# Deterministic and Stochastic Generation of Evaporation Data for Long-Term Mine Pit Lake Water Balance Modelling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{lake}(mm day

^{−1}) is the evaporation rate, U (ms

^{−1}) is wind speed 1 m above the lake surface, e*

_{lake}(kPa) is the saturation vapour pressure at the lake surface temperature, e

_{a}(kPa) is the vapour pressure of the air 1 m above the lake surface, and f(U) is a wind function identified empirically for the lake in question. Identifying the wind function requires in situ measurements of lake evaporation, wind speed, water temperature and humidity, as well as an equilibrium surface temperature model to estimate e*

_{lake}. Where such measurements are not available, a generalised wind function that uses data from a nearby weather station and which accounts for the area of the lake may be sufficient [5]. Comparable generalised aerodynamic equations have been used in other pit lake applications [11]. Using this type of model provides an opportunity to reconstruct lake evaporation over the period for which nearby weather station data are available. A primary limitation of this wind function approach and other aerodynamic and combination equations is lack of explicit representation of heat storage, which may limit accuracy for deep lakes [12]. While there are many evaluations of deterministic evaporation models for lakes generally [9,10,11,12,13], there are very few [5,6] that critically evaluate the accuracy and applicability of models of mine pit lake evaporation, a gap that is addressed by this paper.

- Assess deterministic models for reconstructing historical daily evaporation data set for a mine pit lake in semi-arid Australia.
- Develop stochastic models of evaporation and rainfall at daily, monthly and annual time steps, and assess their accuracy including the rainfall-evaporation interdependence.
- Discuss options for improving the accuracy of open water evaporation data used in long-term projections of pit lake water balances.

## 2. Materials and Methods

#### 2.1. Case Study Description

#### 2.2. Historical Data Reconstruction

_{a}, and the function f(U) was then optimised by least squares regression of the daily E

_{lake}/(e*

_{lake}-e) values against the measured average daily wind speed. e*

_{lake}was calculated using the measured water temperature. The 62 less reliable measurements of evaporation were not used in the fitting, but the model was then employed to in-fill these missing data.

^{2}) is the lake surface area, introduced to allow for water vapour entrainment effects. In this reconstruction, e*

_{lake}was calculated using a water temperature equilibrium model [20,21,22]. The equilibrium temperature is the temperature towards which the water temperature is driven by the net heat exchange. The equilibrium temperature model allows an estimate of the temperature of a water body to be derived using standard meteorological measurements. Modelled water temperatures are dependent on the equilibrium temperature, T

_{w}(°C) and a time constant, τ (days). The time constant reflects the time which would be required to reach equilibrium between air and water temperatures if conditions remained stable. Using this method, T

_{w}is calculated as:

_{w}

_{0}(°C) is the water temperature at the previous time step. The full methodology for deriving T

_{e}and τ is given in [5].

#### 2.3. Stochastic Models

_{t}is a 3 × 1 matrix of standardised climate data (annual averages of rainfall, daily maximum air temperature and evaporation) for year t, A and B are 3 × 3 coefficient matrices that preserve the correlations between the three variables, and ε

_{t}is a random number with zero mean and unit variance presenting the random variability of the three variables. A = M

_{1}M

_{0}

^{−1}and B = M

_{1}M

_{0}

^{−1}M

_{1}

^{T}, where M

_{0}and M

_{1}are 3 × 3 matrices that describe the lag zero and lag one cross-correlations between the three modelled variables [14].

_{h}that has similar annual climate variable values to year Y, and monthly values in December of year Y

_{h}-1 similar to those modelled for year Y-1 (which are known because year Y-1 has already been modelled). To calculate similarity, these two criteria are weighted as explained in the Stochastic Climate Library guide [14] (page 38).

## 3. Results

#### 3.1. Observed In Situ Weather Data

^{−1}and 7.4 ms

^{−1}. Total rainfall for the study period was 310 mm and the highest rainfall in one day was 42 mm on 6 March 2018. On average, the surface temperature of the lake was approximately 0.8 °C warmer than that of the pan.

#### 3.2. Identified Wind Function

_{lake}and wind speed obtained from the in situ weather station are shown in Figure 2 along with the optimised wind function (Figure 2F):

_{lake}/(e*

_{lake}-e

_{a}) is explained by the function, with a root mean square error of 0.44 mm day

^{−1}kPa

^{−1}. Residual analysis did not identify any other relations with climate variables or patterns over time except the slight overestimation E

_{lake}/(e*

_{lake}-e

_{a}) at low wind speeds evident in Figure 2F. This model was used to infill the missing 62 days of evaporation data.

#### 3.3. Constructing Long-Term Evaporation Data

^{2}value is 0.87 and the measured evaporation is underestimated on average by 6.5%. Figure 3C shows that applying the measured pan evaporation from the nearby Mount Isa weather station (location in Figure 1) gives a much higher error variance, while Figure 3D shows that applying a commonly used pan coefficient of 0.7 to that weather station data leads not only to the high variance but to an average bias (underestimate) of 45%.

#### 3.4. Stochastic Model Results

_{0}and M

_{1}matrices obtained from running the annual model are in Table 1. Air temperature is not considered further here but is included in Table 1 for completeness. M

_{0}indicates that rainfall has a strong negative correlation with evaporation. M

_{1}shows that rainfall has a weak positive lag-1 correlation with itself and evaporation has a moderate positive lag-1 correlation with itself. The lag-1 cross-correlations reflect the nature of these other correlations.

## 4. Summary and Discussion

^{2}value was 0.005 and for rainfall the p-value was 0.46. Hence, there is some evidence of a potential linear relation between evaporation residual and ENSO; however, if the relation exists, it is weak and it does not explain the magnitude of the residuals observed in Figure 5. Further research could explore the effect of other synoptic climate variables and teleconnection controls that are known to influence weather in this region [26]. Such an extension to the model may also permit the stochastic model to be used for downscaling climate model outputs, or alternative stochastic downscaling frameworks could be employed [27,28].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Infilling of Historical Wind Speed Data

## Appendix B. Performance of Daily Stochastic Model Conditional on whether the Day Is Wet or Dry

**Figure A1.**Stochastic model performance for evaporation on dry days: (

**A**) Evaporation mean; (

**B**) Evaporation standard deviation; (

**C**) Evaporation skewness coefficient.

**Figure A2.**Stochastic model performance for evaporation and rainfall on wet days (rainfall > 10 mm/day): (

**A**) Evaporation mean; (

**B**) Rainfall mean; (

**C**) Evaporation standard deviation; (

**D**) Rainfall standard deviation; (

**E**) Evaporation skewness coefficient; (

**F**) Rainfall skewness coefficient; (

**G**) Evaporation–rainfall correlation coefficient.

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**Figure 2.**Data measured at the pit lake floating weather station: (

**A**) average daily temperature; (

**B**) wind speed; (

**C**) relative humidity; (

**D**) rainfall; (

**E**) evaporation; (

**F**) E

_{lake}/(e*

_{lake}-e

_{a}) against wind speed.

**Figure 3.**Scatter plots of: (

**A**) surface water temperature modelled using the equilibrium model vs. measured surface water temperature; (

**B**) evaporation using the long-term model vs. evaporation measured using the floating evaporation pan; (

**C**) evaporation measured at the Mount Isa weather station vs. evaporation measured using the floating evaporation pan; (

**D**) applying a pan coefficient of 0.7 to evaporation measured at the Mount Isa weather station vs. evaporation measured using the floating evaporation pan.

**Figure 4.**Comparison of reconstructed evaporation and measured rainfall with results of annual stochastic simulations The box plots show 10th, 25th, 50th, 75th and 90th percentiles of model realisations with outliers shown as dots.

**Figure 5.**Comparison of reconstructed evaporation and measured rainfall statistics with results of monthly stochastic simulations (Month 1 = January): (

**A**) evaporation mean; (

**B**) rainfall mean; (

**C**) evaporation standard deviation; (

**D**) rainfall standard deviation; (

**E**) evaporation skewness coefficient; (

**F**) rainfall skewness coefficient; (

**G**) proportion of months that are dry; (

**H**) evaporation–rainfall correlation coefficient.

**Figure 6.**Comparison of reconstructed evaporation and measured rainfall statistics with results of daily stochastic simulations for each month (Month 1 = January): (

**A**) evaporation standard deviation; (

**B**) rainfall standard deviation; (

**C**) evaporation skewness coefficient; (

**D**) rainfall skewness coefficient; (

**E**) proportion of days that are dry; (

**F**) evaporation–rainfall correlation coefficient. Plots of mean values not shown because they are identical to those in Figure 5.

**Figure 7.**Relationship between evaporation and rainfall: (

**A**) a representative sample realisation from the daily stochastic model (correlation coefficient = −0.31); (

**B**) reconstructed evaporation and measured rainfall data (correlation coefficient = −0.70).

Matrix M_{0} | |||
---|---|---|---|

Variable | Rainfall | Evaporation | Max Temperature |

Rainfall | 1.000 | −0.771 | −0.679 |

Evaporation | −0.771 | 1.000 | 0.778 |

Max Temperature | −0.679 | 0.778 | 1.000 |

Matrix M_{1} | |||

Variable | Rainfall | Evaporation | Max Temperature |

Rainfall | 0.079 | −0.033 | −0.069 |

Evaporation | −0.403 | 0.311 | 0.277 |

Max Temperature | −0.228 | 0.215 | 0.383 |

**Table 2.**Annual stochastic model results: comparison of statistics of modelled and measured rainfall and evaporation.

Simulated Percentiles using 500 Replicates | |||||||
---|---|---|---|---|---|---|---|

Value Observed during 1974–2017 | 10% | 25% | 50% | 75% | 90% | ||

Rainfall | Mean (mm) | 467 | 403 | 441 | 465 | 490 | 534 |

Standard deviation (mm) | 212 | 161 | 191 | 209 | 228 | 268 | |

Skewness coefficient | 0.89 | 0.07 | 0.48 | 0.72 | 1.01 | 1.81 | |

Evaporation | Mean | 3056 | 2961 | 3027 | 3061 | 3091 | 3146 |

Standard deviation | 232 | 179 | 210 | 226 | 245 | 278 | |

Skewness coefficient | −0.141 | −0.758 | −0.331 | −0.086 | 0.119 | 0.532 | |

Rainfall-evaporation | Cross-correlation | −0.681 | −0.807 | −0.718 | −0.665 | −0.611 | −0.498 |

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**MDPI and ACS Style**

Mandaran, K.; McIntyre, N.; McJannet, D.
Deterministic and Stochastic Generation of Evaporation Data for Long-Term Mine Pit Lake Water Balance Modelling. *Water* **2022**, *14*, 4123.
https://doi.org/10.3390/w14244123

**AMA Style**

Mandaran K, McIntyre N, McJannet D.
Deterministic and Stochastic Generation of Evaporation Data for Long-Term Mine Pit Lake Water Balance Modelling. *Water*. 2022; 14(24):4123.
https://doi.org/10.3390/w14244123

**Chicago/Turabian Style**

Mandaran, Kristian, Neil McIntyre, and David McJannet.
2022. "Deterministic and Stochastic Generation of Evaporation Data for Long-Term Mine Pit Lake Water Balance Modelling" *Water* 14, no. 24: 4123.
https://doi.org/10.3390/w14244123