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.