Probabilistic%20modelling%20of%20drought%20characteristics

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Probabilistic modelling of drought
characteristics
SIMPOSIO Gli eventi estremi alla ricerca di un
paradigma scientifico Alghero, 24-26 Settembre
2003

• G. Rossi, B. Bonaccorso, A. Cancelliere
• Department of Civil and Environmental Engineering
  
• University of Catania

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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

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DROUGHT PROCESS AND DEFINITIONS
Precipitation deficit PD
Meteorological drought
Unsaturated Soil Storage
Soil Moisture Deficit (SMD)
Agricultural drought
Surface Water Storage
Groundwater Storage
Groundwater Deficit (GWD)
Surface Flow Deficit (SFD)
Hydrological Drought
Measures for increasing resources and/or reducing
demands
Water Supply Systems
Water Resource Drought
Water Supply Shortage (SFS)
Measures for mitigating drought impacts
Socio-economic Systems
Economic and Intangible Impacts (EII)
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DROUGHT DEFINITIONS (1/2)

• Meteorological drought
• precipitation deficit (drought input) caused by
  atmospheric fluctuations related to
• solar energy fluctuations (?)
• earth processes (geophysical oceanographic
  interactions)
• biosphere feedbacks
• Agricultural drought
• soil moisture deficit deriving from
  meteorological drought routed trough soil storage
  mechanism (time delay and amount change)

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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

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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

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Review of drought characterization methods (1/9)
IDENTIFICATION OF AT-SITE DROUGHT
Threshold level method (original run analysis)
(Yevjevich, 1967)
Threshold level and inter-event time criterion
to identify independent drought for Lsurpluslt
Lc ? LdLd iLd i1 DcDc iDc
i1 (Zelenhasic and Salvai, 1987)
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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

• Madsen and Rosbjerg (1995) use a threshold level
  and both inter-event timeand
• inter-event volume criteria to identify
  independent droughts

Tallaksen et al. (1997) use a modified method
where LdLd iLd i1Ls i and DcDc iDc
i1-si Cancelliere et al. (1995) applied run
analysis to moving average series to take into
account the recovery concept
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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)

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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, (Matalas,
  1963)
• Pearson type III and type IV
• Gamma and Weibull (Joseph, 1970)
• Weibull distribution (Gustard et al., 1992)

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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)

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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

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Review of drought characterization methods (6/9)
LIMITS OF THE INFERENTIAL APPROACH
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

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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)

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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)

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PROBABILITY MASS FUNCTION OF DROUGHT DURATION LD
For stationary and time independent or Markov
lag-1 series Ld geometric (p1)
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DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc
(1/4)
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DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc
(2/4)
Probability distribution of Dt
1 per 0 ltdt lt ?
con p0Pxt?x0 e I(dt)
0 per dt ? 0
rth moment of Dt
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DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc
(3/4)
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DERIVATION OF THE PROBABILITY DISTRIBUTION OF Dc
(4/4) VALIDATION OF DC CDF ON GENERATED DATA
Lognormal series of 10,000 years
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DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION
OF Dc AND Ld (1/3)
JOINT PDF
For i.i.d. series
Hp DcLd gamma (r, b)
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DERIVATION OF THE JOINT PROBABILITY DISTRIBUTION
OF Dc AND Ld (2/3)
Hp.1 For Xt normal (mx, sx), lognormal (my, sy)
or gamma (rx,bx) Hp.2
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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
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RETURN PERIOD OF DROUGHT EVENTS
It can be defined as the average interarrival
time Td between two critical events
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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
• with Li the interarrival time between two any
  successive drought events

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ASSESSMENT OF DROUGHT RETURN PERIOD BIVARIATE
CASE
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Applications of probabilistic models to
precipitation series normal distributed
BIVARIATE CASE
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Applications of probabilistic models to
precipitation series lognormal distributed
BIVARIATE CASE
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Applications of probabilistic models to
precipitation series gamma distributed BIVARIATE
CASE
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Applications of probabilistic models to lognormal
and gamma streamflow series UNIVARIATE CASE
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Applications of probabilistic models to lognormal
and gamma streamflow series BIVARIATE CASE
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COMPARISON BETWEEN THE INFERENTIAL APPROACH AND
THE PROPOSED MODEL (1/3)
Log-normal series of 10,000 years
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COMPARISON BETWEEN THE INFERENTIAL APPROACH AND
THE PROPOSED MODEL (2/3)
Log-normal series of 10,000 years
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COMPARISON BETWEEN THE INFERENTIAL APPROACH AND
THE PROPOSED MODEL (3/3)
Log-normal series of 10,000 years
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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
  (x0mx-asx)
• 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)