824548

I t lfi-l 13 1 0 5 rlt U

--

-

ThG unedited intcncl note rccords the author B ViLJ for p~UpOSGB of

xcyi ~ v (ud CCtalllGllt This note d)06 net clIr r y the uthoX-ity of the Scicntific

bull

R HlJSTPN H J DAR~~--S

JUNpound 197amp ) I

bull

bull

bull

Ti2 noto d~8crib()13 n proposed modol lhich Hy be uGed to 1epl csent the

rc1ntionBhip bcten the G1odth of n fought fir Hnd tlw nnler of jeto hich

0 flpplied rho model can be used to predict the effect et applyine jcto at

()i~fcrent times and hence can be u6cd in the 6tlldy of thc nlOnber of Dpplanc~G

rcqlired at caell fire atation

ll8 model Io dermiddotivcd b~ conGidcrine 3impe E0thcmaticul form of the artgt

gimiddotth curve hich can be used to d~8cribe the observed bc)ioUl of 1(81

fi18( and which arc comptiblt Iith tn6 observed charnctel5tics of exp-imenO

firos Tllo different forms of 8 oyarhatic g1olth model al0 derived cnd Game

eS~iLlates of the coefficientn are given

~h- model desclibed her s deteri1l)n16tic cnd it lemaillG to investigate tha

propertics of DJl 1-quivalent stochastic model and to find SOn0 means of vcl idutirmiddots

01 displpving this proposed model bull

bull

A PROPOSM) MODEL FOH IIIE GH(J OF FUUCHT FIRE

Contentsshy

1 In troduction

2 Obsclvlltione from the DDto

4 Thooretical Hodol of Firo Grmmiddotth

6 EsUmates of the Parllmotcrs of the Conctullt ~ioratic Hodd

7 Alternativo Forms of the Qaadratic Il ooel

8 The Uso of the Quadratic Ilodcl

9 ]imitations of the Data

10 Alternative Methods of Estimating th~ ParCJTleters

11 Further Workmiddot

IHE GP-OH1Il OF A FOUGHT Fl[

1 II1rnOllUCrION

In 01 study of the numbJx of Plllf per GcCtion 0) a otudy of the filst attendance

rules it io eGsentiJl to have uu UlldcrGicndinG of the Hay in which fires amprolt

after the arival of the first pump ThLo Horldne noto iD a deoc6ptiol1 of a

proposed modol

Thin model is ~pplicle to buEdinC fir ~ c IIhich lulte Bprend beyond ch room

of ori-gin

I 2 OBSEllVATIONS FHOH THE DAfA

I I

AI1Y model which rcplaquosents the [oOltth of a fought filo must be copable of

exploining the foJloHb effects 11hich arc ob6cnuc in the 5AF2 data

i ~he middotaverace amount of Bpread for fir8 IIhich amp10 of tll~ snIDe nile Rt

arriVD1 middotincreDsell with th(l number of jets UGeurod This is due to the

differing fiercenccs of the fires An examination of the number of jets

UBed ugDinst the iinrl oren elso shown that spjcndinl3 fires use molS jeto

ii The proportion of fires which spread after Dlli val increases middotd th the

size of fire at 8rrival

iii But the average Mount of Gpread (for opreading and non-spreadJng

fires) is approxima tely constant for all sizES of fire

iv ii and Hi imply that for fires gthich spread tho average spread

(AfArgt ) io leGS for larGe fires

v Fires for which all the pumps errive srnuJutnncolsly are still likely

to spread nfter arrival

vi The conttol time (as recorcied on the K433 f orms) increuses ith

increaGing final urea

3 RFl3RESSION ANALYSIS

An onalJsis of the rclRtionship neblcen the ollnt f d d ~h t lth spree en le lm~s il fo

1

bull

bull

bull

Ti2 noto d~8crib()13 n proposed modol lhich Hy be uGed to 1epl csent the

rc1ntionBhip bcten the G1odth of n fought fir Hnd tlw nnler of jeto hich

0 flpplied rho model can be used to predict the effect et applyine jcto at

()i~fcrent times and hence can be u6cd in the 6tlldy of thc nlOnber of Dpplanc~G

rcqlired at caell fire atation

ll8 model Io dermiddotivcd b~ conGidcrine 3impe E0thcmaticul form of the artgt

gimiddotth curve hich can be used to d~8cribe the observed bc)ioUl of 1(81

fi18( and which arc comptiblt Iith tn6 observed charnctel5tics of exp-imenO

firos Tllo different forms of 8 oyarhatic g1olth model al0 derived cnd Game

eS~iLlates of the coefficientn are given

~h- model desclibed her s deteri1l)n16tic cnd it lemaillG to investigate tha

propertics of DJl 1-quivalent stochastic model and to find SOn0 means of vcl idutirmiddots

01 displpving this proposed model bull

bull

A PROPOSM) MODEL FOH IIIE GH(J OF FUUCHT FIRE

Contentsshy

1 In troduction

2 Obsclvlltione from the DDto

4 Thooretical Hodol of Firo Grmmiddotth

6 EsUmates of the Parllmotcrs of the Conctullt ~ioratic Hodd

7 Alternativo Forms of the Qaadratic Il ooel

8 The Uso of the Quadratic Ilodcl

9 ]imitations of the Data

10 Alternative Methods of Estimating th~ ParCJTleters

11 Further Workmiddot

IHE GP-OH1Il OF A FOUGHT Fl[

1 II1rnOllUCrION

In 01 study of the numbJx of Plllf per GcCtion 0) a otudy of the filst attendance

rules it io eGsentiJl to have uu UlldcrGicndinG of the Hay in which fires amprolt

after the arival of the first pump ThLo Horldne noto iD a deoc6ptiol1 of a

proposed modol

Thin model is ~pplicle to buEdinC fir ~ c IIhich lulte Bprend beyond ch room

of ori-gin

I 2 OBSEllVATIONS FHOH THE DAfA

I I

AI1Y model which rcplaquosents the [oOltth of a fought filo must be copable of

exploining the foJloHb effects 11hich arc ob6cnuc in the 5AF2 data

i ~he middotaverace amount of Bpread for fir8 IIhich amp10 of tll~ snIDe nile Rt

arriVD1 middotincreDsell with th(l number of jets UGeurod This is due to the

differing fiercenccs of the fires An examination of the number of jets

UBed ugDinst the iinrl oren elso shown that spjcndinl3 fires use molS jeto

ii The proportion of fires which spread after Dlli val increases middotd th the

size of fire at 8rrival

iii But the average Mount of Gpread (for opreading and non-spreadJng

fires) is approxima tely constant for all sizES of fire

iv ii and Hi imply that for fires gthich spread tho average spread

(AfArgt ) io leGS for larGe fires

v Fires for which all the pumps errive srnuJutnncolsly are still likely

to spread nfter arrival

vi The conttol time (as recorcied on the K433 f orms) increuses ith

increaGing final urea

3 RFl3RESSION ANALYSIS

An onalJsis of the rclRtionship neblcen the ollnt f d d ~h t lth spree en le lm~s il fo

1

Ti2 noto d~8crib()13 n proposed modol lhich Hy be uGed to 1epl csent the

rc1ntionBhip bcten the G1odth of n fought fir Hnd tlw nnler of jeto hich

0 flpplied rho model can be used to predict the effect et applyine jcto at

()i~fcrent times and hence can be u6cd in the 6tlldy of thc nlOnber of Dpplanc~G

rcqlired at caell fire atation

ll8 model Io dermiddotivcd b~ conGidcrine 3impe E0thcmaticul form of the artgt

gimiddotth curve hich can be used to d~8cribe the observed bc)ioUl of 1(81

fi18( and which arc comptiblt Iith tn6 observed charnctel5tics of exp-imenO

firos Tllo different forms of 8 oyarhatic g1olth model al0 derived cnd Game

eS~iLlates of the coefficientn are given

~h- model desclibed her s deteri1l)n16tic cnd it lemaillG to investigate tha

propertics of DJl 1-quivalent stochastic model and to find SOn0 means of vcl idutirmiddots

01 displpving this proposed model bull

bull

A PROPOSM) MODEL FOH IIIE GH(J OF FUUCHT FIRE

Contentsshy

1 In troduction

2 Obsclvlltione from the DDto

4 Thooretical Hodol of Firo Grmmiddotth

6 EsUmates of the Parllmotcrs of the Conctullt ~ioratic Hodd

7 Alternativo Forms of the Qaadratic Il ooel

8 The Uso of the Quadratic Ilodcl

9 ]imitations of the Data

10 Alternative Methods of Estimating th~ ParCJTleters

11 Further Workmiddot

IHE GP-OH1Il OF A FOUGHT Fl[

1 II1rnOllUCrION

In 01 study of the numbJx of Plllf per GcCtion 0) a otudy of the filst attendance

rules it io eGsentiJl to have uu UlldcrGicndinG of the Hay in which fires amprolt

after the arival of the first pump ThLo Horldne noto iD a deoc6ptiol1 of a

proposed modol

Thin model is ~pplicle to buEdinC fir ~ c IIhich lulte Bprend beyond ch room

of ori-gin

I 2 OBSEllVATIONS FHOH THE DAfA

I I

AI1Y model which rcplaquosents the [oOltth of a fought filo must be copable of

exploining the foJloHb effects 11hich arc ob6cnuc in the 5AF2 data

i ~he middotaverace amount of Bpread for fir8 IIhich amp10 of tll~ snIDe nile Rt

arriVD1 middotincreDsell with th(l number of jets UGeurod This is due to the

differing fiercenccs of the fires An examination of the number of jets

UBed ugDinst the iinrl oren elso shown that spjcndinl3 fires use molS jeto

ii The proportion of fires which spread after Dlli val increases middotd th the

size of fire at 8rrival

iii But the average Mount of Gpread (for opreading and non-spreadJng

fires) is approxima tely constant for all sizES of fire

iv ii and Hi imply that for fires gthich spread tho average spread

(AfArgt ) io leGS for larGe fires

v Fires for which all the pumps errive srnuJutnncolsly are still likely

to spread nfter arrival

vi The conttol time (as recorcied on the K433 f orms) increuses ith

increaGing final urea

3 RFl3RESSION ANALYSIS

An onalJsis of the rclRtionship neblcen the ollnt f d d ~h t lth spree en le lm~s il fo

1

A PROPOSM) MODEL FOH IIIE GH(J OF FUUCHT FIRE

Contentsshy

1 In troduction

2 Obsclvlltione from the DDto

4 Thooretical Hodol of Firo Grmmiddotth

6 EsUmates of the Parllmotcrs of the Conctullt ~ioratic Hodd

7 Alternativo Forms of the Qaadratic Il ooel

8 The Uso of the Quadratic Ilodcl

9 ]imitations of the Data

10 Alternative Methods of Estimating th~ ParCJTleters

11 Further Workmiddot

IHE GP-OH1Il OF A FOUGHT Fl[

1 II1rnOllUCrION

In 01 study of the numbJx of Plllf per GcCtion 0) a otudy of the filst attendance

rules it io eGsentiJl to have uu UlldcrGicndinG of the Hay in which fires amprolt

after the arival of the first pump ThLo Horldne noto iD a deoc6ptiol1 of a

proposed modol

Thin model is ~pplicle to buEdinC fir ~ c IIhich lulte Bprend beyond ch room

of ori-gin

I 2 OBSEllVATIONS FHOH THE DAfA

I I

AI1Y model which rcplaquosents the [oOltth of a fought filo must be copable of

exploining the foJloHb effects 11hich arc ob6cnuc in the 5AF2 data

i ~he middotaverace amount of Bpread for fir8 IIhich amp10 of tll~ snIDe nile Rt

arriVD1 middotincreDsell with th(l number of jets UGeurod This is due to the

differing fiercenccs of the fires An examination of the number of jets

UBed ugDinst the iinrl oren elso shown that spjcndinl3 fires use molS jeto

ii The proportion of fires which spread after Dlli val increases middotd th the

size of fire at 8rrival

iii But the average Mount of Gpread (for opreading and non-spreadJng

fires) is approxima tely constant for all sizES of fire

iv ii and Hi imply that for fires gthich spread tho average spread

(AfArgt ) io leGS for larGe fires

v Fires for which all the pumps errive srnuJutnncolsly are still likely

to spread nfter arrival

vi The conttol time (as recorcied on the K433 f orms) increuses ith

increaGing final urea

3 RFl3RESSION ANALYSIS

An onalJsis of the rclRtionship neblcen the ollnt f d d ~h t lth spree en le lm~s il fo

1

IHE GP-OH1Il OF A FOUGHT Fl[

1 II1rnOllUCrION

In 01 study of the numbJx of Plllf per GcCtion 0) a otudy of the filst attendance

rules it io eGsentiJl to have uu UlldcrGicndinG of the Hay in which fires amprolt

after the arival of the first pump ThLo Horldne noto iD a deoc6ptiol1 of a

proposed modol

Thin model is ~pplicle to buEdinC fir ~ c IIhich lulte Bprend beyond ch room

of ori-gin

I 2 OBSEllVATIONS FHOH THE DAfA

I I

AI1Y model which rcplaquosents the [oOltth of a fought filo must be copable of

exploining the foJloHb effects 11hich arc ob6cnuc in the 5AF2 data

i ~he middotaverace amount of Bpread for fir8 IIhich amp10 of tll~ snIDe nile Rt

arriVD1 middotincreDsell with th(l number of jets UGeurod This is due to the

differing fiercenccs of the fires An examination of the number of jets

UBed ugDinst the iinrl oren elso shown that spjcndinl3 fires use molS jeto

ii The proportion of fires which spread after Dlli val increases middotd th the

size of fire at 8rrival

iii But the average Mount of Gpread (for opreading and non-spreadJng

fires) is approxima tely constant for all sizES of fire

iv ii and Hi imply that for fires gthich spread tho average spread

(AfArgt ) io leGS for larGe fires

v Fires for which all the pumps errive srnuJutnncolsly are still likely

to spread nfter arrival

vi The conttol time (as recorcied on the K433 f orms) increuses ith

increaGing final urea

3 RFl3RESSION ANALYSIS

An onalJsis of the rclRtionship neblcen the ollnt f d d ~h t lth spree en le lm~s il fo

1

~h the jotc Hero vppllcd HOulcl PlOVC the oimplcst )KmiddotCi blo model of

or the follming eGGumptions

(j) Th rate of cromiddotth of firec is cxponcntiel

(ii) Each pump oupplicd 001) jet

(iii) Thc arrival timeG of the pumps are the times a c hich hc jete

f le

rlI model UGccl to reprctlcnt the grolth of a j jet fire as A ~ Ao e

rot~ jtpound1 -0 V j- ~OOCl~ e J

Hh0n Ap- is the oregtt dllmagccl Dtext in ction

Ao is the alea dam~ecl at arril8l

f is the exp(luentiRl growth rata

ti~1 le the i-1 th interarrival time ie the interval beteen the arrival

of tho itll and (i_1)th pUOlp k e represents finy croHth not ficcounted for by intc)$rrivlll groth

This regresGion analysiB RS tried on multi-storey non-chJellil1g firen l-Ihicb hUo

spraad bGyond the room of ori(in

1 jGt Ures Af 111f Hean Ar- 111

(n =598) AQ Ao

2 jet Ures Ap ooltt1 Hean Af ~ 116 116(n 366) A 0 Aa

All the R v81u(~ Hera very low and the regression coampfficiGnts woro -hot sienifcn

3 jet firos (n 11+6)

4 jet fires (n 95)

5 jot fires (n 42)

or only sigliificaut at the 10 level Host of the spread is cccounted fo by

dclfy in lntcrnrrival timer l1j)U providEl8 on order o ~l lnW t itn rj- ecirlto of

dOOG not provide u reasonable rD oil3 for D mod(Il of fh0 ijr-o-rcli(o

4 A THlWUhTICAL HCDFJ OF FIRE GliOI-lTH

If the firc size is roduced by applying Iter to tll p imetcr of the Hre

the tillle tden to extinguish the fire gtlDl be P)~OPGlltmiddot io middot i to At hel A is

the size of the fir( (Se Bsldlir 11olinok ThoCilflO Fire Re~car(h Note No 8811)

ell - at ob (42)dt

A _ AD - ~ 2 + bt (43) 2

The only inforruation El have about a fire iB tho urea at alrival (Ao) the

arell damcged (A) and the extinction tile (t ) The information defiueD twoE

points nnd a tangent on the graph of fire sizI] against time The quadratic

equation 11 42 43 is the simplest flUlction which ill fit a curve defiMd

by threo points

t -gt

EqlUltion 42 6h0l8 that the maximum a)ea i8 reached at time t ba a[tol

arrival This maxirutlIa fire size ill b thlt ni t damDged at extinction

3

lUU tbo ebsolut u DOlOunt ()f

rhe extinctioi1 ti(o js the tilT) c1 lhich 0

ba

a detGrmineB hOil qui0kly the fie si7o iD roduced 8 wUl c~epend on ~middothc

char[cteristic[J of the firo (ie Bnd filtrceu~Gs ) and lIil nlso reflec t the

nIgtJnba of j ~ tB uGed to figt 1ho firc As more j~t5 are apoJied the fire GheI

Hill be rGduccd or e quickly and the pa1rgtJueter a llill iucrcace

The parameter b represents the colltinucd grogtlth of the firo UFo it it being

fouGht If tho fire is not being fought a must hamp zero 81ulto Md the fir

groVlth will be dAdt b 0lthotlSh b ItlPy not be constant in gtIhich CO6 b is a tanGent to the grogtth curve at tho point A h2dAdt

Considet first tile 5impleot GittoCion ~hcre 1J1l thu pUlps middothich ay bo needed

arc available at tho time of the fhst alltrivnl and [lS5ume the parameters lgt and b

are constantt

The firEls HHl be charactcrised by their inItial 5izG Ao and their fierceness

The brigade in apply the appropiintc number of jets accolding to their

lisspoundsment of the fire size Dnd severity Fires of tho Sml0 initial size

Ao Hill folloll different groHth curvet according to their severity

As the severity increascs the ~ount of spread and control tiGe also incr~eso

lUU tbo ebsolut u DOlOunt ()f

rhe extinctioi1 ti(o js the tilT) c1 lhich 0

ba

a detGrmineB hOil qui0kly the fie si7o iD roduced 8 wUl c~epend on ~middothc

char[cteristic[J of the firo (ie Bnd filtrceu~Gs ) and lIil nlso reflec t the

nIgtJnba of j ~ tB uGed to figt 1ho firc As more j~t5 are apoJied the fire GheI

Hill be rGduccd or e quickly and the pa1rgtJueter a llill iucrcace

The parameter b represents the colltinucd grogtlth of the firo UFo it it being

fouGht If tho fire is not being fought a must hamp zero 81ulto Md the fir

groVlth will be dAdt b 0lthotlSh b ItlPy not be constant in gtIhich CO6 b is a tanGent to the grogtth curve at tho point A h2dAdt

Considet first tile 5impleot GittoCion ~hcre 1J1l thu pUlps middothich ay bo needed

arc available at tho time of the fhst alltrivnl and [lS5ume the parameters lgt and b

are constantt

The firEls HHl be charactcrised by their inItial 5izG Ao and their fierceness

The brigade in apply the appropiintc number of jets accolding to their

lisspoundsment of the fire size Dnd severity Fires of tho Sml0 initial size

Ao Hill folloll different groHth curvet according to their severity

As the severity increascs the ~ount of spread and control tiGe also incr~eso

Yo- a ghcn Ao [0 tmiddotJle fJc1rc6 ty incrcaceo the paXarncteZmiddot Cl will accrclt)cc

MCor the parumetcL b middotill incrcace

It 5s u66umed thampt all th~ dHfcrcncf)s n the cromiddot th of th6 fhcc ll 0 due

to the differences bchloen fires He a60UC1( thnt the brlgad Dpp1y tIle right

number of jctll and do net allovl for a fercc effoct by 6uGcel3ting that th

fhmiddotc may be put out more guielly by usinf more jet

5 A middotLfHOD OF ESTIHATING TIlE PAllAJmiddot~EW~S 1011 AN AVERAGE FIRE

~ ue still asouraLng that the puametels a und b a rc COllGtents and conBiceriq

onJy those fireG for Hhieh all the pumpG 10)C availo1gtle Ilimul tallcouGly If

take o eroup of fil06 of the BMa initial Gize Ao fought by the GBruG num0T

of jets He can observe avcraee values for the sprpad and control timo ftom

thr SAF2 data If Wc flUther aSGume tbat thil group of fires IJaG the same

(or at 16ilGt simila) Il and b 1uGO then Ie can use thu hlo obs epved avcr6r~B

to eShD1ilto the two porameters

2-

A rrrangcmcllt of the cqttation6 AI - Ao b 2a nnd tE ba +Ga giVeuro6

a ~ z[fiL~JAF J2

b J 2ak

2where l~ il3 the absolute amow1t of spreild) b 2a

If e can make soma assumption about the ay in vhich u changes RG edditioncl

jets are brought into we then can generalise the growth equation to include

thoGe fircl3 in lIhich there wae a delay in the arrival of the second and subsequent c-t-b

jets In a study of wood ~ firos (0 Donghcrty and Young Fire Research Note e

No 603) it HUS fOWld thut the maxiGm] amount of fire spreld IWS propottional lo

) -121 ( Bate of Hater Application bull If this result is used it Iltould irnplJ th~t

5

being fouht by J jetl3 In tlliii proUminnry anlys1gtj huv rcatlOned

that in reDl fireo th additional jets ill havo le e cUed thn they oula

huVG in experimental firco and hovc used the cinlplcr re1laquo t iomiddotl ip aJ Jl1

If D fhe ia etinguishcd by J jets hich arrivG Ht 11 t 2 bullbull t _ afterJ 1

middotthe firot j0t tho e10llth curve lIill be of th9 fOlon chmn ueoo

JI i~--- ~I I

I

I

Thlt3 equation of the groVlth in the filml soction 111 be

A Aa + 0 I (b - 1t) dt +jt2 (b - D t) dt + bullbullbullbull2f t1

A bull Ao -l- bt - e ) +1

If baJ gt t J~1 the maxirJuu Hill occur in the final section ond the area

damaged nt extinction F will be

2 (t( - Cl ) AI = Ao b +~ ll2a i 1J

Tlw extinction time te i s obtainod from (5 bull3) b) solving fOl t Ilhen

A =o J-1

o = Aa - bt + C (0 - u )E 4+1 li=1

If we use the reI tioaship 1I = it a then [1 nrd b can be estilLO ted trolni 1

the cilllultlillcoUS oquntions

[

6

I

being fouht by J jetl3 In tlliii proUminnry anlys1gtj huv rcatlOned

that in reDl fireo th additional jets ill havo le e cUed thn they oula

huVG in experimental firco and hovc used the cinlplcr re1laquo t iomiddotl ip aJ Jl1

If D fhe ia etinguishcd by J jets hich arrivG Ht 11 t 2 bullbull t _ afterJ 1

middotthe firot j0t tho e10llth curve lIill be of th9 fOlon chmn ueoo

JI i~--- ~I I

I

I

Thlt3 equation of the groVlth in the filml soction 111 be

A Aa + 0 I (b - 1t) dt +jt2 (b - D t) dt + bullbullbullbull2f t1

A bull Ao -l- bt - e ) +1

If baJ gt t J~1 the maxirJuu Hill occur in the final section ond the area

damaged nt extinction F will be

2 (t( - Cl ) AI = Ao b +~ ll2a i 1J

Tlw extinction time te i s obtainod from (5 bull3) b) solving fOl t Ilhen

A =o J-1

o = Aa - bt + C (0 - u )E 4+1 li=1

If we use the reI tioaship 1I = it a then [1 nrd b can be estilLO ted trolni 1

the cilllultlillcoUS oquntions

[

6

I

room of oriGin gtCl t3ed to test the es timatioll methods The fires Here gnngtpcu

according to the size at Glrivols the nur~bor of jets used ono the intenlrriv~l

timeD if more than o)a jet an -Gcti the inte-ulrival timeD ~Jcre calculated

by wl3UJlline that ith jet D9 applied llt the time the i th pump llldvcd For each

g~OlP of firoB p-n averagc (xtil~tion time and avcage GpreHd has c81c tloted and

the flVe~agc8 used tc caticoato v and b CG de6cribed in section 50 Tbe results1

are GhoHn id toble 1 I

oIithin tlie limitations of the lteta (sec ocction 8) Tho leonlt6 OhOlI

(i) Jar n giell Aa pound1 derrcnses and b incNases slightly with inereaning1

munbcgt of 1eta

(ii) for the two jet fires of n given Aa the values of a and bare

npployir1ate ly constant for IllrioUB valueo of t 1

0

(ii) a and b both increall6 I th incre~Bine Aa1

Figures 1 and 2 sholtl the l~elatioBhip bet-een deg 1

b and Aa The Glopes of

the log alog Aa and log blog AD elationships are about the 6Bllle and both

a 8m b are proportional to Amiddot 75 1

7 ALTERNATIVE FORlS OF THE QUADRATIC middotODEL

Hodel lsing P[oportional P=Dmet~2

The parameters b nnd a represent the absoluto rates of increaso nnd decreasD

of the fire It Ifotlld seem reanonllble to suggest that a and b might be

functions of the firo siz~ If a and b are directly proportional to the fire

size then the nodel beco01eo

bull2log A ~ log Aa - o( + pt (72) 2

In the more general ccse where there are delays bobcen the arrival of Gubneqtwnt

jets the sproad will be

7

bull

~shy

= 2r

rtud the control timo Clt1ll be estimated fl~om the cquntion J -I ~ 2 2o = log A + j3tE + Ii~n t (C(i+1 - 0( ) - tr1 oC (7~)i=1 12 J

2

11 this model the parameter 3 ia the exponential eruVJh fnetol The cctimatcs

of~J 1 and p [10 derive d ill on equivalent Hoy to that oec clibcd in section 6

Tho esti0l9to arc shOlm ill table 2

lhc ootimntcs oJlaquo 1 and ~ 010 quito doble with )7(Spcct to chanGinG Ao

decrea60s Ii th incI(ocing ritJllo C- of jets but there iD nstlfficiont eidc~c

to decide whether the grmlth a te f3 vories

~ehie model jmplies that the relationGhip between spread and arrival timo is

of the form

+ pound( 0( - c( ) t 2 i+1 t ~

2

This lelation8hip call be tested by applying this form of the recsroG6ion model

to the SAr2 dc-tu In the limited llUJober of regressions Imiddothich VleC t imed it

US found that this relationship gp-ve a bet t er fit or at least no OlSe a fit

than the intolarrivul time r1IodGl fbst tried ( section 3)

An Intermedi a te Hod2l

The relationship bob-lcen the constant palametOls )1 band Ao suggest a lliQdel

in which a and b aro propoltional to 75

1dA - 8 dt

A more genelaquol model and DJ] estimation procedure can be del iYei in a similar

fOlill to the o t her two models considered

8 (

~shy

= 2r

rtud the control timo Clt1ll be estimated fl~om the cquntion J -I ~ 2 2o = log A + j3tE + Ii~n t (C(i+1 - 0( ) - tr1 oC (7~)i=1 12 J

2

11 this model the parameter 3 ia the exponential eruVJh fnetol The cctimatcs

of~J 1 and p [10 derive d ill on equivalent Hoy to that oec clibcd in section 6

Tho esti0l9to arc shOlm ill table 2

lhc ootimntcs oJlaquo 1 and ~ 010 quito doble with )7(Spcct to chanGinG Ao

decrea60s Ii th incI(ocing ritJllo C- of jets but there iD nstlfficiont eidc~c

to decide whether the grmlth a te f3 vories

~ehie model jmplies that the relationGhip between spread and arrival timo is

of the form

+ pound( 0( - c( ) t 2 i+1 t ~

2

This lelation8hip call be tested by applying this form of the recsroG6ion model

to the SAr2 dc-tu In the limited llUJober of regressions Imiddothich VleC t imed it

US found that this relationship gp-ve a bet t er fit or at least no OlSe a fit

than the intolarrivul time r1IodGl fbst tried ( section 3)

An Intermedi a te Hod2l

The relationship bob-lcen the constant palametOls )1 band Ao suggest a lliQdel

in which a and b aro propoltional to 75

1dA - 8 dt

A more genelaquol model and DJ] estimation procedure can be del iYei in a similar

fOlill to the o t her two models considered

8 (

8 THE sE OF QUADRtrIC 1lonEL

of the brieade dc-ponds on the charucteribticG of the Ere nuel a130 all the

action tken by the brilgtde A modd Ihich iu to be uB~d to In ~dic~_ tho

fire ard the brirgt2pound fhe (lqu~tions discussed in thiG note represent the

fire grovth for a given l3equence of arrivlll times t2 tJ Tile - Bequenco of arrival times will depend on the nlUnber of jotH Ihich arc applied

initially (from the first attendance or the aS6ist~nce Hl1ich ie roquested) and

0160 on the suboquent decisions to brlrlg on morc jets as tilt fire g1ol-l8

The relationship betlleen the area dRr~aged at oxtillction ilnd tIle totol number

of jets used has Blready been stndied Tbe relatiollship Ciltn bc explensed u

the form shmm beloH

C~~lo -K

p 0 ro

~ v~~ ~_r b 4 1egt

lt I rG T ~

o

A fire of a given siZQ can be charactelised by ito lelat~e sever-Ht - ie

a number between Omiddot and 100 which dctermilles its p0sHion on the cumulativ~

scale of utLnbors of jets required

s~~(-The followiug genenl principle could be used to FuiiCgt the growth of an

individual fire which is of size Ao at arrival

i Generate a random nUJlber to lepre6cnt the relative 6everity of the f51Clt

9

H Apply tho Il ~r of iClgt nc terrnincu by tho pocitiOh of middottile fire

on the jet aroD gl ph Tu () j e ts QuId be provideu by the firc t

attenuance or if necessary fJI3Glstanc() would ba roqvcf ted 1113 GOOa llll

the firt pump arrives

Hi ))termine ~ll8 a and h pawmoters of the fire hich dll u[o in be

a func tion of a)e aml rcl~t) v e severHy

iv PlGoict the (loHth of the fire and check wheth cl the f i re lI its

mrolmvm sie anu Hith the ~me soverity index would requiro Rdditiol19l

jets I v Reeucst additional jets if roquired and continue ~xtrapoliJtlng the

I grol-lth of the firo Hi th th~ chaneing a value

1

90 LDlITNIWNS OF TilE DATA

the pumps ani we hllt e lkd to ass ume Goroe sirJpl rule about the number cmiddotf jetbull (cAt- L tCr

provided by ~ach pump For any illliividurmiddotl fi) e He do not know hOgtl many jets

came fIOIl each pump or eveu hether the jets Cle appljed immedia tely the

pump arrived

he Extinction Time He have taken the extinction time to be the control

time as recorded on the Klf33 fOl-inS The control time is the tirJe bcteen tho

arrival of the fhst pWllP and the stop lIlessoge

For the smaller firos the time of the stop message 11 approxiroate to the

time the firo is out but for the larger fires th) stop message may be

wnt Holl before the fire is Ol1t

The AmOlU1t et S)re~ _ The runOWlt of spread is taken from the area at alriva)

and final area damaee RS recorded on the SAF2 forms The area at arrival Hill

usuRlly not be knoa with any accuracy and is ofteu recorded as beillg the GarJe

as the aleo at extinction The aroas are recod~d to two significnnt fieureG

10

an~ thir roulldjlC error introducej cltdd i jonal illHCCUlDCY

~Y~~i~t~ Ill uictributions of cplefld Incl control tillle ar hiGhly

Gk~cd Dncl if one V0lY llu(o value iG incluurccl in - OBlTIplo thio can conpletcly

outlmiddoteiCh 011 the other obnervationso

le ALTBIIHArIVE tlETlIODS OF ESTII1ApoundlIW rifE PJH111jETERS

1 EstiMlto the pllr ~ tcrs for- o11ch homOeCne01J8 eronp of flres g~ch C)OP

C011Biflt5 of fins IIhich hLwa the some ar~e at In )ival nutrbOl of jota and

irtcrarrival time6 Obtain an array of param2telS for eclt jet-area group

by interp012tior~ and smoothing of the eroup cnt)~ tes

2 Estim~te the parameter velues for each individual fire possibly iatroducir

II -ando01 clement desi~n~cl to put back the ronnciing erroro Dotermine the

parIJlJletels foX (middot8ch jet-area group by taking the aveNGe of the indivmiddot i(~ual vaJucc

in the group ThB re(son for puttilllS back the rounding error is to get a

continuouo distlibution of spread Rnd hence a continuous dictribution of estiulDtes

of a ond b

3 Obtain a r~gre6sion r~latiomgthip bebeen sYead and interarrival times anu

extinction tirue and interarXiva1 times Use the legression to estiOlatlt the spred

and extinction time for fires of a givltm area and number of jets Estimate the

a b parameters for this group using the) regrcssion estiMates of Gprcad and

control time

4 Use the regression ro1a tionLihip bettecn splcad contlol time number

of jets and area at arrival to estimate a b directly by considering the

correspondence between the coefficieJlts of the rlresGion model and the quadlat)c

modal

11 ~Eng2~iORK

The RelaJj2poundEJip betHeen ~ini5tic and ~chastic Bode

tIe have l[sumed that the pa rDJIleters a Iind b arc constant for each grol1p of

fires Ihelcas in Ienlity there Hill be a distribution of a and b values lhe

11

c_ b ~p viour of 90 fire Hith avor u b valueo mDy not bo the l3ame aB the

oTilEC of u11 fheEl dth individual 0 b Vaht0130 Tho properbol3 of the

GtnGhostic mOc1 Gl need to bE p-yllmincu anu in partiGuler it moy bo Horthhilu

consiuering a mpdcl in Hhich GOlDe fircn hccve zero b VllUClO (no growth) Dnd

tho remo inlng fheB luwe pos itivo b lues bull

Yilidcc ti~2 It should be possible to validate (or to diGprove) tllio Pl~opce J

mcdol by a more thorough study of the lorr6sion rclatioTJl3hips obtainable

flDm the SAF2 data

reotrictionc on groNth In practico lDany fires will bl) confined by the buHoillg

or ut lcal3t some fire compartment Hith i n the buildjng The effect of builuine

sie must be Qxamined

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