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Reliability Analysis for Drought Early Warning in an Ecological System Arman Ganji

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Reliability Analysis for Drought Early Warning in an Ecological System

Arman Ganji

Ecological System Behavioursunder drought condition

• Ecological systems are likely to be continually in atransient state, however their behaviours are affectedby random events, such as droughts. It is important,therefore, to determine how their behaviours may bemodified as a result of these random events.

02

46

810

12

23

45

6

3

4

5

Rainfall

Time

Population/Intensity Old state

New state

Objective: Finding a way to predict the possibility 

of failure in system behaviour

Solution:System Reliability Analysis

Reliability

Reliability is the ability of a system to perform its required functions under stated conditions for a specified period of time.

An Example in water resource

The number of times that thesystem works at a satisfactory stateover the planning horizon. Thisnumber of times represents a kind ofpredictive uncertainty or randomnessin system performance.

Reliability: Vague/Crisp index ?• A number of systems are neither fully functioning nor completely failed,

but in some intermediate state. Due to this vagueness, the crisp /binarystate assumptions for system failure /success are not always appropriate.

The natural systems involve vaguely definedfailure which cannot be specified as eitherpartial or complete to a certainty.

Vagueness & Fuzzy• Vagueness is a type of uncertainty it is not related to randomness. In the

case of vagueness in system failure, the fuzzy sets theory which means thesystem failure can not be defined in a precise way, but in a fuzzy way,should be considered.

Fuzzy sets permits the gradual assessment of system failure

Failure and success in classic reliability analysis Failure and success in 

classic reliability analysis Failure and success inFuzzy environment

Failure and success inFuzzy environment

1‐P1=P2 P2Failure Success

Crisp Bound

µ=1

µ=0

Failure            Success

Profust reliability theoryCAI Kai‐Yuan and WEN Chuan‐Yuan (1990)

Profust reliability theory is based on the classic reliabilityanalysis (probability assumption) and the fuzzy-state assumption:

• Probability assumption: – the system failure behaviour is fully characterized in the context of

probability measures.

• Fuzzy-state assumption: – the system success and failure are characterized by fuzzy states. At any

time the system can be viewed as being in one of the two fuzzy states to some extent. That is, the meaning of system failure is not defined in a precise way, but in a fuzzy way.

Failure evaluation in a system using Profust

Suppose that a population in an ecological system has n topological (non-fuzzy) states {S1,...,S2}. Let U = {Sl,..., S2} be the universe of discourse. Theprofust interval reliability is defined as:

Failure evaluation Using Profust Reliability: 1‐ failure possibility 

(from a fuzzy success state to a fuzzy failure state)Failure = 1‐ Reliability

Failure evaluation Using Profust Reliability: 2‐ Randomness in failure

The second term on the right hand side of reliability equation shows theprobability that no failure occurs over the time interval [t0, t+t0].

Cai and Chuan‐Yuan (1990) showed the above equation can be representedas:

}Sinissystem+tttimeP{at+t)=t, (tR i

n

jTFS 0

1

*~00

~

FSFS TT ~*~ 1

Profust in Literature• Cai, et al., (1991,1993,1995,1996) gave a different insight by introducing the

possibility assumption and fuzzy state assumption to replace the probabilityand binary state assumptions (profust reliability).

• Cia, et al., (1992) used profust reliability theory in typical systems (series,parallel, coherent).

• Chuan, et al., (1995) used mixture models for profust reliability.

• Cornelia (2000) used the profust reliability for linear and parallel system.

• Mohanta et al., (2004) used profust model for the generating unit.

• Jiang, et al.,(2005) studied membership function of profut reliability.

• Pandey, et al., (2007) used the profust reliability for degradable systems.

• Sunil(2007) evaluated reliability of engineering systems by profust theory.

Now we show how the Profust theory  can be adopted to evaluate the reliability of ecological systems 

under drought condition

Case study

1‐Gareh‐Bygone Rangeland

Coordination: 28° 38' N, 53° 55' E Area: 6000 haMean annual precipitation:260 mm

In 1994, 8 floodwaterspreading systems wereestablished in some partsof this area.

Parameters evaluation

}Sinissystem+tttimeP{at i0

1‐ failure possibility for the rangelandMixture model for plant species in non‐grazingrangeland• Suppose that there are n plant species with specificgross primary production (GPP) in a rangeland. Let Pmand Pi be the maximum level of GPP and the GPPproduced at year i during a drought. Then, we definefailure and success membership functions of a plantspecies for ith year as:

• However, the total GPP by the multiple plant speciessystem at time t is

• and the maximum total GPP is

• So we may reasonably define the system failure andsuccess membership functions

RYR-Precipitation relationship results

StatusRYR(simulation)RYR(observation)Production(kg/ha)precipitationYear

Moderate0.560.56317.81651994

Moderate0.360.4432.22851995

Good00721.55131996

Good0.490.19581208.51997

Good0.380.37457273.51998

Moderate0.590.63501.9141.51999

Moderate0.690.65252.5832000

Bad0.510.811341932001

StatusRYR(simulation)RYR(observation)Production(kg/ha)precipitationYear

Very bad0.880.88871651994

Very bad0.850.88902851995

Bad0.810.83125.65131996

Bad0.870.85105208.51997

Bad0.860.8698273.51998

Rangeland  with Floodw

ater spreading system

NoFloodw

ater spreading system

Typical failure function for some plant species 

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Relative yeild reduction (kg/kg)

µ fµ f

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.19 0.36 0.4 0.56 0.63 0.65 0.814

F

Relative yeild reduction (kg/kg)

The possibility of failure/success (Normal to other states)

Rangeland with floodwater

spreading systems Transition

0.83 Normal to Sever

0.55 Normal to Moderate

StatesRelative Yield

Reduction indexIntervals

State index

for rainfall

SEVERE 0.72 P < 122.8 61.4

MODERATE 0.52 122.8< P <191.1 156.95

NORMAL 0.21 191.1 < P 252.5

2‐Randomness in failureConsidering the form of random part of profust reliability equation, asemi‐Markov chain (Nolau, 1981) is used to describe the sequence ofseverity states.We also assumed a linear relationship between yield and rainfall forthe current rangeland, and defined some discrete states of the systemand drought severity.

The semi‐Markov model consists of two basic parts:• the state transitions;• the duration of each state

}Sinissystem+tttimeP{at i0

Definition of Semi‐Markov processLet represent the states of the semi‐Markov process. Also arerandom variables taking values in [0, +∞). These represent the duration ofstate . Now, consider to be the total duration of the first n+1drought events; which could be defined as (Bardossy, 1991):

where the right hand side of above equation represents the transitiontime to a future state depends on the present drought severity

1tS Nt1td

1n

0andd 0

n

0t1t1n

Ai)S|Td,sS(P)d...,,d;S...,,S|)1t(d,sS(P t1ti

1t1t1t1t1ti

1t1t

1tS

Model simplification)...,,;...,,|)1(,( 11111 ddSStdsSP ttt

itt

:

)|)1(( 111ittt sStdP

),|)1(( 111j

ttittt sSsStdP

)())1((

11

111itt

ittt

sSPsStdP

ddifsSP

yyyyPti

tt

it

it

idt

idt

1

11

121

)()1,0...,,0,1(

ddif)sS(P

P.P).sS(P1ti

1t1t

1tl2dtl

lii

1dtij

jtt

)(

.).(

11

1 12

itt

tij

jtt

sSP

PPsSP tldtl

lii

)|( 11j

ttitt sSsSP

jiAji ,,

1tijP

1- Assuming 1td is independent of j

tt sS and using conditional probability

2.Using definition of Bernoulli variable

3. Using Markov chain structure of drought severity

Model verification for Probability of transition

The probability of the next drought duration, given normal state of severity in the present state

)...,,;...,,|)1(,( 11111 ddSStdsSP tttitt

Model verification for Probability of transition

)...,,;...,,|)1(,( 11111 ddSStdsSP tttitt

Profust lifetime(Normal statelower states)

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Year

Rel

iabi

lity

Increase in rangeland reliability by Floodwater spreading system 

The reliability has been improved to 20% for 1 year duration after constructingeight water spreading systems. This finding is congruent with the findings ofBakhtiar et al. (1997), Rahmati and Sharifi (1997), and Ahmadvand and Karami(2009) regarding crop yield production impacts of project in the Gareh‐Bygonplain.

0

5

10

15

20

25

1 2 3 4 5 6

Incr

ease

d re

liabi

lity

Year

Summary and Conclusions • A new methodology of reliability evaluation is developed for ecological

systems to evaluate possibility of change in the state of system during adrought event.

• To determine the randomness in failure a new formulation is developed.This formulation determines the probability of the next drought duration,given the amount of severity in the present state.

• The ability of this model is evaluated for a real case study of rangelandsystem. The result shows that the rangeland system reliability has beenincreased by 20% after constructing eight floodwater spreading systems.

• The proposed reliability index can be used as drought early warning indexthat shows system resistance to water stress in terms of duration andseverity.

• The new approach provides a comprehensive framework that can beapplied to reliability analysis of a variety of environmental issues involvingsoil and water resources systems degradation.

Bernoulli