Compatibility of Drought Magnitude Based Method with SPA for Assessing Reservoir Volume: Analysis Using Canadian River Flows ()

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

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

**DOI: **10.4236/jwarp.2022.141001
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Department of Civil Engineering, Lakehead University, Thunder Bay, ON, Canada.

The traditional sequent peak algorithm (SPA) was used to assess the reservoir volume (*V _{R}*) for comparison with deficit volume,

Keywords

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

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Sharma, T. and Panu, U. (2022) Compatibility of Drought Magnitude Based Method with SPA for Assessing Reservoir Volume: Analysis Using Canadian River Flows. *Journal of Water Resource and Protection*, **14**, 1-20. doi: 10.4236/jwarp.2022.141001.

1. Introduction

A considerable amount of research can be traced to the hydrologic drought models utilizing the river flow data that focus on the estimation of drought duration and magnitude (previously termed as severity). Two major elements of the hydrologic drought studies have been the truncation level approach and the analysis by simulation and/or analytical methods. The analytical methods are pursued by the use of the frequency analyses of drought events in terms of duration and deficit volumes. The noteworthy contributions in this area of frequency analyses are that of [1] [2] [3] [4] , among others. The other route in the domain of analytical methods is the use of the theory of runs, which is well documented in [5] - [13] . Several hydrologic drought indices have been suggested such as standardized runoff index (SSI) [14] , streamflow drought index (SDI) [15] , and standardized hydrologic index (SHI) [12] [13] . These indices are essentially standardized (statistically) values of historical stream flows or in some transformed version (normalization in a probabilistic sense) at the desired time scale. The standardized hydrological index (SHI) is the standardized value (statistical) of river* *flows with the mean 0.0 and the standard deviation equal to 1.0, unlike the standardized precipitation index (SPI), which* *is normalized after standardization [16] . On the monthly time scale, it is the month-by-month standardization and so on at the weekly time scale.

The major application of the SPI refers to drought monitoring which is an essential element in the process of drought early warning and preparedness. Applications of SPI are amenable because of the widespread availability of precipitation data. Though some attempts have been made to classify the hydrological drought [15] on the lines of SPI, yet such uses of* *hydrological drought indices are limited. However, there have been investigations to use the SPI to relate the propagation of meteorological droughts to hydrological droughts in Spanish catchments [17] and for the U.K. catchments [18] , among others. Despite such limitations, hydrological drought indices have potential in the estimation of drought magnitude that plays an important role in the assessment of shortage of water in rivers and consequently in reservoirs. Even with the aforesaid studies, a few investigations other than Sharma and Panu [19] [20] have been made to link the deficit volume to reservoir volume, and also how and at what time scale of analysis would be aptly meaningful in this regard.

The term drought magnitude has been variously defined in the earlier literature such as the drought severity [5] [6] and the deficit volume [2] . In this paper, the term deficit volume (denoted by *D*) represents the deficiency or the shortage of water below the truncation level in a river flow sequence, and the drought magnitude (*M*) refers to the deficiency in terms of the SHI (standardized flow) sequences. The deficit volume and drought magnitude are related by the linkage relationship: *D* = *σ* × *M* [5] , in which *σ* is the standard deviation of the flow sequence. The analyses are usually conducted in the standardized domain to assess deficit volume, *D* through the above linkage relationship.

To the best of the authors’ knowledge, no research investigations other than those of authors [19] [20] have been reported in the literature on the application of drought indices and magnitude-based analyses, and models for sizing of reservoirs. This paper represents one of the pioneering attempts to address and bridge the above gap and demonstrate the utility of such analyses of drought magnitude in assessing the size of reservoirs. The standardized hydrological index (SHI) has been used in this analysis utilizing streamflow data from Canadian rivers. The data on annual, monthly and weekly flow sequences were analyzed using the draft at the mean annual flow for sizing of reservoirs. However, the authors’ preliminary investigations indicate that the detailed analysis related to the sizing of reservoirs be conducted at an annual scale in view of ease and simplicity in handling annual streamflow data.

2. Preliminaries on Methods for Sizing the Reservoirs

The two textbook-based methods [21] [22] for sizing the reservoirs are the Rippl graphical procedure and the sequent peak algorithm (SPA). In the Rippl method, the graphical plot of cumulative inflows as well as outflows is used to derive an estimate of the reservoir size. In the SPA, the calculations are conducted numerically using the cumulative or residual mass curve methods to obtain the estimate of reservoir volume, *V _{R}*. In the drought magnitude-based method, SHI sequences are obtained after standardization. It corresponds to truncating the annual river flow sequences at the mean level or SHI value = 0. Since the drought lengths and corresponding drought magnitudes yield conservative values for the design of reservoirs, therefore SHI = 0 as a truncation level is preferred and has been used in the analysis. The SHIs

Estimation of Deficit Volumes by DM Model

A majority of models for the estimation of drought magnitude implicitly involves the use of the frequency distribution of drought events [2] [4] . It is in this regard that the moving average (MA) and sequent peak algorithm (SPA) form the important tools for analysis [2] [9] . In other approaches, the probability-based relationships are hypothesized for estimating drought magnitude (*M*) using the relationship: drought magnitude = drought intensity × drought length [23] . As mentioned above, drought intensities essentially are deficit spikes and are derived by truncating a SHI sequence. The deficit spikes have a negative sign because each spike lay on the downside (negative side) of the truncation level, with the lower bound as −∞ and an upper bound as truncation level such as *z*_{0}, which is also a negative number with a maximum value of 0. It is tacitly assumed that the SHI* *sequences obey standard normal probability density function (pdf) which after truncating at the desired level (*z*_{0}) shall result in a truncated normal pdf, whose mean and variance would be different from 0 and 1. One can develop a probabilistic relationship for *M _{T}*, using the extreme number theorem [7] [24] that implicitly involves drought intensity and drought length (

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

At the annual level, the flow sequences in Canadian rivers have been found to follow the normal pdf [13] , leading SHI sequences to obey standard normal pdf. Therefore, the assumption of deficit spikes to obey truncated normal distribution is reasonably justified. Based on the above premises, a detailed derivation has been tracked by Sharma and Panu [19] [20] and is not reproduced here for brevity. The concluding expressions for the present paper are described as follows.

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

To compute
$\left[P\left({M}_{T}\le {Y}_{j+}{}_{1}\right)\u2013P\left({M}_{T}\le {Y}_{j}\right)\right]$ in Equation (2), the integration of the normal probability function is numerically performed as described in Sharma and Panu [12] . Theoretically, the upper limit of summation (*n*_{1}) in Equation (2) is ∞, but for numerical integration purposes, a finite value is chosen. For drought magnitude analysis based on annual flows, a value of *n*_{1} = 30 (with an increment in *j* = 0.05 was found to be large enough to ensure sufficient accuracy in the process of numerical integration. For brevity, henceforth *E*(*M _{T}*) shall be written as
${M}_{T\text{-}e}$ ,

A particular version of *M _{T}*

${M}_{T\text{-}e}={\mu}_{d}\cdot {L}_{T}=abs\left[-\frac{\mathrm{exp}\left(-0.5{z}_{0}^{2}\right)}{q\sqrt{2\pi}}-{z}_{0}\right]\cdot {L}_{T}$ (*abs* means absolute) (3)

where, *L _{T}* is the largest drought length obtained using Markov chain based algorithm,

3. Data Acquisition and Calculations of Reservoir Volumes

Fifteen rivers from prairies to Atlantic Canada (Table 1, Figure 1) were involved

Table 1. Summary of statistical properties of annual flows of the rivers under consideration.

Note: The *cv*,* γ*, *ρ* respectively represent the coefficient of variation,* *skewness, lag-1 autocorrelation. Small values of skewness indicate the normal pdf of the annual flow sequences.

Figure 1. Location of the river gauging stations used in the analysis across Canada (not to the scale) [Source: Environment Canada].

in the analysis. The rivers encompassed drainage areas ranging from 97 to 56,369 km^{2} with the data bank spanning from 38 to 108 years. The flow data for these 15 rivers were extracted from the Canadian hydrological database [25] . To increase the number of samples, some of the rivers with large data sizes such as the Bow, English, Lepreau, Bevearbank, and North Margaree were also analyzed by forming 2 - 4 subsamples with the data size of 40 years or more. This type of analysis created around 30 samples from 15 rivers to obtain a robust and reliable estimate of the performance statistics. Based on the above premises, the results of various analyses are described in the sections to follow.

The first step in the analysis was to discern the role of time scale in influencing the reservoir size. Therefore, reservoir volumes (*V _{R}*) were assessed using the SPA at the demand level equivalent to the mean flows at the annual, monthly, and weekly scales. The procedure advanced in Linsley

Table 2. Calculations of storage volumes at the mean level of flows for varying time scales.

Note: The italicized values in parentheses are calculated at the mean levels (variable means) of the respective months and weeks without standardization of the flow sequences.

For each time scale, the drought magnitudes, *M _{T}*

The MA sequences can be formed from flows or the SHI sequence, alike. However, it is convenient to apply flow sequences to compute the *V _{R}* using SPA, whereas the DM based method explicitly requires SHI sequences. When the annual SHI (or flow) sequence is used without involving any moving average operations then such a sequence is designated as moving average 1 (MA1) sequence. In other words, a non-averaged value of SHI (or flow) is essentially the annual SHI (or flow). When consecutive 2 or 3 or annual SHIs

Figure 2. Redistribution of drought lengths and magnitudes with varying MA smoothing.

Table 3. Summarized *V _{R}* (SPA) and

Note: *SPA denotes sequent peak algorithm, ***V _{R}* reservoir volume (bold letter) closest to SPA based value.

standard deviation, the MA1, MA2 and MA3 flow sequences were converted to respective SHI sequences. In the process of analysis, the number of drought spells (*Ns*) dropped from the MA1 through MA3 sequences and are presented in column 5 of Table 3. After a few MA smoothing, Ns attained nearly an equilibrium state and thus suggesting no further MA smoothing were warranted. For example, in Table 3, *Ns* values for MA3 smoothing marginally deviate from MA2 but significantly drop from MA1.

For a comparative analysis on *V _{R}*, the counting procedure was applied to the MA1 sequences. The

When the *D _{T}*

Parallel to the counting procedure, the *M _{T}* values (denoted as

4. Results

4.1. Role of the Time Scale on the Estimates of Reservoir Size

To discern the role of the annual, monthly and weekly time scales, *V _{R}* (SPA) and

Figure 3. Flow diagram for computing deficit volumes under various options in the DM method.

*D _{T}*

The above numbers displaying the large discrepancies between the SPA and the DM based (MA1) estimates highlight that either the SPA yields excessive values of reservoir volume or the DM method yields too small estimates at the draft level of mean annual flow (MAF, 1*µ*). At the annual scale, however, when the draft was lowered to 0.90*µ* or less, the estimates by the SPA and the DM method converged to the same value [20] . In other words, a region with a draft level between 0.90*µ* and 1*µ* requires special consideration for the estimation of reservoir volumes by the DM method. The SPA based estimates can be construed as fixed with a little scope to lower them in view of the inherent algorithm imbued in it. But the DM based estimates can be boosted by utilizing the MA procedure to attain parity with SPA based estimates. The SPA has been in vogue since the 1960s [26] and is universally accepted to design the reservoir capacity, therefore, the focus in this study is to arrive at a suitable MA smoothing that should yield *D _{T}* comparable to

The DM based estimates (*italicized*, Table 2) at the monthly and weekly scales without standardization were found to be slightly different (mostly smaller) in comparison to the standardization based values (SHI sequences). On average, the standardization based estimates were found about 12% larger than those based on the non-standardized values. This discrepancy can be perceived to arise because of
${\sigma}_{av}$ , which has been taken as the representative value of the standard deviation to convert the magnitude in deficit volume* *(*D _{T} *=

4.2. Comparison of Reservoir Sizes Using the SPA and the DM Based Counting Procedure

In computing the *D _{T}*

In assessing the efficacy of various smoothing, the values of *D _{T}*

Based on aforesaid calculations, it was found that *M _{T}*

Table 4. Performance statistics for comparison of SPA based *V _{R}* with DM based

be significantly smaller. Since the discrepancies in values of *M _{T}*

Firstly, the MA2 based *M _{T}*

Thus, MA3 smoothing was undertaken (flow chart—Figure 3) and values of *M _{T}*

In the process of moving from the MA1 smoothing to the MA2 smoothing, there has been a considerable reduction in the number of drought spells (column 5, Table 3). Such a reduction suggests that there is a significant increase in the drought length (Figure 2) and in turn, there is also a significant increase in the drought magnitude. In other words, the smoothing procedure led to the amalgamation of smaller drought episodes with the larger ones which resulted in enhanced values of the *D _{T}*

Figure 4. Comparison of SPA based *V _{R}* with (A)

4.3. Comparison of Reservoir Sizes Using the SPA and DM Based Model

The drought magnitudes (*M _{T}*

In view of the abysmal values of the performance statistics by Equation (3), Equation (2) was used to estimate *M _{T}*

It was observed that the MA2 smoothing resulted in similar values of NSE for both the counted *D _{T}*

Table 5. Summary of the
${{V}^{\prime}}_{R}$ , *M _{T}*

Note: *(asterisk) means the value is based on comparing *M _{T-o}* and

viz. NSE and MER, the
${{V}^{\prime}}_{R}$ and *M _{T}* (larger between

The foregoing selection criteria of the larger estimate between the *M _{T}*

5. Discussion

From the foregoing analysis, it was observed, that the discrepancy between the *V _{R}* (SPA) and

It turned out that at the demand level equivalent to the long term mean of the river flow, the *V _{R}* values are fairly constant no matter what time scale is chosen. In contrast, the drought magnitude-based methodology resulted in significant discrepancy among these estimates with the annual time scale yielding much higher values compared to those at the monthly and weekly time scales. In such a scenario, one can even be tempted to limit the analysis with the annual flow sequences only as it is trivial and the annual flow data can easily be synthesized or generated. The MA1 based estimates were found to be significantly smaller than the SPA based values but adequately catered for deficiencies arising in the wake of severe droughts over a return period of

In a bid to attain the same *D _{T}* values as

In the present analysis, a draft at the level of mean annual flow was chosen, with the sole objective of demonstrating the application of the drought magnitude-based methodology for estimating the storage capacity of reservoirs both under independent and dependent (Markovian) river flow conditions. In practice, the majority of rivers worldwide are designed, based on the draft of 75% of the mean annual flow [22] , under such a situation the above-described methodology can be applied. Using the above methodology, Sharma and Panu [19] noted that at such a draft level, the DM method with no moving averaging (MA1) of SHI sequences turn out to be equal to SPA based estimates meaning that no higher-order averaging *i.e.* MA2 or MA3 is required for evolving storage estimates. This is an important observation, which speaks the worth of the DM based methodology as a viable method in tandem with SPA. The method can be extended to monthly SHI sequences, which yields more accurate estimates of reservoir capacity [19] while considering the higher level of dependence (autocorrelation) and skewness (gamma probability distribution) in monthly flows.

6. Conclusions

The analysis was carried out to compare the *V _{R}* and

The estimation of* D _{T}*

Acknowledgements

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

NOTES

*Corresponding author.

Conflicts of Interest

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

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