The two-component hydrograph separation method with
conductivity as a tracer is favored by hydrologists owing to its low cost
and easy application. This study analyzes the sensitivity of the baseflow
index (BFI, long-term ratio of baseflow to streamflow) calculated using this
method to errors or uncertainties in two parameters (BFC, the
conductivity of baseflow, and ROC, the conductivity of surface runoff)
and two variables (yk, streamflow, and SCk, specific conductance of
streamflow, where k is the time step) and then estimates the uncertainty in
BFI. The analysis shows that for time series longer than 365 days, random
measurement errors in yk or SCk will cancel each other out, and their
influence on BFI can be neglected. An uncertainty estimation method of BFI
is derived on the basis of the sensitivity analysis. Representative
sensitivity indices (the ratio of the relative error in BFI to that
of BFC or ROC) and BFI′ uncertainties are determined
by applying the resulting equations to 24 watersheds in the US. These
dimensionless sensitivity indices can well express the propagation of errors
or uncertainties in BFC or ROC into BFI. The results indicate that
BFI is more sensitive to BFC, and the conductivity two-component
hydrograph separation method may be more suitable for the long time series
in a small watershed. When the mutual offset of the measurement errors in
conductivity and streamflow is considered, the uncertainty in BFI is reduced by half.
Introduction
Hydrograph separation (also called baseflow separation), aims to identify
the proportion of water in different runoff pathways in the export flow of a
basin, which helps in identifying the conversion relationship between
groundwater and surface water; in addition, it is a necessary condition for
optimal allocation of water resources (Cartwright et al., 2014; Miller et
al., 2014; Costelloe et al., 2015). Some researchers indicated that
tracer-based hydrograph separation methods yield the most realistic results
because they are the most physically based methods (Miller et al., 2014; Mei
and Anagnostou, 2015; Zhang et al., 2017). Many hydrologists have suggested
that electrical conductivity can be used as a tracer in hydrograph
separation (Stewart et al., 2007; Munyaneza et al., 2012; Cartwright et al.,
2014; Lott and Stewart, 2016; Okello et al., 2018). Conductivity is a
suitable tracer because its measurement is simple and inexpensive, and it
has distinct applicability in long-series hydrograph separation (Okello et al., 2018).
The two-component hydrograph separation method with conductivity as a tracer
(also called conductivity mass balance (CMB) method; Stewart et al., 2007)
calculates baseflow through a two-component mass balance equation. The
general equation is shown in Eq. (1), which is based on the following assumptions:
contributions from end-members other than baseflow and surface runoff are negligible;
the specific conductance of runoff and baseflow are constant (or vary in
a known manner) over the period of record;
in-stream processes (such as evaporation) do not change specific
conductance markedly;
baseflow and surface runoff have significantly different specific conductance.
bk=ykSCk-ROcBFc-ROc,
where b is baseflow (L3 T-1), y is streamflow (L3 T-1), SC is the
electrical conductivity of streamflow, and k is time step number. The two
parameters BFC and ROC represent the electrical conductivity of
baseflow and surface runoff, respectively.
Stewart et al. (2007) conducted a field test in a drainage basin of
12 km2 in southeast Hillsborough County, Florida, and showed that
the maximum conductivity of streamflow can be used to replace BFC and
the minimum conductivity can be used to replace ROC. However, Miller et
al. (2014) pointed out that the maximum conductivity of streamflow may
exceed the real BFC; therefore, they suggested that the 99th percentile
of the conductivity of each year should be used as BFC to avoid the
impact of high BFC estimates on the separation results and assumed that
baseflow conductivity varies linearly between years. The determination of
the parameters (BFC, ROC) of the conductivity two-component
hydrograph separation method involves some uncertainties (Miller et al.,
2014; Okello et al., 2018). Therefore, sensitivity analysis of parameters
and quantitative analysis of the uncertainties will contribute towards
further optimization of the CMB method and improving the accuracy of hydrograph separation.
Most existing parameter sensitivity analysis methods are empirical methods
that usually substitute varying values of a certain parameter into the
separation model and then compare the range of the separation results
produced by these varying parameter values (Eckhardt, 2005; Miller et al.,
2014; Okello et al., 2018). Eckhardt (2012) indicated that “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”. Eckhardt (2012) derived sensitivity indices of
equation parameters by the partial derivative of a two-parameter recursive
digital baseflow separation filter equation. Until now, the parameters'
sensitivity indices of the CMB equation have not been derived.
At present, the uncertainty in the separation results of the CMB method is
mainly estimated using an uncertainty transfer equation based on the
uncertainty in BFC, ROC, and SCk (Genereux, 1998; Miller et
al., 2014). See Sect. 3.1 for details. In this uncertainty estimation
method, the uncertainty in the baseflow ratio (fbf, the ratio of
baseflow to streamflow in a single calculation process) is estimated, and
the average uncertainty in multiple calculation processes is then used to
estimate the uncertainty in the baseflow index (BFI, long-term ratio of
baseflow to total streamflow). This method can neither directly estimate the
uncertainty in BFI nor consider the randomness and mutual offset of
conductivity measurement errors, and, thus, it does not provide accurate
estimates of BFI uncertainty.
The main objectives of this study are as follows: (i) analyze the
sensitivity of long-term series of baseflow separation results (BFI) to
parameters and variables of the CMB equation (Sect. 2); (ii) derive the
uncertainty in BFI (Sect. 3). The derived solutions were applied to 24 basins
in the US, and the parameter sensitivity indices and BFI uncertainty
characteristics were analyzed (Sect. 4).
Sensitivity analysisParameters BFC and ROC
In order to calculate the sensitivity indices of the parameters, the partial
derivatives of bk in Eq. (1) with respect to BFC and ROC are
required (the derivation process is expressed as Eqs. A1 and A2):
∂bk∂BFc=-ykSCk-ROcBFc-ROc2,∂bk∂ROc=ykSCk-BFcBFc-ROc2.
For the convenience of comparison, the baseflow index (BFI) is selected as
the baseflow separation result for long time series to analyze the influence
of parameter uncertainty on BFI,
BFI=∑k=1nbk∑k=1nyk=by,
where b and y denote the total baseflow and total streamflow, respectively,
over the whole available streamflow sequences, and n is the number of
available streamflow data.
Then, the partial derivatives of BFI to BFC and ROC should be
calculated (the derivation process is presented in Eqs. A3 and A4):
∂BFI∂BFc=yROc-∑k=1nykSCkyBFc-ROc2,∂BFI∂ROc=∑k=1nykSCk-yBFcyBFc-ROc2.
The definition of the partial derivative suggests that the influence of the
errors in the parameters (ΔBFC and ΔROC) in
Eq. (1) on the BFI can be expressed by the product of the errors and its partial
derivatives. Then the errors in BFI caused by small errors in BFC
and ROC can be approximated by
ΔBFcBFI=∂BFI∂BFcΔBFc=yROc-∑k=1nykSCkyBFc-ROc2ΔBFc,ΔROcBFI=∂BFI∂ROcΔROc=∑k=1nykSCk-yBFcyBFc-ROc2ΔROc.
The dimensionless sensitivity indices (S) can be obtained by comparing the
relative error in BFI caused by the small errors in BFC and ROC
with that of BFC and ROC (see Eqs. B1 and B2):
SBFI|BFc=ΔBFcBFIBFIΔBFcBFc=BFcyROc-∑k=1nykSCkyBFIBFc-ROc2,SBFI|ROc=ΔROcBFIBFIΔROcROc=ROc∑k=1nykSCk-yBFcyBFIBFc-ROc2,
where S(BFI|BFc) represent the dimensionless
sensitivity index of BFI (output) with BFc (uncertain input) and
S(BFI|ROc) with ROc.
The dimensionless sensitivity index is also called the “elasticity index”,
and it reflects the proportional relationship between the relative error in
BFI and the relative error in parameters (e.g., if S(BFI|BFc)=1.5 and the
relative error in BFc is 5 %, then the relative
error in BFI is 1.5 times 5 % = 7.5 %). After determining the values
of BFC, ROC, BFI, y, yk, and SCk, the sensitivity indices
S(BFI|BFc) and S(BFI|ROc) can
be calculated and compared.
Variables yk and SCk
In addition to the two parameters, there are two variables (SCk and
yk) in Eq. (1). This section describes the sensitivity analysis of BFI
to these two variables. Similar to Sect. 2.1, the partial derivatives
of bk in Eq. (1) to SCk and yk are obtained (see Eqs. A5 and A6),
and the partial derivatives of BFI to SCk and yk are further obtained
(see Eqs. A7 and A8):
∂BFI∂SCk=1BFc-ROc,∂BFI∂yk=∑k=1nSCk-ROc-nBFIBFc-ROcyBFc-ROc.
According to previous studies (Munyaneza et al., 2012; Cartwright et al.,
2014; Miller et al., 2014; Okello et al., 2018) and this study (Table 1),
the difference between BFC and ROC is often greater than
100 µs cm-1. Therefore, ∂BFI/∂SCk is
usually less than 0.01 cm µs-1. Appendix C shows that the value of
∂BFI/∂yk is usually far less than 1 day m-3.
Basic information, parameter sensitivity analysis, and uncertainty
estimation results for 24 basins in the US. The asterisk in the “area” column
indicates that the values are estimated based on data from adjacent sites.
Small errors in SCk and yk cause errors in BFI:
ΔSCkBFI=∂BFI∂SCkΔSCk=ΔSCkBFc-ROc,ΔykBFI=∂BFI∂ykΔyk=∑k=1nSCk-ROc-nBFIBFc-ROcyBFc-ROcΔyk.
The errors in BFI caused by SCk and yk are summed up to obtain the
error in BFI caused by ∑k=1nSCk and
∑k=1nyk in the whole time series:
Δ∑k=1nSCkBFI=∑k=1nΔSCkBFI=∑k=1nΔSCkBFc-ROc=1BFc-ROc∑k=1nΔSCk,Δ∑k=1nykBFI=∑k=1nΔykBFI=∑k=1n,∑k=1nSCk-ROc-nBFIBFc-ROcyBFc-ROcΔyk=∑k=1nSCk-ROc-nBFIBFc-ROcyBFc-ROc∑k=1nΔyk.
Wagner et al. (2006) reported that the uncertainty in instruments is usually
less than 5 % for SCk (<100µs cm-1) and less than 3 %
for SCk (>100µs cm-1). According to Hamilton
and Moore (2012), streamflow data from the US Geological Survey's (USGS) are often assumed by analysts to be
accurate and precise within ±5 % at the 95 % confidence interval.
In this study, the error ranges of SCk and yk are all considered to
be ±5 %. The errors in SCk and yk mainly comprise random
measurement errors which mostly follow a normal distribution or a uniform
distribution (Huang and Chen, 2011). Considering the mutual offset of random
errors, when the time series (n) is sufficiently long,
∑k=1nΔSCck in Eq. (15) and
∑k=1nΔyk in Eq. (16) will approach zero.
The analysis of ∑k=1nΔSCk and
∑k=1nΔyk under different time series (n) and
different error distributions (normal distribution or uniform distribution)
of a surface water station (USGS site number 0297100) showed that the random
errors in daily average conductivity and streamflow have a negligible effect
on BFI when the time series is greater than 365 days (see Supplement S1 for detail).
Uncertainty estimationPrevious attempts
According to previous studies, in the case where a variable g is calculated
as a function of several factor x1, x2, x3, …, xn
(e.g., g=G(x1, x2, x3, …, xn)) and based on the assumptions that the
factors are uncorrelated and have a Gaussian distribution, the transfer
equation (also known as Gaussian error propagation) between the uncertainty in the independent factors and the uncertainty in g is
Wg=∂g∂x1Wx12+∂g∂x2Wx22+…+∂g∂xnWxn2,
where Wg, Wx1, Wx2, and Wxn are the same type of uncertainty
values (e.g., all average errors or all standard deviations) for g, x1, x2,
and xn, respectively. A more detailed description of this equation can
be found in Taylor (1982), Kline (1985), and Ernest (2005).
According to Genereux (1998), “While any set of consistent uncertainty (W)
values may be propagated using Gaussian error propagation, using standard
deviations multiplied by t values from the Student's t distribution (each t for
the same confidence level, such as 95 %) has the advantage of providing a
clear meaning (tied to a confidence interval) for the computed uncertainty
would correspond to, for example, 95 % confidence limits on BFI”.
Based on the above principle, Genereux (1998) substituted Eq. (18) into
Eq. (17) to derive the uncertainty estimation equation (Eq. 19) of the CMB method:
fbf=SCk-ROcBFc-ROc,Wfbf=fbfBFc-ROcWBFC2+1-fbfBFc-ROcWROc2+1BFc-ROcWSC2,
where fbf is the ratio of baseflow to streamflow in a single calculation
process, Wfbf is the uncertainty in fbf at the 95 % confidence
interval, WBFC and WROC are the standard deviations of the BFC
and ROC multiplied by the t value (α=0.05; two-tail) from the
Student's distribution, and WSC is the analytical error in conductivity
multiplied by the t value (α=0.05; two-tail) (Miller et al., 2014).
Better estimates of the uncertainty in fbf within a single calculation
step can be obtained using Eq. (19). Hydrologists usually approximate the
uncertainty in BFI by averaging the uncertainty in all steps (Genereux, 1998;
Miller et al., 2014). However, this method does not consider the
mutual offset of the conductivity measurement errors and cannot accurately
reflect the uncertainty in BFI. In this study, an uncertainty estimation
equation of BFI is derived on the basis of the parameter sensitivity analysis.
Uncertainty estimation of BFI
BFI is a function of BFc, ROc, SCk, and yk. In
addition, the uncertainties in BFc, ROc, SCk, and
yk are independent of each other. As explained earlier (Sect. 2.2), the random
errors in daily average conductivity and streamflow have a negligible effect
on BFI when the time series (n) is greater than 365 days (1 year);
therefore, the uncertainty in BFI can be expressed as
WBFI=∂BFI∂BFcWBFC2+∂BFI∂ROcWROC2,∂BFI∂BFc=SBFI|BFcBFIBFc,∂BFI∂ROc=SBFI|ROcBFIROc.
Then, Eq. (20) can be rewritten as
WBFI=SBFI|BFcBFIBFcWBFC2+SBFI|ROcBFIROcWROC2,
where WBFI, WBFC, and WROC are the same type of uncertainty values
for BFI, BFC, and ROC, respectively, as described above.
ApplicationData and processing
The above sensitivity analysis and uncertainty estimation methods were
applied to 24 catchments in the US (Table 1). All basins used in
this study are perennial streams, with drainage areas ranging from
10 to 1 258 481 km2. Each gage has about at least 1 year of
continuous streamflow and conductivity for the same period of records. All
streamflow and conductivity data are daily average values retrieved from the
USGS National Water Information System (NWIS) website: http://waterdata.usgs.gov/nwis
(last access: September 2018).
The daily baseflow of each basin was calculated using Eq. (1). The
99th percentile of the conductivity of each year was used as BFC, and linear
variation in baseflow conductivity between years was assumed. The
first percentile of the conductivity of the whole series of streamflow in each
basin was used as the ROC. The total baseflow b, total streamflow y, and
baseflow index BFI of each watershed were then calculated. According to the
results of the hydrograph separation, the parameter sensitivity indices of
BFI for mean BFC (S(BFI|BFc))
and ROC (S(BFI|ROc)) were calculated using
Eqs. (9) and (10), respectively.
Finally, the uncertainty in fbf in each step was calculated using
Eq. (19) and averaged to obtain the mean Wfbf for each basin. The
uncertainty (WBFI) in BFI was directly calculated using Eq. (23), and
then the values of mean Wfbf and WBFI were compared. For each basin,
WBFC is the standard deviation of the BFC of the whole series
multiplied by the t value (α=0.05; two-tail) from the Student's
distribution, WROC is the standard deviation of the lowest 1 % of
measured conductivity multiplied by the t value (α=0.05; two-tail)
from the Student's distribution, and WSC is the analytical error in the
conductivity (5 %) multiplied by the t value (α=0.05; two-tail).
Scatterplots of sensitivity indices vs. time series (n) and drainage
area of the 24 US basins. The watershed area uses a logarithmic axis, while the
others are linear axes.
Results and discussion
The calculation results are shown in Table 1. The average baseflow index of
the 24 watersheds is 0.34, the average sensitivity index of BFI for mean BFC
(S(BFI|BFc)) is -1.40, and the average sensitivity index of BFI for ROC
(S(BFI|ROc)) is -0.89.
The negative sensitivity indices indicate negative correlations
between BFI and BFC (ROC). The absolute value of the sensitivity
index for BFC is generally greater than that for ROC, indicating
that BFI is more affected by BFC – for example, if there are 10 %
uncertainties in both BFC and ROC, then BFC leads to -1.40
times 10 % of uncertainty in BFI (-14.0 %), while ROC leads
to -0.89 times 10 % (-8.9 %). Therefore, the determination of BFC
requires more caution, and any small error may lead to greater uncertainty
in BFI. Miller et al. (2014) reported that anthropogenic activities over
long periods of time or year to year changes in the elevation of the water
table may result in temporal changes in BFC. They recommended taking
different BFC values per year based on the conductivity values during
low-flow periods to avoid the effects of temporal fluctuations in BFC.
The sensitivity index of BFI for BFC shows a decreasing trend with the
increase in time series (n) (Fig. 1a) and an increasing trend with
increasing watershed area (Fig. 1b), with correlation coefficients
of 0.1492 and 0.3577, respectively. Although the correlations are not obvious,
they have important guiding significance. Large basins comprise many
different subsurface flow paths contributing to streams (Okello et al.,
2018), each of which has a unique conductivity value (Miller et al., 2014).
Furthermore, it is difficult to represent the conductivity characteristics
of subsurface flow with a special value. Therefore, the CMB method has
higher applicability to long time series for small watersheds.
Scatterplot of uncertainty in BFI (WBFI) and mean
uncertainty in fbf (mean Wfbf).
The sensitivity index of BFI for ROC did not change significantly with
the increase in time series and watershed area (Fig. 1c and d).
During rainstorms, the conductivity of streams became similar to that of the
rainfall (Stewart et al., 2007). The electrical conductivity of rainfall
varies slightly by region, is usually at a fixed value, and has no
significant relationship with the basin area and year (Munyaneza et al.,
2012). Therefore, the temporal and spatial variation characteristics of BFI
for ROC are not obvious.
Estimation results of (a) baseflow index and uncertainty (b)
sensitivity indices, under different low-conductivity soil flow ratios, assuming
high-conductivity baseflow remains unchanged. An asterisk indicates the results of
the estimation considering the low-conductivity soil flow.
Genereux's method (Eq. 19) estimates the average uncertainty in BFI in the
24 basins (average of mean Wfbf) to be 0.20, whereas the average
uncertainty in BFI (average of WBFI) calculated directly using the
proposed method (Eq. 23) is 0.11 (Table 1). Mean Wfbf in each basin is
generally larger than WBFI (WBFI is about 0.51 times of
mean Wfbf), and there is a significant linear correlation (Fig. 2). This
shows that the two methods have the same volatility characteristics for BFI
uncertainty estimation, but Genereux's method (Eq. 19) often overestimates
the uncertainty in BFI. This also means that when the time series is longer
than 365 days, the measurement errors in conductivity and streamflow will
cancel each other out and thus reduce the uncertainty in BFI (about half of the original).
The conductivity of shallow subsurface and soil flow in real watersheds is
sensitive to climatic conditions and usually shows obvious fluctuations
(Miller et al., 2014). The CMB method classifies high-conductivity flow
(e.g., deep subsurface flow) as baseflow and low-conductivity flow (e.g.,
local shallow soil flow) as surface runoff (Cartwright et al., 2014).
Therefore, in the watershed containing a large number of low-conductivity
soil flows, the BFI calculated by the CMB method comprised only the baseflow
index of the deep subsurface flow. The parameter sensitivity indices and
uncertainty in the deep subsurface flow were also calculated by the methods
of this paper. Cartwright et al. (2014) showed that the ratio of
low-conductivity soil flow to high-conductivity subsurface flow in the
Barwon Basin in southeastern Australia is close to 1. If only the BFI
doubles and other parameters remain unchanged, then the sensitivity indices
calculated by Eqs. (9) and (10) are halved, whereas the uncertainty
calculated by Eq. (23) remains unchanged. Therefore, nonconstant soil flow
conductivity may lead to an overestimation of sensitivity, but it has less
impact on uncertainty estimates.
To better understand the effects of low-conductivity soil flow on BFI,
parameter sensitivity, and the uncertainty estimation results, this study
assumed that the high-conductivity baseflow is constant and the ratio of
low-conductivity soil flow to high-conductivity baseflow (SF / BF) is
between 0 and 1. Based on the average values of the 24 watersheds mentioned in
Table 1, the estimation results with and without consideration of low-conductivity
soil flow were analyzed (Fig. 3). The CMB method used in this study neglects
low-conductivity soil flow; thus, the BFI, sensitivity indices, and
uncertainty do not change with a change in SF / BF (orange line in Fig. 3). When
the low-conductivity soil flow is considered in the estimations, the BFI
value is found to increase linearly with the increase in SF / BF (blue solid line
in Fig. 3a); the absolute values of sensitivity indices decrease
nonlinearly with the increase in SF/BF; and the difference between
S(BFI|BFc) and
S(BFI|ROc)
decreases gradually (blue line in Fig. 3b). The uncertainty in BFI does
not fluctuate with changes in SF / BF (blue dashed line in Fig. 3a). In
general, the deviation between BFI, sensitivity indices, and the “true
values” gradually increases with an increase in the low-conductivity soil flow of a basin.
Conclusions
This study analyzed the sensitivity of BFI, calculated using the CMB method,
to errors or uncertainties in the parameters BFC and ROC and the
variables yk and SCk. In addition, the uncertainty in BFI was
calculated. The equations derived in this study (Eqs. 9 and 10) could
calculate the sensitivity indices of BFI for BFC and ROC. For time
series longer than 365 days, the measurement errors in conductivity and
streamflow exhibited a mutual offset effect, and their influence on BFI
could be neglected. Considering the mutual offset, the uncertainty in BFI
would be halved. From this perspective, Eq. (23) could estimate the
uncertainty in BFI for time series longer than 365 days. The application of
the method to 24 basins in the US showed that BFI is more
sensitive to BFC. Future studies should dedicate more effort to determining the value of BFC. In addition, the CMB method may be more
suitable for long time series of small watersheds.
Systematic errors in specific conductance and streamflow as well as temporal
and spatial variations in baseflow conductivity may be the main sources of
BFI uncertainty. Better rating curves are probably more important than
better loggers, and understanding the specific conductance of baseflow is
likely more important than understanding that of surface runoff.
The above conclusions were drawn only from the average of the studied
24 basins, and further research in other countries or in more watersheds is thus required. This study focused on the two-component hydrograph separation
method with conductivity as a tracer, but parameter sensitivity analysis and
uncertainty analysis methods involving other tracers are similar. Therefore,
similar equations can easily be derived by referring to the findings of this study.
Data availability
All streamflow and conductivity data can be retrieved from
the US Geological Survey's (USGS) National Water Information System (NWIS)
website using the special gage number: http://waterdata.usgs.gov/nwis (NWIS, 2018).
∂BFI∂yk=∑k=1nSCk-ROc-nBFIBFc-ROcyBFc-ROc(seeEq.A8).
Because of n>0, BFI > 0 (BFC- ROC)>0,
the above formula can be simplified:
∂BFI∂yk<∑k=1nSCk-ROcyBFc-ROc.
Since BFC is usually much larger than SCk, the above formula
can be rewritten as
∂BFI∂yk<∑k=1nBFc-ROcyBFc-ROc=nBFc-ROcyBFc-ROc=ny=1y‾.
The daily average streamflow (y‾) is usually much larger than
1 m3 day-1, so ∂BFI/∂yk is
far less than 1 day m-3.
The supplement related to this article is available online at: https://doi.org/10.5194/hess-23-1103-2019-supplement.
Author contributions
WY, CX, and XL developed the research train of thought. WY and
CX completed the parameters' sensitivity analysis. XL completed the uncertainty
estimate of BFI. WY carried out most of the data analysis and prepared the
manuscript with contributions from all coauthors.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (41572216),
the Provincial School Co-construction Project Special – Leading Technology
Guide (SXGJQY2017-6), the China Geological Survey Shenyang Geological Survey
Center “Changji Economic Circle Geological Environment Survey” project (121201007000150012),
and the Jilin Province Key Geological Foundation Project (2014-13). We thank
the anonymous reviewers for useful comments to improve the manuscript.
Edited by: Markus Hrachowitz
Reviewed by: three anonymous referees
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