Methods

3.1 Time-Trend Analysis Method

Time trend analysis method can be applied in single catchment studies based on the rainfall-runoff relationships in different periods without the need of a control catchment (Bosch & Hewlett, 1982). This method is mainly used for detecting streamflow changes with vegetation variations due to anthropogenic activities such as plantations or natural variability such as bushfire. This method first constructs and evaluates the regression relationship between rainfall and runoff during the calibration period (the pre-bushfire period in this study), and then estimates streamflow using observed precipitation during the testing period (the post-bushfire period in this study). The differences of observed and predicted streamflow represent the vegetation change impacts on streamflow in the testing period. The equation can be expressed as follows (Lee, 1980):
During the calibration period:
\(Q_{1}=f(P_{1})\) (1)
During the testing period:
\(Q_{2}^{{}^{\prime}}=f\left(P_{2}\right)\) (2)
\({Q}^{\text{veg}}=\overset{\overline{}}{Q_{2}}-\overset{\overline{}}{Q_{2}^{{}^{\prime}}}\)(3)
where \(P_{1}\) and \(P_{2}\) represent annual precipitation (mm) during calibration and testing period respectively. \(Q\) represents measured streamflow (mm) and \(Q^{{}^{\prime}}\) represents predicted streamflow (mm).\({Q}^{\text{veg}}\) is the bushfire induced change of average annual streamflow (mm). The function expressed in equation (1) and (2) can be either linear or nonlinear depending on the rainfall-runoff relationships in the specific catchments.

3.2 Statistics for Evaluation of the Regression relationships

As mentioned above, the regression model between rainfall and runoff in a catchment should be evaluated once it has been constructed. Following Legates and McCabe Jr (1999), four statistics generally used were applied to indicate the accuracy of the regression relationship between rainfall and runoff, which are: the coefficient of determination (\(R^{2}\)), The modified coefficient of efficiency (\(E_{1}\)), the modified index of agreement (\(d_{1}\)), and the mean absolute error (MAE). They are defined as
\(R^{2}=\left\{\frac{\sum_{i=1}^{N}{(O_{i}-\overset{\overline{}}{O})(P_{i}-\overset{\overline{}}{P})}}{{[\sum_{i=1}^{N}{(O_{i}-\overset{\overline{}}{O})}]}^{0.5}{[\sum_{i=1}^{N}{(P_{i}-\overset{\overline{}}{P})}]}^{0.5}}\right\}^{2},\)(4)
\(E_{1}=1.0-\frac{\sum_{i=1}^{N}\left|O_{i}-P_{i}\right|}{\sum_{i=1}^{N}\left|O_{i}-\overset{\overline{}}{O}\right|}\ ,\)(5)
\(d_{1}=1.0-\frac{\sum_{i=1}^{N}\left|O_{i}-P_{i}\right|}{\sum_{i=1}^{N}\left(\left|P_{i}-\overset{\overline{}}{O}\right|+\left|O_{i}-\overset{\overline{}}{O}\right|\right)}\ ,\)(6)
\(\text{MAE}=N^{-1}\sum_{i=1}^{N}{|O_{i}-P_{i}|},\ \) (7)
where \(O\) and \(P\) are the observed and predicted data,\(\overset{\overline{}}{O}\) and \(\overset{\overline{}}{P}\) are the mean value of observed and predicted data, and N is the number of paired observations.
Confidence limits for the mean (Snedecor & Cochran, 1989) were designed to indicate a confidence interval for the mean. In our study, a confidence coefficient of 95% is utilised to represent the confidence limits, which are calculated as:
\(\overset{\overline{}}{Y}\pm t_{(\frac{\alpha}{2},\ N-1)}\sigma/\sqrt{N}\)(8)
where \(\overset{\overline{}}{Y}\) is the mean of the sample, \(\sigma\)is the standard deviation of the sample, N is the size of the sample,\(\alpha\) is the significance level, and\(t_{(\frac{\alpha}{2},\ N-1)}\) is the upper critical value of the\(t\) distribution with \(N-1\) degree of freedom. The confidence coefficient is \(1-\alpha\).

3.3 Estimating the Effects of Climate Variability and Bushfire Impact on Streamflow

The total mean annual streamflow changes in a given catchment can be estimated as following:
\({Q}^{\text{tot}}={\overset{\overline{}}{Q_{2}}}^{\text{obs}}-{\overset{\overline{}}{Q_{1}}}^{\text{obs}}\)(9)
where \({Q}^{\text{tot}}\) is the total changes of mean annual streamflow (mm), \({\overset{\overline{}}{Q_{1}}}^{\text{obs}}\) and\({\overset{\overline{}}{Q_{2}}}^{\text{obs}}\) is the measured mean annual streamflow in the calibration period and testing period respectively.
Climate variables (i.e. rainfall and potential evaporation) and catchment characteristics (i.e. vegetation change) are key aspects that alter hydrological processes. However, the interaction mechanism in the water cycle is complex. For catchments with bushfire impacts, it is assumed that the total changes of mean annual streamflow can be composed of the climate variability induced and the vegetation change induced, which are calculated as
\({Q}^{\text{tot}}={Q}^{\text{clim}}+{Q}^{\text{veg}}\) (10)
where \({Q}^{\text{clim}}\) and \({Q}^{\text{veg}}\) is the mean annual streamflow change (mm) caused by climate variability and vegetation change (e.g., bushfire) respectively.
The mean annual streamflow change due to vegetation change can be calculated by time-trend analysis mentioned above while a sensitivity-based method (Dooge et al., 1999; Jones et al., 2006; Schaake & Liu, 1989) is used to estimate the mean annual streamflow change caused by climate variables. This method assumed the changes in catchment water balance are partially attributed to perturbations in climate variables such as precipitation and potential evaporation (PET). Over a sufficiently long time scale, change in mean annual streamflow due to climate change can be calculated as (Koster & Suarez, 1999; Milly & Dunne, 2002):
\({Q}^{\text{clim}}=aP+bPET\) (11)
where \(P\) and \(PET\) represent the changes in precipitation (mm) and potential evaporation (mm), and a, b are the sensitivity coefficients of streamflow to precipitation and potential evaporation, which can be obtained from the Budyko-curve model proposed by Zhang et al. (2001) as following description (Li et al., 2007):
\(a=\frac{1+2x+3wx^{2}}{{(1+x+wx^{2})}^{2}}\) (12)
\(b=\frac{1+2wx}{{(1+x+wx^{2})}^{2}}\) (13)
where x is the dryness index (PET/P) and \(w\) is a model parameter associated with catchment characteristics (Zhang et al., 2001). The value of \(w\) was determined by climatic condition and vegetation cover in a given catchment before bushfire and was kept constant in the whole period.

3.4 Determination of the Calibration and Testing Periods

In this study, we are focusing on catchments affected by the 2009 Victoria bushfire event. The calibration period was determined according to the data quality and length of streamflow data in each catchment (See Table 1). Once the calibration period was determined, annual rainfall and streamflow data were utilized to develop a regression relationship for the catchment. The testing period of all catchments was set to be 2009-2015 since the bushfire occurred in early February, 2009 and the dataset of climate variables and streamflow until pre-2016 were collated. Annual precipitation, PET and streamflow were divided into two parts to represent the calibration and testing periods.