# Entropy Parameter M in Modeling a Flow Duration Curve

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Derivation of FDC

_{m}is the mean discharge. Entropy maximizing is done using the method of Lagrange multipliers:

#### 2.2. Study Area

## 3. Results and Discussion

#### 3.1. Flow Duration Curve Estimation

^{2}) between the observed and estimated FDCs was 0.969, which showed a good agreement, as shown in Figure 3 and Figure 4.

^{2}values of the predicted and observed FDCs were 0.979, 0.969, and 0.960, respectively. Figure 10, Figure 11 and Figure 12 show that 95% intervals covered most of the observed data. The same was done for other stations in the basin.

^{2}was not always good. Figure 13 showed a good fit for the relationship with the ratio of $\overline{Q}$ and ${Q}_{max}$. When $\overline{Q}$/${Q}_{max}$ ≥ 0.10, R

^{2}≥ 0.90. Further investigation could focus on making adjustments for better FDC prediction.

#### 3.2. Time Variability of M

_{1}times ${Q}_{max}$, as expressed by Equation (6), which relates it to ${Q}_{max}$, ${Q}_{min}$, and $\overline{Q}$. Though Equation (6) is slightly complicated, it can be simplified by setting ${Q}_{min}$ equal to zero, which can usually be assumed to be near zero (it is true at most of the stations in the Brazos River basin). Then we found that M had an inverse relation with the ratio of $\overline{Q}$ and ${Q}_{max}$, as shown in figures plotting M and the ratio (Figure 16 and Figure 17)

#### 3.3. Spatial Variability

#### 3.4. Test for Ungauged Stations

^{2}for both sides. Using the calculated M value led to a mean R

^{2}= 0.91, which ranged from 0.70 to 0.99, and simulated M led to mean R

^{2}= 0.89 which ranged from 0.68 to 0.95 which had a 2.20% difference with the calculated one. At last, we applied Equation (12) to all the stations in the basin and got simulated M for all the stations. The mean R

^{2}= 0.86 for the basin ranged from 0.58 to 0.93, while the calculated M from the records led to an R

^{2}= 0.88 and ranged from 0.61 to 0.95, which showed a mean difference between the results from the calculated and simulated M of 2.32%. Those test results indicated that the function can be applied to other ungauged stations.

## 4. Conclusions

^{2}+ 0.388x + 1.567, which can be used in other basins. For most of the years, the average M yielded a good agreement between predicted and observed FDCs, where the mean R

^{2}was 0.92. Some years did not have good fit, especially in large discharge parts of the FDC; the reason why this occurred should be studied further. The procedure of applying the entropy parameter M for modeling the FDC can be extended to other basins. Further studies such as the adaptation to other basins, and improvement for the goodness of fit should be investigated.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 12.**Estimation of the FDC for year 1994 using average M, a, and b values of station 08093100.

**Figure 14.**Locations of reservoirs and stations (middle part of the basin, the scalar applies to the basin panel).

**Figure 15.**(

**a**) M dynamics at station 08093100; (

**b**) M dynamics at station 08099500; and (

**c**) M dynamics at station 08093360.

**Figure 16.**Correlation between M values and the ratio of $\overline{Q}$ and ${Q}_{max}$ at station 08089000.

**Figure 17.**Powered relationship between M value and ratio of $\overline{Q}$ and ${Q}_{max}$ at station 08089000.

Water Year | Year | Q_{max} | Q_{min} | LI Q_{max} | LI Q_{min} | UI Q_{max} | UI Q_{min} | a | b | M | R^{2} |
---|---|---|---|---|---|---|---|---|---|---|---|

1.3 | 2003 | 121.26 | 0.21 | 44.89 | 0.08 | 302.58 | 0.54 | 1.02 | 0.89 | 9.88 | 0.979 |

1.4 | 2009 | 169.09 | 0.3 | 68.16 | 0.12 | 395.75 | 0.7 | 0.969 | |||

1.8 | 1994 | 257.14 | 0.46 | 114.31 | 0.2 | 558.27 | 0.99 | 0.96 |

^{3}/s.

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**MDPI and ACS Style**

Zhang, Y.; Singh, V.P.; Byrd, A.R.
Entropy Parameter *M* in Modeling a Flow Duration Curve. *Entropy* **2017**, *19*, 654.
https://doi.org/10.3390/e19120654

**AMA Style**

Zhang Y, Singh VP, Byrd AR.
Entropy Parameter *M* in Modeling a Flow Duration Curve. *Entropy*. 2017; 19(12):654.
https://doi.org/10.3390/e19120654

**Chicago/Turabian Style**

Zhang, Yu, Vijay P. Singh, and Aaron R. Byrd.
2017. "Entropy Parameter *M* in Modeling a Flow Duration Curve" *Entropy* 19, no. 12: 654.
https://doi.org/10.3390/e19120654