Probabilistic modelling of drought characteristics
G. Rossi, B. Bonaccorso, A. Cancelliere
Department of Civil and Environmental Engineering University of Catania
SIMPOSIO “Gli eventi estremi: alla ricerca di un paradigma scientifico”
Alghero, 24-26 Settembre 2003
Outline• DROUGHT PROCESS AND DEFINITIONS
• MAIN STEPS OF PROBABILISTIC APPROACH TO DROUGHT ANALYSIS
• REVIEW OF DROUGHT CHARACTERIZATION METHODS- identification of drought events (at-site and over a region)– fitting of probability distributions to duration and accumulated deficit– data generation techniques through stochastic models– analytical derivation of probability distributions of drought characteristics
• PROPOSED PROCEDURE FOR ANALYTICAL DERIVATION OF PROBABILITY DISTRIBUTIONS OF DROUGHT CHARACTERISTICS
– Univariate case– Bivariate case
• ASSESSMENT OF DROUGHT RETURN PERIOD • APPLICATION OF PROBABILISTIC MODELS TO PRECIPITATION AND
STREAMFLOW SERIES
• CONCLUSIONS
SurfaceWater Storage
Socio-economicSystems
Water Supply Systems
Groundwater Storage
AgriculturalAgriculturaldroughtdrought
Soil MoistureDeficit (SMD)
GroundwaterDeficit (GWD)
Water SupplyShortage (SFS)
Economic andIntangible Impacts (EII)
Surface FlowDeficit (SFD)
HydrologicalHydrologicalDroughtDrought
MeteorologicalMeteorologicaldroughtdrought
Water ResourceWater ResourceDroughtDrought
Measures for mitigating drought impacts
Measures for increasing resources
and/or reducing demands
UnsaturatedSoil Storage
DROUGHT PROCESS AND DEFINITIONS
Precipitation deficit PD
Meteorological drought :precipitation deficit (drought input) caused by atmospheric fluctuations related to:i) solar energy fluctuations (?)ii) earth processes (geophysical oceanographic interactions)iii)biosphere feedbacks
Agricultural drought :soil moisture deficit deriving from meteorological drought routed trough soil storage mechanism (time delay and amount change)
DROUGHT DEFINITIONS (1/2)
Hydrological drought :surface flow deficit and groundwater deficit deriving respectively from precipitation deficit and soil moisture deficit routed trough the storage mechanism in natural water bodies
Water Resources drought :water supply shortage (drought output) influenced by artificial storage features (reservoir capacity and operation rules) and by different drought mitigation measures
DROUGHT DEFINITIONS (2/2)
1. SELECTION OF : • the variable of interest (precipitation, streamflow)• the time scale (year, month ,day)• the spatial scale (at-site or regional analysis)
2. SELECTION OF THE METHOD FOR DROUGHT IDENTIFICATION:• threshold level method (TLM) for at-site drought analysis:
- original run-method- modified run-methods
• TLM plus critical area for regional drought analysis
3. SELECTION OF THE METHOD FOR ESTIMATING THE PROBABILITY DISTRIBUTION OF DROUGHT CHARACTERISTICS • fitting parametric/non parametric probability distribution to drought characteristics identified on historical series (inferential approach)• data generation techniques• analytical derivation of drought cdf by using the parameters of the underlying variable distribution
4. ASSESSMENT OF DROUGHT RETURN PERIOD
MAIN STEPS OF PROBABILISTIC APPROACH TO DROUGHT ANALYSIS
Review of drought characterization methods (1/9):
IDENTIFICATION OF AT-SITE DROUGHT
Threshold level and “inter-event time” criterion to identify independent drought: for Lsurplus< Lc Ld=Ld i+Ld i+1
Dc=Dc i+Dc i+1
(Zelenhasic and Salvai, 1987)
Threshold level method(original run analysis)
(Yevjevich, 1967)
)(1
0 t
L
tc XxD
d
Accumulated deficit
1 ifd ttL
dc LDI /
Duration
Intensity
400
500
600
700
800
900
1000
1100
1200
Time (years)
Prec
ipita
tion
(mm
)
Ld=4 Ld=1 Ld=1Ld=4 Ld=5 Ld=1
Dc=338 mm Dc=90 mmDc=456 mm
Dc=74 mmDc=189 mm
Dc=197 mm
Time (days)
Dis
harg
e (m
3 /s)
Ld3Ld2Ld1 Ld4
Ls3
Dc3
Dc3*=Dc3+Dc4
Dc4Ld3*=Ld3+Ld4
d3*=d3+d4for Ls3<L0
Madsen and Rosbjerg (1995) use a threshold level and both “inter-event time”and “inter-event volume” criteria to identify independent droughts
Tallaksen et al. (1997) use a modified method where: Ld=Ld i+Ld i+1+Ls i and Dc=Dc i+Dc i+1-si
Cancelliere et al. (1995) applied run analysis to moving average series to take into account the recovery concept
Correia et al. (1987) apply a recovery criterion which defines the drought termination when the surplus volume is equal to a percentage of the previous cumulated deficit, both computed with reference to a threshold different from that one used to identify drought onset
Review of drought characterization methods (2/9):
IDENTIFICATION OF AT-SITE DROUGHT
Time (days)
Dis
harg
e (m
3 /s)
Ld2 Ld3
Ls2
Dc2
Dc2*=Dc2+Dc3
Dc3Ld2
*=Ld2+Ld3
d3*=d3+d4
s2
for Ls2<L0 ands2/Dc2<s0
- Use of a threshold level, equal for all the stations, on standardized monthly series to identify deficit intervals and of a critical area on a regular grid to identify regional drought (Tase, 1976)
- Use of a threshold level equal to a given percentage of the mean precipitation at each station and of a critical area by using Thiessen polygons to identify regional drought characteristics (deficit area, weighted total deficit) (Rossi, 1979)
- Use of a truncation level equal to a given nonexceedence probability and of a critical area identified by Thiessen polygons; derivation of approximate expressions for pdf of drought duration, intensity and areal extension of regional droughts, assuming multivariate normal precipitation independent in time (Santos, 1983)
Review of drought characterization methods (3/9):
IDENTIFICATION OF REGIONAL DROUGHT
- Gumbel distribution (Gumbel, 1963)
- Gumbel, 3 parameters log-normal, (Matalas, 1963) Pearson type III and type IV
- Gamma and Weibull (Joseph, 1970)
- Weibull distribution (Gustard et al., 1992)
Review of drought characterization methods (4/9): FITTING OF PROBABILITY DISTRIBUTIONS TO LOW-FLOW
(minimum annual n-day average disharge)
Drought characteristics (duration and accumulated deficit) identified by run analysis:
- Exponential distribution to fit both duration and accumulated deficit FD identified on daily discharge series with a
constant threshold (Zelenhasic and Salvai, 1987)
- Geometric distribution to fit duration FD and exponential distribution to fit drought accumulated deficit FD identified on
monthly precipitation series with periodic threshold (Mathier et al., 1992)
Review of drought characterization methods (5/9): FITTING OF PROBABILITY DISTRIBUTIONS TO DROUGHT
CHARACTERISTICS FREQUENCY DISTRIBUTION
WHAT IS THE DIFFERENCE BETWEEN LOW FLOW AND DROUGHT ANALYSIS ?
- Different time scale of the phenomena:days for low flows, months or years for drought events
- Low flow analysis aims to assess the annual minimum flows corresponding to a fixed probability or return period
- Droughts can span over several years: an adequate time interval for drought analysis cannot be adopted
- Drought return period cannot be assessed by the formula Drought return period cannot be assessed by the formula generally applied either for flood or low flow analysisgenerally applied either for flood or low flow analysis
]P[1T
tt xX
The inferential approach is often unsuitable due to the limited number of historical droughts
POSSIBLE SOLUTIONS•Data generation techniques through stochastic models to fictiously increase sample length
•Analytical derivation of probability distribution (or return period) of drought characteristics based on the probability distribution of the underlying hydrological variable
Review of drought characterization methods (6/9): LIMITS OF THE INFERENTIAL APPROACH
0
10
20
30
40
50
60
Valguarnera (75 anni)
Caltanissetta(118 anni)
Padova (164 anni)
Milano Brera(234 anni)
No.
sic
cità
xx 0
Review of drought characterization methods (7/9): DATA GENERATION TECHNIQUES
- Log-normal distribution to fit FD of the longest negative run length and the largest run sum obtained by lag-one autoregressive generated samples (Millan and Yevjevich, 1971)
- Negative Binomial distribution to fit FD of run length and Pearson distribution to fit FD of run sum obtained by a bivariate lag-one autoregressive model (Guerrero and Yevjevich, 1975)
- Beta distribution to fit the FD of regional drought characteristics (deficit area, areal deficit and intensity) obtained by generating monthly precipitation series (time independent but space dependent variable) (Tase, 1976 )
- Gamma distribution to fit the conditional distribution of drought accumulated deficit given drought duration (Shiau and Shen, 2001)
1967 Downer et al. (distribution and moments of run-length and run-sum derived for i.i.d. random variables)
1969 Llamas and Siddiqui (distribution function and moments of run-length, run-sum and run-intensity derived for independent normal and gamma series)
1970 Saldarriaga and Yevjevich (exact and approximate expressions of probabilities of run of wet and dry years for either independent or dependent stationary series of variables following the 1st order linear autoregressive model)
1976 Sen (probability of run-length for stationary lag-1 Markov process)1977 Sen (moments of run-sum for independent and two-state Markov process)1980 Sen (distribution of max deficit for stationary Markov process)1983 Guven (approximate expressions of the probabilities of critical droughts assuming the
deficit sum gamma distributed and the underlying variable normally distributed and generated by a lag-one Markov process)
1985 Sharma (expected value of max deficit for a fixed T return period)1998 Cancelliere et al. (drought accumulated deficit exponential distributed by assuming single
deficit independent and exponential distributed) 2003 Bonaccorso et al. (parameters of accumulated deficit cdf, assumed gamma, derived as
functions of the coefficient of variation of Xt and the threshold level) 2003 Cancelliere and Salas (exact probability distribution and related moments of drought
duration for periodic two-state lag-1 Markov process)
Review of drought characterization methods (8/9):ANALYTICAL DERIVATION OF DROUGHT
CHARACTERISTICS PROBABILITY DISTRIBUTION
PROBABILITY MASS FUNCTION OF DROUGHT DURATION LD
For stationary and time independent or Markov lag-1 series Ld ~ geometric (p1):
1l11d
d1)(lf ppdL
p1=P[xt>x0]
1
1Ep
Ld Expected value
21
11Var
pp
Ld
Variance
DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc (1/4)
For i.i.d. events : tdc DLD EEE
dttdc LDDLD VarEVarEVar 2
dttd
td
LDDL
DL
VarEVarE
EE2
22
r
td
dttd
DLLDDL
EEVarEVarE 2
β
βrE cD
2βrVar cD
Hp: Dc ~ gamma (r, )
c
c
dr
ccD ed
rΓdf
11
Probability distribution of Dt
)(0,X0
D )I(fp1f
tt tt0t d-dxd con p0=P[xtx0] e I(dt)1 per 0 <dt <
0 per dt 0
)d(fpd 1r
X0 0
rt
t tt0r ddxDt
E
rth moment of Dt
DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc (2/4)
x
f(x)Distribuzione
troncata
soglia xo
Valore atteso dei deficit
Distribuzione della x
DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc (3/4)
vα,Cfr vα,Cgβ x
Hp.1 Xt normal (x, x), lognormal (y, y) or gamma (rx, x)
Hp.2 vxxx0 C1μσμx Coeff. of variation of Xt
v
*c
v
1
0cD C,
d,C,Gd
(r)1dF
c
gfzez
zrdc
Incomplete Gamma Function dc/x
DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc (4/4) VALIDATION OF DC CDF ON GENERATED DATA
Lognormal series of 10,000 years
DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION OF Dc AND Ld (1/3)
For i.i.d. series : tcdc DlL|D EE
tcdc DlL|D VarVar
cdLccldL|cDccdL,cD lfdfl,df )(JOINT PDF
t
t
DD
rVarE
l2
c
t
t
DD
EVar
rdc L|DE
2dc rL|D Var
Hp: Dc|Ld ~ gamma (r, )
c
dc
drc
cLD ed
rΓdf
1
|1
DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION OF Dc AND Ld (2/3)
vc C,l r vx C,δ β
Hp.1 For Xt normal (x, x), lognormal (y, y) or gamma (rx,x)
Hp.2 vxxx0 C1μσμx
v
*
v1l
11 Cα,δ
d,Cα,lGp1pF cc ccc,LD ,ld
dcJoint cdf
xcc dd /*
lc= 1 year
lc=3 years
lc=5 years
lc=7 years
DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION OF Dc AND Ld (3/3)
VALIDATION OF JOINT CDF ON GENERATED DATA
Lognormal series of 10,000 years
It can be defined as the It can be defined as the average interarrival time Td between two critical events
Td 1 Td j Td j+1
Hydrological process xt
Time t
Time t
Characteristic Qj
Interarrival time between events with(Q>Q0)
adapted from
Fernandez and Salas (1999)
RETURN PERIOD OF DROUGHT EVENTS
ASSESSMENT OF DROUGHT RETURN PERIOD
01
wd pp1LLL EEE
1nAP1APnNP
Critical droughts
01d pp
11T P[A]
E
Let Let NN be the number of droughts between two critical droughts be the number of droughts between two critical droughts The interarrival time The interarrival time TTdd between these two critical droughts is:between these two critical droughts is:
with Li the interarrival time between two any successive drought events
N
iid LT
1
E[L]E[N]EE
i.i.dN
1iid LTReturn period
ASSESSMENT OF DROUGHT RETURN PERIOD: BIVARIATE CASE
I) A = {D>dc and Ld= lc (lc=1,2,…)}:
II) A = {D>dc and Ld lc (lc=1,2,…)}:
III) A = {I > i and Ld = lc (lc=1,2,…)}:
IV) A = {I > i and Ld lc (lc=1,2,…)}:
1cl11
cc
cdcdLcDcdcc p1p
dl1zlzflLdDP
δ
,d),(,*
, G
cll
1l11
c
cd clldLcDcdcc p1p
dl1zlzflLdDP
δ,d),(,
*
, G
1cl11
cc
icLI,cd p1p
ill1z)l(z,flLi,IP
δ
,d*
G
cll
1l11
c
i clldLIcd p1p
ill1zlzflLiIP
δ,d),(,
*
, G
Petralia(116 years)
dc=1.00
dc=0.50dc=0.00
ic=0.30ic=0.20
ic=0.00
A = {D>dc and Ld= lc} A = {D>dc and Ld lc}
A = {I>ic and Ld= lc} A = {I>ic and Ld lc}
Applications of probabilistic models to precipitation series normal distributed: BIVARIATE CASE
Milano Brera(234 years)
Applications of probabilistic models to precipitation series lognormal distributed: BIVARIATE CASE
Agrigento(111 years)
Applications of probabilistic models to precipitation series gamma distributed: BIVARIATE CASE
Applications of probabilistic models to lognormal and gamma streamflow series: UNIVARIATE CASE
(82 years)
(51 years)
(100 years)
(53 years)
Applications of probabilistic models to lognormal and gamma streamflow series: BIVARIATE CASE
(82 years)
(51 years)
(100 years)
(53 years)
0.20d*c
COMPARISON BETWEEN THE INFERENTIAL APPROACH AND THE PROPOSED MODEL (1/3)
Log-normal series of 10,000 years
0.40d*c
COMPARISON BETWEEN THE INFERENTIAL APPROACH AND THE PROPOSED MODEL (2/3)
Log-normal series of 10,000 years
0.60d*c
COMPARISON BETWEEN THE INFERENTIAL APPROACH AND THE PROPOSED MODEL (3/3)
Log-normal series of 10,000 years
CONCLUSIONS• Probabilistic drought analysis can be carried out by three main approaches:
- fitting of probability distributions to historical drought characteristics;- data generation techniques through stochastic models;- analytical derivation of probability distribution of drought characteristics
• A methodology to derive the probability distribution of both drought characteristics (duration and accumulated deficit) by using the parameters of the underlying variable distribution has been presented
• The parameters of the cdf of Dc and the joint cdf of Dc and Ld have been determined as functions of Cv of the variable Xt and the threshold level (x0=x-x)
• The proposed methodology enables one to overcome the difficulties related to estimation based on historical records alone and results adequate for several hydrological series (precipitation, streamflow)