This section reviews techniques commonly used in estimating chemical substance loads in rivers. Calculation of these loads is a necessary part of analyzing the response of a catchment to rainfall events (Letcher et al., 1999; Newham et al., 2002). Hence, to predict organic carbon loads, a
combination of observed instream concentration and streamflow discharge is needed. For a general case, the continuous time load over a time period, t, can be calculated through Eq. (5.1): Load =
∫
t
CQdt
0(5.1 Where C is the concentration value and
Q
is the streamflow discharge.All stations have prefix ‘5000’ Cotter River
Tributary
Stations which were used for component load estimation
Stations which were not used for component load estimation due to paucity of data (site 05) and Dam impacts (site 07)
Figure 5.1: Sampling sites and their connected regions in the Cotter River Catchment
The discrete form of the load equation can be written as:
Bendora Dam Cotter Dam 07 06 04 05 01 02 03
Load =
ξ
ξ i t i iQ C∑
= / 1 (5.2Where sampling interval,
ξ
, is short compared to the period of time over which the streamflow and concentration vary (Letcher et al., 1999).Most techniques for load estimation are based on equation 5.2 (Letcher et al., 1999). Loads calculated in this way can be expected to be very close to the true load, assuming the streamflow discharge and organic carbon data are sampled at sufficiently high frequency. Unfortunately, due to the relatively high costs involved with measuring concentrations and the paucity of streamflow data, high frequency flow and concentration data are rarely available (Littlewood, 1992).
There are many methods used to predict load estimates. In order to select the best option for each case study, a number of criteria should be met, including data resolution (flow and concentration). The list of the criteria is discussed in section 5.3.1.
Five methods for estimating component loads are considered here:
• Regression methods (Walling and Webb, 1985; Ferguson, 1986; Ferguson, 1987; Preston et al., 1989; Littlewood, 1992; Kronvang and Bruhn, 1996; Letcher et al., 1999; Schilling, 2000; Croke, 2002; Newham et al., 2002);
• Ratio estimators or flow-weight mean concentration (Preston et al., 1989; Letcher et al., 1999);
• Interval concentration and discharge methods (Lesack, 1993; Letcher et al., 2002); • Arithmetic mean methods (Walling and Webb, 1985; Letcher et al., 2002);
• Averaging techniques (used in this case study) (Walling and Webb, 1981; Walling and Webb, 1985; Preston et al., 1989; Letcher et al., 1999; Letcher et al., 2002).
5.2.1 Comparison of methods
5.2.1.1 Regression methods
Regression estimators, referred to as rating curves, have been applied to calculate suspended sediments and nutrient loads (Letcher et al., 1999; Schilling, 2000; Croke, 2002). This technique is
related variables that are measured at high frequency. Streamflow discharge is generally used as the independent variable, using a simple power law rating curve (Littlewood, 1992; Letcher et al., 1999). The annual load is given by Kronvang and Bruhn (1996) as:
Load = 365
[
exp(ˆ0 ˆ1ln )]
1 i i i b b q q +∑
= (5.3Where bˆ0 and bˆ1 are fitted regression parameters and qi is the discharge value.
The above relationship can be applied using two regression methods, as presented by Schilling (2000), to estimate concentration loads. These two functional forms were used to relate suspended sediment concentration to streamflow: [SS] = aQ + c and [SS] = aQb (where [SS] is suspended sediment concentration, Q is streamflow and a, b and c are constants).
This technique extrapolates a limited number of concentration values over the entire period of interest through developing a mathematical linear relationship between concentration and streamflow (Letcher et al., 1999). Hence, parameter values for such power functions are derived using a linear regression to the logarithm of the input data (Croke, 2002). As a consequence of this, it was found that the regression curve estimates based on this log-log relationship were biased and usually underestimated suspended sediment and overestimated nutrient loads (Ferguson, 1986; Preston et al., 1989; Letcher et al., 1999; Newham et al., 2002).
To address this problem, bias correction factors have been suggested by a number of researchers (Ferguson, 1987; Walling and Webb, 1988) in combination with the two regression methods of Letcher et al. (1999) and Schilling (2000). The two regression techniques were tested by Newham (2002) to calculate nutrient and suspended sediment loads using a daily time-step of concentration results and daily streamflow outcomes to obtain reliable results. Unfortunately, this condition could not be met in this case study due to the small quantity of data collected. Organic carbon samples were only collected three or four times per month for each sampling site. Moreover, due to the lack of streamflow data observed during the sampling program after the wildfire of 2003, this necessary data set was not available.
5.2.1.2 Ratio estimators or flow weighted mean concentration
This technique integrates streamflow and concentration as a means of averaging the concentration data. In other words, this method assumes the concentrations are sufficiently constant and independent of flow (Letcher et al., 1999). This ratio estimator is identified as:
Load = (y/q)* Q (5.4 Where y and q are the sample average of concentration and flow discharge, respectively, and Q is the streamflow value. This relationship is represented by Preston et al. (1989) as:
Load = Q * ( cqi n n i i / 1
∑
= ) / ( q n n i i / 1∑
= ) (5.5 Where Q is total discharge, n is the number of days sampled and ci & qi are concentration and discharge during the sampling interval, respectively.Based on Preston et al. (1989) and Letcher et al. (1999), the above method requires the two following conditions to be met:
• The relationship between the average concentration and streamflow is a straight line passing through the origin;
• The variance of mean of concentration about the line is proportional to the average of streamflow.
Unfortunately, these conditions could not be met in this case study. Streamflow measurements could not be related to the observed concentrations due to some technical problems and paucity of collected data. Hence, the ratio estimator was biased as well. Moreover, because of limitations in relation to the sampling program in the Cotter River Catchment, especially the instability of roads after the wildfire, collecting concentration data during the flood events was impossible. As a result of these limitations, this method was not used for this case study.
5.2.1.3 Interval concentration and discharge methods
This method estimates the concentration for the interval between samples through averaging the sample concentration at the start and end of the interval (Letcher et al., 2002). Consequently, to estimate the chemical substance loads, concentration for a sample interval should be multiplied by discharge for a sample interval (Lesack, 1993; Letcher et al., 2002). This technique is usually estimated based on:
Load =
∑
(
)
= + + n i i i i c c q 1 1 /2 (5.6Where ci and ci+1 indicate sample concentration at the start and end of the interval and qi is
discharge for a sample interval.
According to Lesack (1993), this method is used if concentration data are collected during flood discharge with an adequate sampling interval. This condition could also not be met in this case study and subsequently this method was not considered to be an appropriate method.
5.2.1.4 Arithmetic mean concentration
As indicated by Letcher et al. (2002), the arithmetic average technique is acceptable if the concentration data set presents the true range of concentrations:
Load = Q * c n n i i/ 1
∑
= (5.7 Where ci /n is the average sample concentration and Q is the annual discharge.This method is not normally considered to be a reliable method for estimating concentration load in streams because combining average concentration with annual streamflow value generates a bias (either negative or positive) when calculating loads using routinely collected data (weekly, fortnightly or monthly). For this case study, a routine weekly program was selected because of the relatively high costs associated with analyzing organic carbon samples and the restrictions placed on doing a field survey following the 2003 wildfire in the Cotter River Catchment.
5.2.1.5 Arithmetic mean loads (averaging methods)
Average techniques are usually identified as the simplest methods for calculation of load, and are used because of a lack of the data needed to apply more sophisticated approaches. According to Walling and Webb (1985), load over a time period is calculated using the mean of streamflow values, and concentrations for a given subinterval and summing these results over a different time period such as a month or a year. Considering the assumptions considered through this method including an independent and identically distributed data set, the results of this calculation method are biased. These biased calculations cause over or under-estimation in the calculation of loads if the sampling plan does not allow data to be collected from the entire range of streamflow and component variability (Letcher et al., 1999). To address this, organic carbon data was collected roughly weekly, with three or four samples per month, for this case study.
Consequently, it was assumed that averaging methods were more likely to generate accurate load estimates in this case study. Similar conclusions were reached by Preston et al. (1989) who indicated that average methods using daily discharge with monthly average concentration frequently have the lowest mean squared error. However, the precision achieved in this method is less than that of other methods and the standard deviation is higher (Letcher et al., 1999).
As a result, the average method used in this study was calculated through average sample load, scaled for time:
Load = k * ( cqi n n i i / 1
∑
= ) (5.8 Where k is correcting for the temporal resolution of the data and ciand qi are the average ofconcentration and streamflow respectively, during the sampling period and n is the number of days sampled.
5.2.2 Summary of methods
In summary, many methods exist for estimating loads; these employ different sampling strategies, but the difficulty of integrating continuous streamflow data (observed or simulated) with non- continuous concentration data (Littlewood, 1992; Kronvang and Bruhn, 1996; Letcher et al., 1999;
calculation method depends on data availability, the hydrological characteristics of the catchment, an acceptable accuracy of predictions, and the complexity of the relevant load technique.
The averaging method used in this case study was not the best method in terms of accuracy and precision. However, it was chosen based on the characteristics of the Cotter River Catchment, the availability of data, and the optimum consistency and compatibility between organic carbon data and streamflow outcomes. Consequently, organic carbon (TOC, DOC, and POC) loads for the seven sampling sites (ungauged stations or sub-catchments areas) in the Cotter River Catchment were generated using linear interpolation and a relevant averaging technique on a one-day time step for each month from July 2003 to June 2004.