaou, P.
A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes. *Hydrology* **2021**, *8*, 59.
https://doi.org/10.3390/hydrology8020059

**AMA Style**

Dimitriadis P, Koutsoyiannis D, Iliopoulou T, Papanicolaou P.
A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes. *Hydrology*. 2021; 8(2):59.
https://doi.org/10.3390/hydrology8020059

**Chicago/Turabian Style**

Dimitriadis, Panayiotis, Demetris Koutsoyiannis, Theano Iliopoulou, and Panos Papanicolaou.
2021. "A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes" *Hydrology* 8, no. 2: 59.
https://doi.org/10.3390/hydrology8020059