# On Rigorous Drought Assessment Using Daily Time Scale: Non-Stationary Frequency Analyses, Revisited Concepts, and a New Method to Yield Non-Parametric Indices

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Data for the Case Study

_{0}) rate and river flow data from the Blue Nile basin of Ethiopia and Sudan in Africa were used. The Blue Nile basin (Figure 1) has a catchment area of about 325,000 km

^{2}. The Blue Nile, which emanates from the Ethiopian Highlands based on the two main tributaries, the Dinder and Rahad Rivers, flows into and out of Lake Tana. The climate of the basin is characterized by seasonal migration of the inter-tropical convergence zone. According to the International Water Management Institute IWMI [18], the majority of the population where the Blue Nile basin is located depends on rain-fed cropping system to support livelihoods.

_{rad}, MJ/m

^{2}/day) as well as the minimum (T

_{min}, °C) and maximum (T

_{max}, °C) temperature were obtained from the National Centers for Environmental Prediction NCEP [20] database. These meteorological time series which were from 1979 to 2000 were extracted over the region covering latitude from 5° to 17° N and longitude from 30° to 42° E. Based on the S

_{rad}, T

_{min}and T

_{max}, ET

_{0}(mm/day) was computed at each grid point using the Food and Agriculture Organization (FAO)-Penman-Monteith method [21]. At each grid point, the daily precipitation insufficiency was computed by subtracting the ET

_{0}from precipitation.

#### 2.2. Stationarity versus Non-Stationarity

#### 2.2.1. Pre-Requisites for Frequency Analysis

#### Data Transformation

#### Independence of the Extreme Events

- (i)
- the time in between the two events should not be less than the stipulated value;
- (ii)
- the extracted event should not be less than the specified threshold; and
- (iii)
- the independency ratio should not be greater than a stipulated value.

^{3}/s. The minimum number of days to characterize a hydrological dry spell was set to 175 days.

#### Significance of Trend in the Data

_{0}(no trend) can be rejected using trend direction [30,31]. Conversely, the H

_{0}(no trend) cannot be rejected for a linear trend whose slope or magnitude is so huge that it may not be disregarded for decision making on the planning and management of water resources [30,31]. Therefore, there is need to assess the significance of both trend magnitude and direction before extreme value analysis of hydrological extremes, especially if non-stationarity is to be considered.

_{j}and x

_{i}are the jth and ith observations, respectively. Because the RTOs of the POTs can be unevenly spaced, the denominator (j − i) of Equation (1) should be replaced by their corresponding jth and ith RTOs, respectively. However, for the annual maxima time series, Equation (1) can be used as it is i.e., using the actual j and i values since the years of observations remain evenly spaced.

_{0}(no trend) at the selected significance level α

_{s}%. Several non-parametric methods exist for trend detection including the Mann–Kendall (MK) [35,36], Spearman’s Rho (SMR) [37,38,39], and Cumulative Sum of rank Difference (CSD) [30,34,40] tests. The CSD trend test was recently developed by the author of this paper and applied to assess changes in the hydrometeorology of the River Nile basin. It is well-known that the MK and SMR tests rely purely on statistical results. However, the use of purely statistical trend results might be meaningless sometimes [41]. Eventually, the CSD makes use of both graphical diagnoses and statistical analyses. The graphical component of the CSD test is based on the pattern of the partial terms of the trend statistic that eventually are statistically used to test the H

_{0}(no trend) in the data [30,34,40]. When data have no ties and are not auto-correlated, the trend detection methods are comparable in performance for various circumstances of sample size, variation, slope, etc., as shown by Yue et al. [29] for MK and SMR, and Onyutha [30] for CSD and MK. However, the differences among these methods are not negligible when applied to detect trends in time series with persistent fluctuations [34]. In trend analyses, the use of one method leads to uncertainty in the results due to the influence from the choice of the method [34,42]. Therefore, CSD and MK tests were adopted in this paper.

_{CSD}is computed using Onyutha [30]:

_{CSD}indicates an upward/downward linear trend. The distribution of T

_{CSD}is approximately normal with the mean of zero and variance (V

_{1}) given by Onyutha [34,40]:

_{2}(y

_{j}− x

_{i}) is as defined in Equation (4).

_{αs/2}as the standard normal variate at α

_{s}%, while Z

_{CSD}denotes the standardized trend statistic which follows the standard normal distribution with mean of zero and the variance equals to one. If |Z

_{CSD}| ≥ |Z

_{αs/2}|, the H

_{0}(no trend) is rejected at α

_{s}%, otherwise the H

_{0}is not rejected at the α

_{s}%. Generally, the Z

_{CSD}can be computed using

_{s}%). The ${r}_{k}^{\alpha \mathrm{s}}$ can be computed using detrended time series Q obtained by Q

_{i}= X

_{i}− m × i based on m from Equation (1). Considering Q

^{#}as the mean of Q

_{i}, the values of the r

_{k}can be computed using [43]:

_{s})% confidence intervals limits (C

_{L}) for testing the significance of the r

_{k}can be computed using Anderson [44]:

_{i}from Equation (2) versus the time of observation (for details, see Appendix B). To implement the CSD trend test, a freely downloadable tool CSD-NAIM can be obtained from https://sites.google.com/site/conyutha/tools-to-download (accessed on: 14 September 2017).

#### 2.2.2. Frequency Analyses of Stationary Extremes

_{t}such that:

_{t}) parameters.

_{0}do not show upper limits), and therefore, the heavy and normal-tailed cases are more common than the light-tailed GPD distribution. To identify the case γ > 0, Pareto quantile plot (i.e., (−ln{1 − G(h)}) in abscissa versus the ln(h) in ordinate) can be used. In the Pareto quantile plot, the heavy-tailed GPD (Equation (12)) appears as a straight line. Eventually, the γ (which approximates to the slope of straight line) can be computed by the least square weighted linear regression technique (Equation (14)), and for the case of γ ≠ 0, the parameter α = γ × h

_{t}. To identify the normal-tailed (or exponential) case (i.e., γ = 0) of the GPD, the exponential quantile plot i.e.; (−ln{1 − G(h)}) in the abscissa is plotted against h as the ordinate. The GPD class with γ = 0 appears as a straight line in the exponential quantile plot. It is self explanatory why a straight line is expected in the quantile plot. In fact, using Equation (13), −ln{1 − G(h)} = (h − h

_{t})/α implying a straight line with the slope (1/α) and intercept (−h

_{t}/α). Eventually, when γ = 0, the parameter α can be computed using Equation (15) based on the weighting factor proposed by Hill [46] constrained to t number of events above h

_{t}.

_{i}for i = 1, 2, ..., n′ is defined in terms of the inverse distribution G

^{−1}(1 − a

_{i}) where a

_{i}= i/(n′ + 1) and corresponds to the Weibull plotting position of a quantile plot. The simplest function that is independent of the parameter values of G(h) but instead linearly depends on G

^{−1}(1 − a

_{i}) is called a quantile function M(a). Thus, M(a) = −ln(a) = −ln{1 − G(h)}. This is the basis upon which the plot of empirical versus theoretical quantiles (also called quantile-quantile plot) can be used to visualize the tail shape of the particular class of the GPD.

_{t}is the h above which the extreme value distribution fitted to the t observations yields the minimum average bias. In the calibration procedure, for every selected value t, how well the fitted extreme value distribution appears in the tail of the distribution can be visually checked and confirmed through statistical computation of average bias, e.g., of the theoretical from empirical return periods.

_{T}is theoretical quantile corresponding to a selected T.

_{T}can be computed using Equations (17) and (18).

_{T}should be back-transformed using (1/H) and (-H) when analysis is being done using low flow events and precipitation insufficiency, respectively.

#### 2.2.3. Frequency Analyses of Non-Stationary Extremes

_{s}% to obtain a scaling factor Ω for modifying the observed extreme events. The α

_{s}% can be specified based on its relevance for the intended application or the expert judgment of the practitioner. Normally, α

_{s}= 5% is commonly used for hydro-meteorological research. Alternatively, the significance of the trend slope can be used. Thus, the α

_{s}% can be taken as the p-value (probability value) computed based on the trend magnitude. To compute Ω, rank the extracted POT events of size n′ from the highest to the lowest and select λ as the ({0.005 × α

_{s}%} × n′)th highest value. Next, the scaling factor is computed using Ω = λ − h

_{t}where h

_{t}is as defined in Equations (12) and (13). The quantile

## Method 1: The use of significance of quantiles

${R}_{T}^{\#}$ can be computed using ${R}_{T}^{\#}=\Omega +{R}_{T}$ where R_{T}is obtained using Equation (17) or (18) depending on the shape parameter γ.

#### Method 2: The use of statistical simulation of extreme events

- (1)
- Obtain the plot of POTs versus the RTOs and compute the intercept of the linear trend line.
- (2)
- Detrend the POTs to obtain Q
_{i}= POT_{i}− (m × RTO_{i}+ intercept) for i = 1, 2, ..., n′. Obtain Q_{min}i.e., the absolute value of the minimum residual time series Q_{i}using Q_{min}= |min(Q_{i})|. - (3)
- Investigate which correlation model the extreme events follow. The persistence model can be used for generating the synthetic extreme events. Assuming that the remaining persistence in the POTs following the application of the independence criteria for extracting the extreme events is of the lag-1 autoregressive AR(1) process, compute the AR(1) serial correlation coefficient r
_{1}of the detrended time series Q_{i}using Equation (10). The AR(1) was assumed for illustration purpose. Persistence may be of the forms characterized by the fractional Gaussian noise, fractionally integrated autoregressive moving average, autoregressive integrated moving average, discrete fractional Brownian motion, fractionally differenced process, etc. - (4)
- Obtain W and N such that N = βn′ and W (in year) = βw, where β determines the length of the synthetic events, while w and n′ are as defined shortly before. For instance, in this paper, β was set to 2 as a precaution to minimize the possible introduction of uncertainty in the extreme value analysis due to finite sample size.
- (5)
- Let θ
_{1}denote the RTO of the first POT event. Generate the serial number Φ for the synthetic time series using Φ_{i}= θ_{1}+ Δ × (i − 1) for i = 1, 2, ..., N, where Δ = (365.25 × W)/N. The value of Φ should be rounded to a whole number.Obtain two trend lines based on the values of Φ. Using the trend slope m from Equation (1) and the intercept computed from Step (1), obtain the first trend line L_{1}using L_{1,i}= (m × Φ_{i}+ intercept) where i = 1, 2, ..., N. Using the Q_{min}from Step (2), obtain the second line L_{2,i}= L_{1,i}− Q_{min}. The second line L_{2}is to ensure that the synthetic events do not go below the threshold used in the independence criteria for extraction of the POTs from the original or full time series. - (6)
- Generate large number, say, N
_{sim}, of synthetic time series using Equation (19) based on the relevant persistence model identified in Step (3). In this case, for illustration, based on first-order (Markov) AR stochastic process assumed in Step (3), the synthetic extreme events were generated using:$${G}_{i}=\begin{array}{c}{G}_{o}\mathrm{for}i=1\hfill \\ E\left(G\right)+{r}_{1}\left({G}_{i-1}-E\left(G\right)\right)+{\epsilon}_{i}\mathrm{for}i=2,3,\dots ,N\end{array}$$_{i}(i.e., the POT events), G_{0}denotes the first POT event, ε_{i}is the white-noise process with mean μ_{ε}= 0 and standard deviation σ_{ε}. Based on the standard deviation S_{D}of the POT events, σ_{ε}= S_{D}× (1 − ${r}_{1}^{2}$)^{0.5}and r_{1}is as defined in Step (3).Superimpose, for i = 1, 2, ..., N, the linear trend onto the time series using G_{1,i}= Gi + L_{1,i}− E(G). Eventually, for i = 1, 2, ..., N, the final synthetic events G_{sys}= G_{1,i}if G_{1,i}≥ L_{2,i}or G_{sys}= (G_{1,i}+ L_{2,i}+ Q_{min}) when G_{1,i}< L_{2,i}. - (7)
- The simulation procedure will yield synthetic extreme events in the form of a matrix of N rows and N
_{sim}columns. For each row, rank, in descending order, the N_{sim}synthetic events. Construct the upper and lower limits of the (100-α_{s})% confidence interval on the synthetic event of each row using (0.005 × α_{s}% × N_{sim})th and ({1 − [0.005 × αs%]} × N_{sim})th ranked values, respectively. Similarly, compute the mean of the N_{sim}values in each row. Rank the computed mean values, as well as the (100-α_{s})% confidence interval limits from the highest to the lowest. Finally, compute the return period T (in years) of the synthetic values as the ratio of W (in years) to the rank j of the synthetic events (ordered such that j = 1 is for the highest ranking value).

#### 2.2.4. Non-Parametric Indices (NPIs) for Drought Assessment

_{agg}), e.g., 1 day, 7 days, 14 days, 30 days, 60 days, 120 days, 180 days, 270 days, etc. such that:

_{k}and e

_{k}are, respectively, the mean and number of the x

_{i}’s in the kth time slice. By varying k from 1 to n, the determination of e

_{k}and the computation of a

_{k}can be done in a step-wise way. Firstly, the term v is computed, for the selected A

_{agg}, using Equation (21),

_{k}using Equations (22)–(24).

_{k}(from Equation (20)) to obtain c

_{k}. Actually, c

_{k}should look mirrored to the original data and this is purposeful for the correctness e.g., of the analyses of trend directional sign. However, for drought analyses, negation is applied to the values of c

_{k}to obtain d

_{k}; expressly, d

_{k}= −1 × c

_{k}for 1 ≤ k ≤ n.

^{2}in Equation (25) becomes n(n − 1)(n + 1)/3. In other words, the bounds for the values of NPI denoted by NPI

_{bound}for untied data can be given by Equation (26) in terms of n only.

## 3. Results and Discussion

#### 3.1. Statistical Trend Analyses

_{s}= 0.05. Thus, for low flow, the H

_{0}: m = 0 was rejected at α

_{s}= 5%. However, for the precipitation insufficiency, the H

_{0}: m = 0 was not rejected at α

_{s}= 5%. Although both low flow and precipitation insufficiency were shown to reduce with time, the difference in the significance of their trend magnitudes could be thought of with respect to the data periods, which, for the river flow (1965–2002) and precipitation insufficiency (1979–2000) were different. To verify this explanation, the significance of the trend magnitude in the low flow over the period 1979–2000 as that of the precipitation insufficiency was assessed. Indeed, it can be seen in Table 4 that, just like for the precipitation insufficiency, the H

_{0}: m = 0 was also not rejected at α

_{s}= 5% for the low flow of the period 1979–2000. Notwithstanding the insignificance of the trend magnitude in both the low flow and precipitation insufficiency over the period 1979–2000, for the purpose of demonstration of the new approach being introduced, further consideration of non-stationarity for low flow over the period 1965–2002 (for which the H

_{0}: m = 0 was rejected at α

_{s}= 5%) was taken. Moreover, this consideration was in line with the fact that, in trend analysis, the longer the data, the more reliable the results. If both the flow and the catchment-wide precipitation insufficiency are of the same data long-term record period but show contrasting magnitudes of the linear trends, further investigations are required. In such a case, some of the questions which may need to be answered include the following: Can the variation in precipitation insufficiency explain the variability in flow? If not, could such a contrast be due to the questionable data quality? Could it be that apart from the meteorological input into the catchment system, changes in other factors (e.g., those due to human influence, say, land use change, transition in forest or land cover, abstraction or diversion of water, urbanization, etc.) are significantly impacting on the behavior of the catchment?

_{0}: m = 0 was not rejected at α

_{s}= 5% in both flow and precipitation insufficiency. For (1/H) low flow, the trend magnitudes m based on monthly data over both periods 1965–2002 and 1979–2000 were less than those based on daily data. This is because for daily data only the extreme events which occur in an independent and identically distributed way are considered compared to the monthly data in which all the daily values in the month under consideration are averaged and used. The magnitude of the monthly mean depends on the number of the daily extreme events in each month. The extreme events may be few in number while their increase more strongly depends on time than that of the monthly mean value of the variable. Other factors that would influence the difference between the trends based on daily time scale and that from monthly data include the CV, sample size, persistence, etc. The influence of these factors on trend results could also be in a synergistic way. Separation of the various layers of such synergistic influences while comparing results obtained from daily and monthly data requires meticulous simulation and analyses and this was out of the scope of this study.

_{0}(no trend) in the (1/H) low flow as well as (−H) precipitation insufficiency was not rejected at α

_{s}= 5% for both tests. The rejection of the H

_{0}(no trend) in the (1/H) low flow was for both the periods 1965–2002 and 1979–2000. However, the significance of the trend direction was more considerate for the (1/H) low flow of the period 1965–2002 than that of 1979–2000. Again, given the results of the significance of trend direction, in reality, the frequency analyses for both low flow and precipitation insufficiency would be conducted considering stationarity. However, as already shown before that the H

_{0}: m = 0 was rejected at α

_{s}= 5% for low flow over the period 1965–2002, decision was made to do conduct frequency analysis of low flow considering non-stationarity following the information from Table 2. Moreover, for frequency analysis of precipitation insufficiency, stationarity was considered.

_{0}(no trend) in the monthly (1/H) low flow or (−H) precipitation insufficiency was also not rejected at α

_{s}= 5% for both tests (Table 5). However, the p-values from monthly data were far much larger than those from the POTs of daily time series. For a careful decision regarding the use of non-stationarity for frequency analyses of the POTs characterizing dry conditions, it is recommended that the analyses of trend direction be based on extreme events extracted from data of daily time series.

#### 3.2. Frequency Analyses of Stationary Extremes

_{0}: m = 0 was accepted at α

_{s}= 5%. With respect to trend direction, H

_{0}(no trend) was also not rejected at α

_{s}= 5%. Given that there should be more concern for risk related to hydrological drought when there is an increasing than decreasing trend in dry spells, the frequency of the dry spells was analyzed assuming they are analogous outcomes of a stationary process. It is vital to note that, for ephemeral rivers, the result from Figure 2 which is based on the number of dry days may not be directly comparable with those when actual flow values are to be used. For instance, there can be a decrease in the number of dry days over time (indicating change from a very dry to less dry condition) while the amplitude of the variable (i.e., low flow in this case) is also decreasing (i.e., change from less dry to a very dry hydrological condition).

^{3}/s, the dry spell of T of 100 years was found to go up to about 270 days. A related point to consider is that the threshold as well as the aggregation level (or duration) to obtain results as presented in Figure 3 should depend on the purpose of the application for which the analysis is being conducted.

_{t}. In Figure 4, the exponential plot was used; however, log-transformed T was considered in the abscissa to linearize the quantiles. Extrapolation of the quantiles can be made using the calibrated extreme value distributions. Although not implemented in the results shown in Figure 4, the actual quantiles can be obtained by back-transformation of the values in Figure 4a,b using (1/H) and (−H), respectively.

^{3}/s)

^{−1}estimated from monthly data was comparable to 1.3-year return level (i.e., 0.00727 (m

^{3}/s)

^{−1}) obtained based on daily data. For T of 20 years, the daily (1/H) low flow quantile was 0.0099 (m

^{3}/s)

^{−1}. Such differences in the return levels from data of daily and monthly time series can lead to disparity in quantile-based drought categorization (as will be seen in Section C.2.4) when different temporal resolutions are used.

#### 3.3. Frequency Analyses of Non-Stationary Extremes

#### 3.4. Non-Parametric Indices (NPIs) for Drought Assessment

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Performance of the CSD Trend Statistic Variance Correction

_{CSD}variance correction in Equation (9) for series characterized by lag-1 serial correlation was already demonstrated by Onyutha [40]. However, given its generality for various forms of persistence, the suitability of Equation (9) was demonstrated in this paper using synthetic series characterized by Fractional Gaussian Noise (FGN), Discrete Fractional Brownian Motion (DFBM), Fractionally Differenced Process (FDP), and Autoregressive Integrated Moving Average (ARIMA). The ARIMA models are also known as Autoregressive Fractionally Integrated Moving Average (ARFIMA) models and they comprise three parameters AR, MA, and FD for defining the autoregressive, moving average, and fractional differencing components, respectively.

_{0}(no trend) was tested with and without T

_{CSD}variance correction. Since there was no trend imposed on each generated series, variance correction was done without detrending. Of course, for real time series, detrending should be done in the variance correction procedure. The rejection rate (%) or type I error was computed as the ratio of the number of simulations for which the H

_{0}was rejected to 2000 (expressed in percentage).

_{CSD}variance correction for the various persistence models. It can be noted that the rejection rates are lower for the series with (Figure A1b,d,f,h,j,l) than without variance correction (Figure A1a,c,e,g,i,k). The rejection rates are influenced by both the persistence and the sample size. Except for ARFIMA (MA) (Figure A1i), as the parameter of the persistence model increases, the rejection rate also increases. For ARFIMA, the less negative the value of the parameter MA, the higher the persistence in the series. Generally, this results show that large variance inflation (i.e., spread of the tail of the distribution of T

_{CSD}) is obtained with high than low persistence. It is vital to note that for H = 0.5 and DP = 0, the FGN and FDP processes respectively correspond to the case of white noise. Eventually, it is evident that rejection rates for parameters H = 0.5 and DP = 0 of FGN and FDP, respectively, were close to the nominal significance level of 5% considered for the trend test (Figure A1a,e). By applying the variance correction, it is expected that the rejection rates get close to the nominal significance level i.e., 5% in this case. This is evident for the various persistence models; thus, the acceptability of the above variance correction procedure. However, in some cases, the rejection rates were slightly higher or lower than 5%. This was due to the difficulty in obtaining the exact measure of persistence from the series generated by the various models. In fact, the inherent nature of, especially, high persistence could not accurately be captured by the various model structures in generation of synthetic series. Besides, an accurate estimation of persistence requires large samples i.e., perhaps n larger than those used for the synthetic series.

## Appendix B. Graphical Diagnoses of Changes Based on CSD Trend Test

_{sum}) of c (Equation (2)) can be obtained using:

_{sum,i}can be plotted against i or the time of observation to identify changes in the series.

_{sum,i}= 0 line is taken as the reference (the case with all the data points tied up i.e., a tie with 100% extent). The deviation of the values of the C

_{sum,i}from the reference characterizes temporal changes in the series. If the series is from purely a non-stationary process, the temporal variation can be described in a cumulative way by C

_{sum,i}= (ni − i

^{2}) or C

_{sum,i}= (i

^{2}− ni) for maximum positive and negative linear trends, respectively. The entire curve in a plot of C

_{sum,i}versus i (or time of observations) for series with negative and positive trend will fall below and above the reference, respectively (see Figure A2c,d and Figure 3 ) and Figure 4 ). Sometimes, it can be difficult to notice from time series plots that both positive and negative directions of trend exist. Because the effects of the positive and negative sub-trends cancel out each other, when statistical trend test is applied to such time series, the long-term time series can show no trend. However, the plot of C

_{sum,i}versus i can show which part of the series has positive or negative sub-trends based on the curves formed below or above the reference (see Figure A2a,b,1,2). For time series with a sub-period characterized by a random variation of the values while in the other part there is a linear trend, the tendency to form a curve will be obtained over the section with linear trend (see Figure A2h,8). For the case of step jump in the mean for data that has no trend in the sub-series before and after the change point, two lines with opposite slope signs intersect with the vertex at the change point (see Figure A2e,f,5,6). This vertex occurs above (below) the reference for a step upward (downward) jump. It is possible that there can be a step jump in the intercept of linear trend. In other words, the sub-series before and after the step jump have linear trends that are in the same direction. In this case, the two curves will be formed with the intersection at the point of the step jump (see Figure A2i,9). If the series characterizes a white noise process, the series will cross the reference in a random way (see Figure A2g,7).

_{sum,i}from Equation (A1). If the step jump is in the intercept of the linear sub-trends before and after the change, the change-point can be detected graphically as time of observation where the two curves intersect (Figure A2i and Figure 9). If a time series has two sub-trends of opposite directional or slope sign, the change-point will actually be the time of observation where the first (second) curve ends (starts), i.e., where, apart from i = 1 and i = n, the overall C

_{sum,i}curve crosses the reference (Figure A2a,b and Figure 1 and Figure 2). If a series has no trend in the first part but a linear increase or decrease in the second portion, the change point is where the curve over the last sub-period begins (Figure A2i and Figure 8).

## Appendix C. Revisiting Concepts on Drought Assessment through Shift from Coarse to Fine Temporal Scale

#### C.1. Introduction

#### C.2. Definitions of Key Drought Terms Based on Daily Time Scale

#### C.2.1. Threshold

#### C.2.2. Deficit, Deficit Period, Drought Intensity and Extremity

#### C.2.3. Deficit Sum versus Days of Deficit Period

#### C.2.4. Drought Quantile and Incidence

^{3}/s) and precipitation insufficiency (H, mm/day), transformation can be done using (1/H) and (−H), respectively. At the end of the analyses, back-transformation is done to obtain the actual return levels. For analyses using dry spell, transformation may not be needed.

#### C.2.5. Relationships between Deficit or Incidence with Threshold and Duration

- (1)
- Select aggregation levels (A
_{agg}). For illustration in this paper, the values of A_{agg}were set to 1, 7, 15, 30, 60, 90 and 150 days. In reality, the range of A_{agg}to be selected should be relevant for the intended application. Next, pass an overlapping moving averaging window of length equal to each A_{agg}using Equations (20)–(24). - (2)
- Select thresholds for determining the deficit period. In this study, 20, 30, 40 and 50% of the long-term mean of daily river flow, and 120, 130, 140 and 150% of the long-term mean of daily precipitation insufficiency were selected.
- (3)
- For each threshold, compute the historical LDP and incidence based on the a
_{j}’s from each value of A_{agg}from Step 1. The last step is the compilation of the LDP and incidence for the various thresholds and values of A_{agg}to constitute LDP-DT and IDT relationships respectively.

#### C.2.6. Amplitude–Duration–Frequency Relationships

_{agg}to constitute Low Flow–Duration–Frequency (LFDF) and Precipitation Insufficiency–Duration–Frequency (PIDF) relationships. The following steps can be used to construct LFDF and PIDF relationships:

- (1)
- Transform river flow (H) and precipitation insufficiency (H) using (1/H) and (-H), respectively.
- (2)
- Select various values of A
_{agg}with the range relevant for the intended application. - (3)
- For each of the A
_{agg}values, perform:- (a)
- temporal aggregation of the series using Equation (20);
- (b)
- extraction of the POT events using the independence criteria from Section 2.2.1; since aggregation may bring about auto-correlation, the specifications of these criteria can be in a relaxed or stringent way based on whether small or large values of A
_{agg}have been used to smoothen the data; - (c)
- selection of various T’s relevant for the intended application; and
- (d)
- derivation, for the corresponding T’s, the return levels assuming H is from stationary or non-stationary process using the details of methodology presented in Section 2.2.2 and Section 2.2.3.

- (4)
- Compile back-transformed quantiles for the various T’s and A
_{agg}values.

#### C.2.7. Relationship between Dry Spell or Deficit Period, Duration and Frequency

- (1)
- Values of A
_{agg}were set to 15, 60 and 90 days and Equation (20) was applied to the time series. - (2)
- For each A
_{agg}, hydrological dry spells or deficit periods were independent (or nearly so). Expressly, daily flow threshold was set to 800 m^{3}/s, and the minimum number of days to characterize a hydrological dry spell was set to 175 days. - (3)
- For each A
_{agg}, empirical T was computed for the independent dry spells. Theoretical dry spell D_{T}for a given T was estimated using D_{T}= η × {log(T) − log(T_{h})} + D_{h}where D_{h}is the threshold dry spell, T_{h}is the return period of D_{h}, and η is the slope of the line fitted above the threshold. To estimate η, log-transformed empirical T is plotted against the dry spell. The slope of the logarithmic line fitted to the events above the D_{h}gives the η estimate.

#### C.2.8. Hydrological Response to Meteorological Drought

- (1)
- Set the relevant hydrological or flow thresholds. In this study, for illustration, daily flow thresholds were set to 200, 500, 750 m
^{3}/s, etc. - (2)
- Determine the hydrological deficit periods.
- (3)
- For each deficit period or drought event, obtain the absolute sum of the hydrological deficits (m
^{3}/s). Multiply each daily hydrological deficit by (86.4/C_{A}) where C_{A}is the catchment area in km^{2}. This converts all the obtained deficits from m^{3}/s to mm/day to match the unit of the precipitation insufficiency. - (4)
- For each corresponding daily hydrological deficit period, obtain the absolute sum of the precipitation insufficiency (mm/day).
- (5)
- By trial and error, look for a constant (hereinafter taken as the HRC) by which when all the sums of the hydrological deficits are multiplied, the difference between the meteorological and hydrological deficits becomes minimal. This can be done by mean squared error minimization technique. Expectedly, the scatter plots of the hydrological versus meteorological deficits should exhibit a linear relationship. For an ideal system, the line passes through the origin and has a slope equal to 1.
- (6)
- Repeat Steps 2 to 5 for various A
_{agg}values. Finally, compile the HRCs from the various A_{agg}values and thresholds to comprise the HRC–DT relationships.

#### C.3. Applications of Daily Time Scale for Drought Analyses

#### C.3.1. Threshold and Deficits

^{3}/s, the hydrological deficit periods of drought events marked I-VI (Figure A3a) were of length 198, 277, 274, 281, 279, and 262 days, respectively. The corresponding hydrological drought deficits when converted from m

^{3}/s to mm/day were 41, 61, 62, 128, 63 and 25 mm/day, respectively. Thus, the hydrological LDP and mean deficit were 281 days and 63 mm/day respectively. Based on the total number of days from the selected data period 1979–1984 (i.e., 2192 days), the hydrological drought extremity was found to be 0.128. This is typical of a hydro-meteorological condition influenced by seasonality of hydrological processes. The threshold for the precipitation insufficiency that corresponded to the flow of 1200 m

^{3}/s was found to be −5 mm/day. It means the hydrological deficits for drought events I–VI (Figure A3a) reflect the propagation of the meteorological dry conditions marked 1–6 (Figure A3b). The mean meteorological deficit was 2103 mm/day.

#### C.3.2. Deficit Sum versus Days of Deficit Period

#### C.3.3. Drought Quantile and Incidence

#### C.3.4. Relationships between Deficit or Incidence with Threshold and Duration

^{3}/s and −6.4 mm/day, respectively. Generally, as the threshold increases for a particular A

_{agg}, the drought of each type is characterized by an increase in both the incidence and deficit period. It is also noticeable that as the A

_{agg}increases, the incidence or LDP reduces. However, due to randomness in data of low A

_{agg}values; (e.g., increasing A

_{agg}from one day up to about a month; see Figure A7d) the incidence may even increase. This is an indication in the anomaly of drought propagation from daily to monthly time scale which cannot be revealed by using coarse time scale e.g., monthly data. The duration of a particular drought event was (though not shown) found to increase as the value of A

_{agg}was also increased. The slopes of the lines in Figure A7 show the influence of temporal aggregation of series on the drought incidence and LDP. In drought assessment using monthly data, the detailed and compressed information on incidence and dry spell (as shown in Figure A7 for daily time scale) cannot be obtained.

#### C.3.5. Amplitude–Duration–Frequency Relationships

_{agg}values and T’s. A small slope of the T-year curve of the LFDF relationships shows high temporal homogeneity in the river flow. However, a large slope of the T-year curve of the LFDF relationships shows a strong variation in the wet-dry conditions in the variable, e.g., river flow, precipitation insufficiency, etc. Besides, the parameters of the selected extreme value distribution used in the compilation of the LFDF and PIDF relationships can also be indicative of the variations in the flow and meteorological conditions, respectively. For instance, a high value of the scale parameter of exponential distribution, would indicate large variation in the extreme low events from river flow or precipitation insufficiency. In summary, the use of high temporal resolution allows a possible attachment of physical meaning to the results of drought analyses at a more refined relevant information than if monthly series would be used. This follows the need to characterize the functionality of the catchment or watershed as a system based on a fine temporal scale.

#### C.3.6. Relationship between Dry Spell or Deficit Period, Duration and Frequency

_{agg}), the lower the curve. The results in Figure A9 were based on daily flow time series from 1965 to 2002. As a side note, the minimum number of days to characterize a hydrological dry spell depends on the threshold to define a dry day and the A

_{agg}. The dry spells with T’s of 5, 10, 50 and 100 years indicate hydrological dry spells with quantile in the category which can be described as mild, moderate, extreme and exceptional, respectively. The conventional analyses of drought based on monthly series cannot yield such an important information for catchment-based monitoring of drought.

#### C.3.7. Hydrological Response to Meteorological Drought

^{3}/s, there were 63 hydrological drought events over the data period 1979–2000. It was noted (though not shown) that by increasing threshold, the number of drought events reduced. When converted from m

^{3}/s to mm/day, the hydrological deficits ranged from 0.0012 to 2.6912 mm/day. The absolute sum of the meteorological deficit (mm/day) for the corresponding hydrological deficit periods can be seen in Figure A10b. The difference in the orders of the magnitude of the sums of hydrological and meteorological deficits is noticeable (Figure A10a,b). When the values in Figure A10a were multiplied by 426 (i.e., the HRC), the difference between the sums of the hydrological and meteorological deficits (Figure A10a,b) became minimal (Figure A10c). Scatter plot of hydrological versus meteorological deficits exhibited a linear behavior (Figure A11a). Relationship between the hydrological deficits and the deficit periods can be seen in Figure A11b.

_{A}is small). The response from a catchment of a large size can be slow. This is because during rainy season, the catchment stores enough of the net rainfall in the form of sub-surface water or soil moisture which can continuously replenish the river flow during dry period. For a catchment with slow response i.e., high HRC, it can be possible that the effect of meteorological drought of a particular season can be reflected in the hydrological drought over a different season occurring later than the period of actual meteorological drought. Spatial heterogeneity in the distribution of rainfall and ET

_{0}can lead to difference in meteorological deficit period and wet-dry variation across the catchment. The hydrological response to meteorological drought can be indicative of the net effect of the intermittency in meteorological inputs. Other physiographic factors on which the HRC can depend include geology, topography, vegetation cover, etc.

_{agg}value, the amount of explained variability was found to be 89, 93, 95, and 97% for hydrological thresholds of 200, 500, 750, and 1000 m

^{3}/s, respectively. The unexplained variability in hydrological drought may be due to other factors apart from precipitation insufficiency.

_{agg}follows a power function (Figure A12a). A large value of HRC indicates large difference between hydrological and meteorological deficits. It means that, meteorological droughts of short-duration (or low A

_{agg}values) have reduced influence on hydrological drought. The larger the A

_{agg}value, the more reduced is the variation of flow or precipitation insufficiency. Therefore, large A

_{agg}reduces the difference between the hydrological and meteorological deficits. For a particular catchment, prolonged meteorological drought (i.e., drought detected using large A

_{agg}values) has wide spatial extent and leads to larger hydrological deficit (Figure A12b) than that of short-duration.

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**Figure 1.**The Blue Nile Basin. The Digital Elevation Model (DEM) used as the background map was obtained online from the International Centre for Tropical Agriculture, CIAT-CSI SRTM website, http://strm.csi.cgiar.org/ (accessed: 20 October 2010). The station numbers are consistent with those in Table 1.

**Figure 2.**Hydrological dry spells after passing 15-day smoothing or averaging window through daily flow time series at Khartoum from 1965 to 2002. The dotted line is a linear trend.

**Figure 3.**Theoretical distribution fitted to hydrological dry spell of 15-day aggregation level in flow observed from 1965 to 2002 at Khartoum. The horizontal axis is based on logarithmic scale.

**Figure 4.**Exponential distribution calibrated to: (

**a**) (1/H) low flow events over the period 1965–2002 at Khartoum; and (

**b**) (−H) precipitation insufficiency (Prec. ins.) for the period 1965–2002 over the Blue Nile basin. Comparison of: (

**c**) (1/H) low flow; and (

**d**) (−H) precipitation insufficiency quantiles are based on daily and monthly data. The horizontal axis of each chart is based on logarithmic scale.

**Figure 5.**Selected drought event based on the catchment-wide: (

**a**) daily; and (

**b**) monthly (H) precipitation insufficiency (Prec. ins.) from 1979–2000.

**Figure 6.**POT events of (1/H) low flow from 1965 to 2002 at Khartoum based on: (

**a**) daily; and (

**b**) monthly data; as well as the POTs from basin-wide (−H) precipitation insufficiency for the period 1979–2000 based on: (

**c**) daily; and (

**d**) monthly data. The dotted line is a linear trend.

**Figure 7.**POT events of (1/H) low flows versus: (

**a**) RTO; and (

**b**) return period. In the legend of (

**b**), “CI” stands for confidence interval on the T-year events from “Method 2”. For chart (

**b**), the horizontal axis is based on logarithmic scale.

**Figure 8.**Return period T against: (

**a**) T-year low flow (H) after back-transformation from (1/H); and (

**b**) bias on the T-year low flow (H).

**Figure 9.**Meteorological dry and wet conditions in terms of NPI and SPEI applied to catchment-wide: (

**a**) daily; and (

**b**) monthly precipitation insufficiency. Chart (

**c**) shows NPI derived from daily river flow (H) and precipitation insufficiency (H). In each chart, the label text in “( )” shows the aggregation level.

**Figure 10.**Meteorological dry and wet conditions in terms of: (

**a**,

**c**) SPEI; and (

**b**,

**d**) NPI applied to catchment-wide daily precipitation insufficiency. The label text in “( )” of the legend shows the aggregation level.

**Figure 11.**Skewness of SPEI when applied to daily precipitation insufficiency based on the aggregation level of: (

**a**) one day; and (

**b**) 180 days. The label text in “( )” of the legend shows the aggregation level.

**Figure 12.**Extreme dry and wet conditions in terms of NPI and SPEI based on catchment-wide: (

**a**) daily; and (

**b**) monthly precipitation insufficiency over the period 1979–2002; and (

**c**,

**d**) the skewness of the indices based on daily and monthly precipitation insufficiency, respectively.

**Figure A1.**Rejection rate for (

**a**,

**b**) FGN; (

**c**,

**d**) DFBM; (

**e**,

**f**) FDP; and (

**g**–

**l**) ARFIMA models. Trend results obtained: (

**a**,

**c**,

**e**,

**g**,

**i**,

**k**) without; and (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**) with T

_{CSD}variance correction for persistence. All the charts share the same legend as in (

**b**).

**Figure A2.**Plots of: (

**a**–

**i**) synthetic series X

_{syn}of n = 200; and (

**1**–

**9**) the cumulative effects (C

_{sum}, Equation (A1)) of the temporal variations for the corresponding time series.

**Figure A3.**Hydro-climatic deficit periods based on daily: (

**a**) flow; and (

**b**) precipitation insufficiency.

**Figure A5.**Meteorological deficit of: (

**a**,

**b**) mild; (

**c**) severe; and (

**d**) extreme drought quantile category. In the legend, e.g., T5 stands for return period of 5 years.

**Figure A6.**Historical: (

**a**,

**b**) incidence (%); and (

**c**,

**d**) LDP (day) of the precipitation insufficiency based on threshold of: (

**a**,

**c**) −10 mm/day; and (

**b**,

**d**) −5 mm/day, considering the period 1979–2000.

**Figure A7.**IDT relationships for: (

**a**) hydrological drought; and (

**c**) meteorological drought. LDP-DT relationships for: (

**b**) hydrological drought; and (

**d**) meteorological drought.

**Figure A8.**T-year events comprising: (

**a**) LFDF relationships; and (

**b**) MIDF relationship. For chart (

**a**), the T-year events considering stationarity and non-stationarity are presented in the form of lines and markers, respectively. In the legend, e.g., T5 denotes T-year curve for T = 5 years.

**Figure A10.**(

**a**) Hydrological drought; (

**b**) sum of meteorological deficit; and (

**c**) comparison of the sum of hydrological and meteorological deficits.

**Figure A11.**Hydrological deficit versus: (

**a**) sum of precipitation insufficiency; and (

**b**) deficit period.

**Figure A12.**Hydrological response constant (HRC) versus: (

**a**) hydrological deficit; and (

**b**) duration.

Station | Period | Statistics | ||||
---|---|---|---|---|---|---|

Number | Identity | Name | From | To | CV | Skewness |

1 | ET000063331 | Gondar | 1952 | 1988 | 2.30 | 3.63 |

2 | ET000063332 | Bahar Dar | 1952 | 1988 | 2.53 | 4.57 |

3 | ET000063333 | Combolcha | 1953 | 1988 | 2.54 | 3.98 |

4 | ET000063334 | Debremarcos | 1953 | 1987 | 1.94 | 2.93 |

5 | ET000063402 | Jimma | 1952 | 1988 | 1.92 | 3.11 |

6 | ET000063403 | Gore | 1952 | 2014 | 1.95 | 3.40 |

7 | SU000062721 | Khartoum | 1957 | 1987 | 8.31 | 13.66 |

8 | SU000062752 | Gedaref | 1957 | 1976 | 4.37 | 8.21 |

9 | SU000062762 | Sennar | 1961 | 2013 | 6.17 | 11.22 |

Decision | Trend Magnitude | ||
---|---|---|---|

m ≠ 0 | m ≈ 0 | ||

Trend direction | |Z*| ≥ |Z_{αs/2}| | Non-stationarity | Stat * |

|Z*| < |Z_{αs/2}| | Non-stationarity | Stationarity |

_{s}%, but for practical reason there is need to take the trend magnitude into decision making perspective, non-stationarity should be considered; otherwise, frequency analysis should be conducted assuming analogy to realizations from a stationary process. Furthermore, m ≠ 0: means that for (1/H) flow or (−H) precipitation insufficiency, m > 0. Thus, after back-transformation of the data, i.e., for (H) flow or (H) precipitation insufficiency, the trend slope becomes m < 0. However, in flood frequency analyses for which data transformation is not required, the concern is on m > 0 for the significance of both trend direction and magnitude detected using (H) flow or (H) precipitation insufficiency. Finally, Z*: means Z

_{CSD}or Z

_{MK}or the standardized trend statistic of the method used.

Daily Time Scale | Monthly Time Scale | Category | Return Period T (year) | ||
---|---|---|---|---|---|

From | To | From | To | ||

0 | −1.72258 | 0 | −1.44421 | Near normal | ≤1 |

−1.72259 | −1.73112 | −1.44422 | −1.70391 | Mild | 2–9 |

−1.73113 | −1.73159 | −1.70392 | −1.71833 | Moderate | 10–19 |

−1.73160 | −1.73188 | −1.71834 | −1.72169 | Severe | 20–49 |

−1.73189 | −1.73197 | −1.72170 | −1.72988 | Extreme | 50–99 |

−1.73198 | <−1.73198 | −1.72989 | <−1.72989 | Exceptional | ≥100 |

POT | Data Period | Daily Data | Monthly Data | ||
---|---|---|---|---|---|

m | p-Value | m | p-Value | ||

(1/H) low flow | 1965–2002 | 9.57 × 10^{−8} | 0.007 | 6.58 × 10^{−10} | 0.843 |

(1/H) low flow | 1979–2000 | 1.64 × 10^{−7} | 0.152 | −3.85 × 10^{−9} | 0.622 |

(-H) precipitation insufficiency | 1979–2000 | 2.77 × 10^{−5} | 0.685 | 2.94 × 10^{−5} | 0.480 |

POT | Data Period | p-Value From | p-Value From | ||
---|---|---|---|---|---|

Daily Data | Monthly Data | ||||

CSD Test | MK Test | CSD Test | MK Test | ||

(1/H) low flow | 1965–2002 | 0.151 | 0.136 | 0.998 | 0.985 |

(1/H) low flow | 1979–2000 | 0.375 | 0.413 | 0.914 | 0.937 |

(-H) precipitation insufficiency | 1979–2000 | 0.497 | 0.511 | 0.977 | 0.963 |

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**MDPI and ACS Style**

Onyutha, C. On Rigorous Drought Assessment Using Daily Time Scale: Non-Stationary Frequency Analyses, Revisited Concepts, and a New Method to Yield Non-Parametric Indices. *Hydrology* **2017**, *4*, 48.
https://doi.org/10.3390/hydrology4040048

**AMA Style**

Onyutha C. On Rigorous Drought Assessment Using Daily Time Scale: Non-Stationary Frequency Analyses, Revisited Concepts, and a New Method to Yield Non-Parametric Indices. *Hydrology*. 2017; 4(4):48.
https://doi.org/10.3390/hydrology4040048

**Chicago/Turabian Style**

Onyutha, Charles. 2017. "On Rigorous Drought Assessment Using Daily Time Scale: Non-Stationary Frequency Analyses, Revisited Concepts, and a New Method to Yield Non-Parametric Indices" *Hydrology* 4, no. 4: 48.
https://doi.org/10.3390/hydrology4040048