Compatibility of drought magnitude based method with SPA for assessing reservoir volumes: analysis using Canadian river flows

: The traditional sequent peak algorithm (SPA) was used to assess the reservoir volume (V R ) for comparison with deficit volume, D T , (subscript T representing the return period) obtained from the drought magnitude (DM) based method with draft level set at the mean annual flow on 15 rivers across Canada. At an annual scale, the SPA based estimates were found to be larger with an average of nearly 70% compared to DM based estimates. To ramp up DM based estimates to be in parity with SPA based values, the analysis was carried out through the counting and the analytical procedures involving only the annual SHI (standardized hydrological index, i.e. standardized values of annual flows) sequences. It was found that MA2 or MA3 (moving average of 2 or 3 consecutive values) of SHI sequences were required to match the counted values of D T to V R . Further, the inclusion of mean, as well as the variance of the drought intensity in the analytical procedure, with aforesaid smoothing led D T comparable to V R . The distinctive point in the DM based method is that no assumption is necessary such as the reservoir being full at the beginning of the analysis - as is the case with SPA.


Introduction
A considerable amount of research can be traced to the hydrologic drought models that focus on the estimation of drought duration and magnitude (previously termed as severity) using the river flow data. 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 These indices are essentially standardized (statistically) values Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 July 2021 doi:10.20944/preprints202107.0301.v1 of historical stream flows or in some transformed version (normalization in a probabilistic sense)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 and the standard deviation equal to 1, 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 on the use of 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, 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 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 the 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.

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, VR. 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 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 below the truncation level are referred to as the deficit (dubbed as d), whereas above the level are referred to as the surplus (dubbed as s). In a historical record of N (= T) years, there shall emerge several spells of deficit and surplus, and the longest spell length of deficits (representing LT) is recorded. Likewise, the corresponding deficits are added to represent the largest magnitude (MT). These deficits are being referred to as drought intensities and represent truncated values of SHIs below the truncation level. The foregoing approach of calculation of LT and MT is dubbed as the counting procedure in the ensuing sections. The largest deficit volume (DT) during the drought period is computed as DT = σ × MT [5].
It is noted that the unit of DT is the same as that of σ because MT is a dimensionless entity. It is stated that the quantity DT obtained using either the DM based counting or analytical procedure is perceived equivalent to VR calculated by the SPA method. In the counting procedure, the entities MT and DT are respectively obtained from the historical or observed data and hence are denoted as MT-o and DT-o, where subscript "o" stands for the observed.

Estimation of deficit volumes by DM model
The majority of models for estimation of drought magnitude are built on the frequency distribution of drought events [2], [4]. The moving average (MA) and sequent peak algorithm (SPA) form the important tools for analysis [9], [2]. In the other approach, probability based relationships are hypothesized for estimating drought magnitude (M) using the relationship: drought magnitude = drought intensity × drought length [23]. As mentioned above in the text, 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 z0, which is also a negative number with maximum value as 0. It is tacitly assumed that the SHI sequences obey standard normal probability density function (pdf) which after truncating at the desired level (z0) 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 MT expressed as follows, through the use of the extreme number theorem [7], [24] that implicitly involves drought intensity and drought length (LT).
In which, q represents the simple probability of drought and qq represents the conditional probability that the present period is drought given the past period is also drought and T is equivalent to return period; M stands for the drought magnitude which takes on non-integer values represented by Y, and P (.) represents the notation of cumulative probability. Since Y's (such as Y1, Y2, Y3, Y4, ..…) correspond to values of M, thus the largest of them will represent MT. In the above expression, M is construed to follow a normal pdf with mean and variance related to the mean and variance of drought intensity and a characteristic drought length. The characteristic drought length is related to mean drought length and the extreme drought length, LT.
At the annual level, the flow sequences in Canadian rivers have been found to follow the normal pdf [13], leading SHI sequences to obey the 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 are not reproduced here for the sake of brevity. The final expressions for the purpose of the present paper are described as follows.
To compute [P (MT ≤ Yj+1) -P (MT ≤ Yj)] 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 (n1) in Where, LT is the largest drought length obtained using Markov chain based algorithm, q is drought probability at the truncation level z0. For example, a standard normal pdf respectively can be truncated at z0 = 0.0 (mean level) and z0 = -0.30 (which is 70 % of a mean level) and the corresponding drought probability q can be found from standard normal probability tables to be 0.5 and 0.38.

Data Acquisition and Calculations of Reservoir Volumes
Fifteen rivers from prairies to Atlantic Canada ( Figure 1, Table 1) were involved in the analysis. The rivers encompassed drainage areas ranging from 97 to 56369 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   Table 1 Summary of statistical properties of annual flows of the rivers under consideration Name, location, and the numeric identifier in Figure 1   Therefore, flows were standardized at the above three time scales to obtain SHI sequences. In all three time scales, the values of the drought probability, q, were obtained by the counting procedure in which an SHI sequence was chopped at level 0 (mean level). In general, the annual flows tend to follow a normal pdf in the Canadian settings so, at the mean level, q values cluster around 0.50 (Table 2,  sequences, at the monthly and weekly scales [13], the q values are significantly larger than 0.50 (Table 2, columns 3 and 4).   turned out to be the best estimator [12]. The DT-o values were also estimated without standardization and truncating the flow series at the variable mean levels corresponding to the respective monthly and weekly time scales. In the standardized domain, the variable means are homogenized with a common mean = 0.

Figure 2
Redistribution of drought lengths and magnitudes with varying MA smoothing.
The MA sequences can be formed from flows or the SHI sequences, alike. However, it is convenient to apply flow sequences to compute the VR using SPA, whereas 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 MA1 sequence (flows) was subjected to analysis to compute the mean (µ), standard deviation (σ) and lag-1 autocorrelation (ρ). The aforesaid statistics were also evaluated for the MA2 and MA3 (flows) sequences and are shown in Table 3. Using the above values of mean and 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.

Role of the time scale on the size of reservoirs
To discern the role of the annual, monthly and weekly time scales, VR (SPA) and DT-o (counting procedure in the DM method) from the respective flows and SHI sequences were computed and are summarized for 6 rivers in 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 σav, which has been taken as the representative value of the standard deviation to convert the magnitude in deficit volume (DT = σav× MT). Needless to mention that σav is an estimator representing 12 values of monthly σ's and is unlikely to be the best for all situations, but is construed to be a better option compared to other options mentioned earlier. Since standardization is purely statistical in this operation, the pdf of monthly sequences is less likely to play the role in explaining the aforesaid discrepancy.
At the annual scale, however, it should be noted that the standard deviation has only one value, thus estimates by both routes turned out to be identical.
Therefore, only one value of DT-o is reported in Table 2. Since these estimates (i.e. without standardization) do not require the use of standard deviation in the calculations and thus can be deemed more accurate. However, the standardization procedure is better amenable to statistical analysis, and estimates based on this approach are more conservative (i.e. higher compared to the non-standardization, Table 2). Based on the foregoing reasoning, the route involving standardization (i.e., the use of SHI sequences) for the estimation of DT-o has been preferred in subsequent analyses and the annual scale is considered as a first choice.  (Table   3). It is apparent from  Firstly, the MA2 based MT-o2 values were compared to the VR' on a 1:1 basis. It was discovered that with the MA2 smoothing, the matching to VR' significantly improved resulting in NSE ≈ 72% and MER ≈ -13% (Table 4) Table   4). The important point to be noted is that at every smoothing, a new value of MT-o will emerge, which is multiplied by the MA1 smoothing based value of σ (= σ1) to arrive at the new estimate of DT-o. 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

Comparison of reservoir sizes using the SPA and DM based Model
The drought magnitudes (MT-e) in the standardized domain were estimated using Equations (2) and (3). At the annual scale, the characteristic drought length was found equivalent to extreme drought length, LT [12],  Table 4 ). The NSE for the MA3 smoothing was also low with the highest value nearly equal to 57%.
In view of the abysmal values of the performance statistics by Equation  (Table 5, Figure   4B).

Discussion
From the foregoing analysis, it was observed, that the discrepancy between the VR (SPA) and DT (DM) is significant at the draft level of mean In the case of SPA, the difference between the full reservoir level (reservoir is assumed to be full at the beginning) and the lowest level reached during the sampling period, is taken as the reservoir volume. During this intervening period, several droughts including the longest one may occur and the recovery in the water levels in the reservoir may succeed. Conversely in the DM based method, no such assumption is invoked and thus the total shortfall of water below the long term mean flow in a river during the longest drought period is regarded as the required reservoir volume. The reservoir should be designed to store the above volume of water during the period of excess flow in the river.
In a bid to attain the same DT values as VR using the DM based method, the drought lengths and in turn, the magnitudes were amplified by a moving average procedure that resulted in the MA2 and MA3 sequences. between the VR and DT values, provided MA smoothing such as 3-, 4-, 5-, 6-or higher monthly SHI sequences are utilized [28]. This is an area for further research justifying the use of monthly based analysis in the design of reservoirs.
It turned out that at the demand level at the mean annual flow of a river, the VR values tend to be not significantly different from each other irrespective of what time scale is chosen. In contrast, the drought magnitude based method resulted in significant discrepancy among these estimates (DT) with the annual time scale yielding much higher values compared to those at the monthly and weekly scales. In such a scenario, one can even be tempted to limit the analysis to the annual flow sequences only as it is trivial and the annual flow data can easily be synthesized or generated at any desired location on a river. The aim is to store enough water to meet any exigency or unaccounted episodes. There is also a need to examine other methods of estimating the reservoir capacity and compare them with the DM based estimates. Finally, all the estimates may be averaged out to arrive at the final design value of the reservoir capacity.

Conclusions
The analysis was carried out to compare the VR and DT respectively,