# Application of Asymmetrical Statistical Distributions for 1D Simulation of Solute Transport in Streams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

^{−3}), D

_{x}is the dispersion coefficient in the longitudinal direction (m

^{2}s

^{−1}), v

_{x}is water velocity in the longitudinal direction (m.s

^{−1}), M

_{s}is a function representing the sources of solute (kg.m

^{−3}s

^{−1}), x is the spatial coordinate distance (m).

_{s}is the concentration of the substance in the storage zone (kg.m

^{−3}), m is the mass exchange coefficient between the main stream and the storage zone (s

^{−1}), and n is the ratio between the storage zone and the main stream cross-sectional area (–). If m or n becomes zero, the (2) reduces to (1).

^{2}) and f is the unknown function (“similarity solution “) [19]. The unknown function f can be determined in two different ways:

- Based on experiment derive a curve fit to real data.
- By analytically solving the (4).

#### 2.1. Analytical Solution

**f**could be interpreted as having a form of Gaussian normal distribution with parameters of normal distribution (e.g., standard deviation σ, etc.). In fact, this curve does not have the exact shape of a Gaussian normal distribution, rather it has an asymmetric form, caused by substitution of the standard deviation σ (which is constant in Gaussian distribution) by the time-dependent term $\sqrt{2{D}_{x}t}$.

**f**from (4). Instead a different statistical distribution with asymmetric shape should be used.

#### 2.2. The GumbelD Approximation Method

**f**from the Equation (4), the formula of the Gumbel distribution is proposed. The general equation of the Gumbel distribution is [31]

_{x,G}is the dispersion coefficient in the longitudinal direction (m

^{2}s

^{−1}) used in the proposed model. To be dimensionally consistent, z is a dimensionless parameter and σ has the dimension of length (m). To get the results in concentration units (kg m

^{−3}) the term M/A must be included into (6). Substituting parameters from (8) and (9) into (6), the one-dimensional analytical solution has the form:

#### 2.3. Three-Parametric LogNormD Approximation Method

**f**from the (4), the expression of the three-parametric log-normal distribution. The general equation of this distribution is [32]

_{max}, in which the maximum (peak) concentration occurs is defined as

_{max}is known from field measurements, Equation (12) yields

_{0}(start time of nonzero concentrations) can be expressed as a proportion of the peak (maximal concentration) time, i.e., t

_{0}= k t

_{max}, where 0 < k < 1.

#### 2.4. Generalized Extreme Value Distribution (GEVD) Method

**f**from (4) is the generalized extreme value (GEV) distribution. The general equation of the GEV distribution is [33]

_{max}is defined as

## 3. Methods

#### 3.1. Literature Data Use

^{−3}), c

_{m,t}is the measured value (concentration) at the time t (kg.m

^{−3}), c

_{a,t}is the approximated value in the time t (kg.m

^{−3}), t

_{1}is the measurement start time (sec) and t

_{2}is the measurement end time (sec), NRMSE (Eq. 24) is the normalized root mean square error (–), and c

_{max}and c

_{min}are the maximal and minimal concentration in each particular experiment (kg.m

^{−3}).

^{3}.s

^{−1}(17–241000 cfs—cubic feet per second) and from 0.07 up to 1.86 m.s

^{−-1}, respectively. All the data are listed in Table 1.

#### 3.2. Field Experiments Data Use

^{3}.s

^{−1}. The hydraulic roughness was n = 0.035 (Manning coefficient) and the water level slope was found to be approximately 1.5‰. The stream in the considered reach is prismatic with a width of 5.5 m and a water depth in the range from 0.4 to 0.6 m. The range of the determined longitudinal dispersion coefficient was from 0.5 to 2.5 m

^{2}.s

^{−1}. The experiments were performed in high summer (July 2012), when submerged vegetation was present to a large extent.

^{3}.s

^{−1}, 4 to 5.5 m, 0.4 to 0.8 m, and 0.245 to 0.282 m.s

^{−1}, respectively. The range of the determined longitudinal dispersion coefficient was from 0.63 to 0.98 m

^{2}.s

^{−1}. Field experiments were performed in spring (March 2014) when the presence of submerged vegetation was very low.

^{3}s

^{−1}. The water level slope, specified by levelling measurements, was about 0.45‰. The stream shape in the examined stream section can be considered prismatic, the width was around 5 m, the average depth was 0.88 m with a maximum value of about 1 m. The dispersion coefficient was 0.95 m

^{2}.s

^{−1}. The experiment was performed in high summer (August 2016), with up to approximately one meter of emerged vegetation on the banks, as well as submerged vegetation in the central part of the channel. This vegetation on the stream banks and bed created dead zones for the flow. A picture of the vegetation and hydraulic conditions in this stream is shown on Figure 4.

_{mix-vert}= 100*h [20], where h is the water depth. In the three Slovakian rivers, water depth h was always < 1 m (on average, 0.5, 0.6, and 0.88 m). So, L

_{mix-vert}< 100 (from 50 to 88 m). This means in the field measurement the assumption of vertical mixing holds.

_{mix-transv}is more difficult because no established theory exists to predict transverse mixing rate [36]. This rate is related to several boundary conditions of the mixing process (riverbank/riverbed irregularities, vegetation, meandering, bends, islands, confluences). The vegetation is expected to affect Manning coefficient [37], which is related to the friction factor, the shear velocity, and, ultimately, to the transverse mixing rate.

_{mix-transv}= c

_{1}*(W²/*h), where W is channel width and c

_{1}is a numerical constant related to the rate of the process of transversal mixing [38]. In the three Slovakian rivers, c

_{1}could be in the order of 20–40, so the L

_{mix-transv}should be in the order of 150–200 m, which is quite a lot shorter than the minimum distance of the field experiments (Table 2). Hence, the assumption of complete mixing in the cross-section should be valid.

## 4. Results

#### 4.1. Literature Data

_{max}− c

_{min}) [39]. Since the minimal concentration in all experiments was set up to be zero, the NRMSE = RMSE/c

_{max}. For all experiments, the NRMSE was determined to evaluate the fit between the measured and approximated values. Results are listed in Table 3.

#### 4.2. Field Experiments

## 5. Discussion

#### 5.1. Literature Data

^{2}> 0.99), whereas all the correlations of the LogNormD method were quite weak (r

^{2}was about 0.2). This means that the dispersion coefficients for the GumbelD and GEVD methods were very close to our understanding of the role of the dispersion coefficient, as identified from the standard Gauss–based analytical solution of the ADE (Equation (5)). Even the values of the dispersion coefficient between the GaussD and GEVD methods were quite similar, probably due to a linear dependency between them.

_{0}(see (15), whereas the dispersion coefficient is very difficult to estimate due its relatively small range (average 0.413, std. dev. 0.1). On the other hand, the parameter t

_{0}can be very easily measured in tracer experiments as it is the time of the first nonzero concentration measured in the stream.

_{x}, D

_{x,G}, and D

_{x,GEV}(Gauss, Gumbel, and GEV methods) on stream distance from injection point, stream velocity, and stream width (see Table 8). The parameter ξ, used in the GEV method (see (22)) has only a weak correlation to the stream hydraulic parameters, but graph analysis revealed that higher values of this parameter correspond with the long “tail” of the time-concentration curve. Hence, it is possible to assume that this parameter characterizes the extent of dead zones in the stream. Very weak (almost none) correlations were found between the dispersion parameters obtained using the LogNorm approximation method, which can be regarded as a serious disadvantage of this method (difficulty with parameters prediction).

#### 5.2. Field Experiments

## 6. Application of the Proposed Approximation Method to Inverse Tasks

## 7. Conclusions

- The Gumbel method is very easy to compute, but is less accurate because this method does not allow justification of the time concentration curve shape expressing the extent of dead zones along the examining stream reach;
- The LogNormD approximation method is more precise than Gumbel and Gauss methods, but its parameters are not directly determinable and there is considerable uncertainty in their determination (there is a simple estimation of the t
_{0}parameter, but it has difficult predictable dispersion and location parameters (µ, D_{x,LN}), which have no direct physical meaning); - The GEVD approximation method is the most precise method, the dispersion coefficient (D
_{x,GEV}) has similar meaning and way of determination as the dispersion coefficients D_{x}and D_{x,G}. It also has relatively predictable shape coefficient ξ (higher values correspond with longer “tails” of the time concentration curves, i.e., it expresses the extent of the dead zones). The correlation between the dispersion coefficient used in this method (D_{x,GEV}) and the dispersion coefficient D_{x}, used in Equation (5) can be a subject of further research, with the aim of determination of the D_{x,GEV}value, as well as the shape coefficient (ξ) value.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Graphical evaluation of approximation methods—case 28, Red River, March 13, 1972, site 1, distance 5 miles (8.045 km), see [34].

**Figure 2.**Graphical evaluation of approximation methods—case 15, Salt Creek, September 6, 1972, site 6, distance 37.62 miles (60.5 km), see [34].

**Figure 3.**Graphical evaluation of approximation methods—case 20, Comite River, November 4, 1968, site 5, distance 49 miles (78.8 km), see [34].

**Figure 4.**Emerged and submerged vegetation in the Malina stream, West Slovakia, near the village of Zohor (GPS 48.334830, 16.967463).

**Figure 5.**Graphical evaluation of approximation methods—Malina Stream, August 3, 2016, site 6, distance 1.435 km.

**Figure 6.**Graphical evaluation of approximation methods—Šúrsky Stream, March 20, 2014, experiment 12-II, distance 0.406 km.

**Table 1.**Main characteristics of the considered rivers/streams [34].

Nr. | Case Nr. * | River/Stream Name | Date | Min. Distance | Max. Distance | Min. Discharge | Max. Discharge | Min. Velocity | Max. Velocity |
---|---|---|---|---|---|---|---|---|---|

(km) | (km) | (m^{3}.s^{−}^{1}) | (m^{3}.s^{−}^{1}) | (m.s^{−}^{1}) | (m.s^{−}^{1}) | ||||

1 | 1 | Anteniam Creek | 27 May 1969 | 2.57 | 29.61 | 1.16 | 1.78 | 0.12 | 0.22 |

2 | 2 | Anteniam Creek | 24 March 1970 | 2.57 | 66.69 | 5.10 | 12.18 | 0.35 | 0.51 |

3 | 7 | Monocacy River | 7 June 1968 | 10.30 | 34.27 | 15.15 | 18.55 | 0.37 | 0.38 |

4 | 9 | Monocacy River | 25 September 1968 | 7.48 | 33.79 | 3.06 | 3.20 | 0.08 | 0.11 |

5 | 10 | Conococheague Creek | 6 May 1969 | 4.42 | 19.87 | 6.82 | 6.94 | 0.35 | 0.41 |

6 | 12 | Conococheague Creek | 30 April 1970 | 4.42 | 33.87 | 29.45 | 30.58 | 0.62 | 0.87 |

7 | 13 | Chattahoochee River | 20 April 1971 | 16.01 | 76.59 | 107.60 | 179.53 | 0.73 | 0.97 |

8 | 14 | Chattahoochee River | 4 May 1971 | 10.49 | 104.60 | 140.17 | 140.17 | 0.75 | 1.09 |

9 | 15 | Salt Creek River | 6 September 1972 | 9.25 | 51.81 | 2.46 | 4.08 | 0.29 | 0.34 |

10 | 19 | Bayou Anacoco River | 15 June 1969 | 11.42 | 37.97 | 2.01 | 2.71 | 0.15 | 0.16 |

11 | 20 | Comite River | 4 November 1968 | 6.76 | 78.84 | 0.76 | 1.13 | 0.08 | 0.15 |

12 | 21 | Bayou Bartholomew | 25 June 1971 | 3.22 | 117.46 | 4.11 | 8.10 | 0.12 | 0.15 |

13 | 22 | Amite River | 21 October 1968 | 9.98 | 148.03 | 5.66 | 9.91 | 0.18 | 0.33 |

14 | 24 | Tangipahoa River | 24 April 1969 | 8.21 | 93.97 | 5.78 | 18.69 | 0.22 | 0.34 |

15 | 25 | Tangipahoa River | 15 September 1969 | 8.21 | 93.97 | 3.45 | 10.85 | 0.16 | 0.27 |

16 | 26 | Red River | 7 April 1971 | 5.74 | 193.08 | 230.22 | 249.47 | 0.57 | 0.63 |

17 | 27 | Red River | 14 April 1971 | 8.05 | 159.29 | 139.60 | 184.91 | 0.46 | 0.64 |

18 | 28 | Red River | 13 March 1972 | 8.05 | 159.29 | 187.46 | 251.74 | 0.43 | 0.52 |

19 | 29 | Red River | 12 June 1972 | 12.07 | 199.52 | 107.60 | 166.50 | 0.45 | 0.51 |

20 | 30 | Sabine River | 9 September 1969 | 7.88 | 79.81 | 110.72 | 130.26 | 0.40 | 0.80 |

21 | 31 | Sabine River | 4 February 1972 | 17.22 | 209.65 | 311.49 | 433.25 | 0.57 | 0.84 |

22 | 32 | Sabine River | 16 April 1972 | 22.20 | 85.92 | 0.48 | 1.33 | 0.09 | 0.10 |

23 | 33 | Sabine River | 16 April 1972 | 17.06 | 121.00 | 1.02 | 6.91 | 0.07 | 0.11 |

24 | 34 | Sabine River | 16 April 1972 | 21.08 | 113.43 | 6.17 | 7.08 | 0.12 | 0.15 |

25 | 35 | Mississippi River | 15 September 1965 | 35.40 | 204.34 | 3398.02 | 6796.04 | 0.62 | 0.81 |

26 | 36 | Mississippi River | 11 March 1968 | 54.71 | 294.45 | 2633.47 | 2973.27 | 0.95 | 1.10 |

27 | 37 | Mississippi River | 7 August 1968 | 54.71 | 294.45 | 6824.36 | 6824.36 | 1.36 | 1.48 |

28 | 38 | Wind/Righorn River | 21 March 1971 | 9.17 | 181.33 | 54.93 | 68.81 | 0.83 | 0.95 |

29 | 39 | Wind/Righorn River | 29 June 1971 | 9.17 | 181.33 | 215.21 | 254.85 | 1.51 | 1.86 |

Min. value | 2.57 | 19.87 | 0.48 | 1.13 | 0.07 | 0.10 | |||

Max. value | 54.71 | 294.45 | 6824.36 | 6824.36 | 1.51 | 1.86 |

Nr. | Stream/ Channel Name | Date | Min. Distance | Max. Distance | Min. Discharge | Max. Discharge | Min. Velocity | Max. Velocity |
---|---|---|---|---|---|---|---|---|

(km) | (km) | (m^{3}.s^{−1}) | (m^{3}.s^{−1}) | (m.s^{−1}) | (m.s^{−1}) | |||

1 | Malá Nitra | 3–4 July 2012 | 0.785 | 1.340 | 0.230 | 0.235 | 0.131 | 0.163 |

2 | Šúrsky stream | 19–20 March 2014 | 0.397 | 0.506 | 0.380 | 0.430 | 0.245 | 0.282 |

3 | Malina | 3 August 2016 | 1.435 | 1.435 | 0.408 | 0.408 | 0.076 | 0.076 |

Statistical Parameter | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

% | % | % | % | |

NRMSE—Average | 6.40 | 3.40 | 2.60 | 2.44 |

NRMSE—Min | 1.62 | 0.71 | 0.69 | 0.70 |

NRMSE—Max | 15.55 | 11.00 | 6.80 | 6.42 |

Std. dev. | 2.49 | 1.78 | 1.28 | 1.15 |

**Table 4.**Relative comparison of approximation methods’ accuracy (relative to Gaussian distribution (GaussD) = 100%).

Statistical Parameter | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

% | % | % | % | |

Relative NRMSE—Average | 100 | 53.1 | 40.6 | 38.2 |

Relative NRMSE—Min | 100 | 40.6 | 42.5 | 42.8 |

Relative NRMSE—Max | 100 | 70.7 | 43.7 | 41.3 |

Std. Dev. | 100 | 67.6 | 51.3 | 46.2 |

Limit | GumbelD | LogNormD | GEVD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|---|---|

- | - | - | % | % | % | |

0% | 0 | 0 | 0 | 0% | 0% | 0% |

10% | 0 | 1 | 0 | 0.0% | 0.7% | 0.0% |

20% | 2 | 15 | 13 | 1.4% | 10.6% | 9.2% |

40% | 38 | 63 | 76 | 27.0% | 44.7% | 53.9% |

50% | 70 | 97 | 109 | 49.6% | 68.8% | 77.3% |

60% | 98 | 123 | 125 | 69.5% | 87.2% | 88.7% |

80% | 120 | 137 | 137 | 85.1% | 97.2% | 97.2% |

100% | 126 | 140 | 138 | 89.4% | 99.3% | 97.9% |

Method | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

GaussD | - | 0.9988 | 0.2131 | 0.9983 |

GumbelD | 0.9988 | - | 0.1897 | 0.9967 |

LogNormD | 0.2131 | 0.1897 | - | 0.2225 |

GEVD | 0.9983 | 0.9967 | 0.2225 | - |

NRMSE of Method | Q–Discharge | X–Distance | V–Velocity | W–River Width | D–Mean Depth | S_{0}–Slope | U^{*}–Shear Velocity | A–Unitless Disp. Coef. |
---|---|---|---|---|---|---|---|---|

GaussD | 0.277 | 0.173 | 0.088 | 0.212 | 0.049 | –0.280 | –0.158 | 0.401 |

GumbelD | 0.232 | 0.144 | –0.065 | 0.126 | 0.038 | –0.201 | –0.230 | 0.378 |

LogNormD | 0.248 | 0.272 | –0.028 | 0.182 | 0.096 | –0.202 | –0.154 | 0.271 |

GEVD | 0.250 | 0.309 | –0.023 | 0.241 | 0.136 | –0.228 | –0.166 | 0.285 |

^{−1})

**Table 8.**Correlation (r

_{xy}) between the approximation method parameters and hydraulic parameters of examined streams.

Parameter | Equation | Q—Discharge | X—Discharge | V—Velocity | W—River Width | D—Mean Depth | S_{0}—Slope | U^{*}—Shear Velocity | A—Unitless Disp. Coef. |
---|---|---|---|---|---|---|---|---|---|

D_{x} | (5) | 0.248 | 0.416 | 0.600 | 0.593 | 0.357 | –0.121 | 0.224 | 0.269 |

D_{x,G} | (10) | 0.231 | 0.416 | 0.604 | 0.609 | 0.368 | –0.109 | 0.240 | 0.239 |

D_{x.LN} | (16) | 0.227 | –0.096 | 0.061 | 0.065 | –0.047 | –0.189 | –0.105 | 0.348 |

k=t_{0}/t_{tmax} | (17) | 0.118 | 0.215 | –0.016 | 0.060 | –0.057 | –0.382 | –0.342 | 0.166 |

D_{x,GEV} | (22) | 0.232 | 0.409 | 0.594 | 0.568 | 0.341 | –0.121 | 0.212 | 0.305 |

ξ | (22) | 0.261 | –0.019 | 0.077 | 0.083 | –0.030 | –0.214 | –0.146 | 0.371 |

Dispersion Parameter | Equation | Average | Min. Value | Max. Value | Std. Deviation |
---|---|---|---|---|---|

D_{x} | (5) | 92.27 | 0.97 | 3353.66 | 299.88 |

D_{x,G} | (10) | 151.66 | 1.65 | 5733.00 | 507.40 |

D_{x.LN} | (16) | 0.413 | 0.100 | 1.140 | 0.181 |

k = t_{0}/t_{tmax} | (17) | 0.835 | 0.134 | 0.975 | 0.102 |

D_{x,GEV} | (22) | 163.48 | 1.68 | 6314.14 | 560.83 |

ξ | (22) | 0.030 | –0.354 | 0.860 | 0.182 |

Statistical Parameter | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

% | % | % | % | |

NRMSE—Average | 9.68 | 4.81 | 3.52 | 3.01 |

NRMSE—Min | 5.32 | 1.72 | 1.39 | 1.08 |

NRMSE—Max | 15.70 | 11.59 | 10.74 | 11.00 |

Std. Dev. | 3.19 | 2.69 | 2.50 | 2.68 |

Statistical Parameter | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

% | % | % | % | |

Relative NRMSE—Average | 100 | 49.7 | 36.4 | 31.1 |

Relative NRMSE—Min | 100 | 32.2 | 26.2 | 20.3 |

Relative NRMSE—Max | 100 | 73.8 | 68.4 | 70.1 |

Std. Dev. | 100 | 84.5 | 78.4 | 83.9 |

Limit | GumbelD | LogNormD | GEVD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|---|---|

- | - | - | % | % | % | |

0% | 16 | 16 | 16 | 100.0% | 100.0% | 100.0% |

10% | 14 | 10 | 5 | 87.5% | 62.5% | 31.3% |

20% | 9 | 1 | 1 | 56.3% | 6.3% | 6.3% |

40% | 3 | 1 | 1 | 18.8% | 6.3% | 6.3% |

50% | 2 | 0 | 0 | 12.5% | 0.0% | 0.0% |

60% | 0 | 0 | 0 | 0.0% | 0.0% | 0.0% |

80% | 0 | 0 | 0 | 0.0% | 0.0% | 0.0% |

100% | 0 | 0 | 0 | 0.0% | 0.0% | 0.0% |

Method | GaussD | GumbelD | LogNormD | GEVD |
---|---|---|---|---|

GaussD | - | 0.9965 | 0.2675 | 0.9993 |

GumbelD | 0.9965 | - | 0.2288 | 0.9981 |

LogNormD | 0.2675 | 0.2288 | - | 0.2687 |

GEVD | 0.9993 | 0.9981 | 0.2687 | - |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sokáč, M.; Velísková, Y.; Gualtieri, C.
Application of Asymmetrical Statistical Distributions for 1D Simulation of Solute Transport in Streams. *Water* **2019**, *11*, 2145.
https://doi.org/10.3390/w11102145

**AMA Style**

Sokáč M, Velísková Y, Gualtieri C.
Application of Asymmetrical Statistical Distributions for 1D Simulation of Solute Transport in Streams. *Water*. 2019; 11(10):2145.
https://doi.org/10.3390/w11102145

**Chicago/Turabian Style**

Sokáč, Marek, Yvetta Velísková, and Carlo Gualtieri.
2019. "Application of Asymmetrical Statistical Distributions for 1D Simulation of Solute Transport in Streams" *Water* 11, no. 10: 2145.
https://doi.org/10.3390/w11102145