The mathematical variable names used in this section and throughout the paper were chosen with the goal of straightforward interpretation. The names were constructed by a combination of the following base “alphabet”: n is for “number”, v is for “value”, b is for “bin”, s is for “(time) slice”, e is for “entropy”, t is for “time”, and w is for “width”. For example, variable nvb is formed by a combination of three symbols and represents the “number of value bins”. To avoid confusion of combined variable names with multiplication (e.g. nvb could be falsely interpreted as the product of variables n, v and b), we explicitly indicate each multiplication with the “⋅” symbol.

Applying the method to a given time series with overall nt time steps consists of a number of steps and related choices. At first, for each variate involved, a suitable discretization (binning) scheme is chosen. The bins must cover the entire value range, and their total number (nvb) can be chosen according to a user's demands regarding data resolution. Next, the time series is divided into a number of ns time slices. The slices must be mutually exclusive and together must cover the time series. The slices are preferably, but not necessarily, of uniform width. Next, separately for each slice, a discrete probability distribution (histogram) is calculated using the data in the slice and the chosen binning scheme. From the histogram obtained in this way, the Shannon information entropy H (Shannon, 1948) is calculated following Eq. (1): 1H(X)=-∑vb=1nvbp(xvb)⋅log⁡2(p(xvb)), where X is all sample data within the slice, p(xvb) refers to the probability of variate value x falling into bin vb, and nvb is the total number of value bins. Entropy measures data variability or uncertainty in bit, with the intuitive interpretation as “the minimum number of binary (Yes/No) questions needed to be asked to correctly guess values drawn from a known distribution”. Cover and Thomas (2006) provide an excellent introduction to information theory, and applications to hydrology and hydro-meteorology are e.g. presented in Singh (2013) and Neuper and Ehret (2019). Neuper and Ehret (2019) also describe the relation of entropy and variance: “As with the variance of a distribution, entropy is a measure of spread, but there are some important differences: while variance takes the values of the data into account and is expressed in (squared) units of the underlying data, entropy takes the probabilities of the data into account and is measured in bit. Variance is influenced by the relative position of the data on the measure scale and is dominated by values far from the mean: entropy is influenced by the distribution of probability mass and is dominated by large probabilities. Some welcome properties of entropy are that it is applicable to data that cannot be placed on a measure scale (categorical data) and that it allows comparison of distributions from different data due to its generalized expression in bit”.

As entropy values may differ between slices, an overall uncertainty estimate for all slices is calculated as the expected value of all slice entropies. For equal-width slices, this is mean entropy according to Eq. (2). 2Uncertainty=E(H(X))=H(X)‾=1ns⋅∑s=1nsHs(X), where s refers to a particular slice of all ns time slices. The uncertainty defined in this way measures the average within-slice variability of the data, i.e. uncertainty of the time series as seen through the lens of the chosen time-slicing scheme.

Next, we consider the variability of entropy across all the slices, and as before we measure variability by entropy. In order to calculate this higher-order “entropy of entropies”, a suitable binning scheme for entropy values must be chosen, which can be based on the same criteria as outlined above. It is then used to calculate a histogram of the ns entropy values. We thus define complexity as the entropy of entropy values, which is calculated following Eq. (3). 3Complexity=H(H(X))=-∑eb=1nebp(Heb)⋅log⁡2(p(Heb)), where neb denotes the total number of entropy bins, eb denotes a particular entropy bin, and p(Heb) denotes the probability of a time slice entropy Hs falling into bin eb. Complexity measures how uncertain we are about the uncertainty in a particular time slice when all we know is the distribution of uncertainties (entropies) across all time slices in the time series. The question may arise why complexity is calculated as the entropy rather than the variance (second moment) of entropies, which would seem a logical extension of uncertainty being calculated as the mean (first moment) of entropies. There are three reasons for this choice, i.e. consistency, interpretability and robustness. “Consistency” refers to the idea that, when expressing the variability of the distribution within a time slice by entropy, we think that it is also a natural choice to express the variability of the variabilities by entropy. Thus, variability is always expressed in the same unit of bit, which increases comparability among the values and upper bounds of uncertainty and complexity. “Interpretability” refers to the fact that entropy has the intuitive interpretation of the “number of binary yes–no questions to ask to move from a prior to posterior state of knowledge”, while variance lacks this straightforward interpretation. “Robustness” refers to the previously discussed property of variance being more sensitive to outliers in the data than entropy. While variance is a good choice for extreme-value statistics with a focus on the tails of a distribution, we think that, for a characterization of the overall variability of a data set, entropy is more appropriate.

The entire procedure of calculating uncertainty and complexity is repeated for many different choices of ns (time-slicing schemes). For each choice of ns, for equal-width slices the width of a time slice is sw=nt/ns. In principle, ns can be chosen to take any value in the range [1,nt]. For ns=1, the entire time series is contained in a single slice of width sw=nt. For ns=nt, each time slice contains only a single time step. However, it is recommended to choose ns – and with it sw – from a smaller range: if we require that, for a robust estimation of a time slice histogram, each of its nvb bins should on average be populated by a minimum number of m values, then the width sw of a time slice (i.e. the number of values within) must be at least nvb⋅m (see Eq. 4). This means that, for robust estimates of uncertainty, the time series should be split into only a few but wide time slices. For robust estimates of complexity, however, it is the opposite: the histogram of uncertainty values is populated by a total of ns values (the entropies of all time slices). If for the sake of a robust estimation we again require that each of the histogram bins should be populated by at least m values, then at least neb⋅m time slice entropy values are needed. This means that the time series should be split into many – and hence narrow – time slices. These two antagonistic constraints lead to upper and lower limits for the choice of sw, which is formalized in Eq. (4). For a (subjective) user's choice of m, Eq. (4) yields the range of time slice widths sw satisfying the “m criterion” for both the uncertainty and complexity histogram as a function of time series length nt and the number of bins for both uncertainty (nvb) and complexity (neb). 4ntneb⋅m≥sw≥nvb⋅m

For example, for a time series with nt=30000 time steps, choices of m=3 and nvb=neb=10 (all histograms resolved by 10 bins), the range of suitable time slice widths is [30,1000]. It should be noted that Eq. (4), through the choice of sw, provides one possible guideline for robust histogram estimation, but a user can also resort to other binning guidelines, such as the methods suggested by Sturges (1926), Scott (1979), Freedman and Diaconis (1981), Pechlivanidis et al. (2016) or Knuth (2019). Throughout all the time-slicing schemes, the number of value and entropy bins must be kept constant to ensure comparability. Together, the set of all time-slicing schemes produces a set of complexity–uncertainty value pairs. Plotting them with uncertainty values on the x axis and complexity values on the y axis is what we call the c-u-curve. It summarizes several interesting properties of the time series under consideration, which will be discussed in Sect. 3.

In this section, we briefly summarize some general properties of the c-u-curve and discuss its limitations and possible generalizations.

Two methods similar to the c-u-curve have been proposed in the literature, which in the following we will briefly explain and discuss. The first, CLMC, was proposed by López-Ruiz (1995), and the second, MSE, by Costa et al. (2002). CLMC is a statistical measure of complexity for physical systems. It is calculated as the product of the system's information content, which is measured by the (normalized) Shannon entropy of the probability distribution of all of its accessible discrete states and disequilibrium, which is measured by the sum – taken over all accessible discrete states – of squared differences between the system's probability distribution and a corresponding uniform (maximum-entropy) distribution. For example, a crystal has high disequilibrium but low information content, and an ideal gas has low disequilibrium but high information content, but for both the product CLMC is small, indicating low complexity. Plotting a system's CLMC over its information content (see Fig. 2 in López-Ruiz, 1995) looks similar to the c-u-curve, including the limit behaviour (complexity approaches zero for systems with very high and very low entropy) and the existence of an upper bound for complexity as a function of entropy. Feldman and Crutchfield (1998) later proposed replacing the somewhat arbitrary measure of disequilibrium in López-Ruiz (1995) by Kullback–Leibler divergence, but the essential differences of CLMC and the c-u-curve methods remain. Firstly, the former defines complexity as the product of two separate system characteristics, of which one is the departure from a benchmark system and the latter derives both characteristics from the system alone. Secondly, the former does not take the order of the data into account, while the latter explicitly does when calculating entropy for data within temporally neighbouring data within time slices.

The MSE method calculates the entropy of a time series for various coarse-grained (time-averaged) versions thereof and then plots the entropy over the size of the averaging window (referred to as a scale factor τ in Costa et al., 2002). MSE shares with the c-u-curve the idea that from the joint display and comparison of various entropy values of a time series much can be learned about the underlying dynamical system. It is also similar in that the temporal order of the data is explicitly taken into account. The main difference is that, in MSE, data in a time window are averaged; i.e. the within-window variability of the data is essentially removed, while in the c-u-curve entropy calculations are always done on the original data. The second difference is that MSE does not provide an objective measure of system complexity; rather, this is visually inferred from the plot: complex systems are those revealing high entropy values across a wide range of scale factors. Obviously, the MSE and the c-u-curve approach can be joined by repeating c-u-curve calculations for various coarse-grained versions of a time series, which seems like a very promising idea for future work.

We discuss the properties of the c-u-curve with the example of six time series as shown in Fig. 1a–f. Time series a–c are synthetic time series: a straight line, random uniform noise and the famous Lorenz attractor (Lorenz, 1963). We chose them for their simple, exemplary and well-known behaviour. The straight line (Fig. 1a) contains no variability whatsoever and should therefore show very little uncertainty and complexity. The random noise (Fig. 1b) contains very high but constant variability and should therefore show high uncertainty and low complexity. The Lorenz attractor (Fig. 1c) is a prime example of complex behaviour arising from feedbacks in dynamical systems. We used the code as provided by Moiseev (2022) with standard parameters to produce a time series of the Lorenz attractor. From its three variates, for clarity, only the first one is shown and discussed, and the results from jointly considering all three variates are similar. All synthetic time series consist of nt=30000 time steps, and for both value binning and entropy binning, 10 bins were used. With a choice of m=3, the range of recommended time slice widths is sw=[30,1000] according to Eq. (4). In addition to the recommended range of time slice widths, we also included the two extreme values sw=1 and sw=30000 for demonstration purposes.

Time series d–f are hydro-meteorological observations taken from the CAMELS US data set (Newman et al., 2015). The first (Fig. 1d) are daily precipitation observations for the South Toe River, NC (short: STR), basin, and the second (Fig. 1e) is the corresponding time series of daily streamflow observations. The basin size is 113.1 km2, and precipitation mainly falls as rain (the fraction of precipitation as snow is 8.5 %). The third time series (Fig. 1f) also contains daily streamflow observations but from the 111.5 km2 Green River, MA (short: GR), basin, which is more snow-dominated (the fraction of precipitation as snow is 22.2 %). We chose the time series for the following reasons. Comparing precipitation and streamflow series from the same basin (STR) allows the effect of the rainfall–runoff transformation process on uncertainty and complexity to be analysed. Here we expect that a basin – by spatio-temporal convolution of precipitation – will mainly reduce precipitation variability and with it uncertainty and complexity. Comparing streamflow from two basins with different levels of snow influence (STR and GR) allows the effect of snow processes on uncertainty and complexity to be analysed. Here we expect that the carryover effect of snow accumulation and the influence of an independent additional driver of hydrological dynamics – radiation – should increase both uncertainty and complexity. All hydro-meteorological time series contain 12 418 daily observations from 1 October 1980 to 30 September 2014 (34 years). As for the synthetic time series, we also used 10 bins to resolve both the range of values and the range of entropies. However, we used a different time-slicing scheme to reflect standard ways of time aggregation of real-world data. In particular, we used the set of sw={1,7,14,21,30,60,91,182,365,730,12418} d, which corresponds to 1 d, 1–3 weeks, 1–6 months, 1–2 years and the entire 34-year period. Please note that, for a choice of m=3, the range of recommended time slice widths is sw=[30,414] d according to Eq. (4). Results for time slices outside of this range should therefore be treated with caution. We included them nevertheless for a more complete assessment of the time series.

For convenience, we normalized all six time series to a [0,1] value range and then calculated uncertainty and complexity according to Eqs. (2) and (3).

In this section, we present and discuss the c-u-curves of all six time series. We start by discussing the three artificial time series, followed by the three hydro-meteorological time series. All the c-u-curves are shown in Fig. 2, and their key characteristics are summarized in Table 1. For clarity, Fig. 3 additionally shows only the hydro-meteorological time series in a subregion of Fig. 2. For further illustration, selected histograms of time series streamflow in GR are shown in Appendix A.

The overall shape of each c-u-curve contains key characteristics of the underlying time series. We start by discussing the c-u-curve plot of the straight line in Fig. 2. It shows – as expected – the simplest behaviour: for all the time-slicing schemes, both within-slice variability and across-slice variability are zero; i.e. the series displays zero uncertainty and complexity throughout (all dots are stacked at the origin). As a consequence, mean uncertainty and complexity across all the time-slicing schemes (indicated by the brown pentagram in the plot and listed in Table 1) are also zero.

The random noise series in Fig. 2 by contrast displays very high uncertainty and low complexity for most of the time-slicing schemes (most dots are stacked in the lower-right corner of the plot), and only for many but narrow time slices of 50, 40 and 30 values per slice does complexity assume non-zero values. This can be attributed to random effects in small samples, where purely by chance both highly and hardly variable samples can occur, thus creating a wide range of time slice entropies and resulting in apparent non-zero complexity. For wider slices, the larger sample size leads to more similarly distributed samples, resulting in a narrow range of time slice entropies and hence low complexity. Overall, mean uncertainty is very high and mean complexity is very low (position of the pink pentagram in Fig. 2 and values in Table 1), which is what we expected from random noise.

The Lorenz attractor in Fig. 2 reveals a more diverse behaviour across the range of time-slicing schemes. We start discussing it for the case of sw=30000, i.e. when a single time slice covers the entire time series. As described in the general properties, for this case uncertainty is always at its maximum and equals the entropy of the time series, and complexity is zero, because only a single entropy value populates the entropy distribution. The actual uncertainty value (3.11 bits), or its distance from the upper limit of uncertainty (3.11/3.32=94%), is a key characteristic of the time series and expresses its overall variability. Decreasing the time slice width sw decreases within-slice variability (uncertainty). Also, it provides the potential for non-zero complexity as more and more entropy values populate the entropy distribution. For the curve shown in Fig. 2, complexity continuously increases and reaches its first maximum value of 2.66 bits (or 2.66/3.32=80%) for sw=200 and at 2.22 bits of uncertainty. This point is another key characteristic of a c-u-curve, indicating at which temporal aggregation the across-slice variability is highest. Further decreasing slice width first leads to a decrease and then another increase in complexity until a second maximum of 2.66 bits is reached at sw=50 (see the values in Table 1). Afterwards, complexity and uncertainty decrease to zero for sw=1, which is a general property of any c-u-curve (see the discussion of general properties above). Taking the uncertainty and complexity mean across all the time slices summarizes the c-u-curve in a single point (purple pentagram in Fig. 2, values in Table 1). For the Lorenz attractor, this reveals medium average uncertainty and high average complexity. In fact, the overall shape of the c-u-curve is close to the upper complexity limit reachable at a given uncertainty (shown in the plot as a black arch). This is in accordance with expectations, as the Lorenz attractor is known for exhibiting complex behaviour on many timescales. Interestingly, apart from revealing its generally complex behaviour, the c-u-curve also reveals at which particular time slice width the complexity of the Lorenz attractor is at a maximum. This can be interpreted as a “characteristic timescale” of the time series.

Next, we discuss the c-u-curves of the hydro-meteorological time series. In Fig. 2, they are indicated by the lines without markers. It is immediately obvious that they all possess low uncertainty, much lower than the theoretical maximum (indicated by the vertical “maximum uncertainty” limit) and the random noise and also lower than the Lorenz attractor. This is in accordance with our expectations and a consequence of the typically high temporal autocorrelation of hydro-meteorological time series, which clearly separates them from purely random time series. For a better view of the details, we re-plotted the hydro-meteorological time series in a subregion of the uncertainty limits in Fig. 3, which we will refer to in the following.

Despite the generally low uncertainties, the precipitation STR time series in Fig. 3 displays considerable complexity (indicated overall by the c-u-curve being close to the upper complexity limit and for mean complexity by the relatively high pentagram position), which can be explained by the existence of meteorological regimes with different levels of precipitation variability, such as dry periods (low variability), periods with alternating dry and wet periods (high variability) and wet times with diverse precipitation amounts (high variability). The highest complexity occurs for a time slice width of sw=14 d, indicating that the greatest variability of within-slice precipitation variability occurs for 2-week periods.

Interestingly, the corresponding streamflow STR time series displays much lower mean and maximum values (see Table 1) for both uncertainty (within-slice variability) and complexity (across-slice variability). This is in accordance with the general hydrological understanding that in the absence of major carryover mechanisms, rainfall–runoff transformation in catchments is mainly by aggregation and convolution, thus reducing the variability of the precipitation signal. It is noteworthy that while this harmonizing effect changes uncertainty and complexity means and maxima, it does not affect the characteristic timescale. For both streamflow STR and precipitation STR, this is 2 weeks. This suggests that precipitation remains the main control of streamflow complexity despite the processes involved in rainfall–runoff transformation.

This is different for the second streamflow GR time series. Here, in addition to the above-mentioned rainfall–runoff transformation, precipitation is partly stored as snow and later released as streamflow by melting. The temporal pattern of snowmelt is not only governed by snow availability, i.e. the precipitation regime, but also by energy availability, i.e. the long-term radiation and temperature regime. Such additional, independent controls of hydrological function can add uncertainty and complexity to streamflow production. Compared to streamflow STR, both uncertainty and complexity are indeed much larger in terms of mean and maximum values, and they are even larger than the corresponding values for precipitation STR (compare the pentagram positions in Fig. 3 and the values in Table 1). The characteristic timescale of streamflow GR is 2–3 months (60–91 d). This is considerably longer than for streamflow STR and can be explained by the carryover effect of snow accumulation and snowmelt acting on timescales of the order of months rather than days or weeks. For further illustration of the c-u-curve method, selected histograms for streamflow GR are shown in Appendix A.