Transcript Document

SIMPOSIO “Gli eventi estremi: alla ricerca di un paradigma scientifico”
Alghero, 24-26 Settembre 2003
Probabilistic modelling of drought
characteristics
G. Rossi, B. Bonaccorso, A. Cancelliere
Department of Civil and Environmental Engineering
University of Catania
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
DROUGHT PROCESS AND DEFINITIONS
Precipitation deficit PD
Meteorological
drought
Unsaturated
Soil Storage
Soil Moisture
Deficit (SMD)
Agricultural
drought
Surface
Water Storage
Hydrological
Drought
Groundwater
Storage
Surface Flow
Deficit (SFD)
Groundwater
Deficit (GWD)
Water Supply Systems
Water Resource
Drought
Water Supply
Shortage (SFS)
Socio-economic
Systems
Economic and
Intangible Impacts (EII)
Measures for
increasing
resources
and/or reducing
demands
Measures for
mitigating
drought
impacts
DROUGHT DEFINITIONS (1/2)
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 (2/2)
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
MAIN STEPS OF PROBABILISTIC APPROACH TO
DROUGHT ANALYSIS
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
Review of drought characterization methods (1/9):
IDENTIFICATION OF AT-SITE DROUGHT
1200
1100
Ld= 4
L d= 1
Ld= 4
Ld= 5
Ld= 1
L d= 1
Threshold level method
(original run analysis)
Ld  t f  ti  1 Duration
900
Ld
Accumulated
Dc   ( x0  X t )
deficit
t 1
800
700
600
I  Dc / Ld
Dc= 74 m m
Dc= 189
500
Dc= 338 m m
mm
Dc= 90 m m
(Yevjevich, 1967)
400
T i m e (y e a r s)
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)
Intensity
Dc= 197 m m
Dc= 456 m m
Disharge (m3/s)
Pre cipita tion (m m )
1000
Ls3
Ld1 Ld2 Ld3 Ld4
Dc3
for Ls3<L0
D
Ld3*=Ld3+Ld4 c4
Dc3*=Dc3+Dc4
d3*=d3+d4
Time (days)
Review of drought characterization methods (2/9):
IDENTIFICATION OF AT-SITE DROUGHT
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
Ls 2
Ld 2
Disharge (m 3 /s)
Madsen and Rosbjerg (1995)
use a threshold level and both
“inter-event time”and
“inter-event volume” criteria to
identify independent droughts
s2
Ld 3
Dc 2
*
fo r L s 2 < L 0 and
Ld 2 =Ld 2 +Ld 3
s 2 /D c 2 < s 0
D c d2 3 =D
* = d 3c+2d+D
4
c3
Dc 3
*
Tim e (d a y s )
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
Review of drought characterization methods (3/9):
IDENTIFICATION OF REGIONAL DROUGHT
- 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 (4/9):
FITTING OF PROBABILITY DISTRIBUTIONS TO LOW-FLOW
(minimum annual n-day average disharge)
- Gumbel distribution
(Gumbel, 1963)
- Gumbel, 3 parameters log-normal,
Pearson type III and type IV
(Matalas, 1963)
- Gamma and Weibull
(Joseph, 1970)
- Weibull distribution
(Gustard et al., 1992)
Review of drought characterization methods (5/9):
FITTING OF PROBABILITY DISTRIBUTIONS TO DROUGHT
CHARACTERISTICS FREQUENCY DISTRIBUTION
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)
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
generally applied either for flood or low flow analysis
1
T
P[ X t  xt ]
Review of drought characterization methods (6/9):
LIMITS OF THE INFERENTIAL APPROACH
60
50
No. siccità
The inferential approach
is often unsuitable due
to the limited number of
historical droughts
x0  x
40
30
20
10
0
Valguarnera
(75 anni)
Caltanissetta
(118 anni)
Padova
(164 anni)
Milano Brera
(234 anni)
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 (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)
Review of drought characterization methods (8/9):
ANALYTICAL DERIVATION OF DROUGHT
CHARACTERISTICS PROBABILITY DISTRIBUTION
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)
PROBABILITY MASS FUNCTION OF DROUGHT
DURATION LD
For stationary and time independent or Markov lag-1 series
Ld ~ geometric (p1):
p1=P[xt>x0]
f Ld (ld )  p1  1  p1 ld 1
Expected value
Variance
ELd  
Var Ld  
1  p1
p12
1
p1
DERIVATION OF THE PROBABILITY
DISTRIBUTION OF Dc (1/4)
For i.i.d. events :
EDc   ELd  EDt 
Var Dc   ELd  Var Dt   EDt 2  Var Ld 
r 1
1  dc 
  e
f Dc d c  
bΓ r   b 
Hp: Dc ~ gamma (r, b)
EDc   r  β

dc
b
VarDc   r  β 2
E Ld  E Dt 
2
r
2
E Ld VarDt   E Dt  VarLd 
2
E Ld VarDt   E Dt  VarLd 
β
E Ld  E Dt 
2
DERIVATION OF THE PROBABILITY
DISTRIBUTION OF Dc (2/4)
Probability distribution of Dt
1
f D t d t  
 f X t x0 -d t  I( d t ) (0,)
p0
con p0=P[xtx0] e I(dt)
soglia xo
f(x)
Distribuzione
troncata
Valore atteso
dei deficit
Distribuzione della x
E[x|x<x0]
x0
rth moment of Dt

  
r
d
r 1
t
E D tr 
 f X t x0  d t  d(d t )
p0
0
x
1 per 0 <dt < 
0 per dt  0
DERIVATION OF THE PROBABILITY
DISTRIBUTION OF Dc (3/4)
Hp.1 Xt normal (x, sx), lognormal (y, sy) or gamma (rx, bx)
Hp.2 x0  μ x    σ x  μ x 1    C v 
r  f α,Cv 
Coeff. of variation of Xt
β   x  g α,Cv 
dc/x
Incomplete Gamma Function
FDc d c  
dc

0
r 1  z

1 z
 
b(r)  b 
e
b
*


d
c

dz  G f  , C v ,

g  , C v  

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)
JOINT
PDF
f Dc ,Ld (d c ,lc )  f Dc |Ld lc d c   f Ld lc 
For i.i.d. series :
EDc | Ld   lc  EDt 
Var Dc | Ld   lc  Var Dt 
Hp: Dc|Ld ~ gamma (r, b)
EDc | Ld   r  b
Var Dc | Ld   r  b 2
1  dc 
 
f Dc |Ld d c  
b  Γ r   b 
r 1  d c
b
e
EDt 2 b  Var Dt 
r  lc 
EDt 
Var Dt 
DERIVATION OF THE JOINT PROBABILITY
DISTRIBUTION OF Dc AND Ld (2/3)
Hp.1 For Xt normal (x, sx), lognormal (y, sy) or gamma (rx,bx)
Hp.2 x0  μ x    σ x  μ x 1    C v 
r  l c    , C v 
β   x  δ , C v 
d c*  d c /  x
Joint cdf
*


d
c
l c 1


FDc ,Ld d c ,l c   p1 1  p1   G l c   α, C v ,

δα, C v  


DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION OF Dc AND Ld (3/3)
VALIDATION OF JOINT CDF ON GENERATED DATA
lc= 1 year
lc=3 years
lc=5 years
lc=7 years
Lognormal series of 10,000 years
RETURN PERIOD OF DROUGHT EVENTS
It can be defined as the average interarrival time Td between two
critical events
Hydrological process xt
Time t
Characteristic Qj
Td 1
Td j
Td j+1
Interarrival time between events with(Q>Q0)
Time t
adapted from
Fernandez and Salas (1999)
ASSESSMENT OF DROUGHT RETURN PERIOD


Let N be the number of droughts between two critical droughts
The interarrival time Td between these two critical droughts is:
N
Td   Li
i 1
with Li the interarrival time between two any successive drought events
Return period
EL   ELd   ELw  
 N  i.i.d
ETd   E  Li   E[N] E[L]
 i 1 
1
p1 p 0
Critical droughts
PN  n  PA 1  PAn1
ETd  
1
1

P [A] p1 p0
ASSESSMENT OF DROUGHT RETURN PERIOD:
BIVARIATE CASE
I) A = {D>dc and Ld= lc (lc=1,2,…)}:


d *c
PDc  d c , Ld  lc    f Dc , Ld ( z, lc ) d z  1  G lc ,

δ

dc



  p1 1  p1 lc 1


II) A = {D>dc and Ld  lc (lc=1,2,…)}:

 d *c
PDc  d c , Ld  lc     f Dc , Ld ( z, l ) d z   1  G l ,

δ
l lc 
dc l lc


 


  p1 1  p1 l 1


III) A = {I > i and Ld = lc (lc=1,2,…)}:

PI  i, Ld  l c   
i


l c i * 
  p1 1  p1 lc 1
f I,L (z,l c ) d z  1  G  l c ,
δ 


IV) A = {I > i and Ld  lc (lc=1,2,…)}:
 
PI  i, Ld  l c    
i l  lc

 l c i * 
  p1 1  p1 l 1
f I , Ld ( z, l ) d z   1  G l ,
δ 
l  lc 



Applications of probabilistic models to precipitation series
normal distributed: BIVARIATE CASE
dc=1.00
Petralia
(116 years)
dc=0.50
dc=0.00
A = {D>dc and Ld= lc}
A = {D>dc and Ld lc}
ic=0.30
ic=0.20
ic=0.00
A = {I>ic and Ld= lc}
A = {I>ic and Ld lc}
Applications of probabilistic models to precipitation series
lognormal distributed: BIVARIATE CASE
Milano Brera
(234 years)
Applications of probabilistic models to precipitation series
gamma distributed: BIVARIATE CASE
Agrigento
(111 years)
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)
COMPARISON BETWEEN THE INFERENTIAL
APPROACH AND THE PROPOSED MODEL (1/3)
d*c  0.20
Log-normal series of 10,000 years
COMPARISON BETWEEN THE INFERENTIAL
APPROACH AND THE PROPOSED MODEL (2/3)
d*c  0.40
Log-normal series of 10,000 years
COMPARISON BETWEEN THE INFERENTIAL
APPROACH AND THE PROPOSED MODEL (3/3)
d*c  0.60
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-sx)
•
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)