13.3.1 Determination of Drought Severity, Duration, and Interarrival Time

The theory of runs, proposed by Yevjevich (1967), was used to determine the drought severity (i.e., total flow deficit in the case of hydrological drought). Following Yevjevich (1967), the threshold equation can be written as follows:

X0i¯=μi+ασi(13.1)

where μ_i, σ_iμi,σi represent the estimated long-term mean and standard deviation (daily or monthly), depending on the streamflow records.

Given the high fluctuation of daily streamflow, we will use the long-term daily average streamflow as a threshold. Additionally, following Zelenhasic and Salvai (1987), (a) minor drought events were ignored under the condition of S_i ≤ 0.005 max (S), i = 1, 2, ..NSi≤0.005maxS,i=1,2,…,N, where N represents the total number of drought events identified and S represents the drought severity; and (b) the two consecutive drought events were pooled into a single drought event if the interevent wet period was relatively short and the ratio of surplus of pervious drought severity was small, the pooled event can be given as follows:

where S_i, S_i + 1Si,Si+1 represent drought severity of two consecutive drought events; D_i, D_i + 1Di,Di+1 represent the drought duration of two consecutive drought events; and SP_{i, i + 1}SPi,i+1 represents the total amount of streamflow above the threshold in the wet period between the two consecutive droughts.

The rules to pool the events are as follows:

1. 
   

   i. The consecutive drought events are pooled into one drought event, if the interevent wet period is less than seven days.

   
   
2. 
   

   ii. To pool the consecutive drought events if the ratio Δ ≤ 0.05Δ≤0.05 in Equation (13.2) such that the total surplus in the interevent period cannot relieve the dry condition.

To further illustrate the process, we show a simple example using daily streamflow from June 20, 1943, to May 2, 1944. Table 13.1 lists observed daily streamflow and its difference from the long-term daily average. The surplus of daily streamflow is in bold Italic. Figure 13.2 graphs the streamflow time series and the differences from daily thresholds. It is seen from Figure 13.2 that there existed flow deficit for most of the time in the period from June 20, 1943, to May 2, 1944.

Figure 13.2 Daily streamflow and its difference from the long-term daily threshold from June 20, 1943, to May 2, 1944.

Without either ignoring or pooling drought events, Table 13.2 lists the drought events by adding the continuous flow deficit (negative flow differences). In addition, before pooling the drought events together, we compute the maximum drought deficit, which is 2.36E+05 cfs.day from all the available daily streamflow data investigated in the case study. With the maximum drought deficit, the deficit less than 0.005 max (deficit) = 0.005(2.36 × 10⁵) = 1181.2 cfs. day0.005maxdeficit=0.0052.36×105=1181.2cfs.day. With this criterion, the minor drought event from March 23, 1944, to March 27, 1944, is ignored, which is in bold Italic (Table 13.2). After ignoring minor droughts, Table 13.3 lists the remaining drought events. These remaining drought events are then further pooled using the rules of pooling discussed earlier. In addition, the last two droughts also need to be pooled, since Δ = 0.03 ≤ 0.05Δ=0.03≤0.05. Finally, all the individual droughts in Table 13.3 need to be pooled into one drought as follows:

• 
  

  Severity = 89257.53 cfs.day; Duration = 293 days; Inter-arrival time=334 days;

  
  
• 
  

  Starting time = 06/20/1943; Ending time = 05/02/1944.

Using the identification procedure for drought events explained previously, we identified a total of 115 drought events. Table 13.4 lists the statistics of these identified drought events. As shown in Table 13.4, the fluctuations of drought variables are significant with a heavy tail. The drought events, such identified, are assumed as independent random variables. To further ensure the assumption, we will apply correlation by lag, as shown in Figure 13.3. The autocorrelation function plot with lag indicates that there is no serial dependence within each individual drought variable. As a result, drought variables can be considered as random variables for analysis. More specifically, drought variables are assumed to be continuous random variables throughout the analysis.

Figure 13.3 Check for independence of drought variables.

13.3.2 Univariate Drought Frequency Analysis

Before we proceed to the bivariate and trivariate drought analyses, we will first perform univariate drought frequency analysis. The fitted parametric univariate distribution will be applied for risk analysis. To study the univariate drought frequency analysis, drought variables are conventionally assumed as continuous random variables. It is also known that there may be ties that commonly exist in drought duration and drought interarrival time variables. To avoid the impact of ties in the univariate analysis, the parametric univariate distribution is fitted to the unique values (e.g., if there is more than one 30-day duration drought, only one 30-day duration will be used for fitting the univariate distribution). We obtain that (i) if there is no tie in drought severity, all 115 drought severity values are applied for univariate analysis; (ii) there are 15 values that are repeated more than once for the drought duration, and 93 unique duration values are applied for univariate analysis; (iii) there are 13 values that are repeated more than once for drought interarrival times, and 98 unique interarrival time values are applied for the univariate analysis.

Based on the previous studies, the following parametric univariate distributions are considered as candidates: gamma, exponential, and Weibull distributions. In addition, the log-normal distribution has been commonly applied to model drought severity (i.e., streamflow deficit), while the Weibull distribution has been commonly applied to model drought duration and drought interarrival time. Table 13.5 lists the fitted univariate distributions as well as the formal goodness-of-fit (GoF) statistics using the Kolmogorov–Smirnov (KS) test. Figure 13.4 compares the parametric marginal distributions with the empirical distributions. From Table 13.5, it is seen that the log-normal (for severity and duration) and exponential (for interarrival time) distributions yield the smallest KS test statistics (i.e., the smallest distance between the parametric and empirical distributions). However, comparisons in Figure 13.4 show that (i) in the case of drought severity and drought duration, the Weibull distribution fits the upper tail better than the lognormal distribution; and (ii) in the case of drought interarrival time, there is minimal difference in fitting the upper tail for log-normal and Weibull distributions. To comply with the conventional univariate drought analysis, we will use the conventional marginal distributions for illustration, i.e., log-normal distribution for drought severity and Weibull distribution for drought duration and drought interarrival time. One other reason of applying the conventional distributions is that both log-normal and Weibull distributions pass the formal GoF KS test.

Figure 13.4 Fitted parametric marginal distributions versus empirical distribution.

13.3.3 Bivariate Drought Frequency Analysis

Fitting Copula Functions to Bivariate Drought Variables

Using the identified drought events, Figure 13.5 plots show the scatter plots of the paired random variables. Figure 13.5 clearly shows the positive dependence between drought severity and drought duration; between drought severity and drought interarrival time; as well as between drought duration and drought interarrival time.

Figure 13.5 Scatter plots of paired drought variables.

As introduced in Chapter 3, we will investigate the dependence separately from the marginals. The empirical marginals (i.e., using the Weibull plotting position formula) are applied to study the dependence among drought variables. The parametric marginal distributions, fitted in the previous section, will be applied for risk analysis through joint (and/or conditional) return periods. In addition, we will choose the Archimedean and meta-elliptical copulas for the bivariate drought analysis, while only meta-elliptical copulas will be chosen for trivariate drought analysis. Given the positive dependence structure between drought variables, the selected Archimedean copulas are the Gumbel–Hougaard (which belongs to the extreme value family), Clayton, and Frank copulas. One may also choose other Archimedean copulas as candidates. The Gaussian and Student t copulas are selected as candidates from the meta-elliptical family.

Applying the pseudo-MLE to the bivariate drought variables, Table 13.6 lists the estimated copula parameters as well as the GoF test results through the improved Rosenblatt transform following the procedures discussed in Section 3.8.3 (i.e., the SnB test; Genest, et al., 2007) with the following steps briefly outlined:

1. 
   

   i. For d-dimensional random variables u = [u₁, u₂, …, u_d]u=u1u2…ud; setting Z = [Z₁, Z₂, …, Z_d],Z=Z1Z2…Zd, where Z1=u1=F1x1;Z2=∂Cu1u2∂u1;…,Zd=∂Cd−1u1…ud∂u1∂u2…∂ud−1∂Cd−1u1…ud−1∂u1∂u2…∂ud−1The null hypothesis is that u = [u₁, u₂, …, u_d]~C_θ ⇒ Z = [Z₁, Z₂, …, Z_d]u=u1u2…ud~Cθ⇒Z=Z1Z2…Zd are independent.

   
   
2. 
   

   ii. Compute the test statistic using

SnB=n3d−12d−1∑i=1n∑k=1d1−Eik2+1n∑i=1n∑j=1n∏k=1d1−maxZikZjk(13.3).

1. 
   

   iii. Generate random variables from the fitted copula function with the same sample size as the sample data.

   
   
2. 
   

   iv. Reestimate the copula parameters from the tested copula function and recompute the test statistics.

   
   
3. 
   

   v. Repeat steps ii–iv for a large number of times (N) and approximate the P-value using

   
   

   Pvalue=1N∑k=1N1Sn,kB∗>SnB, where Sn,kB∗ represents the test statistic from step iv.

From Table 13.6 we obtain that (i) all the copula candidates from Archimedean and meta-elliptical families may be applied to model drought severity (S) and drought duration (D) as well as drought severity (S) and drought interarrival time (INT); (ii) the Gumbel–Hougaard and Guassian copulas are the only two copula functions that may be applied to model drought duration and drought interarrival time.

Table 13.6. Parameters estimated and GoF test for copula candidates.

	S and D		S and INT		D and INT
	(1)	(2)	(1)	(2)	(1)	(2)
GHa	[6.2, 157.31]	[0.17, 0.24]	[3.30, 90.00]	[0.06, 0.81]	[3.97, 109.02]	[0.27, 0.08]
Clayton	[4.11, 92.71]	[0.26, 0.08]	[2.10, 49.23]	[0.21, 0.13]	[2.80, 64.50]	[0.42, 0.02]
Frank	[18.88, 126.33]	[0.17, 0.23]	[11.21, 80.73]	[0.10, 0.56]	[14.76, 80.73]	[0.32, 0.04]
Gaussian	[0.95, 137.11]	[0.33, 0.15]	[0.87, 80.77]	[0.08, 0.71]	[0.91, 99.78]	[0.26, 0.10]
Student t	(0.95, 1.47)b	[0.24, 0.10]	(0.88, 5.79)	[0.10, 0.54]	(0.92, 4.99)	[0.31, 0.04]
	144.68c		83.59		104.73

Notes: (1) estimated parameter and CL; (2) SnB test statistics and P-value;

^a GH represents the Gumbel–Hougaard copula, ^bcorrelation and degree of freedom, ^c CL for Student t copula.

Joint and Conditional Return Period for Bivariate Drought Analysis

In this section, we will apply the Gumbel–Hougaard copula (which belongs to the extreme value family) and the Gaussian copula to investigate risk through joint and conditional return periods. Here, we will only focus on drought severity and drought duration.

Joint Return Period of Bivariate Drought Analysis

As discussed in Section 3.10.2, the bivariate joint return period may be represented with either the “AND” case or “OR” case. Here we will focus on the “AND” case only. Equation (3.139) in Chapter 3 can be revised as follows:

TANDsd=EINT1−FSs−FDd+C12FSsFDd(13.4)

where E(INT) represents the expected drought interarrival time in years, E(INT) = 0.723 year.

Using Equation (3.139), Table 13.7 lists the “AND” case joint return periods with the fitted parametric marginal distributions. To further illustrate the computation, we will show how to compute the return period for T(S>8000 cfs. day, D>30 days)TS>8000cfs.dayD>30days:

• 
  

  F_S(S < 8000) = 0.2035FSS<8000=0.2035 from the fitted log-normal distribution S~LN2(10.1992, 1.4614)S~LN210.19921.4614.

  
  
• 
  

  F_D(D < 30) = 0.1661FDD<30=0.1661 from the fitted Weibull distribution D~Weibull(203.80, 0.8903)D~Weibull203.800.8903.

  
  
• 
  

  F_{S, D}(S ≤ 8000, D ≤ 30) = C(0.2035, 0.1661; θ = 6.2015) = 0.1479FS,DS≤8000D≤30=C0.20350.1661θ=6.2015=0.1479 from the fitted Gumbel–Hougaard copula for drought severity and drought duration.

  
  
• 
  

  The exceedance probability:

  
  
  FS>8000D>30=1−FS8000−FD30+CFSFD=1−0.2035−0.1661+0.1479=0.7783
  
  
  
  
• 
  

  The “AND” case joint return period TS>8000D>30=0.7230.7783≈0.93yr.

Figure 13. 6 shows the Joint return period of the “AND” case for drought severity and drought duration.

Table 13.7. Joint return period of drought severity and duration (“AND” case).

		D (days) Gumbel–Hougaard copula						D (days) Gaussian copula
		30	120	365	520	700	1120	30	120	365	520	700	1120
	8000	0.93	1.35	3.88	7.23	14.52	69.01	0.93	1.35	3.88	7.23	14.52	69.01
	28000	1.48	1.55	3.88	7.23	14.52	69.01	1.48	1.57	3.88	7.23	14.52	69.01
S	92000	3.62	3.62	4.20	7.25	14.52	69.01	3.62	3.62	4.54	7.41	14.54	69.01
(cfs.day)	180000	7.48	7.48	7.51	8.29	14.58	69.01	7.48	7.48	7.71	9.38	15.34	69.03
	300000	14.64	14.64	14.64	14.68	16.47	69.01	14.64	14.64	14.67	15.36	19.48	69.65
	800000	71.44	71.44	71.44	71.44	71.44	79.63	71.44	71.44	71.44	71.45	72.02	103.25

Figure 13.6 Joint return period of the “AND” case for drought severity and drought duration.

Conditional Return Period of Bivariate Drought Variables

There are two commonly applied approaches to study the conditional return period:

T(X₁>x₁| X₂>x₂)TX1>x1X2>x2 and T(X₁>x₁| X₂ = x₂)TX1>x1X2=x2. Here, we will investigate both conditional return periods for drought severity and drought duration, with the use of drought duration as the conditioning variable.

T(S>s| D>d)TS>sD>d In this case, the exceedance conditional probability of S given D exceeding a given duration (d) can be written through the copula as follows:

PS>sD>d=PS>sD>dPD>d=1−FSs−FDd+C12FSsFDd1−FDd(13.5)

The conditional return period can then be written as follows:

TS>sD>d=EINT1−FDd1−FSs−FDd+C12FSsFDd(13.6)

Equation (13.5) may also tell whether there exists the right tail increasing (RTI) property. The RTI property exists if the conditional exceedance probability is a nondecreasing function of drought duration for all drought severity values.

Using Equations (13.5) and (13.6) and the Gumbel–Hougaard copula as an illustrative example, Table 13.8 lists the conditional exceedance probability and conditional return period. Figure 13.7 plots the conditional exceedance probability and conditional return period. From Table 13.8 and Figure 13.7, it is seen that the exceedance probability is a nondecreasing function of duration, i.e., with the increase of drought duration, the exceedance probability of S>s|D>d is nondecreasing. The RTI property indicates that it is more likely for the drought severity exceeding a given threshold conditioned on a higher drought duration than that conditioned on a lower drought duration. Using S > 8000 cfs.day in Table 13.8 as an example, we have the following:

PS>8000D>30<PS>8000D>120=PS>8000D>365=PS>8000D>520=P(S>8000∣D>700=PS>8000D>1120

From Table 13.8 and Figure 13.7, it is also seen that for a given drought duration, the exceedance probability decreases with the increase of drought severity. To illustrate the computation, we will show the procedure to compute P(S>8000| D>30)PS>8000D>30 and T(S>8000| D>30)TS>8000D>30:

• 
  

  Previously we have computed P(S>8000, D>30) = 0.7783PS>8000D>30=0.7783 for the “AND” case.

  
  
• 
  

  The exceedance conditional probability is as follows: PS>8000D>30=PS>8000D>30PD>30=0.77831−0.1661=0.933

  
  
• 
  

  The conditional return period is as follows:

  
  
  TS>8000D>30=EINT1−FD30PS>8000.D>30=0.7231−0.16610.7783≈1.11yr
  
  

Table 13.8. Conditional exceedance probability and conditional return period using drought duration as the conditioning variable.

		D (days) P(S>s| D>d)PS>sD>d						D (days) T(S>s| D>d)TS>sD>d
		30	120	365	520	700	1,120	30	120	365	520	700	1,120
	8,000	0.93	1.00	1.00	1.00	1.00	1.00	1.11	2.52	20.82	72.27	291.60	6586.07
	28,000	0.59	0.87	1.00	1.00	1.00	1.00	1.77	2.88	20.82	72.27	291.60	6586.07
S	92,000	0.24	0.37	0.92	1.00	1.00	1.00	4.34	6.75	22.54	72.46	291.61	6586.07
(cfs.day)	180,000	0.12	0.18	0.52	0.87	1.00	1.00	8.98	13.97	40.30	82.84	292.81	6586.08
	300,000	0.06	0.09	0.27	0.49	0.88	1.00	17.55	27.32	78.54	146.80	330.78	6586.36
	800,000	0.01	0.02	0.05	0.10	0.20	0.87	85.66	133.33	383.31	714.17	1434.63	7600.17

Figure 13.7 Conditional exceedance probability and conditional return period of S>s ∣ D>dS>s∣D>d.

T(S>s| D = d)TS>sD=d. In this case, the drought duration is the fixed conditioning variable. The exceedance conditional probability may be written as follows:

PS>sD=d=1−PS≤sD=d=1−CFSFD=FDd=1−∂C12FSFD∂FDdFDd(13.7)

The conditional return period can then be written as follows:

TS>sD=d=EINTPS>sD=d=EINT1−∂C12FSFD∂FDdFDd(13.8)

According to Nelson (2006), the stochastic increasing (SI) property exists if P(S>s| D = d)PS>sD=d is a nondecreasing function of drought duration for all drought severity values.

Using Equations (13.7) and (13.8) and the Gumbel–Hougaard copula as an illustrative example, Table 13.9 lists the exceedance conditional probability and conditional return period. Figure 13.8 plots the exceedance probability and conditional return period. Figure 13.8 clearly shows that for D = d, the exceedance probability P(S>s| D = d)PS>sD=d is a nondecreasing function of drought duration, i.e., P(S>s| D = d₁) ≤ P(S>s| D = d₂), d₁ < d₂PS>sD=d1≤PS>sD=d2,d1<d2.

Table 13.9. Exceedance conditional probability and conditional return period of S>s ∣ D = dS>s∣D=d.

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