Graphical-statistical method to explore variability of hydrological time series

In the last decade, almost 90% of the scientific articles published on precipitation, hydrology, and extreme climatic conditions contained the word ‘trends’ (Iliopoulou & Koutsoyiannis 2020). A popular practice by hydrologists in the 21st century has been fitting of trends everywhere, based on real data, and projecting them to the future (Iliopoulou & Koutsoyiannis 2020; Koutsoyiannis 2020). To do so, non-parametric or distribution-free methods including the Mann–Kendall (MK) (Mann 1945; Kendall 1975) and Spearman's rho (SMR) (Spearman 1904; Lehmann 1975) tests are conventionally applied to the entire time series for determining whether the null hypothesis H₀ (no trend) is rejected. There are some crucial issues regarding the conventional approach of testing trends in climatic data:

1. Common trend detection methods tend to be applied in a purely statistical way. Results from such purely statistical trend tests can be meaningless in some cases (Kundzewicz & Robson 2000), especially if not supported by inferences from graphical exploration of changes in the data.

2. The idea of testing the H₀ (no trend) using the entire data record to get insight on the future climatic condition is a misleading practice. Prediction of the future using trends fitted to the entire record was recently found to be worse than simply taking the mean of the data (Iliopoulou & Koutsoyiannis 2020). By testing the H₀ (no trend) using an entire long-term record, a great deal of information is missed out about hidden short-durational increase or decrease (hereinafter referred to as sub-trends) in the data. In other words, application of MK or SMR test to a given long-term (for instance, 100-year) series can show that the H₀ (no trend) is not rejected (p > α). This can be achieved because the effects of increasing and decreasing sub-trends within the series cancel each other and the net result of such cancellation may yield trend statistics for which the H₀ (no trend) cannot be rejected (p > α). However, the H₀ (no trend) may be rejected (p < α) for several sub-series from the same dataset when analyzed separately. Therefore, instead of focusing on long-term (like 100-year) trend which may be due to the effect of an external forcing on the system, analyses of sub-trends are important to characterize climate variability, a phenomenon which is crucial for consideration of designs of hydraulic structures or operation of hydrological applications. Here, there are two factors to be considered, including the relevant time scales and length of the sub-series for which the H₀ (no trend) is rejected (p < α). For instance, a 15-year data period is relevant as the design life of some water supply projects or systems for an irrigation scheme.

Importantly, identification of sub-trends can reveal fascinating information on temporal variability in climatic data and this can invite further investigations to maximize our understanding on driving forces of hydrological processes over short-durational or long periods. Large-scale random fluctuations in terms of rising and falling hydrological trends could be associated with specific attributes over the relevant sub-periods. These large-scale random fluctuations may be considered deterministic, especially if they could be conditioned by a physical explanation and predictability (Koutsoyiannis 2000). If the fluctuations defy a deterministic description, a plausible option of hydrological data analysis would be to adopt a stochastic description (Montanari & Koutsoyiannis 2014; Koutsoyiannis & Montanari 2015).

The main aim of this paper was to present a graphical-statistical methodology (based on cumulative sum of differences between exceedance and non-exceedance counts of data points) for identification and analyses of sub-trends in climatic series. The innovative aspect of the presented methodology lies in the tailored partial sums of the new trend statistic to graphically separate temporal clusters of events above or below the long-term mean of the climatic dataset. The method involves testing the H₀ (natural randomness) in the given data using a non-overlapping window of relevant length passed from the beginning to the end of the time series. Depending on whether the H₀ (natural randomness) is rejected (p < α), further investigation can be conducted to confirm if there are persistent fluctuations. This leads to testing on various other hypotheses to determine attributes of a hydrological change. Furthermore, assessment of persistence allows us to decide on which statistics to employ for further analyses of the given climatic or hydrological time series.

For the case when a data point y (or x) is tied n times, each value of d_y will become zero, meaning that for 1 ≤ i ≤ n.

Actually, Equation (15) is so computationally arduous that it can be limited to theoretical consideration based on small samples with n < 10. In practice, analyses of trends in data with persistence require long-term series. Similarly, sub-trends can occur over long periods, of say, greater than ten years. In such situations, V(T) can be corrected from the influence of autocorrelation by a suitable approximation following a slight modification of the formula from Onyutha (2016b) (see Section SM4 of the Supplementary material).

The H₀ (no trend) is rejected (p<α) for ⁠; otherwise, the H₀ is not rejected (p>α). Adequacy of the variance correction approaches summarized in Equation (16) can be found in Figure SM2 of the Supplementary material. To take into account of the influence of seasonality on trend results, seasonal trend test for the new method can be found in Section SM8 of the Supplementary material.

In another step, q_k is plotted against k or time of observations.

Normally, the number of values calculated based on a window of a particular length when moved from the beginning to the end of the series leads to loss of information proportional to the selected window length. However, Equation (24) ensures that there is no loss of information due to the movement of the time slice or window from the start to the end of the series. In other words, the number of values computed from Equation (23) is equal to the sample size of the give dataset. To test the H₀ (natural randomness), thresholds on the variability can be constructed using ±Z_α/2 after plotting Z_j against the corresponding j^th time, such as, data year. If the values of Z go outside the CILs, the H₀ (natural randomness) is rejected (p<α); otherwise, the H₀ is not rejected (p>α). If the H₀ (natural randomness) is not rejected (p>α), it means temporal variation in the sub-trends occurs randomly. On the other hand, if the H₀ is rejected (p<α) we suspect persistence or long-range dependence (LRD).

To reliably assess long-term hydrological changes, natural variation in hydro-climatic variables should adequately be characterized in terms of the statistical dependence (Onyutha et al. 2019). Thus, instead of testing for the H₀ (natural randomness) using a single time scale, we look for an evidence of multiple-scale variability by quantifying LRD or the Hurst exponent H (Hurst 1951).

To assess the attributes of temporal variation in sub-trends of river flow, we can test several hypotheses, such as (i) variability in precipitation, (ii) changes in potential evapo-transpiration (PET), and (iii) land-use changes and/or human factors. Full details of the hypotheses and the procedure to test each of them can be found in Table SM1 of the Supplementary material.

Rejection rates of the new method as well as SMR and MK tests were compared under various circumstances of coefficient of variation (CV), sample size, lag-1 serial correlation coefficient or first-order autoregressive AR(1) process, and trend slope over the ranges 0.1–0.9, 20–200, −0.9–0.9, and 0.001–0.009, respectively. Each simulation experiment was done using 5,000 synthetic series. Rejection rate was computed as the number of times the H₀ (no trend) was rejected divided by 5,000.

Several synthetic series Y_s with n = 200 or n = 300 were generated for the purpose of illustrating how to apply the new method for graphical-statistical diagnoses of sub-trends. To each of the series, the following steps were taken:

1. Equation (2) was applied to the full time series.

2. Sub-trend plot was made in terms of a_j (Equation (20)) versus j (or time of observations).

3. Sub-trends in the data were identified and separated in the form of areas enclosed between curves and the reference (or line).

4. For each sub-trend, the corresponding sub-series (or data points over the corresponding sub-period) were separated from the full time series.

5. Equation (21) was applied to each separated sub-series.

6. A plot of q_k from Step (v) against corresponding time of observations was made.

7. To the plot in Step (vi), CILs were added based on Equation (22).

The new method was applied to long-term annual river flow data observed at three hydrological stations along the White Nile in Africa. Monthly river flows observed at Malakal and Mongalla (labeled Stations 1 and 2, respectively) were obtained from the Global Runoff Data Centre (GRDC), Koblenz, Germany. Monthly river flows observed at Kamdini (Station 3 from 1950 to 2000) and Jinja (Station 4 measured over the period 1900–1995) were obtained from a report submitted to Uganda Electricity board in 1997 by Kennedy & Donkin Power Ltd in Association with Sir Alexander Gibb & Partners and Kananura Melvin Consulting Engineers. Data at Station 4 comprised Lake Victoria outflows. Monthly Lake Victoria level time series from 1900 to 1995 was also available at Station 4. Additional monthly Lake Victoria outflows at Jinja (Station 4) covering the period 1996–2005 were obtained from Ministry of Water and Environment, Uganda. An overview of the data can be seen in Table 1. Also obtained for this study were gridded hydro-climatic datasets over the upper White Nile, especially in the Equatorial region including (i) monthly precipitation and PET of the Climatic Research Unit (CRU) over the period 1901–2019 (Harris et al. 2020) and (ii) monthly precipitation of CenTrends v1.0 time series from 1900 to 2014 (Funk et al. 2015). Figure 1 shows time series plots of annual data after re-scaling or division of the series by their long-term mean (LTM).

To assess co-variation of observed flow and modeled runoff, a simplified procedure for lumped rainfall-runoff generation was adopted. Precipitation and PET were used to estimate soil moisture deficits. Runoff was generated as a function of the antecedent soil moisture when the evaporation demand was met.

Temporal sub-trends in the hydro-climatic variables were computed using Equation (23). Co-variability of the hydro-climatic variables was assessed using correlation analysis. Attribution of the temporal variation in sub-trends of the river flow was done using various hypotheses and procedure presented in Table SM1 of the Supplementary material. The Hurst parameter was also estimated for each series.

Figure 2 comprises graphical illustration on diagnoses of sub-trends and their significance. In sub-trend plots, the horizontal (or a = 0) line is the reference indicating the case when there is completely no trend in the time series. When there is no trend (see Case 1), the reference is crossed several times with no clear area enclosed between the scatter points and the reference (Figure 2(a)). In this case, the significance of the trend in the data is determined using the entire or full time series. Using α = 0.05, it can be noted that the data with no trend yielded values of which were entirely within the 95% CILs and the H₀ (no trend) was not rejected (p > 0.05) (Figure 2(j)). To statistically quantify the significance of the trend in such series, we apply Equation (19) to the entire dataset since there are no sub-trends.

Monotonic trends (Cases 2 and 3) exhibit sub-trend curves described by quadratic polynomials (Figure 2(b) and 2(c)). The sub-trend curve for a dataset with positive trend (Case 2) will exhibit a concave curve (Figure 2(b)). If the dataset has a negative trend (Case 3), a convex curve will be formed in the sub-trend plot (Figure 2(c)). It is worth noting that for the case with monotonic trend, the maximum absolute value of a_j tends to occur around i = n/2 or the mid of the data period. This can easily be shown (by differentiating Equation (SM3) of the Supplementary material with respect to i and equating the derivative to zero) that the point of inflection is at i = n/2. If the datasets are dominated by a monotonic trends as indicated by the entire sub-trend curve falling below or above the reference, the significance of the trends should be determined using the full time series. For Case 2, values of q_k were above the reference and up-crossed the upper limit of the 95% confidence interval. Similarly, for Case 3, values of q_k were below the reference and down-crossed the lower limit of the 95% confidence interval. Thus, for Cases 2 and 3, the H₀ (no trend) is rejected (p < 0.05) (Figure 2(k) and 2(l)). Statistically, we apply Equation (19) to the entire dataset since there are no sub-trends or the series is dominated by a monotonic increase or decrease.

When the dataset has various sub-trends, two or more curves are formed in the sub-trend plot (see Cases 4 and 5). For the part of the data with positive/negative sub-trend, the sub-trend curve will be concave/convex (Figure 2(d) and 2(e)). In Case 4 or Case 5, there are two sub-trends, one positive and the other negative. Here, the significance of each sub-trend should be determined separately using sub-data taken over the corresponding sub-period. For instance, in Case 4 or Case 5, one sub-trend occurred from to and the other over the sub-period ⁠. It can be seen from Figure 2(m) and 2(n) that the H₀ (no trend) is rejected (p < 0.05) for each of these sub-trends. In other words, the negative and positive sub-trends down-crossed and up-crossed the lower and upper and 95% CILs, respectively (Figure 2(d) and 2(e)). In the conventional procedure of trend analyses without separating sub-trends, the series in Case 4 or Case 5 would yield q_k at equal to zero (or ⁠) thereby indicating no trend, something that would be misleading. Value of q_k at can be zero for data with two sub-trends because the positive and negative trends cancel each other. However, in reality the H₀ (no trend) should be tested separately for each identified sub-trend.

For a step jump in mean (see Case 6 and Case 7), the sub-trend plot produces two lines (having opposite slopes) (Figure 2(f) and 2(g)). These lines have their point of intersection below (above) the reference in the case the data is characterized by a step downward (upward) jump in mean (Figure 2(f) and 2(g)). The significance of sub-trends in the sub-series over the two periods (one before and the other after the change point) should be determined separately. As seen from Figure 2(m) and 2(n), the H₀ (no trend) is not rejected (p > 0.05) for each of these sub-series. In other words, increase or decrease in each sub-series is minimal and the scatter points in the sub-trend plot do not cross the 95% CILs (Figure 2(o) and 2(p)).

A given dataset can have no trend over some sub-series while the other part is characterized by an increasing or decreasing sub-trend (see Cases 8 and 9 of Figure 2). In such situations, the sub-series with no trend can be indicated by a line in the sub-trend plot. However, the sub-series with increasing/decreasing sub-trend can exhibit concave/convex curve in the sub-trend plot (Figure 2(h) and 2(i)). For Cases 8 and 9, the sub-series over the periods with sub-trends should be separated from the full time series. Significance of the various sub-series should be assessed separately. For the various sub-trends, since the values of go beyond the 95% CILs, the H₀ (no trend) is rejected (p < 0.05) (Figure 2(q) and 2(r).

Figure 3 shows sub-trend plots for synthetic data each of n = 300 and having three sub-series with either trend or no trend combined. The first section is from to The second and third sub-series are over the sub-periods and ⁠, respectively (see Cases 1–7 of Figure 3). It can be noticed that the sub-trend plots show lines for sub-trends with no trends. However, for sub-series with trends, corresponding sections of the sub-trend plot appear as curves (Figure 3(a)–3(g)). The significance of the various sub-series is determined separately (Figure 3(j)–3(r). It is vital to note that because the sample sizes of the sub-series are the same (or ⁠), the widths of the 95% confidence intervals are uniform in each plot while testing the significance of the various sub-trends. In reality, sub-trends can be of different lengths of sub-periods and, therefore, the widths of the 95% confidence intervals can differ among sections of the sub-trend plot.

An important question to answer is: Can partial sums of MK and SMR tests be applied to produce the equivalent of sub-trend plots such as, those presented in Figures 2(a)–2(i) and 3(a)–3(g)? Application of partial sums of MK and SMR to produce equivalent sub-trend plots as in Figures 2 and 3 has limitations. Partial sums of MK and SMR trend statistics in their current forms cannot directly be applied to separate clusters of events which occur above and below the long-term mean of the time series. This is because partial sums of the MK test statistic decrease from the beginning towards the end of the series while fluctuating randomly regardless of whether there are sub-trends in the data. SMR test statistic comprises squared difference between ranks of X and Y and this makes its partial sums to always increase in magnitude even when there is an apparent negative sub-trend. Further explanations of these limitations can be found in Section SM3 of the Supplementary material.

Table 2 shows differences among methods or approaches for analyzing changes in series. It is noticeable that the conventional application of MK and SMR tests can yield trend statistic values for which the H₀ (no trend) is not rejected (p > α) despite presence of apparent sub-trends. In cases where there are monotonic trends (and no sub-trends), the new methodology is consistent with conventional approaches. Here, it is important to first graphically show using sub-trend plots that the series has no sub-trends or it is dominated by a monotonic increase or decrease over the entire data record. Application of trend tests to the full time series characterized by step jump in means can lead to rejection (p<α) of the H₀ (no trend) when actually there are no apparent sub-trends. Therefore, trends in the sub-series before and after the step jump should be analyzed separately.

The value of H for white noise or series with random fluctuations is 0.5. Here, the H₀ (natural randomness) tested using temporal variation in sub-trends was not rejected (p> 0.05) for the series with H = 0.502. Furthermore, the H₀ (the series is like white noise) was not rejected (p> 0.05) for this dataset with H = 0.502. For series which are not statistically different from white noise, classical statistics can be used for analyses. Application of classical statistical metrics, such as mean, coefficient of variation, and correlation follows the assumption that the sample comprises independent, and identically distributed events. Other values of H in Table 2 are all above 0.5 and for the corresponding series, the H₀ (the given dataset is typical of white noise) was rejected (p < 0.05). This means that these series were characterized by persistent fluctuations. Furthermore, for each of these series, the H₀ (natural randomness) was rejected (p < 0.05) using temporal variation in sub-trends. Results from Table 2 show that occurrences of sub-trends in a given series can characterize statistical persistence or LRD. The case with H > 0.5 indicates that successive values in a given series depend on the previous values. For instance, if a given value in a series is greater than the mean, the next value is expected to exceed the sample mean with a probability greater than 0.5 (Onyutha et al. 2019). Generally, values of H which are greater than 0.5 as shown in Table 2 indicate association with large-scale and/or multiple-scale variability or enhanced natural variability.

Enhanced natural variability means that at large horizons, we can encounter increased unpredictability. Perhaps, at short time scales, temporal clustering can be present and this can yield some potential for predictability (Dimitriadis et al. 2016). Generally, two frameworks exist to characterize and predict persistent processes. The first option is the Markovian framework in which the latest observations are considered to influence future probabilities. Secondly, in the Hurst–Kolomogorov (HK) (Kolmogorov 1940; Hurst 1951) framework, we consider the entire record of past realizations to condition simulations of future climatic conditions (Koutsoyiannis 2011). Application of the HK framework (which admits stationarity) can make us appreciate the coexistence of stationarity with change at all time scales. Normally, the future climatic condition tends to be predicted deterministically (such as, using linear trend). Here, the uncertainty band around the linear regression line may be narrow indicating reduced uncertainty on our prediction. However, when we apply the HK approach, the uncertainty about the future climatic conditions would be very large, and this may be realistic given the possible limited knowledge we always have especially about the (very) distant future.

Results from various tests applied to series without considering the need to separate sub-trends were highly comparable. Details of graphical comparison can be seen in Figures SM3 and SM4 of the Supplementary material.

Figure 4 shows graphical diagnoses of sub-trends in annual hydro-climatic series. Analyses were conducted using the period over which all the hydro-climatic variables had available data records. Hereinafter, White Nile flow means the average of records from Stations 1–3 of Table 1. White Nile flows and Lake Victoria outflows exhibited an upward step jump in mean (Figure 4(a)). White Nile flows majorly come from Lake Victoria. Precipitation directly contributes to not less than 80% of the water in Lake Victoria which acts as the reservoir supplying Victoria Nile and White Nile. Less than 20% of the Lake Victoria water comes from tributaries. In other words, the Lake Victoria water level and flow from White Nile are largely influenced by precipitation over the Equatorial region. Precipitation from both sources (CenTrends and CRU) also exhibited an upward step jump in mean. However, two curves were formed by scatter points of PET over the periods 1912–1960 and 1961–2000 (Figure 4(a)). This means that PET over the Upper Nile (or White Nile) region was characterized by increasing sub-trends. However, the magnitudes of the linear increase of PET with time were different over the periods 1912–1960 and 1961–2000.

Figure 4(b) and Table 3 show the significance of identified sub-trends. Over both sub-periods 1912–1960 and 1961–2000, precipitation and river flows were characterized by decreasing trends. However, PET was characterized by increasing trends. For PET, the H₀ (no trend) was rejected (p < 0.05) for increasing trends over both sub-periods 1912–1960 and 1961–2000. Over both sub-periods, the H₀ (no trend) was not rejected (p > 0.05) for the decreasing precipitation trends. For both White Nile flow and the Lake Victoria outflow, the H₀ (no trend) was rejected (p < 0.05) for decreasing trends over the second (or 1961–2000) sub-period.

Figure 5 shows co-variability of 15-year sub-trends in annual hydro-climatic variables over the White Nile region. Periods over which the new method's trend statistic Z values for PET were consecutively positive (negative) were 1915–1926, 1935–1954, and 1979–2006 (1901–1914, 1927–1934, 1955–1964, and 2007 until recent 2015). The H₀ (natural randomness) was rejected (p < 0.05) over the period 1901–1914. Consecutive positive (negative) sub-trends in the Lake Victoria outflow and White Nile flow occurred over the period 1949–1967 (1967–1993). Precipitation exhibited positive (negative) sub-trends over the periods 1953–1965, 1983–1995, and 2005 until recent 2019 (1966–1980 and 1995–2004). However, the H₀ (natural randomness) was not rejected (p > 0.05) for temporal sub-trends in precipitation. Figure 5(b) shows correlation between hydro-climatic sub-trends shown in Figure 5(a) applied to 15-year time slice or window moved in an overlapping way from the beginning towards the end of the data. Resulting correlation coefficients were plotted against the ending year of each window. Throughout the data record period, the H₀ (no correlation) was rejected (p < 0.05) for the correlation between White Nile flow and Lake Victoria outflow. The H₀ (no correlation) was also rejected (p < 0.05) for the correlation between White Nile flow and precipitation as well as PET and precipitation. Correlation between White Nile flow and precipitation was mainly positive although negative over a small epoch from the late 1930s to mid-1940s. During this period, correlation between PET and precipitation was also positive. Generally, correlation between PET and precipitation was negative except from the late 1930s to mid-1940s. The conclusive points to take from Figure 5 are that: (i) variation in the White Nile flow is significantly (p < 0.05) determined by changes in the Lake Victoria water level; (ii) variance in Lake Victoria water level and eventually White Nile flow can be significantly (p < 0.05) explained by the variability in the precipitation over the Equatorial region; (iii) variation in precipitation on land is negatively correlated with PET variability; and (iv) temporal co-variability among hydro-climatic variables depend on the period selected for analyses.

Table 4 shows results of various hypotheses put forward to explain temporal variation in sub-trends of the White Nile flows with focus on the upward step jump in mean in 1961.

Table 5 shows measures of statistical dependence in hydro-climatic series over the White Nile region. The H₀ (the given dataset is not different from a typical white noise) was rejected (p < 0.05) for each series. Values of H in Table 5 are all above 0.5 (and significantly different from H = 0.5). This means that each of the hydro-climatic series was characterized by persistent fluctuation thereby indicating association with large-scale and/or multiple-scale variability or enhanced natural variability. Results in Table 5 are consistent with the assumption that rejection of the H₀ (natural randomness) from Figure 5(a) was due to persistence in the hydro-climatic series. Conclusively, the temporal changes in the sub-trends of the hydro-climatic variables analyzed in this study characterized large-scale random fluctuations in the hydro-climate of the White Nile region. Since annual data were used, high values of H indicate high climate variability (Koutsoyiannis 2020). Generally, temporal change is an inherent characteristic of each hydrological process and this is due to the persistent stochastic nature of the hydrological processes (Koutsoyiannis 2020).

Conventional trend analysis in which Mann–Kendall and Spearman's rho tests are applied to the entire given series leads to lack of information on sub-trends (or hidden short-durational trends) in the data. Furthermore, common trend detection methods are purely statistical in nature and they can yield meaningless results in some cases if not supported by graphical exploration of changes in the data.

This paper presented a graphical-statistical methodology (which for lack of appropriate name due to its elaborate procedure, can simply be referred to as Onyutha's test (for detecting trends or sub-trends) or Onyutha's method (for other applications such as, variability analyses, and hydrological change attribution)) to identify and separately analyze sub-trends to support attribution of hydrological change. The methodology considers differences between exceedance and non-exceedance counts of data points. This method (among other things it does) can be used to (i) graphically identify sub-trend(s) over unknown period(s) of increase or decrease in the time series, (ii) statistically and graphically test for sub-trends and/or trends, (iii) analyze temporal variability based on sub-trends using time slice or window of fixed length moved from the start to the end of the dataset, and (iv) test various hypotheses for hydrological change. Furthermore, the methodology makes it possible to appreciate that climate variability comprises large-scale random fluctuations in terms of rising and falling hydrological sub-trends which could be associated with specific attributes over the relevant sub-periods. To accurately characterize large-scale random fluctuations in hydrological systems, long-term series should be considered for analyses.

To implement the CSD-based methodology presented in this paper, the reader can download a MATLAB-based tool named CSD-VAT (which stands for variability analyses tool VAT based on cumulative sum of difference CSD between exceedance and non-exceedance counts of data points) via either https://www.researchgate.net/publication/332798309 or https://sites.google.com/site/conyutha/tools-to-download

The steps for applying the presented methodology include the following:

• 1.

  Identify epochs with clusters of data points above and below long-term mean of the time series. To do so:

• • (a)

    Select a number of time scales (for instance, 5, 10, 15, 20, 25, 30). Here, the unit of each scale is the time unit of the series. For instance, when using annual data, the unit of time scale is year.

  • (b)

    For each time scale from Step(a), apply Equation (23) of the main paper with Z_j replaced by A_j, where A_j denotes the average of sub-series within the window running from the u^th to the v^th data point.

  • (c)

    To each of the series generated from Step(b), apply Equation (20).

  • (d)

    Make sub-trend plot to graphically identify sub-trends (if any) in the series of each time scale. Consult Figures 2 and 3 for a guide in graphical diagnoses of sub-trends. The choice of an epoch with a sub-trend should be based on the purpose for which analysis is being conducted. For water resources applications, sub-trend can be considered relevant for analysis if the epoch over which it occurs is equal to or greater than ten years. This means sub-trends over epochs less than ten years can be filtered and left out (unless they are relevant for intended applications). For epochs over which sub-trends exist, one needs to separate sub-series and separately assess the significance of each sub-trend. If there are no sub-trends, test the H₀ (no trend) using the entire data record. For testing the H₀ (no trend), make use of Equations (4), (12), (16), and (19) of this paper.

• 2.

  Test for H₀ (natural randomness) using temporal variation in sub-trends. To do so:

• • (a)

    Considering the time unit of the series, select a number of time scales (for instance, 10, 15, 20, 25, 30 years).

  • (b)

    For each time scale, apply Equation (23) as it is to the given series.

  • (c)

    Plot results from Step (b) against time of observations and add the CILs.

  • (d)

    Separately test the H₀ (natural randomness) using results based on each time scale.

From Step 2(d), if the H₀ (natural randomness) is not rejected (p > α) from results of any time scale, check whether the entire given series is statistically not different from a typical white noise. In other words, check if H is statistically different from 0.5. CSD-VAT makes use of the generalized Hurst exponent H(q^#) based on the scaling of renormalized q^#–moment of the distribution (Di Matteo 2007). For other methods to estimate H, see Section SM6 of the Supplementary material. In further analyses, treat the series using classical statistics if it is not statistically different from a typical white noise.

From Step 2(d), if the H₀ (natural randomness) is rejected (p < α) from results of any time scale, the given time series can be assumed to be characterized by persistent fluctuations. Estimate the scaling exponent H. If the dataset is confirmed to be statistically different from a typical white noise, determine attributes of identified sub-trend(s) in the hydrological time series by testing various hypotheses. Here, make use of the procedure in Table SM1 of the Supplementary material. If we use classical statistics for series characterized by H being statistically different from 0.5, only a portion of the natural uncertainty of the hydrological or hydro-climatic processes would be described stemming from an implicit assumption of a stable climate (Koutsoyiannis 2003). Therefore for further analyses, use a suitable procedure such as the Hurst–Kolomogorov (Kolmogorov 1940; Hurst 1951) framework to characterize and predict the persistent process. Importantly, the prediction should be on a short period.

The long-term river flow series used in this study were obtained from the Global Runoff Data Centre (GRDC), Koblenz, Germany. The entire work in this paper was based on the effort of the sole author. The author declares no conflict of interest and no competing financial interests. Above all, the author is grateful to IWA Publishing for granting him waiver of article processing charges for the open access publication of this paper.

All relevant data are included in the paper or its Supplementary Information.