Estimation of Reservoir Volumes at Drafts of 40% - 90%: Drought Magnitude Method Applied on Monthly River Flows from Canadian Prairies ()

Tribeni C. Sharma^{}, Umed S. Panu^{*}

Department of Civil Engineering, Lakehead University, Thunder Bay, ON, Canada.

**DOI: **10.4236/jwarp.2022.148030
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Department of Civil Engineering, Lakehead University, Thunder Bay, ON, Canada.

The draft ratios for sizing the reservoirs can vary within a wide range (40% - 90% of the mean annual flow, MAF), depending upon the demands for water by various users, and environmental and ecological considerations. The reservoir volumes based on the drought magnitude (DM) method were assessed at aforesaid draft ratios using monthly-standardized hydrological index (SHI) sequences of 10 Canadian rivers located in the Canadian prairies and northwestern Ontario. These rivers are typified by a high level of persistence lag-1 autocorrelation, *ρ*_{1m} ≥ 0.50 and up to 0.94) and coefficient of variation (*cv*_{o}) in the range of 0.42 to 1.48. The moving average (MA) smoothing of monthly SHI sequences formed the basis of the DM method for estimating reservoir volumes. The truncation or cutoff level in the SHI sequences was found as *SHI _{x}* [=(

Keywords

Draft Ratio, Extreme Number Theorem, Markov Chain, Moving Average Smoothing, Reliability, Standardized Hydrological Index, Sequent Peak Algorithm

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Sharma, T. and Panu, U. (2022) Estimation of Reservoir Volumes at Drafts of 40% - 90%: Drought Magnitude Method Applied on Monthly River Flows from Canadian Prairies. *Journal of Water Resource and Protection*, **14**, 571-591. doi: 10.4236/jwarp.2022.148030.

1. Introduction

The reservoir volumes can be estimated involving annual, monthly or weekly flow sequences. It has been shown by Sharma and Panu [1] that the monthly scale of flow sequences is adequate for estimating the reservoir volume (V_{R}) corresponding to a certain draft ratio (draft in proportion to the mean annual flow, MAF). The annual scale may mask the seasonal variability and persistence characteristics needed in the estimation process; whereas, the weekly scale only adds a little refinement over the results derived using the monthly scale and also renders the analysis rigorous and complex [1]. The draft ratio (*α*) is expressed as the proportion to the mean annual flow (MAF) such as 100% (*α* = 1 MAF), 90% (0.9 MAF), 75% (0.75 MAF), 60% (0.60 MAF) etc. Knowing that in terms of magnitude, the mean annual flow (MAF) and the overall mean monthly flow (*µ _{o}*) are equivalent to each other and thus, MAF and

In terms of the hydrological drought analyses and more specifically for the DM method, two main parameters of interest in a drought episode are duration (length, *L*) and magnitude (*M*, also termed as severity), which have been the subject of intensive study [15] - [25]. The term *M* is a standardized entity (dimensionless) such that the drought deficiency volume *D* = *σ* × *M* [16]. For the estimation of reservoir volume, the term *D* can be regarded as a counterpart term to *V _{R}*. The modelling activity for drought magnitude is of paramount importance in terms of management of waters, and consequently for sizing and operation of reservoirs. Using the concept advanced by Dracup

From an earlier study (Sharma and Panu [1]), the analysis using the draft at 100% of the mean annual flow (or *α* = 1) resulted in an excessively high volume of reservoirs. A median value of the draft for design purposes has been identified as 75% of MAF (*α* = 0.75) [3] [5] [8], which was analyzed for sizing reservoirs in Canadian rivers using the monthly SHI sequences by the DM method [2]. In these investigations, the DM method was compared to SPA at the probability of failure (PF) level of 0%. The term PF is defined as the ratio of the number of time units the reservoir became empty to the total number of time units used in the analysis [3]. Accordingly, reliability is defined as (100-PF). The crucial parameter in the DM method was a scaling parameter (Φ) for drought length, which emerged equal to 0 and 0.5. Also, the concept of moving average (MA) of SHI sequences was introduced in the DM method with smoothing steps MA1 and MA2. The MA2 smoothing means the moving average of two consecutive values of SHI and MA1 means the SHI sequence itself. Succinctly, the results of the DM method were based on the aforesaid fixed values of Φ and MA smoothing (MA1 and MA2 steps) and were found to be satisfactory and compatible with SPA based estimates.

In a worldwide study of reservoirs, McMahon *et al.* [14] noted that the draft ratio, *α* varies from 0.90 to 0.10 with the median value of 0.47 in Australia and 0.29 in South Africa. In the US, the *α* displays a large variability between the eastern and western regions because of wide variability in hydrologic conditions. However, *α* has been reported [14] to vary from 0.40 to 0.90 in the eastern region. In view of the proximity of Canada to the USA, and the hydrologic conditions of the Canadian rivers resembling the eastern USA, this paper attempts to demonstrate the DM method in assessing the size of reservoirs for a range of *α* from 0.90 down to 0.40. Three levels of the probability of failure (*i.e.*, 5%, 2.5% and 0% respectively corresponding to the reliability of 95%, 97.5% and 100%) are considered. The motivation for the current study is to apply the DM method in the above range of draft ratios, which are used in the North American settings and evaluate its promise. In particular, the modus operandi of implementation of the DM method concerning the fixed or variable nature of Φ values in combination with MA smoothing steps was the main focus of the study.

A set of ten rivers from the Canadian prairies and northwestern Ontario exhibiting a high degree of persistence (*ρ*_{1m} ranging from 0.50 to 0.94) and a wide range (0.42 to 1.48) of *cv _{o}* (coefficient of variation) are the focus of analysis. In addition, such a wide range in autocorrelation and coefficient of variation offered an opportunity to identify the exclusive role of these parameters in influencing the reservoir volume by the DM method.

2. Preliminaries on Methods for Sizing Reservoirs

The popular method for sizing reservoirs is the sequent peak algorithm (SPA) which is well documented in [3] [6]. The SPA requires the historical or synthesized river flow data as inflows and demand levels as outflows, and data are analyzed using the original flows (or without standardization) in a sequential format. Computations are numerically conducted using the differences between cumulative inflows and demand values to assess the size of a reservoir. The procedure is fully amenable to computerized computations to arrive at the required reservoir volume (*V _{R}*) for a given situation. The SPA hinges on the reservoir water balance equation in which the reservoir is assumed to be initially full and thus does not allow the determination of the storage at a prefixed level of probability of failure (PF). However, the PF can be evaluated for the SPA based storage volume, which would turn out to be zero. The other variant of the SPA is the Behavior analysis, in which various storage levels in the reservoir (not initially full) are evaluated at the desired level of PF by a trial and error procedure.

In the DM method, SHI sequences are obtained after standardization (month-by-month) and in turn are chopped at a suitable truncation level corresponding to draft ratio, *α*. The truncation (or cutoff) level considered are equivalent to
$SH{I}_{o}=\left(\alpha {\mu}_{o}-{\mu}_{o}\right)/{\sigma}_{o}=\left(\alpha -1\right){\mu}_{o}/{\sigma}_{o}$. The other variants of the truncation level *SHI _{x}* are considered as:
$SH{I}_{\mathrm{max}}=\left(\alpha {\mu}_{o}-{\mu}_{o}\right)/{\sigma}_{\mathrm{max}}=\left(\alpha -1\right){\mu}_{o}/{\sigma}_{\mathrm{max}}$,
$SH{I}_{av}=\left(\alpha -1\right){\mu}_{o}/{\sigma}_{av}$,
$SH{I}_{gm}=\left(\alpha -1\right){\mu}_{o}/{\sigma}_{gm}$, and
$SH{I}_{har}=\left(\alpha -1\right){\mu}_{o}/{\sigma}_{har}$ in which

The SHIs below the truncation level are referred to as the deficit (*d*), whereas above the level are referred to as the surplus (*s*). In a historical record of *N* (=*T*) months, several spells of deficit and surplus shall be encountered and the length of the longest spell (representing *L _{T}*) is recorded. Likewise, the corresponding SHIs are added to represent the largest magnitude (

While using models for estimating a value of *M _{T}*, the following probabilistic relationship through the use of the extreme number theorem [17] [23] provides the basic conceptual link.

$P\left({M}_{T}\le Y\right)=\mathrm{exp}\left[-Tq\left(1-{q}_{q}\right)\left(1-P\left(M\le Y\right)\right)\right]$ (1)

In which, *q* represents the simple probability of drought and *q _{q}* represents the conditional probability that the present period is a drought given the past period was also a drought, and

1) For *M _{T}*

${M}_{T\text{-}e}=\text{abs}\left[-\left\{\frac{\mathrm{exp}\left(-0.5{z}_{0}^{2}\right)}{q\sqrt{2\pi}}\right\}-{z}_{0}\right]{L}_{c}$ (2)

In which, *z*_{0} is the standard normal variate at the drought probability, *q*.

2) For *M _{T}*

${M}_{T\text{-}e}={\displaystyle {\sum}_{j=0}^{{n}_{1}}\frac{{Y}_{j+1}+{Y}_{j}}{2}}\left[P\left({M}_{T}\le {Y}_{j+1}\right)-P\left({M}_{T}\le {Y}_{j}\right)\right]$ (3)

The reliability criteria generally adopted for the design of reservoirs are based on the probability of failure (PF) defined earlier in the text [3], *i.e.* PF is the ratio of the number of months the reservoir went dry to the total number of months used in the analysis. The crucial elements in the process of sizing reservoirs in the DM method are: 1) selection of the cutoff level (*SHI _{x}*) for truncating the monthly SHI sequences at the aforesaid levels of PF and 2) selection of a correct value of the parameter Φ.

3. Data Acquisition and Calculations of Reservoir Volumes

Seven rivers ({1} to {8}) from Canadian prairies (Table 1, Figure 1) and 2 rivers ({9} and {10}) from northwestern Ontario bordering the prairies were involved in the analysis. The rivers encompassed drainage areas ranging from 97 to

Figure 1. Location of the river flow gauging stations used in the analysis [source: Environment Canada].

Table 1. Summary of statistical properties of monthly flows of the selected ten rivers.

A value indicated by an asterisk (*) is the overall coefficient of variation (*cv _{o}* =

119,000 km^{2} with the database spanning from 35 to 108 years. The chief criterion for the selection of these river-gauging stations is the high level of persistence (*ρ*_{1m} ≥ 0.50). The flow data for these 10 rivers were extracted from the Canadian Hydrological database, Environment Canada [26]. The statistical parameters viz. *µ _{o}*,

The analysis required the moving averages of the monthly SHI sequences for the evaluation of *M*, particularly at high draft ratios. A monthly SHI sequence, without any moving average, is designated as moving average 1 (MA1), which essentially is the monthly SHI sequence itself. At times, there arose a need to perform the moving averages of two consecutive values or three consecutive values of monthly SHIs, which are respectively designated as MA2 or MA3. The MA1 sequence of SHI was used for counting as well as estimating values of *L _{T}* and

The aforesaid statistics were also evaluated for MA2, MA3 and MA4 sequences, whenever such a need arose. The MA2 sequence is characterized by mean = 0 with *σ* < 1 and lag-1 autocorrelation (designated as *ρ*_{1m2}) > than *ρ*_{1m}. The MA2 sequence was further standardized and all computations were accomplished in this new standardized (designated as MA2’) domain along with deriving of *M* values. A similar strategy would also apply to MA3 and MA4 sequences. For all MA smoothing (MA1 through MA4), the multiplier to *M* for obtaining D would be *σ _{av}* (

It may be highlighted that the Beaver River flows tend to be highly variable with *cv _{o}* = 1.48 and

Table 2. Values of parameters used in relevant equations for calculating *M*_{T-e} in the DM method.

Asterisk (*) with *SHI _{m}* means

*cv _{o}* = 0.42 but with high

Computational Strategy

In the DM method, SHI sequences were chopped at the level of *SHI _{x}* and the

Towards achieving the desired PF, one needs to consider three variables, *i.e.* 1) cutoff level (*SHI _{x}* =

Table 3. Estimation of reservoir volumes for the Beaver River (*cv _{o}* = 1.48,

Table 4. Estimation of reservoir volumes for the English River (*cv _{o}* = 0.74,

Table 5. Estimation of reservoir volumes for the Churchill River (*cv _{o}* = 0.49,

Asterisk (*) in Tables 3-5 means values in parentheses are the standardized values of *V _{R}* (=

that fulfilled the criterion of meeting the desired PF is listed in Tables 3-5 for *α* ranging from 0.40 to 0.90. Three typical computational cases are presented viz. the Beaver River (Table 3) with the highest *cv _{o}* = 1.48, the English River (Table 4) with modest

In the case of the Beaver River, the *SHI*_{max} proved to be the best cutoff level in combination with step MA1 (except, *α* = 0.40). However, the computed PF values were not exactly 5% and 2.5% (column 9) but were within the close vicinity *i.e.*, 5.07% and 2.58% etc. level with MA steps from MA1 to MA4 (Table 2).

The Churchill River performed well up to *α* = 0.50 but it encountered a problem at *α* = 0.4, where it could not reach the level of PF = 5%. Therefore, at this level of PF, the Behavior analysis was conducted to obtain a value of *D _{T}*

In short, for highly correlated flows with *cv _{o}* of 0.42 at

4. Results and Discussion

The results of analyses indicated that rivers, in general, can be categorized into two groups: 1) exhibiting ρ_{1m} within the range of 0.50 to 0.78 viz. Bow, Athabasca, Beaver, South Saskatchewan, English and Pipestone rivers and 2) exhibiting *ρ*_{1m} ≥ 0.87 such as the Churchill, Sturgeon, Gods and Island Lake rivers (Table 1). The DM method can easily handle the estimation detail of sizing reservoirs for rivers in category (1) along with draft conditions considered. Out of all 10 rivers, the characteristics of the Beaver River (Alberta, Canada) differ significantly from the remaining nine rivers. For example, the Beaver River originates on the boreal plains rather than on the eastern slopes of the Rocky Mountains and as a result, the runoff is not subject to the stabilizing influence of mountain snowmelt and thus displays a considerable variability from year to year and within the year. The river rises to a peak flow in the latter part of April because of spring snowmelt and occurrences of rainstorms during the snowmelt period, and thereafter the flow, generally, recedes through the remainder of the year. Unlike, the prairie streams of southern Alberta, the Beaver River tends to respond to summer rainstorms with dramatic increases in flows. Indeed, in some years, the summer flows may be greater than the peak flow during the spring runoff (Beaver River Alliance, 2013) [27]. Such unique flow characteristics are ramified in *cv _{o}* = 1.48, requiring the cutoff level =

At high draft ratios, in a majority of cases, the variance-based relationship (Equation (3)) was required whereas at low draft ratios only the mean-based relationship (Equation (2)) served the purpose. In the same vein, on several occasions at low draft ratios for the modestly correlated SHI sequences, the MC0 performed well with Equation (2). In the computations of reservoir volumes for rivers (with high values of *ρ*_{1m}) and belonging to category (2), the MA1 step with cutoff level as *SHI _{o}* was found to easily handle

Since a majority of reservoirs on a global basis are designed for the draft conditions within the range of *α* = 0.70 and 0.80, where the DM method has performed satisfactorily with the truncation level equal to *SHI _{x}* (=

The DM method is capable of assessing the effect of coefficient of variation (*cv _{o}* and

Table 6. Values of volume ratios, *R _{s}* at different draft ratios (

The rivers affixed by an asterisk (*), (**) and sign (^{†}) refer to as pairs used for comparison. The bold italic numbers indicate the commonality of parameters in the pair.

(a) (b)

Figure 2. Effect of lag-1 autocorrelation (*ρ*_{1m}) in the monthly SHI sequences on reservoir volume at increasing draft ratios and PF levels: (a) 2.5% and (b) 0%.

(a) (b)

Figure 3. Effect of coefficient of variation (*cv _{o}*) in the monthly flow sequences on reservoir volumes at increasing draft ratios and PF levels: (a) PF = 2.5% and (b) PF = 0%.

(a) (b)

Figure 4. Effect of coefficient of variation (*cv _{av}*) in the monthly SHI sequences on the reservoir volumes at increasing draft ratios and PF levels: (a) 2.5% and (b) 0.0%.

consistently with an increase in the value of *α* for all rivers. The *R _{s}* values are primarily affected by 1) the autocorrelation,

4.1. Assessing the Effect of Autocorrelation and Coefficient of Variation

4.1.1. Effect of Autocorrelation on the Size of Reservoir Storage

To discern the effect of autocorrelation (*ρ*_{1m}) in river flows, a pairwise approach was adopted. The advantages of the pairwise approach can be cited as 1) the ease of comparison of the results based on the existing framework of contrasting data, 2) a quick visual display of the crucial information, and 3) empirically or experimentally confirming the behavior of the anticipated process outputs. Therefore, the pair of the Bow River [{1}, Table 6] and the South Saskatchewan River [{4}, Table 6] was chosen in which both rivers display nearly the same level of *cv _{o}* (≈1.05 to 1.06, Table 6) but

To further validate the effect of autocorrelation, two additional rivers from an earlier publication [2], the Upper Humber River (Newfoundland, Canada) and the North Margaree River (Nova Scotia, Canada) were analyzed. The statistical parameters of these rivers are summarized in Table 6. These rivers are characterized by low but significant values of *ρ*_{1m} (=0.13 and 0.17) [24]. These values of *ρ*_{1m} are far below compared to rivers from Canadian Prairies (Table 1 and Table 6). The *R _{s}* values of these rivers were computed by estimating

4.1.2. Effect of Coefficient of Variation on the Size of Reservoir Storage

The effect of the coefficient of variation on reservoir volumes was investigated by choosing a pair of rivers in which values of *ρ _{1m}* are close to each other while the values of

The foregoing impact of variability in river inflows on the reservoir storage is further investigated by considering the coefficient of variation (*cv _{av}*) of monthly SHI sequences for two rivers as a pair viz. the Sturgeon River ({6}, Table 6),

There are two measures of variability between the monthly data sets considered in this paper, viz. *cv _{o}* and

Succinctly, the analysis amply demonstrates that the DM method is applicable over the entire range of draft ratios commonly encountered in the design of reservoirs at the desired PF levels. However, in the implementation of the DM method, the parameter Φ is variable (it is not confined to 0 and 0.5 as is the case with *α* = 1 and 0.75) and has to be calibrated for individual cases of a river and *α* being considered. Further, the MA smoothing from MA1 to MA4 steps are essential ingredients of the method with a greater number of MA steps needed at draft ratios equal to and greater than 0.7 associated with high *cv _{o}*. The results thus allude to the fact that though the role of coefficient of variation is well understood in the design of reservoir capacity, the impact of autocorrelation is also no less important. This finding underscores the significance of autocorrelation in the design of reservoirs and the DM method handles the role of autocorrelation lucidly through the use of the Markov chain, the extreme number theorem and MA smoothing based concepts.

In the paper, all the parameters (probabilities) were determined by the counting method after choosing a suitable cutoff level. No consideration was given to the gamma pdf of the flow and SHI sequences in the calculations. In fact, calculations are strictly applicable to the normal pdf of flows and SHI sequences. However, the procedural detail can be applied to the gamma pdf of flows and subsequently to SHI sequences by transforming SHI’s into the normal probability domain [24]. From the latter with rigorous computations involving the gamma pdf, it was observed that the outcome was only marginally improved from the one adopted in this paper. In other words for gamma probability distributed flows the calculations can be performed directly in terms of statistically standardized SHI sequences, as was conducted and demonstrated in this paper. In general, the DM method satisfactorily computes the reservoir volumes with *α* > 0.5, whereas at *α* ≤ 0.5, the DM method is less efficient, particularly with highly auto-correlated river flows. Under this situation, the Behavior analysis can be used to provide the needed results on the reservoir volumes while meeting the target PF criterion.

5. Conclusion

The DM method can be successfully utilized to estimate the reservoir volumes equivalent to drought deficiency volumes, *D _{T}*

Acknowledgements

The partial financial support from the Natural Sciences and Engineering Research Council of Canada for this paper is gratefully acknowledged.

Appendix A

The following probability-based relationship from the first principles can be used to estimate the expected value of the largest drought-magnitude, *E*(*M _{T}*).

$E\left({M}_{T}\right)={\displaystyle {\sum}_{j=0}^{\infty}\left({M}_{T}={Y}_{j}\right)\cdot p\left({M}_{T}={Y}_{j}\right)}$ (A1)

The notation *P*(.) stands for the cumulative probability and *p*(.) stands for the simple probability. Since *M _{T}* is a continuous random variable, therefore

$E\left({M}_{T}\right)={\displaystyle {\sum}_{j=0}^{{n}_{1}}\frac{{Y}_{j+1}+{Y}_{j}}{2}\left[P\left({M}_{T}\le {Y}_{j+1}\right)-P\left({M}_{T}\le {Y}_{j}\right)\right]}$ (A2)

The upper limit of integration, ∞ in Equation (A1), is replaced by some finite number *n*_{1}. The general equation for evaluating *P* (*M _{T}* ≤

$P\left({M}_{T}\le {Y}_{j}\right)=\mathrm{exp}\left[-Tq\left(1-{q}_{q}\right)\left(1-P\left(M\le {Y}_{j}\right)\right)\right]$ (A3)

In which *M* takes on non-integer values represented by *Y _{j}*. Since

The truncated normal pdf version will have a mean and variance respectively different from 0 and 1. Applying the basic axioms for the evaluation of moments, expressions for the mean (denoted by *μ _{d}*) and variance (denoted by
${\sigma}_{d}^{2}$ ) of the truncated normal pdf version can be deduced [17] [24] as follows.

${\mu}_{d}=-\frac{\mathrm{exp}\left(-0.5{z}_{0}^{2}\right)}{q\sqrt{2\pi}}-{z}_{0}$ (A4)

${\sigma}_{d}^{2}=1-\frac{{z}_{0}\mathrm{exp}\left(-0.5{z}_{0}^{2}\right)}{q\sqrt{2\pi}}-\frac{\mathrm{exp}\left(-{z}_{0}^{2}\right)}{{q}^{2}2\pi}$ (A5)

Because drought episodes lay below the desired truncation level, therefore an absolute value of the term *μ _{d}* is an estimator of the mean value of drought intensity (

Given the central limit theorem and since *M* is the sum of deficit spikes, its probability structure can be approximated by a normal pdf with mean*μ _{M}* and variance
${\sigma}_{M}^{2}$ [17]. Such a consideration reduces the expression for the term

$P\left(M\le {Y}_{j}\right)=\frac{1}{{\sigma}_{M}\sqrt{2\pi}}{\displaystyle {\int}_{0}^{{Y}_{j}}\mathrm{exp}\left[-0.5{\left(\frac{M-{\mu}_{M}}{{\sigma}_{M}}\right)}^{2}\right]\text{d}M}$ (A6)

It was noted that the parameters *μ _{M}* and
${\sigma}_{M}^{2}$ are related to the extreme drought length,

${L}_{C}=\Phi {L}_{M}+\left(1-\Phi \right){L}_{T}$ (A7)

The parameter Φ can be designated as a weighing parameter as it weighs the mean drought length *L _{M}* and the longest drought length

${L}_{M}=\frac{1}{1-{q}_{q}}$ (A8)

The expression for the expected value of *L _{T}* at MC1 situation in drought periods can be obtained as follows [23]

${L}_{T}=1-\frac{\mathrm{log}\left[FT\left(1-q\right){q}_{p}\right]}{\mathrm{log}\left({q}_{q}\right)}$ (A9)

where,*F* is the factor to account for the plotting position in the empirical estimation of the exceedance probability. That is, in the Hazen plotting position formula, the exceedance probability = 0.5/*T* (*T* = sample size), so the return period is equal to *T*/0.5 = 2*T* or *F* = 2. Likewise in the Weibull plotting position formula *F* = 1. In this analysis, the plotting position formula [31] as developed for Canadian rivers has been used. The formula evaluates the exceedance probability = 0.75/(*T* + 0.25), so *F* = 1.33(1 + 0.25/*T*) ≈ 1.33 as *T* is generally large. The term *q _{q}* stands for the conditional probability of the present period being drought given the previous period was also a drought and likewise,

The expressions for *μ _{M}* and
${\sigma}_{M}^{2}$ are related as follows [17].

${\mu}_{M}={L}_{C}{\mu}_{d}$ (A10)

${\sigma}_{M}^{2}={L}_{C}{\sigma}_{d}^{2}\left[\frac{1+\rho}{1-\rho}-\frac{2\rho \left(1-{\rho}^{{L}_{C}}\right)}{{L}_{C}{\left(1-\rho \right)}^{2}}\right]$ (A11)

Once a proper value of *L _{C}* has been determined, Equation (A6) is integrated numerically to evaluate

A particular version for the estimation of *E*(*M _{T}*) can be taken as (A10) itself and Equation (A10) takes the following form

$E\left({M}_{T}\right)={\mu}_{d}{L}_{c}=\text{abs}\left\{-\left[\frac{\mathrm{exp}\left(-0.5{z}_{0}^{2}\right)}{q\sqrt{2\pi}}\right]-{z}_{0}\right\}{L}_{c}$ (A12)

Note Equation (A2) involves both mean and variance of drought intensity (*I _{d}*) for the estimation of

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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