Global climate change can have impacts on characteristics of rainfall–runoff events and subsequently on the hydrological regime. Meanwhile, the catchment itself changes due to anthropogenic influences. However, it is not easy to prove the link between the hydrology and the forcings. In this context, it might be meaningful to detect the temporal changes of catchments independent from climate change by investigating existing long-term discharge records. For this purpose, a new stochastic system based on copulas for time series analysis is introduced in this study.

A statistical tool like copula has the advantage to scrutinize the dependence structure of the data and, thus, can be used to attribute the catchment behavior by focusing on the following aspects of the statistics defined in the copula domain: (1) copula asymmetry, which can capture the nonsymmetric property of discharge data, differs from one catchment to another due to the intrinsic nature of both runoff and catchment; and (2) copula distances can assist in identifying catchment change by revealing the variability and interdependency of dependence structures.

These measures were calculated for 100 years of daily discharges for the Rhine River and these analyses detected epochs of change in the flow sequences. In a follow-up study, we compared the results of copula asymmetry and copula distance applied to two flow models: (i) antecedent precipitation index (API) and (ii) simulated discharge time series generated by a hydrological model. The results of copula-based analysis of hydrological time series seem to support the assumption that the Neckar catchment had started to change around 1976 and stayed unusual until 1990.

In order to understand the water cycle behavior of a region, it is important to determine its characteristics, but this is difficult to achieve due to the diversity of the system response at different time and space scales. In particular, temporal variability makes parameter estimation difficult and the assessment of model uncertainty essential. As a part of the endeavor to understand the hydrological system, the objective of this research, assessing the anthropogenic impacts on the catchment characteristic independent of the climate change, is therefore important, yet hard to accomplish.

The first possible approach is to statistically test the existence or change of trend in hydrological time series which can be related to climate changes or anthropogenic impacts. Mann–Kendall's test was performed to confirm the existence of a trend in the annual discharge, precipitation and sediment loads, then the human intervention and climate impacts based on the available information of the catchments were discussed (Wu et al., 2012). Pettitt's method (Pettitt, 1979) can be used to detect the time point of trend alternation and analyze the impacts based on a double mass curve (Gao et al., 2013) or a hydrological model (Karlsson et al., 2014). These nonparametric methods for detecting the signal seem, however, not capable enough of explaining when and how much the system had changed, thus making it still difficult to relate the change to human activities.

On the other hand, runoff events are initiated by precipitation, then modified by the state and physical features of the catchment. This implies that the integrated information of catchment status might be retrieved by analyzing the discharge time series itself. Focusing on this property, the attempts can be made for capturing the temporal dependence structure of runoff by time series models. The classical time series model, autoregressive integrated moving average (ARIMA), is designed to describe a stationary stochastic process based on the temporal correlation structure of Gaussian random variables (Box and Jenkins, 1976). However, the stationarity of the data is not guaranteed in reality, thus a number of alternative approaches have been suggested. While the application of Fourier analysis is basically for stationary processes, the analysis using empirical mode decomposition (Huang et al., 1998) overcomes the restriction of stationarity by allowing the frequency and local variance of a time series to vary within a component and to separate the signals adaptively by scale. Autoregressive conditional heteroskedasticity (ARCH) models lose the assumption of stationarity to a certain extent so that variance is not constant; however, they model the variance in a similar way to ARIMA. Although inventions and efforts to overcome the limitation of stationarity have been made, it seems still inadequate to model dynamic changes of hydrological processes with these time series models.

Alternatively there is a statistical concept, the copula, which has advantages to model the multivariate dependence independently from marginals and recently adopted in the field of hydrology. A copula (Sklar, 1959) is a multivariate probability distribution designed to flexibly model dependence structure in the uniform (quantile) domain. The use of copulas in hydrology can be found for the assessment of extreme events by considering flooding as a joint behavior of peak and volume (De Michele and Salvadori, 2003). Copulas have been applied to describe the spatiotemporal uncertainty of precipitation (Bárdossy and Pegram, 2009) or the inhomogeneity of groundwater parameters (Bárdossy and Li, 2008). Asymmetry of dependence in a time series can be tested in the framework of a finite-state Markov's chain transition probability matrix (Sharifdoost et al., 2009). Dissimilarity measures can be defined by means of a copula modeling the correlation structure of pairs of discharge time series in order to identify the similarity of catchments with the purpose of transferring catchment properties from one to the other (Samaniego et al., 2010). We aim at utilizing copulas as an alternative to classical time series models and an efficient tool for time series analysis to overcome these hydrological challenges.

The main interest of this study is to assess the human intervention and climate change impacts on hydrological regime for the strategy of future development in the region. For achieving this goal, seven daily discharge gauging stations in southwest Germany (Fig. 1), which have 100 years of daily discharge records, were chosen and extensively analyzed. The gauging stations Andernach, Kaub, Worms and Maxau are located in the main stream of the Rhine, while Kalkofen, Cochem and Plochingen are located on tributaries. For further analysis, daily precipitation and temperature records in the Baden-Württemberg state of Germany for the last 50 years were obtained from the German Weather Service (DWD, 2014). Also, 77 discharge records obtained from the Global Runoff Date Centre in Germany (GRDC, 2012) were utilized.

Locations of seven discharge gauging stations in the upper Rhine region.

The following are the novel aspects introduced in this study:

(1) The catchment characteristics are defined based on copulas and estimated from discharge data. Also, the changes of catchment characteristics are investigated by tracing the temporal change of these statistics.

(2) A method to model systematic changes of dependence structure with the help of copulas is suggested, then its variability and interrelationship with the time series are examined.

(3) Anthropogenic impacts are assessed by the discharge–precipitation relation using API and a hydrological model with copula-based measures.

In this section, the application of copulas to time series is articulated after a brief introduction. The very basics about copulas are presented here; further information can be obtained from Joe (1997) or Nelsen (2006).

In probability theory and statistics, a copula is a multivariate probability
distribution for which the marginal probability distribution of each
variable is uniform.

For the application of copulas to time series analysis, a stochastic system
should be presumed to be similar to the case of spatial copulas (Bárdossy
and Li, 2008): the random variable at time

High and low values might have different dependences in general. Measuring
the asymmetry of copulas could reveal substantial aspects of time series
data, which are not illuminated in the Gaussian approach. We believe
statistics defined by copula shape and calculated from observed discharge
time series to be a new idea. The two types of asymmetry, “asymmetry1” and
“asymmetry2”, are considered for two diagonals on 2-dimensional copulas,
which can be described as a function of time lag

Visualization of

Sketch of the transformation of the values from sample hydrograph
(left) to the points on the scatterplot of ranks (right): empirical copula
calculated from two values separated by time lag

Figure 3 shows the scatterplot of ranked values of a discharge time series
with time lag

Asymmetry functions can be considered as statistics calculated from the observed discharge time series and an important assumption can be made: “asymmetry2 is related to catchment characteristics”. This idea will be discussed and demonstrated in this section. Figure 5 (upper left) shows parts of the hydrographs of seven gauging stations in southwest Germany.

First, an important natural property of discharge seen in this figure is that
the durations of high flow and low flow periods are not symmetric: flood
events, which are initiated by rainfall or snowmelt, do not continue for a
long time because the duration of runoff to rivers is comparatively short. On
the other hand, discharge keeps decreasing and stays low for no rain periods.
This means that, if two consecutive values in a time series are chosen for
small time lag

Annual cycles of mean discharge measured at seven sites in the Rhine basin after smoothing (left) and annual cycle of standard deviation after smoothing (right).

Discharge time series measured at seven sites in the Rhine basin
between 1950 and 1955 before applying normalization (upper left) and after
applying normalization (upper right).

Second, the rates of increase and decrease of discharge are not symmetrical
in the upper limb compared to the lower limb of the hydrograph (Fig. 3): soon
after the rainfall, the river flow rises sharply, but once the rain stops and
peak discharge is observed, then the water level starts to decrease,
typically more slowly on the recession than the rising limb of the
hydrograph. This leads to the negative values of

The change of

In order to reduce such seasonal impacts on the analysis of hydrological time
series, deseasonalization measures can be applied, for example, for daily
stream flow (Grimaldi, 2004). Adopting Grimaldi's method, all the time series
are normalized in this study. First, the mean

The confidence intervals in the figures are gained by calculating

It can be seen that the variation of

Relation between asymmetry of discharge data and catchment
characteristics:

Temporal change of asymmetry2:

This result demonstrates that the information extracted from discharge is
related to the basic information of its catchment to a certain extent. Since
the principal objective is to assess anthropogenic impacts, the idea
introduced now is to use this measure for evaluating the catchment change by
calculating chronological changes of

Temporal change of asymmetry2 is defined

The comparison of

However, it cannot be ascertained whether this is caused by the simultaneous
change of the catchments, the long-term meteorological behavior in the region
or just randomness in the stationary process. To overcome this, temporal
behavior of discharge and temperature were first checked by calculating the
mean, the standard deviation and the minimum in a time window
centered on time

What seems to be a useful outcome from the above exploratory analysis is that
(i) the behavior of

Moving average and standard deviation of the seven daily discharge
records for the window size

Annual minimum (upper panel) and mean of aggregated daily temperature (lower panel) in the Baden-Württemberg state of Germany.

As an alternative to copula asymmetry, which emphasizes the behavior in the corners of copulas, copula distance is suggested here so that the characteristic behavior can be captured in the entire domain of the copula. Calculating this for each time step for different time series and comparing them hopefully exhibits the changes of dependence structure and therefore the catchment change.

The basic idea behind the copula distance is to apply the Cramér–Von
Mises type distance, which by design measures the goodness of fit for two
distribution functions, to two copulas as follows:

In order to apply the concept of copula distance to time series, the
adoption of two copulas in different timescales is considered. An empirical
copula can be obtained from an entire time series which contains the
averaged information of all the time points (global copula). Another empirical
copula can be obtained for a certain time window of width

First of all, the values of

Copula distances of discharge time series in moving time window: variance (top), distance type1 (middle) and distance type2 (bottom); each panel contains the 80 % confidence interval of Gaussian process and one of its realizations (dashed line). The arrows point to 1947, 1982, 2000 and 1977 in which the clear signals of anomalies are detected for all four discharge time series: Andernach (ANDE), Cochem (COCH), Maxau (MAXA) and Plochingen (PLOC).

For copula distance type1, the global copula can be considered as an average
state of the copula, while the local copula can be regarded as a realization
of a possible state of a copula at time step

Variance and copula variance calculated for four discharge time series (ANDE: Andernach, COCH: Cochem, MAXA: Maxau, PLOC: Plochingen).

In the previous section, copula variance was defined as a measure of the
variability characteristic of the copula itself. Here, it is determined
whether covariance can be defined for two copula densities

First, it can be said that the behavior of covariance and correlation in a
moving window are different from

Covariance, correlation, copula covariance and copula correlation between four discharge data (AN: Andernach, CO: Cochem, MA: Maxau, PL: Plochingen).

Copula covariance and copula correlation can be defined similar to copula
variance in order to quantify the overall behavior of two time series
(Sugimoto, 2014).

The measures using copula distance are different from the conventional statistics. This behavior can be explained by the fact that the autocopula has more substantial information about temporal dependence structure than the autocorrelation. Using these measures might enable us to take advantage of a different way of seeing the dependence between time series.

What is new in the analysis of this section is that (i) measures based on copula distance show the different properties of time series in comparison to conventional statistics and (ii) there are significant signals of copula distances for certain time periods common to all the discharge data.

Copula distances of discharge time series in moving time window: covariance (top), correlation (second), copula distance type3 (third) and copula distance type4 (bottom). The arrows point 1947, 1982 and 2000 in which the clear signals of anomalies are detected for the comparisons between four discharge time series: Andernach (ANDE), Cochem (COCH), Maxau (MAXA), Plochingen (PLOC).

Locations of the precipitation gauge stations within Baden-Württemberg (Germany) indicated by colored circles. The upper Neckar catchment (USGS, 2014) is identified by the light green area and the location of the gauging station is indicated by a square.

Copula distances of API time series in moving time window: variance (top), copula distance type1 (middle) and copula distance type2 (bottom), where “C” denotes the central, “SW” denotes the southwest, “NW” denotes the northwest and “NE” denotes the northeast parts of the Baden-Württemberg state of Germany, each containing 80 % confidence intervals of Gaussian process and one of its realizations (dashed line). The arrows indicate the years in which anomalies are detected in the previous analysis (Fig. 10).

The difficulty of analyzing discharge time series in order to detect catchment change is that it is not clear whether the temporal change of stochastic information is caused by catchment change or merely by random behavior of precipitation. To gain an understanding of this process, we attempted to eliminate the influence of precipitation using, first, an antecedent precipitation index (API) for comparison with discharge, second, using a hydrological model with the parameter sets calibrated and fixed for the entire simulation time period.

An API time series, which is generated from observed precipitation time
series and behaves similarly to discharge, is used instead of precipitation.

For this investigation, precipitation data were carefully chosen for four
regions (northwest, northeast, southwest and central) in
Baden-Württemberg (Germany) so that they have several almost continuous
daily records between 1935 and 2005. Figure 12 shows the locations of
measuring stations. The precipitation time series were aggregated into one
for each region by taking their daily average, then four API time series were
calculated in total by Eq. (35). Figure 13 shows the resulting copula
distances

What can be recognized first from Fig. 13 is that the magnitudes of

Variance and copula variance calculated for API time series of four regions in the Baden-Württemberg state of Germany (C: central, SW: southwest, NW: northwest, NE: northeast).

Covariance, correlation, copula covariance and copula correlation between API time series from four regions in the Baden-Württemberg state of Germany (C: central, SW: southwest, NW: northwest, NE: northeast).

Copula distances of API time series in moving time window: covariance (top), correlation (second), copula distance type3 (third) and copula distance type4 (bottom). The arrows indicate the years in which anomalies are detected in the previous analysis (Fig. 11).

Copula distance type3 (top) and type4 (bottom) between four discharge and one API time series which is aggregated for all the daily precipitations depicted in Fig. 12. The arrows indicate the years in which anomalies are detected in the previous analysis (Fig. 11).

For further verification, copula distance type3 and type4 between discharge and API time series were calculated as shown in Fig. 15. This result also shows there is no clear relation between API and discharge time series around 1982.

In this section, simulated discharge time series are generated by a conceptual hydrological model, HBV (Bergström, 1976, 1995), which takes daily precipitation and temperature records as input and simulates discharges for smaller catchments as an example of discharge, to compare with observed discharge, in order to check if differences might occur due to the method.

Thus the idea behind this methodology is similar to the case of API: a hydrological model with the parameters fixed for the entire time period represents the catchment not influenced by anthropogenic impacts. Then, the discharges simulated by this model should not depend on catchment change, while observed discharge is assumed to be influenced by both catchment and precipitation.

For the study area, the upper Neckar catchment was chosen as shown in Fig. 12. One parameter set needed for this model consists of 13 parameters which are calibrated based on the Nash–Sutcliffe model efficiency coefficient using the simulated annealing algorithm for the period between 1960 and 2000. Then, 30 parameter sets are independently calibrated in total and, subsequently, 30 simulated discharge time series are generated to compare with one observed discharge.

Figure 16 shows the result of copula-based analysis calculated for single
time series (

The fact and the notion obtained in this section is that (i) both results from API and HBV based on copula measures indicate that the catchment changed around 1976, and (ii) by comparing the simulated discharge with observed discharge, the origin of the change of stochastical information can be assessed.

Copula asymmetry and copula distances for 30 simulated and 1
observed discharge time series at Plochingen between 1965 and 2000:

In this paper the application of copulas for
hydrological time series data is newly explored for the detection of
catchment characteristics and their temporal changes.

A copula-based measure of asymmetry,

The relation between asymmetry2 and catchment characteristics was tested
for 77 discharge records.

A method based on copula distance was examined for the investigation of temporal behavior of hydrological time series. This measure can detect the time period where dependence structure is unusual and its interdependency between different time series. Clear signals were detected that the dependence structure is unusual for a certain time period and this signal was not found by investigating the time series with variance, covariance or correlation.

API time series were calculated for each region in the Baden-Württemberg state and simulated discharge time series were generated using the HBV model for the upper Neckar catchment. These are the data not influenced by catchment change, thus compared with observed discharge to assess the anthropogenic impacts. The results showed that there was a signal detected only in the observed discharge around 1982, but not in the API or simulated time series, which implies the anthropogenic impacts on the catchment. Also, it was shown in the results of copula asymmetry that the trend clearly changed around 1976.

The results of copula-based analysis of hydrological time series seem to support the assumption that the catchment had started to change around 1976 and stayed unusual until 1990. These changes could correspond to the construction of flood retention basins started around 1982 (Lammersen et al., 2002) and ecological flooding strategy, which allowed small floods to happen for the rehabilitation of ecological systems in the floodplain, introduced in the upper Rhine from 1989 (Siepe, 2006).

Copulas can be seen as an alternative method to analyze hydrological time series data by focusing on the dependence structure, but further exploratory applications and theoretical developments are expected. The copula-based measures introduced in this study can be related to the potential model uncertainty, that is, how much the natural system is varying. Empirical autocopula analysis is a more data-driven approach which retains more information than the copulas estimated with parametric methods, but it is also numerically demanding. The effective way to analyze time series and build up a time series model based on copulas can be further explored.

The 77 discharge records used for this research are provided by Global Runoff
Data Centre

Suppose that a random variable at time

Fundamental research of this paper was initiated by the German Federal Institute of Hydrology with financial support. Special thanks are given to the Global Runoff Data Centre (GRDC) in Germany for offering the discharge data and the German Meteorological Service (DWD) for precipitation and temperature data. The authors thank the reviewers for their care in examining this work. Edited by: S. Archfield