# Short-Term Forecasting of Water Yield from Forested Catchments after Bushfire: A Case Study from Southeast Australia

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study and Data Sources

^{2}(Figure 1). The catchment is covered by native eucalypt forests and soils of the area are derived from highly weathered Ordovician sediments and are acidic and duplex in structure [30]. The underlying rock types are granite, limestone and shale, and the topography is mountainous (steep with rocky outcrops). Summers are characterized as warm and often hot, with dry periods of between six and eight weeks. In winter (July), mean daily maximum and minimum temperatures in sheltered locations (mid-slope) are 14 and −1 °C respectively, while in summer (January) the respective temperatures are 24 °C and 10 °C. Mean annual rainfall is approximately 1150 mm. Annual evaporation and seepage losses from the catchment are estimated to be 630 mm and stream discharge typically peaks between August and September and reaches a minimum between March and May [31] (Figure 2).

**Figure 2.**Annual hydrograph at the inflow of the Corin dam provided by the Bureau of Meteorology of Australia (

**a**); and hyetograph at of the Mount Ginini automatic weather station (

**b**).

#### 2.2. Standardization and Goodness of Fit Criteria

_{stan}is the amount of variable y after standardization; and Max(y) is the maximum value of y within the time series. Following goodness of fit criteria were used for comparing the results of different data driven methods:

- Root Mean Square Error (RMSE)

_{i}and for

_{i}are observed and forecasted value of the dependent variable at time step i, respectively; and n is the total number of time steps.

- Volume Error (VE)

- Correlation (Corr)

- Nash-Sutcliffe Efficiency (NSE)

#### 2.3. Data-Driven Methods

- Non-Linear Multivariate Regression (NLMR)

_{obs}(i) and Q

_{for}(i) are observed and forecasted water yields at time i; x

_{1}to x

_{n}are n predictor variables; c

_{j}and b

_{j}are coefficients of predictors and c

_{n+1}is the constant of the model. Here we used “Lingo optimization package” [40] to optimize the model for coefficients and constant values.

- K-Nearest Neighbor (K-NN)

- (1)
- setting a matrix with m columns (number of predictors) and n + 1 rows (length of time series).
- (2)
- last row of the above-mentioned matrix is assumed as a vector of predictors at current time (x
_{j,t}j = 1:m). - (3)
- remaining rows are assumed as a matrix of predictors at historical time series (x
_{j,t-i}j = 1:m i = 1:n). - (4)
- vector Q is defined with n rows of independent variable values from t − n to t − 1.
- (5)
- using a distance function, distances between x
_{j,t}and x_{j,(t-i)}are calculated.$$Dist(t-i)=f({w}_{j},{x}_{j,(t-i)},{x}_{jt})$$_{j}are weights of predictor variables at the distance function. We chose the Euclidean function as the distance function with equal weights to predictor variables. - (6)
- distance vector (Dist) is sorted from minimum to maximum (SDist) and vector Q is assorted based on SDist.
- (7)
- best number of neighbors (k) are specified based on a variety of methods. Here we have used the empirical equation $K=\sqrt{n}$ in which n is the length of the time series which is used as historical data for calibration and validation stages [41].
- (8)
- a discrete Kernel function is used to give weights to k neighbors [42].$$S(e)=\frac{1/SDist(t-e)}{{\displaystyle \sum _{e=1}^{k}1/SDist(t-e)}}\text{}\text{}e=\mathrm{1...}k$$
- (9)
- forecast value at current time is calculated as:$$Forecast=S\times {Q}^{T}$$

- Nonlinear Autoregressive with External Input Based Artificial Neural Networks (NARX-ANN)

- Symbolic Regression (SR)

## 3. Results and Discussion

^{7}generations were 0.77% and 98.6%, respectively. Ultimately based on the optimum point, the proposed mathematical solution can be presented as (all used variables in Equation (12) are standardized by Equation (1), and therefore are dimensionless):

**Figure 3.**Mathematical solution accuracy vs. model complexity in the SR method. Y-axis shows mean absolute error of the forecast vs. observations for the entire data.

Variable | Sensitivity | % Positive | Positive Magnitude | % Negative | Negative Magnitude |
---|---|---|---|---|---|

NDVI_{i-1} | 1.31 | 31% | 3.81 | 69% | 0.21 |

WY_{i-1} | 0.70 | 55% | 1.16 | 45% | 0.14 |

Preci_{i-1} | 0.08 | 99% | 0.08 | 0% | 0 |

Methods | Calibration-Validation (85% of Data) | Verification (15% of Data) | Entire Data | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Corr | RMSE | VE % | NSE | Corr | RMSE | VE % | NSE | Corr | RMSE | VE % | NSE | |

K-NN | 0.64 | 0.16 | 3.42 | −0.10 | 0.74 | 0.21 | 3.57 | −3.54 | 0.63 | 0.17 | 3.44 | −0.33 |

NLMR | 0.79 | 0.10 | 1.51 | 0.60 | 0.78 | 0.20 | 1.50 | 0.40 | 0.76 | 0.12 | 1.51 | 0.55 |

NARX-ANN | 0.91 | 0.07 | 1.20 | 0.80 | 0.90 | 0.11 | 1.50 | 0.80 | 0.90 | 0.08 | 1.24 | 0.80 |

SR | 0.82 | 0.09 | 1.09 | 0.67 | 0.80 | 0.16 | 1.20 | 0.63 | 0.82 | 0.10 | 1.16 | 0.67 |

_{1}+ cx

_{2}+ dx

_{3}and found its performance very poor (Corr = 0.45) and the technique inappropriate compared to the data-driven models.

**Figure 4.**Final results of standardized forecasts vs. standardized observations using the K-NN method.

**Figure 5.**Final results of standardized forecasts vs. standardized observations using the NLMR method.

**Figure 6.**Final results of standardized forecasts vs. standardized observations using the NARX-ANN method.

**Figure 7.**Final results of standardized forecasts vs. standardized observations using the SR method.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Gharun, M.; Azmi, M.; Adams, M.A.
Short-Term Forecasting of Water Yield from Forested Catchments after Bushfire: A Case Study from Southeast Australia. *Water* **2015**, *7*, 599-614.
https://doi.org/10.3390/w7020599

**AMA Style**

Gharun M, Azmi M, Adams MA.
Short-Term Forecasting of Water Yield from Forested Catchments after Bushfire: A Case Study from Southeast Australia. *Water*. 2015; 7(2):599-614.
https://doi.org/10.3390/w7020599

**Chicago/Turabian Style**

Gharun, Mana, Mohammad Azmi, and Mark A. Adams.
2015. "Short-Term Forecasting of Water Yield from Forested Catchments after Bushfire: A Case Study from Southeast Australia" *Water* 7, no. 2: 599-614.
https://doi.org/10.3390/w7020599