Technical Note: Analytical sensitivity analysis of a two parameter recursive digital baseflow separation filter

A sensitivity analysis for a well-established baseflow separation technique, a two parameter recursive digital filter, is presented. The sensitivity of the calculated baseflow index to errors or uncertainties of the two filter parameters and of the initial baseflow value is analytically ascertained. It is found that the influence of the initial baseflow value is negligible for long time series. The propagation of errors or uncertainties of the two filter parameters into the baseflow index is expressed by a dimensionless sensitivity index, the ratio between the relative error of the baseflow index and the relative error of the respective parameter. Representative index values are derived by application of the resulting equations to 65 North American catchments. In the mean the parametera, the recession constant, has a stronger influence on the calculated baseflow index than the second filter parameter BFImax. This is favourable in that a can be determined by a recession analysis and therefore should be less uncertain. Whether this finding also applies for a specific catchment can easily be checked by means of the derived equations.


The two parameter recursive digital filter
The aim of baseflow separation is to distinguish two streamflow components: baseflow (groundwater discharging into the stream) and quick flow (surface runoff and interflow).In the past, many separation methods have been proposed, amongst them the two parameter recursive digital filter of Eckhardt (2005), which has since been applied in numerous studies, sometimes under the name of "Eckhardt filter".The equation of this low-pass filter is subject to b k ≤ y k , where b is the baseflow, y is the streamflow, and k is the time step number.The filter has two parameters: the recession constant a and the maximum value BFI max of the baseflow index (the long-term ratio of baseflow to total streamflow) that can be modelled by the algorithm.
A key question is how errors and uncertainties in these two parameters affect the results of the separation.A first attempt to answer this question was the empirical sensitivity analysis by Eckhardt (2005).An empirical sensitivity analysis consists of three steps, which are repeated several times: 1. Input of a model (consisting in general of one or more equations) is varied.
2. The model is run.
3. The model output is analysed.
One can also speak of an experimental sensitivity analysis.However, an empirical sensitivity analysis is only a makeshift if an analytical sensitivity analysis, that is an analytical calculation of the error propagation through the model, is not feasible.In the case of Eq. ( 1), such a calculation of the error propagation is possible and will be presented in Sect. 2.

Comparison with other filters
Equation (1) represents a whole class of filter algorithms which are based on the widely accepted linear storage model (Eckhardt, 2005).Examples are the algorithms of Chapman and Maxwell (1996) and Boughton (1993).The filter of Chapman and Maxwell (1996) b Published by Copernicus Publications on behalf of the European Geosciences Union.
is derived from Eq. ( 1) for BFI max = 0.5.The filter of Boughton (1993) b is equivalent to Eq. ( 1) with Filter algorithms which rely more on physics have been presented by Furey and Gupta (2001) and Huyck et al. (2005).
In the algorithm of Furey and Gupta (2001)  Equation ( 1) and the algorithms of Furey and Gupta (2001) and Huyck et al. (2005) exemplify a fundamental problem in hydrology: The sounder the physical basis is, the more complex is the model and the greater is the number of its parameters.Analytical expressions may exist for some or all of the parameters, yet the problem of parameter uncertainty persists.Both Furey and Gupta (2001) and Huyck et al. (2005) empirically analyse the impact of parameter uncertainty on the results of their algorithm.The increasing physical reliability does not diminish the need for a sensitivity analysis, but enhances it.

Parameters a and BFI max
The analytical sensitivity analysis of the filter Eq. ( 1) requires the calculation of the partial derivatives of b k with respect to a and BFI max : In the following, the considered model output is the baseflow index where b denotes the sum of baseflow and y the sum of streamflow over the whole period of the available streamflow measurements.
The error propagation into the model output BFI is described by the partial derivatives of BFI with respect to a and BFI max : In order to get representative BFI values, the filtered hydrographs should be long.In this case the term b 0 − b n in the Eqs ( 8) and ( 9) can be neglected: Now, the question of how an error a in the filter parameter a propagates into the calculated baseflow index BFI can be answered.Small errors in a cause an error in BFI of Correspondingly, small errors BFI max in the filter parameter BFI max cause an error in BFI of As a measure for the sensitivity of the baseflow index BFI with respect to the parameters a and BFI max , a dimensionless sensitivity index S is calculated as the ratio between the relative error of BFI and the relative error of the respective parameter.The sensitivity index for the parameter a is (see Appendix B).In this notation, S stands for "sensitivity index", the first symbol in the parentheses (here BFI) indicates the output that is assessed, and the second symbol (here a) the uncertain input.Sometimes this dimensionless index is also called "elasticity index".The sensitivity index for the parameter BFI max is For specific values of a, BFI max , and BFI, the sensitivity indices S(BFI|a) and S(BFI|BFI max ) can now be calculated and compared.

Initial value b 0
The user of the filter algorithm has to choose an initial baseflow value b 0 .In the present section, the sensitivity of the baseflow index to this value will be ascertained.
Equation ( 1) can be written as and The baseflow b k at the time step k is now traced back to the initial value b 0 : Thus, the partial derivative of b k with respect to b 0 is The partial derivative of BFI with respect to b 0 is For long time series (number n of observations → ∞): Because of |A| < 1, the limit of the geometric series The longer the time series, the greater is the sum y of the measured streamflow values and the smaller is the right side of Eq. ( 22).In other words: for long time series the influence of the initial value b 0 on the baseflow index BFI becomes negligible.

Data and results
An empirical sensitivity analysis requires several runs of the filter over the hydrograph of a specific stream, each one with different values of the two filter parameters.Subsequently, the resulting time series of baseflow have to be analysed to ascertain how the baseflow index varies.This finally allows the calculation of the sensitivity indices.Alternatively, only one filter run and calculation of the baseflow index is sufficient, if Eqs. ( 14) and ( 15) are used for assessing the sensitivity indices.This method has been applied to the 65 catchments whose baseflow indices BFI were calculated by Eckhardt (2008).The results are summarised in Table 1.Two catchment types are distinguished: catchments with a perennial stream and porous aquifer, and catchments with an ephemeral stream and porous aquifer.Eckhardt (2005) suggested attributing a BFI max value of 0.80 to the former and of 0.50 to the latter.The recession constant a of each catchment was determined by a recession analysis as described by Eckhardt (2008), the respective sensitivity indices were calculated with Eqs. ( 14) and (15).

Discussion
The analytical sensitivity analysis shows that the recession constant a influences the calculated baseflow index BFI more than the filter parameter BFI max .In the case of the catchments with perennial stream and porous aquifer, for example, the value S(BFI|a) = −0.77signifies that a relative error of X percent in a causes a relative error of −0.77 times X percent in BFI.A relative error of X percent in BFI max causes only a relative error of 0.26 times X percent in BFI.This is good news because the recession constant a can be determined by a recession analysis whereas an optimal BFI max value cannot be derived from the streamflow measurements alone.Therefore, the value of BFI max will be more uncertain than the value of a.
At first glance, the finding that the parameter a has a stronger influence on the calculated baseflow index than the parameter BFI max seems to contradict the result of the empirical sensitivity analysis of Eckhardt (2005) which used hydrographs of two catchments not belonging to the pool of the 65 catchments analysed for the present paper.
-For a catchment with a perennial stream and porous aquifer, and assuming values of a = 0.925 and BFI max = 0.75, the baseflow index was found to be BFI = 0.72 and sensitivity indices S(b|a) = −0.55 and S(b|BFI max ) = 0.96 were calculated (b: mean value of the baseflow).
-For a catchment with a perennial stream and hard rock aquifer, and assuming values of a = 0.925 and BFI max = 0.25, the baseflow index was found to be BFI = 0.25 and the sensitivity indices were S(b|a) = 0.00 and S(b|BFI max ) = 0.98.
Therefore, the conclusion was that BFI max is the more critical parameter.Indeed, this is confirmed if we insert only the two afore-mentioned sets of values into Eqs.( 14) and ( 15).With a = 0.925, BFI max = 0.75, and BFI = 0.72 one gets S(BFI|a) = −0.10 and S(BFI|BFI max ) = 0.28.With a = 0.925, BFI max = 0.25, and BFI = 0.25 one gets S(BFI|a) = 0.00 and S(BFI|BFI max ) = 0.10.This, however, is obviously a nonrepresentative result.In the case of the first catchment, the main reason is that the value a = 0.925 of the recession constant, which was arbitrarily choosen as a starting point for the empirical sensitivity analysis by Eckhardt (2005), is too small.If the recession analysis of Eckhardt (2008) is applied, it is found that the actual recession constant for this catchment is about 0.96.According to Eq. ( 14), however, the sensitivity index S(BFI|a) is the greater the greater a is.With a = 0.925 the sensitivity index S(BFI|a) was underrated.In the case of the second catchment, the finding S(BFI|a) = 0.00 is a consequence of the special situation that BFI max equals BFI.
We have here a further argument for the analytical sensitivity analysis: because it requires less effort than an empirical sensitivity analysis once the Eqs.( 14) and ( 15) are derived, more catchments can be included and hence a more reliable conclusion can be drawn.

Conclusions
The finding that BFI max is the less critical parameter in the filter Eq. ( 1) is favourable in that BFI max cannot be derived from the streamflow measurements and therefore is more uncertain than the other filter parameter, the recession constant a. Optimal BFI max values have to be found by calibration.Gonzales et al. (2009), for example, have calibrated the filter (Eq. 1) by means of a tracer-based separation using dissolved silica and found an optimal BFI max value of 0.92 for a Dutch catchment.Eckhardt (2005) suggested BFI max = 0.80 for such a catchment with a perennial stream and porous aquifer.Thus there may be an uncertainty of about 0.15 or 19 % in the filter parameter BFI max .The sensitivity index S(BFI|BFI max ) = 0.26 indicates that such an error leads to a mean error in the calculated baseflow index BFI of only 0.26 × 19 % = 5 %.For catchments with ephemeral stream and porous aquifer, the uncertainty is smaller yet.
Of course, these values only characterise mean conditions derived for the 65 North American catchments presented by Eckhardt (2008).The baseflow index of a specific catchment can show another sensitivity to uncertainties in the filter parameters.However, this can easily be checked by means of Eqs. ( 14) and ( 15), which herewith provide important additional information to this baseflow separation technique.Edited by: N. Verhoest

Calculation of the partial derivatives
b k is a function of b k−d−1 and y k−d−1 .Four parameters have to be specified: the time delay d between precipitation and groundwater recharge, the ratio c 1 of overland flow to precipitation, the ratio c 3 of groundwater recharge to precipitation, and the recession constant a. Required are time series of streamflow and precipitation.In the algorithm of Huyck et al. (2005) b k is a function of b k−1 , b k−d , b k−d−1 , y k−d , and y k−d−1 .Twelve parameters have to be specified:d, c1 , c 3 , and nine other parameters describing hydraulic characteristics and the shape of the aquifer.Required are not only time series of streamflow and precipitation, but also a digital elevation model and information on the drainable porosity of the soil.Equation (1) has only two parameters and requires only streamflow data.

Table 1 .
Results of the analysis of 65 North American catchments (mean values and their standard deviation).