DriDanube – Drought Risk in the Danube Region Project co-funded by European Union funds (ERDF, IPA)

Mathematical methodology for meteorological drought risk assessment

Tamás Szentimrey (1), Zita Bihari (2), Sándor Szalai (3),

Mónika Lakatos (2), Andrea Kircsi (2) and Beatrix Izsák (2)

(1) VARIMAX Limited Partnership,

(2) Hungarian Meteorological Service, (3) Szent István University

DriDanube - Drought Risk in the Danube Region

One of the aim of the DriDanube project is the drought risk assessment and the

main responsible partner of this activity is the Hungarian Meteorological Service.

However, unfortunately there is no commonly accepted procedure for this

purpose therefore it was necessary to review and prepare the mathematical

methodology that can be applied for drought risk assessment in this project.

VARIMAX is a subcontractor for the mathematical and software development.

The recommended approach was as it is described in European Commission’s

Risk Assessment and Mapping Guidelines for Disaster Management.

DriDanube - Drought Risk in the Danube Region

EUROPEAN COMMISSION

Risk Assessment and Mapping Guidelines for Disaster Management

5. RISK ASSESSMENT METHODS

5.1. Conceptual Framework and Basic Methodology

According to ISO 31010, risks are the combination of the consequences

of an event or hazard and the associated likelihood of its occurrence.

Risk = hazard impact * probability of occurrence

Qualitative Formulation only!

DriDanube - Drought Risk in the Danube Region

General Mathematical Methodology

(for meteorological drought risk assessment)

Meteorological variables in space (s) and time (t year):

T1 ),(),....,,(),( tXtXt N sssX Ss (e.g. grid) , nt ,..,1

i, Continuous distribution, joint density function: sx;f

ii, Discrete distribution: sip it asX ,P , ,....2,1i

DriDanube - Drought Risk in the Danube Region

Meteorological drought loss (hazard impact): Loss , ts

Meteorological drought loss (hazard impact) function:

E Loss , , ,t t L ts X s X s

tL ,sX that is the conditional expectation (regression) of the

meteorological loss given the meteorological variables.

Meteorological drought risk (depends on location):

E Loss , E ,Risk t L t s s X s

i.e. the expected value of the meteorological loss function.

DriDanube - Drought Risk in the Danube Region

The basic cases

i, Continuous distribution: E , ( ; )NR

Risk L t L f d s X s x x s x

ii, Discrete dstribution: 1

E , ( )i i

i

Risk L t L p

s X s a s

Special case:

If 1aL =hazard impact, )(1 sp =probability of occurence

02 aL , )(1)( 12 ss pp , then

2

1

( )i i

i

Risk L p

s a s =hazard impact*probability of occurence

DriDanube - Drought Risk in the Danube Region

Estimation of meteorological drought risk

1

1ˆ ˆ ,n

t

Risk L tn

s X s

where tL ,ˆ sX is an estimation of loss function tL ,sX ,

since E ,Risk L ts X s

BUT: In general there is no sample for the drought loss values!

DriDanube - Drought Risk in the Danube Region

Loss function derivation from yield function and drought identification

Crop yield values: Y , ts = Yield , ts

Meteorological yield function:

E Yield , , ,t t Y ts X s X s

that is the conditional expectation (regression) of the crop yield

given the meteorological variables.

Drought identification by set D : there is drought if Dt ,sX

The probability of drought: DtPD ,P sX

DriDanube - Drought Risk in the Danube Region

Loss function based on yield function and drought identification

DttL ,if0, sXsX (there is no drought)

tL ,sX tYttY D ,,,E sXsXsX Dt ,if sX (there is drought)

DttY ,,E sXsX : conditional expectation of yield if there is no drought

Then risk can be expressed as,

ERisk L X DD YYPD XXXX EE

DriDanube - Drought Risk in the Danube Region

Special case: definition of set D by an SPI

}{ PCSPID X where DPP PCΦCSPI )(P

and )(xΦ is the standard normal distribution function.

In this project SPI was used for drought identification.

However other optional drought index or drought definition can be used!

DriDanube - Drought Risk in the Danube Region

Estimation of loss function based on estimated yield function and SPI

Let tY ,ˆ sX nt ,..,1 be an estimation of the yield function based on a sample.

Estimation of loss function using estimated yield function and SPI:

PCtSPItL ,if0,ˆ ssX (there is no drought)

tL ,ˆ sX

tYtYnn

PCtSPID

,ˆ,ˆ1

,

sXsXs

PCtSPI ,if s (there is drought)

PCtSPI

Dn,

1s

: frequency of drought

DriDanube - Drought Risk in the Danube Region

Estimation of drought risk based on estimated yield function and SPI

1

1ˆ ˆ ,n

t

Risk L tn

s X s

PP CtSPIDCtSPID

D tYn

tYnn

P,,

,ˆ1,ˆ1ˆ

ss

sXsX

where n

nP D

D ˆ is the estimated probability of drought.

It is an aesthetic formula, thus it must be good!

DriDanube - Drought Risk in the Danube Region

Estimation of the relative yield function that was used

Relative yield in percent:

Y , E Y ,Y , 100%

E Y ,rel

t tt

t

s ss

s

Using the equality E Y , 0rel t s the following regression model was applied,

1 1 ln,1 2 2 2ˆ , ln , ,relY t X t E X t E X s s s s s

where 1 ,X ts is precipitation sum and 2 ,X ts is temperature mean for a given 3

or 6 months period, and ln,1( )E s , 2 ( )E s are the spatial expected values.

The common coefficients 1 2, can be estimated by the method of least squares.

At the risk estimation the 1 ,SPI X ts can be applied. Then we can evaluate the

connection of the different period SPI series with the real drought loss.

DriDanube – Drought Risk in the Danube Region Project co-funded by European Union funds (ERDF, IPA)

Thank you for your attention