Simulation of the occurrence of drought events via copulas

Almeida, Rogério de; Barbosa, Paulo Sérgio Franco

doi:10.1590/2318-0331.252020180182

Rogério de Almeida

Universidade Estadual de Campinas, Campinas, SP, Brasil

http://orcid.org/0000-0002-0469-3233

Paulo Sérgio Franco Barbosa

Universidade Estadual de Campinas, Campinas, SP, Brasil

E-mails: dealmeida135@gmail.com (RA), franco.barbosa@gmail.com (PSFB)

Authors contributions

Rogério de Almeida: This study is part of the author’s doctoral thesis.

Paulo Sérgio Franco Barbosa: Advisor professor of the doctoral thesis of student Rogério de Almeida.

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Figure 1
Region of the Camanducaia River basin.

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Figure 2
Camanducaia River basin and cities supply it.

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Figure 3
Average monthly streamflow of the Camanducaia River from 1944 to 2016 (streamflow gauging station 3D-002).

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Figure 4
Theory of runs applied to define and characterize the drought events.

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Figure 5
Durations and severities of the drought events for the Camanducaia River from 1944 to 2016 considering the threshold levels: (a) P98, (b) P95, (c) P92 and (d) P90.

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Figure 6
Severities’ CDF of the drought events for the Camanducaia River from 1944 to 2016 considering the threshold levels: (a) P98, (b) P95, (c) P92 and (d) P90.

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Figure 7
Durations’ CDF of the drought events for the Camanducaia River from 1944 to 2016 considering the threshold levels: (a) P98, (b) P95, (c) P92 and (d) P90.

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Figure 8
Observed and simulated drought events for the Camanducaia River and joint probabilities (isolines) considering the threshold levels: (a) P98, (b) P95, (c) P92 and (d) P90.

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Table 1
Main statistics of the streamflow data for the Camanducaia River (streamflow gauging station 3D-002).

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Table 2
Main characteristics of the drought events for the Camanducaia River from 1944 to 2016.

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Table 3
p-Values of the Box-Pierce and Ljung tests for observations in series of the severities and durations of the drought events for the Camanducaia River from 1944 to 2016.

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Table 4
Univariate probability distributions tested for the severities and durations of the drought events.

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Table 5
Copulas tested to model the dependence between severities and durations of the drought events for the Camanducaia River.

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Table 6
AIC values and KS test’s p-values for each probability distribution tested to the drought events severities for the Camanducaia River from 1944 to 2016.

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Table 7
AIC values and KS test’s p-values for each probability distribution tested to the drought events durations for the Camanducaia River from 1944 to 2016.

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Table 8
Probability distributions of the severities and durations for the Camanducaia River drought events.

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Table 9
AIC values and GOF test’s p-values for each copulas tested to represent jointly the severities and durations for the Camanducaia River drought events.

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Table 10
Copulas’ parameters selected to represent jointly the severities and durations for the Camanducaia River drought events.

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Table 11
95% confidence intervals for the severity and average duration and Kendall’s

τ

and Spearman’s

ρ

of the drought events of the Camanducaia River based on the copulas.

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Table 12
Ave, Max and Min values of the severities and durations and Kendall’s τ and Spearman’s

ρ

values between the severities and durations of the simulated drought events for the Camanducaia River (pseudo-observations) from the copulas.

(1)

(2)

(3)

(4)

Statistics	Value (m³/s)
Long term average	6.80
Median	5.44
Standard deviation	4.39
Maximum	35.53
Minimum	0.69

Characteristics of the drought events		Threshold level
Characteristics of the drought events		P98	P95	P92	P90
Number of events		10	21	37	40
Severity	Ave	0.82	0.85	1.07	1.41
	Max	4.94	7.97	11.30	13.19
	Min	0.01	0.03	0.08	0.02
Duration	Ave	1.80	2.05	1.89	2.20
	Max	7	9	9	9
	Min	1	1	1	1
Kendall’s $τ$		0.55	0.58	0.78	0.63
Spearman’s $ρ$ .		0.62	0.67	0.77	0.78

Threshold level	p-Values
	Severity		Duration
	Box-Pierce	Ljung	Box-Pierce	Ljung
P98	0.51	0.44	0.57	0.52
P95	0.60	0.68	0.67	0.65
P92	0.78	0.77	0.82	0.81
P90	0.77	0.77	0.91	0.91

Probability distribution	Probability density functions
GAM2	$f (x) = \frac{1}{β^{α} Γ (α)} x^{α - 1} e^{- \frac{x}{β}}$
EXP	$f (x) = λ e^{- λ x}$
WBL	$f (x) = \frac{α}{β} {(\frac{x}{β})}^{α - 1} e^{- {(\frac{x}{β})}^{α}}$
LOGN	$f (x) = \frac{1}{x σ \sqrt{2 π}} e^{- \frac{{(ln (x) - μ)}^{2}}{2 σ^{2}}}$
GP	$f (x) = \frac{1}{σ} {(1 + \frac{ξ (x - μ)}{σ})}^{(- \frac{1}{ξ} - 1)}$
GEV	$\begin{matrix} f (x) = \frac{1}{σ} t {(x)}^{ξ + 1} e^{- t (x)} \\ w h e r e t (x) = \{\begin{matrix} {(1 + ξ (\frac{x - μ}{σ}))}^{- \frac{1}{ξ}} i f ξ \neq 0 \\ e^{- \frac{(x - μ)}{σ}} i f ξ = 0 \end{matrix} \end{matrix}$

Copula	Copula’s function and parameter ranger
CLA	$C (u_{1}, u_{2}) = {(u_{1}^{- θ} + u_{2}^{- θ} - 1)}^{- \frac{1}{θ}}$ where $θ \in (0, \infty)$
FRA	$C (u_{1}, u_{2}) = - \frac{1}{θ} ln (1 + (\frac{(e^{- θ u_{1}} - 1) (e^{- θ u_{2}} - 1)}{e^{- θ} - 1}))$ where $θ \in \bar{R} \ \{0\}$
GUM	$C (u_{1}, u_{2}) = e^{- {({(- ln (u_{1}))}^{θ} + {(- ln (u_{2}))}^{θ})}^{\frac{1}{2}}}$ where $θ \in [1, \infty)$
JOE	$C (u_{1}, u_{2}) = \frac{u_{1} u_{2}}{1 - θ (1 - u_{1}) (1 - u_{2})}$ where $θ \in [1, \infty)$
AMH	$C (u_{1}, u_{2}) = 1 - ({(1 - u_{1})}^{θ} + {(1 - u_{2})}^{θ} - {(1 - u_{1})}^{θ} {(1 - u_{2})}^{θ})$ where $θ \in (- 1,1]$
GAU	$C (u_{1}, u_{2}) = \int_{- \infty}^{Φ^{- 1} (u_{1})} \int_{- \infty}^{Φ^{- 1} (u_{2})} \frac{1}{2 π \sqrt{1 - θ^{2}}} e^{(\frac{- x^{2} - 2 θ x y + y^{2}}{2 (1 - θ^{2})})} d x d y$ where $θ \in [- 1,1]$

Probability distribution	Threshold levels
	P98		P95		P92		P90
	AIC	KS	AIC	KS	AIC	KS	AIC	KS
	AIC	p-value	AIC	p-value	AIC	p-value	AIC	p-value
GAM2	15.77	0.84	38.50	0.78	83.05	0.20	110.33	0.52
EXP	18.03	0.16	37.29	0.19	81.27	0.13	109.25	0.25
WBL	14.90	0.95	33.46	0.98	81.50	0.48	109.21	0.76
LOGN	14.17	1.00	28.79	1.00	71.00	0.75	109.13	0.89
GP	14.65	0.84	30.13	0.97	76.58	0.74	106.75	0.84
GEV	17.37	0.95	31.32	0.89	72.67	0.95	111.28	0.82

Probability distribution	Threshold levels
	P98		P95		P92		P90
	AIC	KS	AIC	KS	AIC	KS	AIC	KS
	AIC	p-value	AIC	p-value	AIC	p-value	AIC	p-value
GAM2	33.69	0.01	68.64	0.05	111.34	0.00	134.95	0.00
EXP	33.76	0.08	74.10	0.18	123.18	0.14	145.08	0.25
WBL	34.95	0.03	72.20	0.05	117.47	0.00	139.07	0.00
LOGN	29.47	0.01	62.21	0.06	100.82	0.00	127.28	0.00
GP	35.75	0.06	75.92	0,09	124.63	0.06	145.49	0.07
GEV	(*)	(*)	(*)	(*)	(*)	(*)	(*)	(*)

Threshold level	Probability distribution and estimated parameters
Threshold level	Severity	Duration
P98	LOGN	EXP
P98	$\hat{μ}$ -1.462 $\hat{σ}$ 1.737	$\hat{λ}$ 0.556
P95	LOGN	LOGN
P95	$\hat{μ}$ -1.127 $\hat{σ}$ 1.347	$\hat{μ}$ 0.498 $\hat{σ}$ 0.588
P92	LOGN	EXP
P92	$\hat{μ}$ -0.569 $\hat{σ}$ 1.057	$\hat{λ}$ 0.529
P90	GP	EXP
P90	$\hat{μ}$ 0.000 $\hat{ξ}$ 0.262 $\hat{σ}$ 1.023	$\hat{λ}$ 0.455

Copula	Threshold level
	P98		P95		P92		P90
	AIC	GOF	AIC	GOF	AIC	GOF	AIC	GOF
	AIC	p-value	AIC	p-value	AIC	p-value	AIC	p-value
CLA	-1.02	^* * Evaluation was not possible.	-11.04	0.10	-9.04	0.00	-12.16	0.00
FRA	-4.52	0.28	-12.96	0.34	-24.82	0.47	-37.62	0.10
GUM	-7.04	0.79	-14.96	0.88	-29.33	0.35	-41.14	0.36
JOE	-8.14	0.90	-12.74	0.45	-29.85	0.17	-43.20	0.28
AMH	-1.65	0.10	-9.80	0.03	-9.32	0.00	-12.31	0.00
GAU	-5.16	0.36	-15.89	0.56	-25.79	0.31	-33.44	0.17

Threshold level	Copula selected	Estimates parameter $(\hat{θ})$
P98	JOE	2.772
P95	GAU	0.729
P92	JOE	2.895
P90	JOE	3.735

Statistics	Threshold level
Statistics	P98	P95	P92	P90
Average severity	0.88 - 1.28	0.73 - 0.88	0.94 - 1.04	1.31 - 1.45
Average duration	1.77 - 1.87	1.90 - 2.00	1.81 - 1.96	2.12 - 2.28
Kendall’s $τ$	0.46 - 0.51	0.50 - 0.54	0.48 - 0.53	0.57 - 0.61
Spearman’s $ρ$	0.63 - 0.70	0.69 - 0.74	0.66 - 0.71	0.76 - 0.80

Statistics		Threshold level
Statistics		P98	P95	P92	P90
Severity	Ave	0.97	0.85	0.99	1.43
	Max	52.49	40.01	32.87	73.08
	Min	0.0008	0.01	0.02	0.001
Duration	Ave	1.77	1.94	1.90	2.25
	Max	12.61	18.62	17.98	16.77
	Min	0.0008	0.26	0.0001	0.003
Kendall’s $τ$		0.46	0.52	0.49	0.59
Spearman’s $ρ$		0.64	0.70	0.67	0.78

[1] E-mails: dealmeida135@gmail.com (RA), franco.barbosa@gmail.com (PSFB)

Brasil

Brasil

Simulation of the occurrence of drought events via copulas

Simulação da ocorrência de eventos de secas via cópulas

ABSTRACT