probabilistic modelling of drought characteristics

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


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)


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


x

f(x)Distribuzione

troncata

soglia xo

Valore atteso dei deficit

Distribuzione della x


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


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


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



0.60d*c



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)

probabilistic modelling of drought characteristics

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