Introduction
Time series analysis
Summary statistics
-
The spring variability coefficient (SVC, Flora 2004), which is the ratio between percentile 0.1 and 0.9. It can be used to describe the responsiveness of karst springs and is less sensitive to spurious data than the ratio between the maximum and the minimum values;
-
The coefficient of variation (CV) has also been used to describe the responsiveness of karst springs (Mangin 1975; Flora 2004). This coefficient is computed as the ratio of the standard deviation of a time series to its mean value. It expresses the level of dispersion around the mean, whatever the frequencies.
-
The memory effect (Mangin 1984), in days, which corresponds to the lag for which the autocorrelation function falls below \(0.2\). The memory effect is automatically computed by XLKarst for lags lower than 125 days using the estimate of the autocorrelation function given by Jenkins and Watts (1968). Memory effects higher than 125 days are not computed, since their estimates are biased by the annual cycle of recharge dynamics.
-
The base flow index (BFI), which is computed as the ratio, in volume, of the baseflow to the total flow. A lot of baseflow separation methods exist in the literature to compute the BFI. Recently, Ladson et al. (2013) proposed a standard approach using the Lyne and Hollick (1979) digital filter. The approach used a reflection of the time series of 30 days to address “warm up” issues, and up to 9 passes of the digital filter with a filter coefficient ranging from 0.9 to 0.98. The result of the BFI is a function of the filter coefficient and the number of passes, which are respectively 0.91 and 3 as default in XLKarst. It is computed on the entire period covered by the time series, which should start and end at the beginning of a hydrological cycle. The daily baseflow time series is also reported on the plot showing the discharge time series;
-
The regulation time, \({T}_{\mathrm{reg}}\), in days, which is only computed for daily time series. \({T}_{\mathrm{reg}}\) corresponds to the half of the spectral density function for the \(0\) frequency, which is computed as the discrete Fourrier transform of the autocorrelation function using a Tuckey lag window (Jenkins and Watts 1968), with a truncation point \(m=125\) days of the correlation function. \({T}_{\mathrm{reg}}\) expresses the relative contribution of long term processes with periods higher than 2 × 125 = 250 days to the total variance of the time series.
-
The ratio \({\sigma }_{250}/\sigma (\mathrm{\%})\), which is a new parameter that can be used to assess \({T}_{\mathrm{reg}}\). XLKarst uses a moving average filter of 250 days to assess the standard deviation of long term processes (\({\sigma }_{250}\)). The result is divided by the standard deviation (\(\sigma\)) of the raw time series over the same period. It is expressed in % and can be related to \({T}_{\mathrm{reg}}\) following Eq. 1:$${T}_{\mathrm{reg}}\approx 125\times {\left(\frac{{\sigma }_{250}}{\sigma }\right)}^{2}$$(1)The ratio \({\sigma }_{250}/\sigma\) can be computed whatever the time step of the time series, assuming that this time step is short enough to capture most discharge fluctuations.
Correlation and spectral analyses
Use of correlation and spectral analyses in karst hydrology
-
the input signal, i.e. the rainfall time series, does not show any autocorrelation. The use of daily rainfall time series is often recommended since shorter time steps would induce a higher autocorrelation in the rainfall time series,
-
the karst system may be conceptualized as a linear system, which can be discussed using the squared coherency spectrum from cross-spectral analyses
Karst spring hydrograph analysis (“Discharge” menu)
Probability plot for cumulative distribution function of hydrologic time series
Recession curves analysis
3.2.1 Method
Automatic selection of flood recession
-
“Min duration” defines the minimum duration in days of a flood recession. The default value is 90 days. As shown by Abirifard et al. (2022), the dynamical volume will be more reliable for aquifers that show a long dry season, which also means that only long flood recessions must be considered to derive baseflow recession parameters.
-
“Min peak” defines the minimum value of the discharge that can be considered as a flood peak. The default value is the percentile 0.9 computed on the whole discharge time series.
-
The “lag” parameter is an optional parameter used when the flood dynamics is controlled by long term variations while short term variations induce many fluctuations of the flow during the recession. By default, a flood peak is defined at time t when \({Q}_{(t)}>{Q}_{\mathrm{Min peak}}\) and \({Q}_{(t-\mathrm{lag})}< {Q}_{(t)}> {Q}_{(t+\mathrm{lag})}\) with \(\mathrm{lag}=1\) time step. This lag parameter can be increased if the automatic selection of flood peaks failed to capture the actual flood peak.
List of parameters derived from recession cure analysis
-
\(t\) is the relative time since the flood peak;
-
\({Q}_{0}\) (m3/s) is the flood peak at the beginning of the recession, i.e. at \(t=0\);
-
\({Q}_{R0}\) (m3/s) is the simulated discharge using \(\phi \left(t\right)\) at \(t=0\). It is a fictive discharge that conceptually represents the discharge coming from storage at the beginning of the recession;
-
\({Q'}_{0}\) is the measured (and the simulated) discharge at \(t={t}_{\mathrm{i}}\);
-
\({q}_{0}\) (m3/s) is the initial infiltration rate, computed as the difference between Q0 and QR0;
-
\({t}_{\mathrm{i}}\) (d) represents the duration of the infiltration after which the baseflow can be modeled by Maillet’s law \(\phi \left(t\right)\);
-
\(\alpha\) (d−1) is the recession coefficient
-
\(\varepsilon\) is the non-dimensional coefficient of heterogeneity used in the infiltration function \(\psi (t)\).
Automatic calibration procedure
-
\({t}_{\upvarepsilon }\), which defines the number of points for \({t\le t}_{\upvarepsilon }\) to take into account for the optimization of the \(\psi\) function (Eq. 2)
-
\({t}_{\mathrm{end}}\), which defines the number of points for \({t}_{\mathrm{i}}<t<{t}_{\mathrm{end}}\) to take into account the optimization of the \(\phi\) function (Eq. 3)
-
\(\varepsilon\) as the slope of the regression line given by the linearization of Eq. 2:$$\frac{{q}_{0}}{Q\left(t\right)-\phi (t)}\times \left(1-{~}^{t}\!\left/ \!{~}_{{t}_{\mathrm{i}}}\right.\right)-1=\varepsilon t$$(7)
Use of the recession curves analysis for karst system classification
-
The \(i\) parameter, called the “infiltration delay”, which is the mean value of \(\frac{1-{~}^{t}\!\left/ \!{~}_{{t}_{\mathrm{i}}}\right.}{1+\varepsilon t}\) for \(t=2\) days;
-
The \(k\) parameter, called the “regulating power”, which is the ratio of the maximum value of \({V}_{\mathrm{d}}\) to the mean annual volume drained by the karst spring. The assessment of \(k\) requires the use of long time series to ensure one captures the seasonal range in the assessment of the maximum value of \({V}_{\mathrm{d}}\)
-
An expanded uncertainty \({\delta }_{i}\) is given by Eq. 8, with \(n\) the number of \(i\) assessments and \(\mathrm{std}(i)\) the standard deviation of the \(i\) values:
-
The \({\delta }_{k}\) uncertainty is computed as the error that would be done on the \({V}_{\mathrm{d}}\) assessment if the recession curve used to compute the value of \({V}_{\mathrm{d}}\) was discarded. A high uncertainty may reflect an inconsistent value for the dynamical volume that is used to compute \(k\). Such a result requires a closer inspection of the corresponding recession curve.
Application to the Fontaine de Nîmes karst system
Site description and dataset
Results
Summary statistics
Name | FdN |
---|---|
Number of data | 8036 |
Date from | 01/09/1999 |
Date to | 31/08/2021 |
Mean | 0.55 |
Median | 0.09 |
σ | 1.27 |
Min | 0.01 |
Max | 16.28 |
Min/Max | 5.53E-04 |
SVC Q10/Q90 | 28.3 |
Q25/Q50 | 1.42 |
Coeff. of variation | 2.32 |
Memory effect (d) | 15 |
Regulation time (d) | 22 |
σ250/σ (%) | 0.41 |
BFI | 0.42 |