# Assessment of Short Term Rainfall and Stream Flows in South Australia

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}values (${R}_{adj}^{2}$), minimum sigma square (σ

^{2}), and a minimum Akaike Information Criterion (AIC). The best performance in the response model is lag rainfall, which indicates at least one day and up to 7 days (past) difference in rainfall, including offset cross products of lag rainfall. With the inclusion of antecedent stream flow as an input with one day time lag, the result shows a significant improvement of the ${R}_{adj}^{2}$ values from 0.18, 0.26 and 0.14 to 0.35, 0.42 and 0.21 at Broughton River, Torrance River and Wakefield River, respectively. A benchmark comparison was made with an Artificial Neural Network analysis. The optimization strategy involved adopting a minimum mean absolute error (MAE).

## 1. Introduction

## 2. Data Collection and Preparation

^{2}. Wakefield River (WR) is an ephemeral river near Rhynie, with a catchment area of approximately 1913 km

^{2}.

Stations name | ID | Location | Elevation | Variables | Data period | % of Missing | ||
---|---|---|---|---|---|---|---|---|

Latitude | Longitude | Start | End | |||||

Broughton River at Mooroola | A5070503 | –33.53 | 138.51 | 196 m | Rainfall | Jun. 1989 | Dec. 2011 | 0.1 |

Stream flow | Jun. 1972 | Dec. 2011 | 0.7 | |||||

Torrance at Mount Pleasant | A5040512 | –34.78 | 139.02 | 414.7 m | Rainfall | Jun. 1989 | Dec. 2011 | 0.6 |

Stream flow | May 1973 | Dec. 2011 | 0.1 | |||||

Wakefield river near Rhyine | A5060500 | –34.13 | 138.63 | 202 m | Rainfall | Sep. 1985 | Dec. 2011 | 0.9 |

Stream flow | Jun. 1971 | Dec. 2011 | 0.2 |

^{n}where n is an integer and n ≥ 0, for assessing the relationship between daily rainfall and stream flow during the period 1990–2012. We observe the discrete sequence of time series {y

_{t}} where {y

_{t}} is an integer ranging in length. We extract multi-level information of observed rainfall and stream flow series in three catchments in South Australia using the Haar wavelet decomposition. We split {y

_{t}} into 10 sub-time series of length power two i.e., 2

^{n}, where n is the level of the time series, starting from 0. We also investigate the correlation between rainfall and stream flow patterns for each sub-series from levels 0 to 8.

## 3. Statistical Analysis

#### 3.1. Assessing the Relationship between Rainfall and Stream Flow

_{x}) to the mean (µ

_{x}). Table 2 shows the degree of variation in rainfall and stream flow patterns.

**Table 2.**Rainfall and stream flow variability at Broughton River, Torrance River and Wakefield River in South Australia (SA) from 1990 to 2011.

Statistics | Broughton River | Torrance River | Wakefield River | |||
---|---|---|---|---|---|---|

Rainfall | Stream flow | Rainfall | Stream flow | Rainfall | Stream flow | |

Mean | 1.653 | 9.817 | 1.530 | 5.396 | 1.282 | 25.333 |

Estimated standard deviation | 0.385 | 4.075 | 0.296 | 4.201 | 0.223 | 21.300 |

Coefficient of variation (CV) | 23.31% | 41.51% | 19.36% | 77.85% | 17.40% | 84.07% |

**Figure 3.**Standard deviations of wavelet coefficients of rainfall and stream flow from level 0 to 8. (

**a**) Rainfall; (

**b**) Stream flow.

#### 3.2. Correlation Structures between Rainfall and Stream Flow

**Table 3.**Constructed correlation pattern for different levels between (a) adjusted rainfall and adjusted stream flow; (b) squared adjusted rainfall and adjusted stream flow; (c) adjusted rainfall and squared adjusted stream flow; (d) squared adjusted rainfall and squared adjusted stream flow.

Days | Broughton River | Torrance River | Wakefield River | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | c | d | a | b | c | d | a | b | c | d | |

1 | 0.71 ** | 0.89 *** | 0.70 ** | 0.86 *** | 0.76 ** | −0.184 | −0.513 | 0.447 | 0.76 ** | −0.53 * | −0.59 * | 0.86 *** |

2 | 0.65 * | 0.264 | 0.469 | 0.449 | 0.72 ** | 0.57 | 0.158 | 0.265 | 0.384 | 0.413 | 0.201 | 0.038 |

4 | 0.56 * | −0.189 | 0.257 | 0.146 | 0.63 * | 0.061 | −0.27 | 0.125 | 0.324 | −0.28 | −0.341 | 0.115 |

8 | 0.087 | 0.369 | −0.296 | 0.103 | 0.032 | 0.55 * | 0.173 | −0.51 * | −0.369 | 0.483 | 0.191 | −0.418 |

16 | 0.233 | 0.08 | −0.009 | −0.326 | 0.275 | −0.15 | 0.009 | −0.311 | 0.306 | −0.652 | 0.081 | −0.393 |

32 | 0.094 | 0.055 | −0.059 | −0.248 | 0.68 * | −0.84 ** | −0.81 *** | 0.97 *** | −0.116 | 0.002 | −0.382 | −0.121 |

64 | 0.411 | 0.036 | 0.292 | 0.238 | 0.488 | 0.67 * | 0.456 | 0.71 ** | −0.091 | 0.166 | −0.005 | −0.301 |

128 | 0.423 | −0.409 | 0.604 * | 0.68 * | 0.299 | −0.162 | 0.186 | −0.223 | 0.279 | −0.575 | 0.128 | −0.51 * |

512 | 0.456 | 0.354 | −0.292 | 0.343 | −0.218 | −0.405 | 0.007 | −0.405 | 0.094 | 0.117 | 0.098 | −0.163 |

#### 3.3. Rainfall-Stream Flow Response Modeling

^{2}and time

^{3}, which allows for possible quadratic and cubic trends; C is cos(2πt/365.25) and S is sin(2πt/365.25) and together these allow for seasonal variation of period one cycle per year; β

_{j}are the unknown coefficients to be estimated; and ε

_{t}are random variations with mean 0 and constant standard derivation.

_{t}) on day t from rainfall (X

_{t}) with corresponding lags k. This is referred to as a Response Model (RM). The regression is defined as:

^{2}) and the Akaike Criterion Information (AIC); The AIC is defined as:

Station | Statistical Summary | Intercept (β_{0}) | Linear Term t | Quadratic Term t | Cubic Term t |
---|---|---|---|---|---|

Broughton River | Estimated rainfall | 1.58 | −0.000042 | −0.000000001 | −0.000000000003 |

Variability of rainfall | 0.106 | 0.00008 | 0.000000017 | 0.000000000008 | |

Estimated stream flow | 52.08 | −0.01244 * | 0.0000031 * | −0.000000000258 | |

Variability of stream flow | 6.18 | 0.004661 | 0.0000009 | 0.000000000485 | |

Torrance River | Estimated rainfall | 1.424 | −0.00007 | 0.000000019 | 0.000000000004 |

Variability of rainfall | 0.077 | 0.00006 | 0.000000012 | 0.000000000006 | |

Estimated stream flow | 3.47 | −0.00174 * | 0.0000003 * | 0.000000000149 * | |

Variability of stream flow | 0.607 | 0.0004573 | 0.000000092 | 0.000000000048 | |

Wakefield River | Estimated rainfall | 1.226 | −0.000123 * | 0.0000000037 | 0.00000000001 * |

Variability of rainfall | 0.067 | 0.000051 | 0.00000001 | 0.000000000005 | |

Estimated stream flow | 15.95 | −0.01144 * | 0.0000007 | 0.0000000008 * | |

Variability of stream flow | 3.576 | 0.002694 | 0.0000005 | 0.000000000280 |

Model | Broughton River | Torrance River | Wakefield River | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${R}_{adj}^{2}$ | Std. Error | AIC | RMSE * | ${R}_{adj}^{2}$ | Std. Error | AIC | RMSE * | ${R}_{adj}^{2}$ | Std. Error | AIC | RMSE * | |

RMO | 0.16 | 333.6 | 1104.9 | 3.5107 | 0.24 | 31.19 | 742.77 | 5.074 | 0.13 | 195.8 | 1023.53 | 1.1777 |

RM_D | 0.18 | 331.3 | 1103.9 | 3.1507 | 0.26 | 31.02 | 741.9 | 5.012 | 0.14 | 195.3 | 1023.1 | 1.1777 |

RMD_AR[1] | 0.35 | 292.9 | 1085.1 | 0.0353 | 0.42 | 27.35 | 722.7 | 0.052 | 0.21 | 187.4 | 1016.9 | 0.11777 |

RMD_AR[2] | 0.36 | 291.7 | 1084.5 | 0.0313 | 0.43 | 27.35 | 722.5 | 0.0452 | 0.22 | 187.4 | 1016.8 | 0.10777 |

RMD_tau | 0.39 | 285.8 | 1081.4 | 0.0035 | 0.42 | 27.32 | 722.1 | 0.0411 | 0.23 | 187.3 | 1016.1 | 0.10178 |

^{3}s

^{−1}.

Stream flow | y1 | y2 | y3 | y4 | y5 | y6 | y7 | y8 | y9 | y10 | y11 | y12 | y13 |

8 | 9 | 0 | 0 | 0 | 2 | 9 | 22 | 3 | 5 | 8 | 8 | 6 | |

Rainfall | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 |

3 | 2 | 5 | 3.2 | 3 | 2.8 | 2.6 | 2.4 | 2.2 | 2 | 1.8 | 1.6 | 1.4 |

_{t}is the observed stream flow, respectively.

^{3}s

^{−1}from RM to RM_D. Adding autoregressive order 1 (AR[1]) with RM_D results in substantially improved ${R}_{adj}^{2}$ values (from 0.18, 0.26, and 0.14 to 0.35, 0.42 and 0.21 for Broughton River, Torrance River and Wakefield River, respectively. Furthermore, when adding autoregressive order 1 (AR[1]) with RM_D, there is evidence of improvement but this may be offset by the increasing number of parameters that affect the complexity of the model. In addition, the RMD_tau model represents a small improvement for two of the three river basins. The best fitted models are RMD_tau for Broughton River, RMD_AR[2] for Torrance River and RMD_tau for Wakefield River, were selected based on the minimum Akaike Information Criterion (AIC) and minimum root mean square error (RMSE) in m

^{3}s

^{−1}. The residuals from the best fitted models were transformed to normalized form by factor multiplication. A factor was calculated, which allows for the fact that the mean of a non-linear function of a random variable is not equal to that function of the mean. The transform series follow an identically normalized form with mean (μ) of zero, standard deviation (σ

^{2}) of 1 and a random disturbance term (ε

_{t}) which is uncorrelated. The transformed series were used to predict the stream flow on day t based on the predicted stream flow influence over the short term, as shown in Figure 4.

**Figure 4.**Predicted stream flow based on dsdt rainfall for (

**a**) Broughton River; (

**b**) Torrance River; and (

**c**) Wakefield River from 1990 to 2010.

#### 3.4. Modeling Stream Flow Using an Artificial Neural Network

_{t+1}= f(S

_{t}, S

_{t-1}, S

_{t-2}, ….., S

_{t-m}, R

_{t}, R

_{t-1}, R

_{t-2},...,R

_{t-n}) where S represents stream flow, R represents rainfall, t is the current day, m = {3,...,8}, n = {3,...,8} and f represents the ANN as a regression function. We investigate necessary lagged inputs of rainfall and river flow for modeling the river flows at three locations in South Australia. We apply an artificial neural network (ANN) technique for modeling river flow. ANN models are developed with all combinations of rainfall and river flow input ranges. In addition, a standard range of nodes in the hidden layer are also considered. Among all models based on inputs and hidden nodes, the best model is selected based on mean absolute error criteria. This entire process is applied to all three locations. ANN models capture the non-linear relationships of rainfall and river flow patterns in modeling river flows from large time series data. For example, if we consider 3 days lag of stream flow and 5 days lag of rainfall, then the total number of input nodes in the ANN structure will be 8 and we consider the number of nodes in the hidden layers ranging from 1 to 10. To achieve the best model using ANN for each location, all inputs not only apply in combination, but we also consider setting a range of parameters, such as different number of nodes in the hidden layer, for each combination of inputs.

^{3}s

^{−1}was minimized through an iteration process that varied the number of nodes in the hidden layer.

**Figure 6.**MAE for training data (1990–2009) using ANN with best lag combinations at each location, units in m

^{3}s

^{−1}.

**Table 7.**Best prediction model based on ${R}_{adj}^{2}$, lowest RMSE and MAE are in m

^{3}s

^{−1}on the training data.

Location | Input Lags | Nodes in Hidden Layer in ANN(H) | ${R}_{adj}^{2}$ | RMSE * | MAE * |
---|---|---|---|---|---|

Broughton River | 3 days rain, 6 days stream flow | 9 | 0.68 | 270.33 | 45.53 |

Torrance River | 3 days rain, 8 days stream flow | 2 | 0.71 | 24.54 | 4.89 |

Wakefield River | 4 days rain, 5 days stream flow | 1 | 0.45 | 179.42 | 19.28 |

^{3}s

^{−1}.

**Figure 7.**Observed and predicted stream flow for (

**a**) Broughton River; (

**b**) Torrance River; and (

**c**) Wakefield River for the year 2010.

^{3}s

^{−1}. We further use this best model for testing and we find the MAE of 32.43 m

^{3}s

^{−1}. For Torrance, the ANN best model in training has 3 days lagged rainfall and 8 days lagged stream flow as inputs with 2 nodes in the hidden layer achieving the MAE of 4.89 m

^{3}s

^{−1}. For testing data, this model gives a MAE of 9.27 m

^{3}s

^{−1}. In case of Wakefield, the best ANN model has 4 days lagged rainfall and 5 days lagged stream flow as inputs with 1 node in the hidden layer achieving the MAE of 19.28 m

^{3}s

^{−1}. For the testing data, this model achieves an MAE of 42.88 m

^{3}s

^{−1}. The reason for the difference in MAE between the training and testing phases could be due to this river’s ephemeral nature, and its substantial dependence on rainfall.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Kamruzzaman, M.; Shahriar, M.S.; Beecham, S.
Assessment of Short Term Rainfall and Stream Flows in South Australia. *Water* **2014**, *6*, 3528-3544.
https://doi.org/10.3390/w6113528

**AMA Style**

Kamruzzaman M, Shahriar MS, Beecham S.
Assessment of Short Term Rainfall and Stream Flows in South Australia. *Water*. 2014; 6(11):3528-3544.
https://doi.org/10.3390/w6113528

**Chicago/Turabian Style**

Kamruzzaman, Mohammad, Md Sumon Shahriar, and Simon Beecham.
2014. "Assessment of Short Term Rainfall and Stream Flows in South Australia" *Water* 6, no. 11: 3528-3544.
https://doi.org/10.3390/w6113528