Scaling Theory of Floods for Developing a Physical

Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s

Oxfor d Resear ch Encyclopedia of N at ur al H azar d Science Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Vijay Gupt a Subject : Risk Assessm ent , Floods Online Publicat ion Dat e: Sep 2017 DOI : 10.1093/acr efor e/9780199389407.013.301

Su m m ar y an d K eyw o r d s Pr edict ion of floods at locat ions w her e no st r eam flow dat a exist is a global issue because m ost of t he count r ies involved don’t have adequat e st r eam flow r ecor ds. The U nit ed St at es Geological Sur vey developed t he r egional flood fr equency (RFF) analysis t o pr edict annual peak flow quant iles, for exam ple, t he 100-year flood, in ungauged basins. RFF equat ions ar e pur e st at ist ical char act er izat ions t hat use hist or ical st r eam flow r ecor ds and t he concept of “ hom ogeneous r egions.” To supplem ent t he accur acy of flood quant ile est im at es due t o lim it ed r ecor d lengt hs, a physical solut ion is r equir ed. I t is fur t her r einfor ced by t he need t o pr edict pot ent ial im pact s of a changing hydr o-clim at e syst em on flood fr equencies. A nonlinear geophysical t heor y of floods, or a scaling t heor y for shor t , focused on r iver basins and abandoned t he “ hom ogeneous r egions” concept in or der t o incor por at e flood pr oducing physical pr ocesses. Self-sim ilar it y in channel net w or k s plays a foundat ional r ole in under st anding t he obser ved scaling, or pow er law r elat ions, bet w een peak flow s and dr ainage ar eas. Scaling t heor y of floods offer s a unified fr am ew or k t o pr edict floods in r ainfall-r unoff (RF-RO) event s and in annual peak flow quant iles in ungauged basins.

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PRI N TED FROM t he OXFORD RESEARCH EN CYCL OPEDI A, N ATU RAL H AZARD SCI EN CE (nat ur alhazar dscience.oxfor dr e.com ). (c) Oxfor d U niver sit y Pr ess U SA, 2016. All Right s Reser ved. Per sonal use only;

Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Theor et ical r esear ch in t he cour se of t im e clar ified sever al k ey ideas: (1) t o under st and scaling in annual peak flow quant iles in t er m s of physical pr ocesses, it w as necessar y t o consider scaling in individual RF-RO event s; (2) a unique par t it ioning of a dr ainage basin int o hillslopes and channel link s is necessar y; (3) a cont inuit y equat ion in t er m s of link st or age and dischar ge w as developed for a link-hillslope pair (t o com plet e t he m at hem at ical specificat ion, anot her equat ion for a channel link involving st or age and dischar ge can be w r it t en t hat gives t he cont inuit y equat ion in t er m s of dischar ge); (4) t he self-sim ilar it y in channel net w or k s plays a pivot al r ole in solving t he cont inuit y equat ion, w hich pr oduces scaling in peak flow s as dr ainage ar ea goes t o infinit y (scaling is an em er gent pr oper t y t hat w as show n t o hold for an idealized case st udy); (5) a t heor y of hydr aulic-geom et r y in channel net w or k s is sum m ar ized; and (6) highlight s of a t heor y of biological diver sit y in r ipar ian veget at ion along a net w or k ar e given. The fir st obser vat ional st udy in t he Goodw in Cr eek Exper im ent al Wat er shed, M ississippi, discover ed t hat t he scaling slopes and int er cept s var y fr om one RF-RO event t o t he next . Subsequent ly, diagnost ic st udies of t his var iabilit y show ed t hat it is a r eflect ion of var iabilit y in t he flood-pr oducing m echanism s. I t has led t o developing a m odel t hat link s t he scaling in RF-RO event s w it h t he annual peak flow quant iles feat ur ed her e. Rainfall-r unoff m odels in engineer ing pr act ice use a var iet y of t echniques t o calibr at e t heir par am et er s using obser ved st r eam flow hydr ogr aphs. I n ungagged basins, st r eam flow dat a ar e not available, and in a changing clim at e, t he r eliabilit y of hist or ic dat a becom es quest ionable, so calibr at ion of par am et er s is not a viable opt ion. Recent pr ogr ess on developing a suit able t heor et ical fr am ew or k t o t est RF-RO m odel par am et er izat ions w it hout calibr at ion is br iefly r eview ed. Cont r ibut ions t o gener alizing t he scaling t heor y of floods t o m edium and lar ge r iver basins spanning differ ent clim at es ar e r eview ed. Tw o st udies t hat have focused on under st anding floods at t he scale of t he ent ir e planet Ear t h ar e cit ed. Finally, t w o case st udies on t he innovat ive applicat ions of t he scaling fr am ew or k t o pr act ical hydr ologic engineer ing pr oblem s ar e highlight ed. They include r eal-t im e flood for ecast ing and t he effect of spat ially dist r ibut ed sm all dam s in a r iver net w or k on r ealt im e flood for ecast ing. Keyw or ds: pow er law s, m ass conser vat ion quest ion in a net w or k , self-sim ilar it y of net w or k s, hillslope-link syst em s

I n t r o d u c t i o n t o t h e Sc al i n g T h eo r y o f Fl o o d s Flood is a geophysical phenom enon. Floods ar e also a nat ur al hazar d because t hey can r esult in m ajor loss of life and subst ant ial dam age t o pr oper t ies and civil infr ast r uct ur e. Accur at e est im at es of t he m agnit ude and fr equency of flood flow s ar e needed for t he design of br idges and r oads, w at er -use and w at er -cont r ol pr oject s, and for floodplain Page 2 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s definit ion and m anagem ent (Daw dy, Gr iffis, & Gupt a, 2012 ). The U nit ed St at es Geological Sur vey (U SGS) developed t he r egional flood fr equency (RFF) equat ions t o est im at e quant iles (e.g., t he 100-year flood) for annual peak st r eam dischar ges, or floods, in ungauged basins, w her e no st r eam flow dat a exist . RFF equat ions ar e pur e st at ist ical char act er izat ions t hat use hist or ical st r eam flow r ecor ds t o car r y out r egr essions. They do not include t he physical pr ocesses t hat pr oduce floods in r ainfall-r unoff (RF-RO) event s (Fur ey, Tr out m an, Gupt a, & Kr ajew sk i, 2016 ). A geophysical under st anding of annual flood quant iles is a long-st anding unsolved pr oblem . The scaling t heor y of floods explained her e developed a physical fr am ew or k t o m ak e flood pr edict ions in ungauged basins. To supplem ent t he accur acy of flood quant ile est im at es due t o lim it ed r ecor d lengt hs— t ypically less t han 30 year s—a physical solut ion t o t he est im at ion of flood quant iles at annual t im e scale is r equir ed. The RFF analysis assum es a st at ionar y hydr o-clim at e, w hich m eans t hat t he fut ur e w ould be st at ist ically sim ilar t o t he past . Gr eenhouse gases and ot her fir st -or der hum an influences ar e changing t he hydr o-clim at e syst em (Pielk e et al., 2009 ). Consequent ly, fut ur e flood fr equencies ar e not expect ed t o be st at ist ically sim ilar t o t he past (Daw dy, 2007 ). A solut ion t o t his pr oblem r equir es t hat annual flood quant iles be under st ood in t er m s of physical m echanism s pr oducing floods. M andelbr ot ’s book ( 1982 ) on t he fr act al geom et r y of nat ur e offer ed self-sim ilar it y as a new scient ific par adigm . Scaling ar ises in self-sim ilar syst em s. A lar ge num ber of st udies had obser ved scaling, or pow er law s, in annual flood quant iles ver sus dr ainage ar eas in r iver basins (L insley, Kohler, & Paulhus, 1982 ). Gupt a and Daw dy ( 1995 ) syst em at ically invest igat ed t he pr esence of pow er law s bet w een flood quant iles and dr ainage ar eas using t he RFF analyses of t he U SGS in t hr ee st at es. A geophysical under st anding of how slopes and int er cept s can be pr edict ed fr om physical pr ocesses r equir ed under st anding t he scaling in RF-RO event s in r iver basins and led t o Gupt a, Cast r o, and Over ( 1996 ). I t abandoned t he concept of “ hom ogeneous r egions” and ident ified t he fundam ent al r ole of self-sim ilar it y in channel net w or k t opology and geom et r y. Ogden and Daw dy ( 2003 ) obser ved scaling r elat ions in peak dischar ges for 226 RF-RO event s t hat spanned hour ly t o daily t im e scales in t he 21 k m 2 Goodw in Cr eek Exper im ent al Wat er shed (GCEW) in M ississippi. They found t hat scaling slopes and int er cept s var y fr om one event t o anot her. Fur ey and Gupt a ( 2007 ) conduct ed a diagnost ic analysis t o under st and t he var iabilit y in t er m s of physical pr ocesses gener at ing floods. Tw o over view paper s (Gupt a, Tr out m an, & Daw dy, 2007 ; Gupt a & Waym ir e,

1998 )

and

subsequent r esear ch have ident ified m any ar eas of science and engineer ing t his ar t icle addr esses, for exam ple, hydr ologic and hydr aulic engineer ing, hydr ologic science, nonlinear geophysics, fr act al geom et r y, fluvial geom or phology, applied pr obabilit y, fluid m echanics, biology, and landscape ecology.

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A B r i ef H i st o r y o f t h e Sc al i n g T h eo r y o f Fl o o d s Kir k by ( 1976 ) and L ee and Delleur ( 1976 ) fir st dem onst r at ed t he unique r ole of channel net w or k w idt h funct ion in est ablishing a connect ion bet w een net w or k geom et r y and st r eam flow hydr ogr aphs. For any given dist ance fr om t he out let , t he w idt h funct ion is defined as t he num ber of st r eam s at t hat dist ance. I f r ainfall is deposit ed inst ant aneously and unifor m ly over a net w or k and t r avels at a const ant velocit y t o t he out let , t hen t he flow hydr ogr aph at t he out let has t he sam e shape as t he w idt h funct ion. The w idt h funct ion is const r uct ed fr om t he link-m agnit ude classificat ion of a net w or k t hat Shr eve ( 1966 , 1967 ) int r oduced. I t is called geom or phologic inst ant aneous unit hydr ogr aph (GI U H ). Subsequent ly, sever al paper s cont r ibut ed t o t his line of t hink ing; for exam ple, Tr out m an and Kar linger ( 1984 , 1998 ), and Gupt a and M esa ( 1988 ). Rodr iguez-I t ur be and Valdes ( 1979 ) and Gupt a, Waym ir e, and Wang ( 1980 ) pr oposed anot her ver sion of a GI U H t hat w as const r uct ed fr om H or t on-St r ahler or der ing and H or t on law s. A subst ant ial lit er at ur e developed in t his line of r esear ch (Rodr iguez-I t ur be & Rinaldo, 1997 ). The above body of published r esear ch clear ly dem onst r at ed t hat channel net w or k t opology and geom et r y have a fundam ent al r ole t o play in t he fut ur e developm ent s of r iver basin hydr ology. M andelbr ot ( 1982 ) published a book on t he fr act al geom et r y of nat ur e, and a union session of t he Am er ican Geophysical U nion w as or ganized ar ound t he book in t he fall m eet ing in 1982. Fr act al geom et r y w as based on t he hypot hesis of self-sim ilar it y, a new for m of invar iance under a change of scale. I t s im pact becam e evident t hr ough m any published paper s and book s w it h applicat ions t o physics, geophysics, geology, and hydr ology (Feder, 1988 ;

Rodr iguez-I t ur be & Rinaldo, 1997 ; Tur cot t , 1997 ).

Scaling, or a pow er law, ar ises in self-sim ilar syst em s, w hich offer ed a new scient ific par adigm t o a lar ge num ber of st udies t hat show ed scaling in annual floods in r iver basins (L insley et al., 1982 , p. 370). Gupt a and Daw dy ( 1995 ) syst em at ically invest igat ed t he pr esence of pow er law s bet w een flood quant iles and dr ainage ar eas using t he RFF analyses of t he U SGS. H ow ever, a physical basis of scaling in RFF r equir ed under st anding scaling r elat ions in individual RF-RO event s, because floods ar e gener at ed at t he scales of event s. I t w as a m ajor shift in focus fr om annual t im e scale t o event scales of hour s and days, and fr om hom ogeneous r egions t o r iver basins. Gupt a et al. ( 1996 ) t ook t he fir st st ep t o develop a physical basis of scaling in peak flow s for RF-RO event s in a Peano basin, an idealized self-sim ilar channel net w or k . I t w as analyt ically show n t hat t he peak flow s exhibit a pow er law r elat ionship w it h r espect t o dr ainage ar ea w it h a flood-scaling exponent , . The par am et er is a fr act al dim ension of t he spat ial r egions of a Peano basin t hat cont r ibut e t o peak flow s at successively lar ger dr ainage ar eas. Gupt a and Waym ir e ( 1998 ) int r oduced an equat ion of cont inuit y for a channel net w or k in t he fir st r eview paper on t he t opic. Tr out m an and Over ( 2001 ) gener alized t he r esult s on scaling exponent s of peak flow s t o a gener al class of net w or k s gener at ed by a det er m inist ic self-sim ilar algor it hm , such as Tok unaga net w or k s. Page 4 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Ot her t heor et ical paper s w er e subsequent ly published on t he t opic (M enabde & Sivapalan, 2001 ). I n or der t o m ak e fur t her pr ogr ess, it becam e appar ent t hat a car eful obser vat ional st udy on scaling in peak flow s for RF-RO event s w as needed. Ogden and Daw dy ( 2003 ) obser ved scaling r elat ions in peak dischar ges for 226 individual RF-RO event s t hat spanned hour ly t o daily t im e scales in t he 21 k m 2 GCEW, in M ississippi. They discover ed t hat t he scaling slopes and int er cept s var y fr om one event t o anot her. The m ean of 226 event slopes (0.82) w as close t o t he com m on slope (0.77) of m ean annual and 20-year r et ur n peak dischar ge quant iles. Fur ey and Gupt a ( 2007 ) car r ied out a diagnost ic analysis of Ogden and Daw dy ( 2003 ) and found t hat event -t o-event var iabilit y in t he scaling slopes and int er cept s is due t o a com binat ion of t em por al r ainfall var iabilit y, spat ial var iabilit y in infilt r at ion, and dischar ge velocit y on a hillslope and in a channel link . Gupt a et al. ( 2007 ) w r ot e a r eview paper on t he scaling t heor y of floods in w hich k ey developm ent s up t o 2006 w er e r eview ed. Scaling t heor y connect s sever al disconnect ed fields for t he pur pose of under st anding floods and solving r elat ed pr oblem s. The r esear ch is fir m ly r oot ed in t he peer -r eview ed lit er at ur e. I t r epr esent s a new par adigm in hydr ologic science and engineer ing. I n t he r em ainder of t his sect ion, foundat ional and not able advances as w ell as cur r ent per spect ives ar e discussed. I n 1956 David Daw dy w as assigned by t he U SGS as chief assist ant t o M anuel Benson in developing t he U SGS RFF Analysis Pr ogr am . Aft er t hat he w as assigned t o develop t he U SGS physically based RF-RO m odel for ext ending RFF r ecor ds t o lar ger scales t han individual st at es and st at ions. As st at e RFF r epor t s began t o appear he cont inued his int er est in RFF analysis. Daw dy obt ained t he base dat a for each of t hese st at e r epor t s and becam e convinced t her e w as m or e infor m at ion cont ained in t he dat a t han just t hose r epor t ed in t he st at e r epor t s, but he could not figur e out how t o use t he dat a on a lar ger scale. I t w as his feeling t hat infor m at ion w as being over look ed, w hich led him t o consult w it h Vijay Gupt a, in t he lat e 1980s. H e found out t hat Gupt a and Edw ar d Waym ir e w er e collabor at ing on fr act als and scaling in hydr ology. Daw dy felt t hat scaling m ight help in m ak ing pr ogr ess. Gupt a and Daw dy m et on sever al occasions in t he ear ly 1990s and t ook t he fir st st eps. Subsequent ly, a fr am ew or k for solving t he pr oblem slow ly developed (Daw dy, 2016 ). Br ent Tr out m an played a subst ant ive r ole t hr ough published r esear ch on t he t opic and co-aut hor ed t he over view paper (Gupt a et al., 2007 ). H e also collabor at ed w it h Gupt a’s gr aduat e st udent s on scaling in floods. H e w as a m at hem at ical st at ist ician in t he U SGS N at ional Resear ch Pr ogr am fr om 1975 unt il his r et ir em ent 2011. Tr out m an’s r esear ch addr essed t he applicat ion of st at ist ical m et hods, pr obabilit y, and st ochast ic pr ocesses t o t he pr edict ion and m odeling of sur face-w at er flow s, par t icular ly quest ions of scale and spat ial/t em por al var iabilit y of pr ocesses influencing flow s (Tr out m an, 2016 ). Waym ir e and Gupt a collabor at ed in under st anding t he m at hem at ics of fr act als and scaling in t he 1980s and 1990s w it h applicat ions t o hydr ology. Waym ir e co-aut hor ed t he fir st over view paper on scaling in floods (Gupt a & Waym ir e, 1998 ) and w r ot e a k ey paper on channel net w or k s (Bur d, Waym ir e, & Winn, 2000 ; Waym ir e, 2016 ).

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D yn am i c Fo r m u l at i o n s f o r R ai n f al l -R u n o f f Even t s i n R i ver B asi n s L et

,

,

be a space-t im e field r epr esent ing r iver dischar ge, or volum e of flow per

unit t im e, t hr ough t he dow nst r eam end of link e in a channel net w or k . L et denot e t he r unoff int ensit y int o link e fr om t he t w o adjacent hillslopes, and let denot e t he com bined ar eas of t he t w o hillslopes. m ay also include r unoff deplet ion per unit t im e due t o channel infilt r at ion or evapor at ion fr om channel sur face in link e. Finally, let denot e st or age in link e at t im e t . Then t he equat ion of cont inuit y for t he syst em of hillslopes link s can be w r it t en as (Gupt a et al., 2007 ): (1)

The t er m s

and

on t he r ight -hand side r epr esent dischar ges fr om t he t w o

upst r eam link s joining link e.

denot es t he r unoff fr om t he t w o hillslopes dr aining int o

link e. Solut ions of Eq. ( 1 ) depend on t he br anching and geom et r ic st r uct ur e of a channel net w or k t hr ough t hese t w o t er m s. Equat ion ( 1 ) as w r it t en is based on an assum pt ion t hat t he channel net w or k is binar y. I t m ay easily be gener alized, how ever, t o non-binar y net w or k s, or t hose for w hich m or e t han t w o upst r eam link s can flow int o a junct ion, for exam ple, a Peano net w or k . H ow ever, t his sit uat ion does not ar ise in act ual r iver net w or k s. Equat ion ( 1 ) assum es t hat no loops ar e pr esent in t he net w or k . A specificat ion of r equir es a hillslope-scale m odel for r epr esent ing r unoff gener at ion fr om a ver y lar ge num ber of hillslopes in a net w or k . For exam ple, GCEW has 544 hillslopes. I t pr esent s a gr eat scient ific challenge because m ost of t he focus on m odel r unoff gener at ion has been in a single hillslope (Cor r adini, Govindar aju, & M or bidelli, 2002 ;

Duffy, 1996 ; Fr eeze, 1980 ). Such appr oaches, called “ bot t om -up,” use a com binat ion of

w ell-k now n unsat ur at ed-sat ur at ed flow cont inuum equat ions. By cont r ast , Fur ey, Gupt a, and Tr out m an ( 2013 ) developed and t est ed a “ t op-dow n” m odel in GCEW, w hich is needed t o m odel r unoff gener at ion

fr om a lar ge num ber of hillslopes in Eq. ( 1 ).

A funct ional r elat ionship bet w een link st or age

and link dischar ge

is needed t o

expr ess Eq. ( 1 ) in t er m s of one dependent var iable. I t can be obt ained fr om t he definit ion of st or age and dischar ge and a specificat ion of t he link velocit y, fr om a m om ent um balance equat ion gover ning t hr ee-dim ensional velocit y field in an open channel at t he sub-link scale. Kean and Sm it h ( 2005 ) m ade pr ogr ess t ow ar d solving t his pr oblem in t he cont ext of pr edict ing a t heor et ical r elat ionship bet w een flow dept h and dischar ge k now n as a r at ing cur ve. They developed a fluid-m echanical m odel t hat r esolves boundar y r oughness elem ent s fr om field m easur em ent s over a nat ur al channel r each and calculat ed t he cr oss-sect ional aver age velocit y t o pr edict t heor et ical r at ing cur ve. Their r esear ch est ablishes a scient ific fr am ew or k t o specify for a link . Jor dan and Kean ( 2010 ) applied t he m odel t o est im at e r at ing cur ves at t en ungagged st r eam r eaches,

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s r anging fr om t hir d t o sixt h or der channels, in t he Whit ew at er basin in Kansas (M ant illa & Gupt a, 2005 ). L et ,

denot e a link lengt h,

t he m ean link w idt h, and

t he m ean link dept h.

Then, (2)

I n view of t he hydr aulic-geom et r ic (H -G) r elat ions for a net w or k (Gupt a & M esa, 2014 ; L eopold & M iller, 1956 ), w idt h, dept h, and velocit y m ay be expr essed as funct ions of dischar ge, or

, w her e f(.) is an ar bit r ar y funct ion. Subst it ut ing t he st or age-

dischar ge r elat ion int o Eq. ( 1 ) gives (3)

w her e

The funct ions

and

in Eq. ( 3 ) var y fr om one hillslope-link pair t o t he ot her. Even

sm all basins lik e GCEW cont ain a lar ge num ber of hillslope-link pair s. A specificat ion of t he physical par am et es in all of t hese pair s is r equir ed for solving Eq. ( 3 ). Gupt a ( 2004A ) st at ed t hat “ st at ist ical scaling is an em er gent pr oper t y of a com plex syst em w hich a-pr ior i is not built int o t he physical equat ions.” Scaling can be used t o t est physical hypot heses (Gupt a, 2004B ). Once a lar ge num ber of par am et er s ar e specified, Eq. ( 3 ) can be solved it er at ively t o obt ain r unoff hydr ogr aph, , , , at t he bot t om of ever y link , and t her efor e in all sub-basins of a r iver basin. I t s solut ions pr oduce st r eam flow hydr ogr aphs at all junct ions in a channel net w or k . They have been used t o obt ain r esult s on spat ial scaling st at ist ics of floods under idealized physical condit ions in idealized channel net w or k s, nam ely M andelbr ot -Vicsek (M enabde & Sivapalan, 2001 ).

Sel f -Si m i l ar R i ver N et w o r k s Gupt a et al. ( 2007 ) r eview ed 20 year s of pr ogr ess on self-sim ilar it y in r eal channel net w or k s t hat included det ails of t he H or t on-St r ahler or der ing and t he H or t on law s. A br ief over view of som e of t he k ey t opics and pr ogr ess since t hen is given her e.

H o r t o n R el at i o n sh i p s f o r R i ver N et w o r k s H or t on-St r ahler or der ing is defined as follow s. A channel w it h no upst r eam inflow s is given or der one. When t w o st r eam segm ent s of t he sam e or der m er ge, t he or der of t he out -flow ing st r eam incr eases by one. I n case t w o m er ging st r eam s have differ ent or der s, t he out -flow ing st r eam has t he higher of t he t w o or der s. The t hr ee m ost w idely st udied H or t on law s ar e for t he num ber of St r ahler st r eam s of or der , , t he aver age lengt h of st r eam s of or der , , and t he aver age upst r eam ar ea of st r eam s of or der , . I t has Page 7 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s been obser ved t hat t he r at ios of t hese quant it ies in successive or der s t end t o be independent of or der, t hat is, , , and , w hich ar e k now n as t he classical H or t on law s. ar e called H or t on r at ios. Self-sim ilar channel net w or k m odels have show n t hat H or t on law s hold in t he lim it as net w or k or der incr eases t o infinit y (M cConnell & Gupt a, 2008 ; Veit zer & Gupt a,

2000 ).

Peck ham and Gupt a ( 1999 ) gener alized t he classical H or t on law s for m ean dr ainage ar eas and m ean channel lengt hs t o full pr obabilit y dist r ibut ions. Specifically, t hey gave obser vat ional and som e t heor et ical ar gum ent s for t he w ell-k now n r andom m odel (Shr eve, 1967 )

t o show t hat pr obabilit y dist r ibut ions of var iables r escaled by t heir m eans

collapse int o a com m on pr obabilit y dist r ibut ion of som e r andom var iable, , t hat is independent of or der. I t is w r it t en as (4)

w her e

is t he H or t on r at io, and

denot es equalit y in pr obabilit y dist r ibut ions of r andom

var iables on bot h sides. The r elat ion in Eq. ( 4 ) is k now n as a gener alized H or t on law . Thus, not only t he m ean value, but any finit e m om ent , if it exist s, obeys H or t on law s. As an exam ple, Figur e 1 show s t he classical and t he gener alized H or t on law s for dr ainage ar eas

in t he Whit ew at er basin in Kansas (M ant illa & Gupt a, 2005 ).

Click t o view lar ger Figur e 1. (L eft ) H or t on law of m ean ar eas. (Right ) Pr obabilit y dist r ibut ions of r escaled ar eas in t he Whit ew at er Basin, KS. (Fr om M ant illa & Gupt a.,

2005 .)

To k u n ag a Sel f -Si m i l ar M o d el Tok unaga ( 1966 ) int r oduced a det er m inist ic m odel of r iver net w or k s t hat w as based on H or t on-St r ahler or der ing r at her t han on link m agnit ude as is t he case for t he r andom t opology m odel. I t w as char act er ized by t he m ean num ber of side t r ibut ar ies. For a st r eam of or der

, let

denot e aver age num ber of side t r ibut ar ies of or der k , k now n as

gener at or s (Tok unaga, 1966 , 1978 ). Gener at or s ar e self-sim ilar if t hey obey t he const r aint , . I t m eans t hat t he gener at or s only depend on t he differ ence independent of . U nder t he addit ional const r aint t hat have t he for m St r ahler or der,

is a const ant , gener at or s

(Tok unaga, 1966 ). The num ber of st r eam s of differ ent H or t onobey a r ecur sion equat ion (Tok unaga,

1978 ),

Page 8 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s (5)

M cConnell and Gupt a ( 2008 ) r igor ously pr oved t he H or t on law of st r eam num ber s as an asym pt ot ic r esult fr om t he solut ion of Eq. ( 5 ) and ext ended it t o a H or t on law for m agnit ude

. Tok unaga also show ed t hat t he gener at or s of t he r andom t opology m odel

(Shr eve, 1967 ) exhibit Tok unaga self-sim ilar it y w it h par am et er s Eq. ( 5 ), w it h t hese values of par am et er s, pr edict s

. The solut ion of

, w hich is t he sam e t hat Shr eve

( 1967 ) obt ained for t he r andom m odel. Gupt a et al. ( 2007 ) r eview ed ot her k ey r esult s for Tok unaga net w or k s.

R an d o m Sel f -Si m i l ar N et w o r k M o d el The geom or phologic and hydr ologic t hink ing w er e gr eat ly influenced by t he r andom m odel of channel net w or k based in a link-m agnit ude classificat ion (Shr eve, 1966 , 1967 ). Jar vis and Woldenber g ( 1984 ) r epr oduced t he k ey paper s fr om 1945 t o 1976 in a book . The paper s w er e divided int o four par t s, and each par t included edit or ial over view, w hich m ak es it an excellent r efer ence. The fluvial geom or phologic r esear ch t ook a new t ur n dur ing t he 1990s. Det ailed em pir ical analysis of lar ge r iver basins w as gr eat ly aided by t he availabilit y of finer esolut ion digit al elevat ion m odels (DEM s). Obser vat ions fr om lar ge basins show ed t hat r andom m odel pr edict ions deviat e subst ant ially fr om em pir ical values (Peck ham , 1995 ). Bur d et al. ( 2000 ) pr oved t hat t he r andom m odel is t he only one am ong finit e, binar y Galt on-Wat son st ochast ic br anching t r ees t hat exhibit s t he m ean self-sim ilar t opology of Tok unaga net w or k s. These findings opened t he door t o develop a new class of st at ist ical channel net w or k m odels called r andom self-sim ilar net w or k s (RSN s), w hich do not belong t o t he class of binar y Galt on-Wat son st ochast ic br anching pr ocesses (Veit zer & Gupt a, 2000 ).

RSN s ar e const r uct ed r ecur sively fr om r andom gener at or s, w hich ar e essent ially

sim ple or der -2 net w or k s w it h a r andom num ber of int er ior nodes. The for m ulat ion allow s for differ ent gener at or pr obabilit y dist r ibut ions for t he r eplacem ent of int er ior and ext er ior link s. Veit zer and Gupt a ( 2000 ) show ed t hat t he gener alized H or t on law, or dist r ibut ional sim ple scaling, holds for t he num ber of link s per st r eam , m agnit ude, and st r eam num ber s as net w or k or der incr eases. The m ean side t r ibut ar y st r uct ur e of a subset of RSN s w er e show n t o obey Tok unaga self-sim ilar it y.

Pr o g r ess o n R an d o m Sel f -Si m i l ar N et w o r k M o d el Tr out m an ( 2005 ) developed an algor it hm t hat allow s a unique ext r act ion of gener at or s fr om act ual net w or k s under cer t ain r est r ict ions. H e gave r esult s show ing t hat a geom et r ic dist r ibut ion for t he num ber of side t r ibut ar ies in bot h int er ior and ext er ior gener at or s fit r easonably w ell t he dat a for t he Flint River basin in Geor gia. M ant illa, Tr out m an, and Gupt a ( 2010 ) t est ed t he RSN m odel in 30 basins fr om diver se clim at ic and geogr aphic set t ings in t he U nit ed St at es. Self-sim ilar it y in a st at ist ical sense w as found t o

Page 9 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s hold in 26 of t he 30 basins, and geom et r ic dist r ibut ion show ed good agr eem ent w it h dat a in all cases. An im por t ant consider at ion in t he analysis of scaling pr oper t ies of r iver basins and t he channel net w or k s t hat dr ain t hem is spat ial em bedding, w hich is also necessar y for sim ulat ing spat ially dist r ibut ed r ainfall int ensit y fields on r iver basins. The idealized basins, Peano and M andelbr ot -Vicsek , ar e spat ially em bedded, w hich enabled t o consider a spat ial dist r ibut ion of r ainfall int ensit ies on t hese basins (Gupt a et al., 1996 ; M enabde & Sivapalan, 2001 ). For net w or k s gener at ed by t he r andom t opology m odel, det er m inist ic self-sim ilar net w or k s lik e Tok unaga, RSN s, t he br anching st r uct ur e is defined but how t he net w or k s fit int o a t w o- or a t hr ee-dim ensional space is not . M ant illa, Gupt a, and Tr out m an ( 2012 ) pr esent ed an algor it hm for em bedding RSN s in a given spat ial r egion.

N u m er i c al St u d i es o n R an d o m Sel f -Si m i l ar Ch an n el N et w o r k M o d el s w i t h A p p l i c at i o n s t o Fl o o d s M ant illa, Gupt a, and M esa ( 2006 ) sim ulat ed peak flow s in t he Walnut Gulch basin, Ar izona, and analyzed t he scaling pr oper t ies of flow hydr ogr aphs by solving t he m ass conser vat ion show n in Eq. ( 3 ) under a const ant flow velocit y assum pt ion. They obser ved t hat t he scaling exponent s for peak flow s ar e lar ger t han t he m axim a of t he w idt h funct ions (Veit zer & Gupt a, 2001 ). This pr oper t y for a r eal net w or k cont r adict s t he pr evious findings for idealized self-sim ilar net w or k s; for exam ple, M andelbr ot -Vicsek , M ant illa, Gupt a, and Tr out m an ( 2011 ) t est ed t his hypot hesis in sim ulat ed RSN s using geom et r ic dist r ibut ions, w it h par am et er s

and

cor r esponding t o int er ior and ext er ior gener at or s, r espect ively.

They com par ed t he num er ical r esult s w it h t he analyt ic expr essions t hat Tr out m an ( 2005 ) had obt ained, w hich ser ved as benchm ar k for t he accur acy of r esult s fr om num er ical sim ulat ions. Their r esult s suppor t ed t he finding of M ant illa et al. ( 2006 ).

A n al yt i c al R esu l t s f o r H yd r au l i c -Geo m et r y i n To k u n ag a Sel f -Si m i l ar Ch an n el N et w o r k s an d I m p l i c at i o n s f o r Sp ec i es R i c h n ess i n R i p ar i an Veg et at i o n I n a classic paper, L eopold and M iller ( 1956 ) discover ed H or t on law s for H -G var iables in dr ainage net w or k s, t her eby ext ending t he H or t on law s fr om t opological and geom et r ic var iables t o H -G var iables (st r eam dischar ge Q, w idt h W, dept h D, velocit y U , slope S, and Page 10 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s M anning’s fr ict ion n’). M ot ivat ed by t he need t o t heor et ically under st and H or t on law s for t he H -G var iables, a long-st anding unsolved pr oblem , Gupt a and M esa ( 2014 ) developed an analyt ical t heor y in self-sim ilar Tok unaga net w or k s. The H -G dat a set s in channel net w or k s fr om t hr ee published st udies and one unpublished st udy w er e used t o t est t heor et ical pr edict ions. Gupt a and M esa ( 2014 ) used dim ensional analysis and t he Buck ingham Pi t heor em t o ident ify six independent dim ensionless r iver -basin num ber s. A m ass conser vat ion equat ion in t er m s of H or t on bifur cat ion and dischar ge r at ios in Tok unaga net w or k s w as der ived. U nder t he assum pt ion t hat t he H -G var iables ar e hom ogeneous and self-sim ilar funct ions of st r eam dischar ge, it w as show n t hat t he funct ions ar e of a w idely assum ed pow er law for m . Asym pt ot ic self-sim ilar it y of t he fir st k ind, or SS-1 (Bar enblat t , 1996 ), w as applied t o pr edict t he H or t on law s for W, D, and U as asym pt ot ic r elat ionships. Pr edict ions of t he exponent s agr eed w it h t hose pr eviously pr edict ed for t he opt im al channel net w or k (OCN ) m odel. Pr edict ed exponent s of w idt h and t he Reynolds num ber w er e t est ed against t hr ee field dat a set s. Ashley basin show ed deviat ions fr om t heor et ical pr edict ions. Test s of ot her pr edict ed exponent s suggest ed t hat H -G in net w or k s does not obey SS-1. I t fails because one of t he dim ensionless r iver -basin num ber s, slope goes t o 0 as net w or k or der incr eases, but it cannot be elim inat ed fr om t he asym pt ot ic lim it . Ther efor e, a gener alizat ion of SS-1, based on t he asym pt ot ic selfsim ilar it y of t he second k ind, or SS-2 (Bar enblat t , 1996 ), w as consider ed. I t int r oduced t w o anom alous scaling exponent s as fr ee par am et er s, w hich enabled t hem t o show t he exist ence of H or t on law s for W, D, U , S, and n’. The t w o anom alous scaling exponent s w er e not pr edict ed. I nst ead, t hey used t he obser ved exponent s of D and S t o pr edict t he exponent of n’ and t o t est it against exponent s fr om t hr ee field st udies m ent ioned above. The Ashley basin show ed som e deviat ion fr om t heor et ical pr edict ions. A physical r eason for t his deviat ion w as given, w hich ident ified an im por t ant t opic for r esear ch. Finally, how t he t w o anom alous scaling exponent s could be est im at ed fr om t he t r anspor t of suspended sedim ent load and t he bed load w as br iefly sk et ched. St at ist ical var iabilit y in t he H or t on law s for t he H -G var iables w as also discussed. Bot h ar e im por t ant open pr oblem s for fut ur e r esear ch.

H yd r o -ec o l o g i c al T h eo r y o f I n t er m i t t en t R i p ar i an D i ver si t y o n St r eam N et w o r k s Dunn, M ilne, M ant illa, and Gupt a ( 2011 ) gener alized t he scaling fr am ew or k t o include ecological var iables coupled t o w at er. Based on digit al land cover m aps, t hey w er e able t o discer n scaling and H or t on law s in t he ar eas occupied by r ipar ian veget at ion in t he Whit ew at er basin. M ilne and Gupt a ( 2017A ) ar e developing a new line of r esear ch t o under st and biological diver sit y and w at er balance in self-sim ilar Tok unaga net w or k s and t est t he developm ent s in t he Whit ew at er basin. The idea or iginat ed t o under st and deviat ion in t he Ashley basin fr om t heor et ical pr edict ion m ent ioned in t he pr evious sect ion. I t r equir es t hat t he Tok unaga gener at or be m odified t o incor por at e int er m it t ency in r unoff gener at ion and t he gr ow t h of r ipar ian veget at ion on hillslopes, w hich is evident fr om t he digit al m aps of r ipar ian veget at ion. Consequent ly, t he r igor ous m at hem at ical Page 11 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s r esult s for Tok unaga net w or k s (M cConnell & Gupt a, 2008 ) ar e used t o der ive H or t on law s for species r ichness. A “ law of hydr o-ecology” is post ulat ed t hat t r eat s w at er balance t hr oughout t he net w or k as a consequence of self-sim ilar non-local int er act ions. I n a com panion paper M ilne and Gupt a ( 2017B ) developed t he ent r opy-H or t on fr am ew or k and dem onst r at ed how it infor m s quest ions of biodiver sit y, r esilience t o per t ur bat ions in w at er supply, changes in pot ent ial evapot r anspir at ion, and land use changes t hat m ove ecosyst em s aw ay fr om opt im al ent r opy w it h concom it ant loss of pr oduct ivit y and biodiver sit y.

Ph ysi c al B asi s o f A n n u al Fl o o d Qu an t i l es i n R i ver B asi n s Building on Ogden and Daw dy (O-D), Fur ey et al. ( 2016 ) conduct ed t he fir st r igor ous analysis t ow ar d a physical under st anding of annual flood quant iles in GCEW. They st at ed t w o hypot heses based on t he r esult s in O-D: (1) scaling slopes of annual peak (AP) quant iles ar e t he sam e for all r et ur n per iods and (2) t he m ean scaling slope of st r eam dischar ge peak s fr om RF-RO event s equals t he m ean slope of AP quant iles. H er e, a “ m ean” r efer s t o an aver age over exceedance pr obabilit ies. H ypot hesis 1 is a for m al st at em ent of sim ple scaling. I t st ands in cont r ast t o m ult iscaling w her e scaling slopes depend on r et ur n per iods (Gupt a, M esa, & Daw dy, 1994 ). H ypot hesis 2 per t ains t o only t he m ean slopes bot h for event s and AP quant iles. Ther efor e, a r eject ion of hypot hesis 1 does not im ply a r eject ion of hypot hesis 2. Result s in O-D suppor t hypot hesis (1), w hile suppor t for hypot hesis (2) is unclear w it hout fur t her analysis. Given t hat hypot hesis (1) holds, t he m ean slope of AP quant iles in hypot hesis (2) w ill coincide w it h t he com m on AP quant ile slope in hypot hesis (1). L et denot e t he com m on slope of AP quant iles and denot e t he m ean scaling slope for RF-RO event s. Result s in O-D pr ovide t he est im at es

and

given in Table 1 . To t est

hypot hesis (2), Fur ey et al. ( 2016 ) evaluat ed t he st at ist ical significance of t he differ ence, . They found t hat t he est im at ed st andar d er r or (SE) of t he differ ence obeys SE > 0.039, yielding a p-value > 0.15. I t indicat es t hat

is not significant at t he 5%

level. Thus, O-D r esult s do not r eject H ypot hesis (2). Result s in O-D r epr esent only one dat aset and r equir ed som e appr oxim at ions in t he above calculat ions due t o lack of infor m at ion. Thus, t o fur t her t est t he t w o hypot heses, Fur ey et al. ( 2016 ) assem bled a second dat aset for 148 RF-RO event s in GCEW. Exam ining dat a fr om a single basin is necessar y t o obt ain a physical under st anding of scaling in dischar ge quant iles. As one goes t o lar ger basins, r ainfall est im at ion r equir es r adar -r ainfall dat a t hat int r oduce r em ot e-sensing er r or s not found in t he r ain gauge dat a in GCEW. A st andar d linear r egr ession m odel w as used t o evaluat e scaling slopes for RF-RO event s

Page 12 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s and for AP quant iles. Figur e 1 show s t he scaling slopes for 148 event s. The m ean and st andar d deviat ion of slopes, 0.78 and 0.16, is sim ilar t o t he cor r esponding values in O-D (Figur e 3 ). A Weibull plot t ing posit ion for m ula w as used t o assign pr obabilit ies t o AP values. For each gauge, t he quant iles w it h em pir ical pr obabilit ies closest t o sever al t ar get pr obabilit ies w er e ident ified. The sm allest pr obabilit y evaluat ed, p = 0.071, r epr esent s a r et ur n per iod of 14 year s, one year beyond t he r ecor d lengt h. The ot her value w as p = 0.5. The r esult s in Figur e 2 show t w o differ ent scaling slopes, suggest ing t he possibilit y of m ult iscaling (Gupt a et al., 1994 ). I t m eans t hat slopes of AP quant ile scaling r elat ions ar e differ ent for differ ent r et ur n per iods. I f such a pr oper t y holds, t he m ean of t he slopes for a given set of r et ur n per iods can be denot ed by . Then hypot hesis 2 is an equalit y bet w een

and μb. Table 1 show s t hat

and

. These values denot e t he m ean

slope of t he quant ile r elat ionships pr esent ed in Figur e 2 and t he m ean slopes of event s given in Figur e 1 . Assum ing t hat hypot hesis 1 holds, t he significance of t he differ ence, , can be assessed as done ear lier for t he O-D r esult s. A p-value of ≥ 0.83 w as obt ained, indicat ing t hat t he differ ence is not significant at t he 5% level, again suppor t ing hypot hesis 2. The sam e r esult s ar e found if quant ile r elat ionships for ot her r et ur n per iods ar e consider ed. I n sum m ar y, r esult s fr om t he second dat a set suppor t hypot hesis 2. To t est for sim ple ver sus m ult iscaling in AP quant iles, indicat ed in Figur e 2 , dat a fr om GCEW t hat include addit ional year s beyond 1994 ar e needed.

Click t o view lar ger Figur e 2. Var iabilit y in scaling slopes for RF-RO event s (Fur ey et al., 2016 ). (Repr int ed w it h per m ission fr om ASCE.)

Page 13 of 35



Click t o view lar ger Figur e 3. Evidence of m ult iscaling in annual peak quant iles (Fur ey et al., 2016 ). (Repr int ed w it h per m ission fr om ASCE.)

Table 1. M ean Scaling Slopes of RF-RO Event s and Annual Peak Quant iles

Sour ce: Fur ey et al. ( 2016 ). The above r esult s clear ly dem onst r at e t hat a connect ion exist s bet w een scaling in event s and in AP quant iles. Fur ey et al. ( 2016 ) char act er ized such a connect ion w it hin a single physical-st at ist ical m odel called a nest ed m ixed-effect s linear (N M EL ) m odel. The N M EL m odel char act er izes event -t o-event var iabilit y in scaling r elat ionships bet w een st r eam dischar ge peak s and dr ainage ar eas. The N M EL m odel leads t o scaling r elat ionships for dischar ge peak quant iles and annual peak (AP) quant iles t hat ar e t he basis for RFF equat ions. Since event -based scaling r elat ionships can be connect ed t o physical pr ocesses, it im plies t hat annual peak quant iles can be connect ed as w ell. To sum m ar ize, t est r esult s of m odel assum pt ions w er e suppor t ive of N M EL m odel st r uct ur e, t hough som e discr epancies w er e found. While t he m odel link s event s t o quant iles, r esult s show t hat t her e ar e im por t ant differ ences bet w een quant ile-based scaling st at ist ics using APs and r elat ed event -based st at ist ics. I n par t icular, quant ile scaling allow s for event m ixing, using dat a fr om differ ent event s t o det er m ine a scaling r elat ionship and pr ovides slope values having a significant ly nar r ow er r ange t han t hat found w it h event s. The lat t er r esult m eans t hat AP-quant ile scaling r elat ionships and RFF r elat ions could under est im at e flooding pot ent ial. Given t hat flood char act er izat ion acr oss a r iver basin should be t ied t o physical pr ocesses and condit ions t hat include pr e-event Page 14 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s soil m oist ur e, infilt r at ion gover ning r unoff gener at ion fr om each hillslope in a basin, and space-t im e var iable hydr aulic-geom et r y gover ning r unoff dynam ics in a net w or k . The findings in Fur ey et al. ( 2016 ) show t hat m et hods for flood char act er izat ion based on event scaling ar e m or e infor m at ive and r obust t han t r adit ional RFF m et hods based on analysis of AP quant iles. A lar ger sam ple st udy fr om GCEW and ot her w at er sheds is needed t o fur t her t est , and possibly r efine, t hese ideas. Test of m ult iscaling is an im por t ant issue t hat needs t o be addr essed in fut ur e r esear ch.

H o r t o n L aw s f o r D i ag n o si n g a R ai n f al l -R u n o f f M o d el W i t h o u t Cal i b r at i o n o f I t s Par am et er s The m easur em ent and collect ion of hydr ologic dat a in channel net w or k s is t edious and expensive. Gupt a and M esa ( 2014 ) illust r at ed t w o field st udies as exam ples (I bbit t , M cKer cher, & Duncan, 1998 ; M cKer cher, I bbit t , Br ow n, & Duncan, 1998 ). As a r esult , t he discover y of t he H or t on law s for hydr ologic var iables has gr eat ly lagged behind geom or phology. The hydr ologic dat a in Gupt a, Ayalew, M ant illa, and Kr ajew sk i ( 2015 ) cam e fr om t he 32,400 k m 2 I ow a River basin locat ed in east er n I ow a befor e it joins t he M ississippi River (Figur e 4 ). The basin is cont inuously m onit or ed by 34 U SGS gauging st at ions. The t op four annual m axim um peak dischar ges obser ved at t he basin out let over t he past 112 year s occur r ed in t he pr evious 7 year s (Sm it h, Baeck , Villar ini, Wr ight , & Kr ajew sk i, 2013 ). Figur e 4 show s t he I ow a River basin and t he locat ion of t he U SGS gauging sit es. A St r ahler or der w as assigned t o each channel link w her e t he U SGS st r eam gauges ar e locat ed, and t he dr ainage ar eas dr aining int o t he gauges w er e est im at ed. The obser ved peak flow s w er e also assigned t he sam e or der. This infor m at ion w as used t o t est t he H or t on law s for dr ainage ar eas and for peak flow s. Ayalew, Kr ajew sk i, and M ant illa ( 2015 ) select ed RF-RO event s t hat occur r ed in t he 12-year per iod fr om 2002 t o 2013. They obser ved a scale-invar iant peak dischar ge w hen t he ent ir e basin got a r unoff-gener at ing r ainfall event at som e point dur ing a t im e w indow equivalent t o t he basin’s concent r at ion t im e (defined as t he t im e r equir ed for a par cel of w at er t o t r avel fr om t he m ost r em ot e hillslope in t he basin t o t he out let ). The concent r at ion t im e is about 15 days for t he I ow a River basin (see Ayalew et al., 2015 , for a det ailed discussion of t he RF-RO event select ion and for t he cr it er ia used t o define peak dischar ge event s). Based on t hese st r ict cr it er ia, 51 RF-RO event s w er e ident ified w hose r esult ing peak dischar ges exhibit scaling w it h r espect t o dr ainage ar ea. Figur e 5 show s a RF-RO event dur ing w hich t he ent ir e basin did not r eceive r ainfall at som e point dur ing t he 15-day t r avel t im e w indow. Figur e 6 illust r at es a RF-RO event dur ing w hich t he ent ir e basin r eceived r ainfall at som e point dur ing t he 15-day t r avel t im e w indow.

Page 15 of 35



Click t o view lar ger Figur e 4. The I ow a River basin and t he locat ion of t he U SGS gauging sit es. St r eam s below or der four ar e not show n for t he sak e of clar it y. (Repr oduced fr om Gupt a et al., of AI P Publishing.)

2015 ,

w it h per m ission

Click t o view lar ger Figur e 5. Exam ple st r eam flow t im e ser ies fr om r epr esent at ive U SGS gauging sit es in t he basin (t op panels) and t he associat ed peak-dischar ge scaling plot (bot t om panel) for t he case w her e only a por t ion of t he basin got r ainfall at som e point dur ing t he 15day t r avel t im e w indow. The st r eam flow t im e ser ies is nor m alized by t he annual m axim um flow for each gauging sit e. (Repr oduced w it h per m ission fr om Ayalew et al., 2015 .)

Page 16 of 35



Click t o view lar ger Figur e 6. Exam ple st r eam flow t im e ser ies fr om r epr esent at ive U SGS gauging sit es in t he basin (t op panels) and t he associat ed peak-dischar ge scaling plot (bot t om panel) for t he case w her e t he ent ir e basin exper ienced r ainfall at som e point dur ing t he 15-day t r avel t im e w indow. The st r eam flow t im e ser ies is nor m alized by t he annual m axim um flow for each gauging sit e. (Repr oduced fr om Gupt a et al., per m ission of AI P Publishing.)

2015 ,

w it h t he

A Co n si st en t T h eo r et i c al Fr am ew o r k t o Est i m at e H o r t o n R at i o s Fur ey and Tr out m an ( 2008 ) developed a consist ent st at ist ical fr am ew or k t hat r esolved im por t ant pr oblem s w it h t he int er pr et at ion and use of t r adit ional H or t on r egr ession st at ist ics. Their appr oach agr ees w it h dist r ibut ional sim ple scaling or t he gener alized H or t on law illust r at ed in Figur e 1 . I t is used in t he dat a analysis of peak flow s. To under st and t he k ey concept s, let be a r andom var iable t hat r epr esent s a geom or phologic pr oper t y such as dr ainage ar ea for basins of or der ω. Assum e t hat (6)

H er e, a and b ar e const ant s and

is a zer o-m ean r andom var iable. Equat ion ( 6 ) is a linear

r egr ession m odel. I t uses individual sam ple values r at her t han t he sam ple m ean t hat is t he st andar d pr act ice in t he classical H or t on analysis. Fur ey and Tr out m an ( 2008 ) t ook t he expect at ion on bot h sides in Eq. ( 6 ) t o get t he expr ession

, and

(7)

Subst it ut ing

w her e

is t he geom et r ic m ean, int o Eq. ( 7 ) gives

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s (8)

Equat ion ( 8 ) gives t he expr ession for t he H or t on r at io,

, as

(9)

Equat ion ( 9 ) is based on geom et r ic r at her t han ar it hm et ic m ean and ser ves as an alt er nat ive expr ession t o est im at e H or t on r at io. Gupt a et al. ( 2015 ) est im at ed t he H or t on ar ea r at io,

, using dr ainage ar ea infor m at ion for

471,101 com plet e or der st r eam s using CU EN CAS (M ant illa & Gupt a, 2005 ). Tw o independent m et hods w er e used t o est im at e

. The fir st m et hod uses t he r egr ession Eq.

( 6 ) and est im at es t he slope, , w hich gives

. The second m et hod uses Eq. ( 7 )

and calculat es geom et r ic m eans for differ ent or der st r eam s. for or der s 7, 8, and 9 w as ignor ed due t o sm all sam ple sizes. The r em aining est im at es of show ed st at ist ical var iabilit y. To com par e t hese est im at es w it h t he value fr om m et hod 1, t he m ean of est im at es for or der s 2, 3, 4, 5, and 6 w as com put ed, w hich gave 4.66. The r esult s show ed t hat t he infor m at ion fr om t he 34 st r eam flow -gauging st at ions can r easonably pr edict t he value t hat is t he sam e as t hat est im at ed using det ailed dr ainage net w or k infor m at ion. I t pr ovided confidence in t he est im at ion of H or t on r at ios fr om t he 34 gauging st at ions and also gave a t em plat e t o est im at e H or t on r at io for peak flow s, w her e sam ple size is a m or e ser ious issue t han for dr ainage ar eas.

Cl assi c al H o r t o n L aw f o r Peak Fl o w s i n R ai n f al l -R u n o f f Even t s Consider a r iver basin w it h st r eam gauges. L et denot e a gauge and it s subbasin, and let i= 1,2, … denot e a RF-RO event . Dat a show t hat peak dischar ges at t he event t im e scale obey a scaling r elat ionship fir st obser ved in GCEW (Ogden & Daw dy, 2003 )

and lat er in t he m edium -sized I ow a River basin (Ayalew et al., 2015 ; Gupt a, M ant illa,

Tr out m an, Daw dy, & Kr ajew sk i, 2010 ). An or dinar y linear r egr ession (OL R) m odel can char act er ize t his r elat ionship (Fur ey et al., 2016 ): (10)

H er e, Q i,j denot es peak dischar ge at gauge j for event i, a(i) and b(i) ar e OL R coefficient s, A j is t he dr ainage ar ea dr aining gauge j, and is a m ean-zer o r andom var iable t hat r epr esent s t he deviat ion of peak dischar ge at gauge j fr om t he aver age linear r elat ion, a(i) + b(i)A j . Random ness in peak flow s is ent ir ely due t o t he er r or t er m . Figur e 7 show s peak dischar ge for four RF-RO event s as exam ples fr om t he 51 event s t hat Ayalew et al. ( 2015 ) analyzed. All of t he t er m s in Eq. ( 10 ) change fr om one RF-RO event t o anot her due t o changes in t he physical pr ocesses t hat pr oduce peak flow s (Ayalew, Kr ajew sk i, & M ant illa, 2014A ; Ayalew et al., 2015 ; Ayalew, Kr ajew sk i, M ant illa, & Sm all, 2014B ; Fur ey & Gupt a, 2005 , 2007 ). I t can be seen t hat t he R 2 values ar e high, w hich suppor t s t he use of OL R t o det ect scaling in peak flow dat a. I t is also evident t hat t he scat t er of peak dischar ges ar ound t he r egr ession line changes fr om event t o anot her event , w hich can be

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s quant ified by t he st andar d deviat ion of t he er r or t er m in Eq. ( 10 ). Physically, it is a m anifest at ion of t he nat ur al var iabilit y of r ainfall and ot her cat chm ent physical var iables t hat cont r ol t he gener at ion of peak dischar ges in space and t im e. To t est t he validit y of t he classical H or t on law for peak flow s, four differ ent event s show n in Figur e 7 w er e select ed. Thr ee differ ent m et hods w er e used t o t est t he st at ist ical sensit ivit y of t he est im at es of t he H or t on r at io for peak flow s,

, i= 1,2,3 … Since t he

U SGS gauges ar e not locat ed at t he end of com plet e St r ahler st r eam s, w hich is necessar y for t he H or t on analysis, it w as assum ed t hat each U SGS st r eam gauge of a given or der is locat ed at t he end of a com plet e St r ahler st r eam of t hat or der. The obser ved peak flow s at t hese gauges for a given RF-RO event const it ut e a r andom sam ple of peak flow s for t hat or der. The r esult of t his assum pt ion pr oduces peak flow s for differ ent St r ahler st r eam s as show n in Figur e 8 w it h gr ey cir cles. The H or t on r at io, RQ (i), for i = 1,2,3,4 w as calculat ed for t he sam e four event s show n in Figur e 7 . I n m et hod 1, Eq. ( 9 ) w as applied t o com put e RQ as t he r at io of t he geom et r ic m eans of t he peak flow s at consecut ive or der s. The r esult s ar e show n in Figur e 8 . I n m et hod 2, peak flow dat a w as ar r anged accor ding t o t he St r ahler or der, and OL R, given in Eq. ( 7 ), w as used t o com put e r elat ion

. I n m et hod 3, t he

der ived in Gupt a et al. ( 2015 ) w as used. The appear ance of t he classical

H or t on law s for peak flow s in RF-RO event s show n in Figur e 8 w as a new hydr ologic discover y. Thr ee est im at ion m et hods t o addr ess t he sm all sam ple size issue for t he peak flow dat a need fur t her invest igat ion and const it ut es an im por t ant ar ea of fut ur e r esear ch.

Click t o view lar ger Figur e 7. Obser ved scaling r elat ions of peak dischar ges w it h dr ainage ar eas in t he I ow a River basin, I ow a. The influence of var iabilit y in t he est im at es of slope b(i) and t he int er cept r eflect s changes in t he physical pr ocesses gener at ing peak dischar ges. (Repr oduced fr om Gupt a et al., per m ission of AI P Publishing.)

2015 ,

w it h t he

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Click t o view lar ger Figur e 8. The classical H or t on law s for t he four peak-dischar ge event s show n in Figur e 7 . The gr ey cir cles show t he obser ved peak dischar ges, w her eas t he black line show s t he est im at ed E[ ln(Q ω )] using Eq. ( 10 ). (Repr oduced fr om Gupt a et al., of AI P Publishing.)

2015 ,

w it h per m ission

Gen er al i zed H o r t o n L aw f o r Peak Fl o w s i n R ai n f al l -R u n o f f Even t s Do peak dischar ges at t he event t im e scale obey a gener alized H or t on law ? Gupt a et al. ( 2015 ) invest igat ed t his issue and der ived (11)

Equat ion ( 11 ) denot es t hat peak flow s divided by it s m ean for a RF-RO event , i = 1,2,3, …, and St r ahler or der, . I t look s lik e a gener alized H or t on law in so far as t he r escaled r andom var iable on t he r ight is independent of t he St r ahler or der, . I t w as explained in t he cont ext of geom or phology (Eq. ( 4 )), but t he pr obabilit y dist r ibut ion of changes w it h each event , . This issue does not ar ise for geom or phologic r andom var iables lik e dr ainage ar eas. I n t his sense, t he feat ur e in Eq. ( 11 ) is new and is pur ely hydr ologic in nat ur e. Gupt a et al. ( 2015 ) hypot hesized t hat t he pr obabilit y dist r ibut ion of

is

com m on for all t he event s and t est ed t heir hypot hesis using dat a. The cum ulat ive dist r ibut ion funct ions (CDF) for t he nor m alized peak dischar ges for each of t he 51 event s w as plot t ed, as show n by t he t hin lines in Figur e 8 . Tw o aut hor s have descr ibed k-sam ple t est s: t he Kolm ogor ov–Sm ir nov (KS) (Conover, 1999 ) and a based on lik elihood st at ist ics (Zhang & Wu,

2007 ).

t est

The condit ion t o per for m a k-sam ple KS

t est applies, since each point in a CDF cor r esponds t o a st r eam gauge, w hich doesn’t change w it h event s. H ow ever, t he t est is m or e pow er ful t han t he KS, and t her efor e it w as adopt ed. The r esult s indicat e t hat t her e is insufficient evidence t o r eject t he hypot hesis t hat

, w hich is t he CDFs show n in Figur e 8 by t hin lines, com e fr om a

com m on dist r ibut ion t hat is independent of event s, , as w ell as of t he St r ahler st r eam or der, . This is a new hydr ologic discover y w it h im por t ant fut ur e applicat ions w it h r espect t o flood pr edict ion in gauged and ungauged basins (Gupt a et al., 2015 ).

One applicat ion is br iefly explained next .

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Click t o view lar ger Figur e 9. CDF plot for t he nor m alized peak dischar ges fr om each of t he 51 RF-RO event s. The bold line r epr esent s t he com m on CDF aft er com bining t he individual CDFs show n in t hin gr ey lines. (Repr oduced fr om Gupt a et al., of AI P Publishing.)

2015 ,

w it h per m ission

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A p p l i c at i o n s o f H o r t o n L aw s f o r Test i n g a R ai n f al l -R u n o f f M o d el W i t h o u t Cal i b r at i o n o f Par am et er s Consider a st at ist ical-m echanical syst em for dr aw ing a for m al analogy w it h physical pr ocesses in a r iver net w or k . I n a st at ist ical-m echanical syst em , t her e ar e t w o scales: m icr oscopic, w it h a ver y lar ge num ber of degr ees of fr eedom (N DOF) t hat cannot be m easur ed, and m acr osopic, w it h a few degr ees of fr eedom t hat can be m easur ed. A w ellk now n exam ple is ideal gas law for a syst em in equilibr ium , w hich has only four par am et er s: pr essur e, num ber of m olecules per unit volum e, t em per at ur e, and Bolt zm an const ant (Reif, 1965 ). The br anch of st at ist ical-m echanics der ives m acr oscopic law s fr om assum pt ions about m olecular dynam ics at t he m icr oscopic scale. The hydr ology pr oblem under discussion r epr esent s a sim ilar sit uat ion, but it is far m or e com plex t han an ideal gas. I n or der t o pr esent t he for m al analogy bet w een RF-RO pr ocesses in a r iver basin and t he afor em ent ioned st at ist ical-m echanical syst em , Gupt a ( 2004A , 2004B ) pr ovides a cont ext . Solut ions of t he coupled or dinar y differ ent ial equat ions r epr esent ing m ass and m om ent um conser vat ion given in Eq. ( 3 ) r equir e specificat ion of dynam ic par am et er s descr ibing r unoff gener at ion in hillslopes and w at er flow dynam ics in channel link s. These par am et er s var y spat ially due t o differ ences in geom et r ic, hydr aulic, and biophysical pr oper t ies am ong individual channel link s and hillslopes, w hich incr ease w it h t he ar ea (size) of a dr ainage basin. Ther efor e, t he num ber of differ ent values t hat t he physical par am et er s can t ak e also incr eases as a basin becom es lar ger. Gupt a ( 2004A ) called it N DOF in or der t o under scor e t he idea t hat t he hydr ological com plexit y in a r iver basin can be com par ed t o t he com plexit y of a st at ist ical-m echanical syst em t hat has a ver y lar ge N DOF. To illust r at e t hat t he N DOF in t he RF-RO syst em is ver y lar ge, Gupt a ( 2004A , 2004B ) consider ed an idealized buck et -t ype r epr esent at ion of r unoff gener at ion and t r anspor t pr ocesses for a single hillslope-link syst em . Four physical par am et er s ar e r equir ed t o specify t he r unoff gener at ion for each hillslope and t hr ee par am et er s gover ning t r anspor t dynam ics in a link . These dynam ic par am et er s var y fr om one hillslope-link pair t o anot her. Typical hillslope has an ar ea of t he or der of 0.05 k m 2 . Ther efor e, a 1 k m 2 basin can be par t it ioned int o 1/0.05 = 20 hillslopes and 10 link s, because each link is dr ained by t w o adjacent hillslopes. The num ber of differ ent spat ial values of t he seven dynam ic par am et er s is 20 · 4 + 10 · 3 = 110. This sim ple calculat ion leads t o t he for m ula N DOF = 110·A. I t show s t hat t he N DOF incr eases linear ly w it h t he ar ea of a basin, A. For t he I ow a River basin, N DOF ~ 3.5 m illion. The flood pr oblem is far m or e com plex t han a st at ist ical-m echanical syst em , w hich is t ypically in equilibr ium . But t he RF-RO syst em is an open syst em t hat is not even in a st eady st at e. Appear ance of a lar ge num ber of physical par am et er s at t he hillslope-link scale r aises t w o issues. Fir st is t he need t o develop a t heor et ical appr oach for specifying a lar ge num ber of dynam ic par am et er s (see Fur ey et al.,

2013 ),

but m or e w or k is needed on t his t opic. The

issue of diagnosis of a physical RF-RO m odel is consider ed her e, because it is fundam ent al t o flood pr edict ion in ungauged basins. A for m al analogy bet w een Page 22 of 35


Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s m icr oscopic and m acr oscopic scales in a hydr ologic syst em allow s us t o illust r at e t he st eps r equir ed t o diagnose dynam ic par am et er izat ions in a hydr ologic m odel. Fir st , for a given RF-RO event , est im at e r ainfall int ensit y field in space and t im e. Select a set of dynam ic par am et er s and r un t he hydr ologic m odel t hat is based on t he decom posit ion of a r iver basin int o hillslope and channel-link unit s (M ant illa, Gupt a, & Tr out m an, 2011 ). Sim ulat e a dischar ge hydr ogr aph at each channel junct ion in a net w or k , w hich gives sim ulat ed peak flow s in t he com plet e St r ahler st r eam s of or der ω = 1, 2, 3 … . Then est im at e

using t he t hr ee m et hods as explained in Gupt a et al. ( 2015 ). The

est im at e of scaling slope for t his event is given as

. Sim ilar ly, t he sim ulat ed

peak flow dat a at each com plet e or der St r ahler st r eam can be used t o com put e t he m ean peak flow. The sam e dat a set can be used t o com put e t he r escaled dist r ibut ion r epr esent ing t he gener alized H or t on law as show n in Figur e 9 for t he 51 event s. These t w o H or t on st at ist ics pr ovide t he m acr oscopic law s for diagnosing par am et er izat ions in a RF-RO m odel. The final st ep can be r epeat ed for differ ent event s as needed. The H or t on st at ist ics can also ser ve as a diagnost ic t ool t o under st and changes in peak flow s over annual and longer t im e scales in differ ent clim at es.

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Cl i m at e Var i ab i l i t y an d t h e Sc al i n g T h eo r y o f Fl o o d s i n M ed i u m t o L ar g e B asi n s o f t h e Wo r l d H ow can t he scaling t heor y of floods be gener alized t o global basins spanning differ ent clim at es at t he scale of t he cont inent s and t he ent ir e planet Ear t h? Poveda et al. ( 2007 ) t ook an im por t ant fir st st ep in t his dir ect ion. Their st udy involved conduct ing annual w at er balances over sever al dr ainage basins for t he ent ir e count r y of Colom bia, w hich has w idely var ying clim at es r anging fr om ext r em ely hum id t o sem i-ar id and ar id. Annual st r eam flow s w er e pr edict ed fr om int er polat ed fields of annual pr ecipit at ion and evapot r anspir at ion using gr ound-based and r em ot ely sensed dat a, finding r easonable agr eem ent w it h obser ved st r eam flow s. Am ong m any alt er nat e m odels t o est im at e evapot r anspir at ion, t he best est im at es w er e given by t he w ell-k now n Budyk o equat ion (Budyk o, 1974 ). Poveda et al. ( 2007 ) show ed t hat pow er law s descr ibe t he r elat ionship bet w een annual flood quant iles and dr ainage ar eas, and t he flood scaling par am et er s can be expr essed as funct ions of annual r unoff obt ained fr om t he w at er balance. This line of invest igat ion pr ovides a new r esear ch dir ect ion for m ak ing flood pr edict ions in a changing global hydr o-clim at e syst em due t o hum an influences (Pielk e et al., 2009 ). Gupt a et al. ( 2007 ) suggest ed specific ideas for fut ur e r esear ch. L im a and L all ( 2010 ) invest igat ed t he r ole of clim at e var iabilit y and change using scaling fr am ew or k for floods. They developed and applied hier ar chical Bayesian m odels t o assess bot h r egional and at -sit e t r ends in t im e in a spat ial scaling fr am ew or k and sim ult aneously pr ovided a r igor ous fr am ew or k for assessing and r educing par am et er and m odel uncer t aint ies. The m odels w er e t est ed w it h r econst r uct ed nat ur al inflow ser ies fr om m or e t han 40 hydr opow er sit es in Br azil w it h cat chm ent s ar eas var ying fr om 2,588 k m 2 t o 823,555 k m 2 . Bot h annual m axim um flood ser ies and m ont hly st r eam flow s w er e consider ed. Cr oss-validat ed r esult s show ed t hat t he hier ar chical Bayesian m odels ar e able t o sk illfully est im at e m ont hly and flood flow pr obabilit y dist r ibut ion par am et er s for sit es not used in m odel fit t ing. The m odels developed can be used t o pr ovide r ecor d augm ent at ion at sit es t hat have shor t r ecor ds or t o est im at e flow s at ungauged sit es, even in t he absence of an assum pt ion of t im e st at ionar it y. Since m odel uncer t aint ies ar e account ed for, t he pr ecision of t he est im at es can be quant ified and hypot heses t est s for r egional and at -sit e t r ends can be for m ally m ade. Viglione, M er z, Viet Dung, Par ajk a, N est er, and Bloschl ( 2016 ) developed a new fr am ew or k for at t r ibut ing flood changes due t o at m ospher ic pr ocesses (e.g., incr easing pr ecipit at ion), cat chm ent pr ocesses (e.g., soil com pact ion associat ed w it h land use change), and r iver syst em pr ocesses (e.g., loss of r et ent ion volum e in t he floodplains) based on r egional analysis. Spat ial scaling of flood changes enabled t hem t o invest igat e at t r ibut ion of m ult iple dr iver s, and Bayesian m et hod allow ed est im at ion of t he at t r ibut ion uncer t aint ies. Flood peak dat a for 97 r iver gauges in upper Aust r ia, w it h ar eas r anging

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s fr om 10 k m 2 t o 79,500 k m 2 and r ecor ds for at least 40 year s aft er 1950, w er e used in a r eal case st udy t o illust r at e t he fr am ew or k . Tw o st udies t hat focus on a physical under st anding of floods on t he planet ar y scale ar e O’Conner, Gr ant , and Cost a ( 2002 ) and Devineni, L all, Xi, and War d ( 2015 ). O’Conner et al. ( 2002 ) dealt w it h paleo flood hydr ology, and scaling in floods is im plicit in t heir ar t icle (see Figur e 7 ). Devineni et al. ( 2015 ) invest igat ed t he scaling of ext r em e r ainfall ar eas at t he planet ar y scale.

En g i n eer i n g A p p l i c at i o n s o f t h e Sc al i n g T h eo r y o f Fl o o d s An im por t ant need is t o apply t he scient ific fr am ew or k t hat t he scaling t heor y has developed for innovat ive applicat ions t o pr act ical hydr ologic engineer ing pr oblem s. Tw o exam ples fr om t he paper s t hat t he I ow a Flood Cent er (I FC), U niver sit y of I ow a, has published ar e discussed her e. The I FC w as est ablished follow ing t he 2008 r ecor d floods and is t he only facilit y of it s k ind in t he U nit ed St at es and t he w or ld conduct ing com pr ehensive scient ific and applied r esear ch on floods r oot ed in t he scaling fr am ew or k .

R eal -T i m e Fl o o d Fo r ec ast i n g Kr ajew sk i et al. ( 2017 ) developed a r eal-t im e flood for ecast ing and infor m at ion dissem inat ion syst em for use by all I ow ans. The syst em com plem ent s t he oper at ional for ecast ing issued by t he N at ional Weat her Ser vice. At it s cor e t he I FC for ecast ing m odel is a cont inuous RF-RO m odel based on landscape decom posit ion int o hillslopes and channel link s. Rainfall conver sion t o r unoff is m odeled t hr ough soil m oist ur e account ing at hillslopes. Channel r out ing is based on a non-linear r epr esent at ion of w at er velocit y t hat consider s t he dischar ge am ount as w ell as t he upst r eam dr ainage ar ea. M at hem at ically, t he m odel r epr esent s a lar ge syst em of or dinar y differ ent ial equat ions or ganized t o follow r iver net w or k t opology. N one of physical par am et er s ar e calibr at ed. I t illust r at es t he im pact of t he scient ific ideas developed in t he scaling t heor y of floods on engineer ing applicat ions. The I FC also developed an efficient num er ical solver suit able for high-per for m ance com put ing ar chit ect ur e. The solver allow s t he I FC t o updat e for ecast s ever y 15 m inut es for m or e t han 1,000 I ow a com m unit ies. The input t o t he syst em com es fr om a r adar r ainfall algor it hm , developed in-house, t hat m aps r ainfall ever y five m inut es w it h high spat ial r esolut ion. The algor it hm uses L evel I I r adar r eflect ivit y and ot her polar im et r ic dat a fr om t he WSR-88DP r adar net w or k . A lar ge libr ar y of flood inundat ion m aps and r eal-t im e r iver st age dat a fr om over 200 I FC “ st r eam -st age sensor s” com plem ent t he I FC

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s infor m at ion syst em . The syst em com m unicat es all t his infor m at ion t o t he gener al public t hr ough a com pr ehensive br ow ser -based and int er act ive plat for m .

I m p ac t o f D am s o n R eal -T i m e Fl o o d Fo r ec ast i n g Dam s ar e ubiquit ous in t he U nit ed St at es, and m or e t han 87,000 of t hem acr oss t he nat ion influence st r eam flow s. The significant m ajor it y of t hese dam s ar e sm all and ar e oft en ignor ed in r eal-t im e flood for ecast ing oper at ions and at -sit e and r egional flood fr equency est im at ions. Even t hough t he im pact of individual sm all dam s on floods is oft en lim it ed, t he com bined flood at t enuat ion effect s of a syst em of such dam s can be significant . Ayalew, Kr ajew sk i, M ant illa, Wr ight , and Sm all ( 2017 ) invest igat ed how a syst em of spat ially dist r ibut ed sm all dam s affect flood fr equency acr oss a r ange of dr ainage basin scales using t he 660 k m 2 Soap Cr eek w at er shed in sout heast er n I ow a, w hich cont ains m or e t han 133 sm all dam s. Result s fr om cont inuous sim ulat ion of t he syst em of sm all dam s indicat e t hat t he peak dischar ges w er e r educed bet w een 20 and 70% but t hat t his effect decr eases as t he dr ainage ar ea incr eases. Consider ing t hat m or e sm all dam s ar e being built acr oss w at er sheds in I ow a and elsew her e in t he count r y, t heir r esult s highlight how t he peak dischar ge at t enuat ion effect s of t hese dam s is an addit ional fact or t hat invalidat es t he st at ionar it y assum pt ion t hat is used in at -sit e and RFF analysis. I n par t icular, t hey show ed t hat neglect ing t he effect s of a syst em of sm all dam s can lead t o an over est im at ion of flood r isk . The r esult s show t hat r eal-t im e flood for ecast ing t hat does not account for t he flood at t enuat ion effect s of t hese dam s m ay suffer fr om t he over est im at ion of flood t hr eat s acr oss a r ange of spat ial scales.

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Co n c l u si o n s Pr edict ion of floods in ungauged basins (i.e., at locat ions w her e no st r eam flow dat a exist ) is a global issue because m ost of t he count r ies involved do not have adequat e st r eam flow r ecor ds. Tw o appr oaches have been developed t o solve t he pr oblem . Fir st is t he RFF analysis by t he U SGS. RFF equat ions ar e pur e st at ist ical char act er izat ions t hat use hist or ical st r eam flow r ecor ds in “ hom ogeneous r egions” t o pr edict annual peak flow quant iles in ungauged basins. The second appr oach is t he scaling t heor y of floods t hat abandoned t he hom ogeneous r egions concept and adopt ed r iver basins inst ead. I t has t he explicit goal of incor por at ing flood-pr oducing physical pr ocesses in under st anding t he obser ved scaling, or pow er law r elat ions, bet w een peak flow s and dr ainage ar eas in r iver basins. The scaling t heor y of floods offer s a unified fr am ew or k t o pr edict floods in RF-RO event s and in annual peak flow quant iles in ungauged basins. The t opics cover ed in t he fir st par t of t he ar t icle include: (1) a br ief hist or y is pr esent ed of t he scaling t heor y of floods; (2) dynam ic for m ulat ions gover ning RF-RO event s in r iver basins ar e descr ibed; (3) self-sim ilar r iver net w or k s (t he Tok unaga m odel and t he RSN m odel) ar e br iefly r eview ed; (4) applicat ions of dynam ic for m ulat ion in RSN ar e sum m ar ized; (5) an analyt ical t heor y of H -G in r iver net w or k s is highlight ed and open pr oblem s m ent ioned; and (6) a hydr o-ecological t heor y of int er m it t ent r ipar ian diver sit y on Tok unaga st r eam net w or k s in pr ogr ess is br iefly sk et ched. The t opics in t he second par t include: (1) highlight s of t he fir st paper on a physical basis of annual peak flow quant iles ar e pr esent ed; (2) highlight s of t he discover y of t he classical and t he gener alized H or t on law s for peak flow s in t he 32,400 k m 2 I ow a r iver basin ar e descr ibed. An applicat ion of t he H or t on law s for diagnosing a RF-RO m odel w it hout calibr at ing it s par am et er s is sum m ar ized; (3) gener alizat ion of t he scaling t heor y of floods in m edium t o lar ge basins and t o longer t im e scales t han annual ar e highlight ed (t w o k ey r efer ences on scaling in r ainfall and paleofloods t o under st and scaling in floods on t he planet ar y scale ar e included); and (4) t w o engineer ing applicat ions of scaling t heor y of floods in t he I ow a River basin ar e sum m ar ized. Sever al significant ar eas of r esear ch r em ain: 1 . Com par at ive st udies ar e needed t o t est differ ent m odel par am et er izat ions w it hout calibr at ion using t he H or t on law s for peak flow s. 2 . St udies ar e needed t o t est analyt ical solut ions of Eq. ( 3 ) under physically r ealist ic assum pt ions in r ealist ic self-sim ilar r iver net w or k s. Ram ir ez (N at ional U niver sit y of Colom bia, per sonal com m unicat ion, 2017) has obt ained analyt ical r esult s on floods using t he physical fr am ew or k pr esent ed her e. I t is t he fir st effor t of it s k ind, and paper s ar e being pr epar ed for publicat ion. 3 . The scaling t heor y of floods needs t o be ext ended t o global basins r epr esent ing differ ent hydr o-clim at ic r egions r anging fr om ver y hum id t o sem i-ar id and ar id.

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s 4 . The scaling fr am ew or k has im por t ant r elevance t o flood pr edict ion in a changing clim at e, because t he self-sim ilar it y of r iver net w or k s is not expect ed t o change. As a r esult , t he pow er law s in peak dischar ges at event and annual t im e scales w ould r em ain int act , but scaling slopes and int er cept s for RF-RO event s and annual flood quant iles w ould change, w hich can be pr edict ed fr om physical pr ocesses. 5 . M any physical pr ocesses gover n floods in upst r eam basins and in flood plains, but all of t hem cannot be included in a m odel. A dynam ic under st anding of scaling can be used t o elim inat e t hose physical pr ocesses t hat play a “ secondar y r ole” in t he over all cont ext .

A c k n o w l ed g m en t s Sever al out st anding gr aduat e st udent s helped m e sust ain t he line of r esear ch descr ibed her e over sever al decades. M y jour ney st ar t ed in 1980 w it h t he fir st gr aduat e st udent , Oscar M esa, and ended in 2007 w it h m y last gr aduat e st udent , Ricar do M ant illa. I n bet w een t his t im e fr am e, sever al ot her s joined t he gr oup, Tom over, Sandr a Cast r o, Scot t Peck ham , Set h Veit zer, Pet er Fur ey, and cont r ibut ed t o t he line of r esear ch. I expr ess m y deep gr at it ude for t heir dedicat ed cont r ibut ions. The N at ional Science Foundat ion, t he Ar m y Resear ch Office and t he N at ional Aer onaut ics and Space Adm inist r at ion funding over t his t im e per iod m ade t he r esear ch possible.

R ef er en c es Ayalew, T. B., Kr ajew sk i, W. F., & M ant illa, R. (2014a). Co n n ec t i n g t h e p o w er -l aw sc al i n g st r u c t u r e o f p eak -d i sc h ar g es t o sp at i al l y var i ab l e r ai n f al l an d c at c h m en t p h ysi c al p r o p er t i es. Advances in Wat er Resour ces, 71, 32–43. Ayalew, T. B., Kr ajew sk i, W. F., & M ant illa, R. (2015). A n al yzi n g t h e ef f ec t s o f exc ess r ai n f al l p r o p er t i es o n t h e sc al i n g st r u c t u r e o f p eak -d i sc h ar g es: I n si g h t s f r o m a m eso sc al e r i ver b asi n . Wat er Resour ces Resear ch, 51(6), 3900–3921. Ayalew, T. B., Kr ajew sk i, W. F., M ant illa, R., & Sm all, S. J. (2014b). Exp l o r i n g t h e ef f ec t s o f h i l l sl o p e-c h an n el l i n k d yn am i c s an d exc ess r ai n f al l p r o p er t i es o n t h e sc al i n g st r u c t u r e o f p eak -d i sc h ar g e. Advances in Wat er Resour ces, 64, 9–20. Ayalew, T. B., Kr ajew sk i, W. F., M ant illa, R., Wr ight , D. B., & Sm all, S. J. (2017). T h e ef f ec t o f sp at i al l y d i st r i b u t ed sm al l d am s o n f l o o d f r eq u en c y: I n si g h t s f r o m t h e So ap Cr eek w at er sh ed . Jour nal of H ydr ologic Engineer ing, 22(7), 1–10. Bar enblat t , G. I . (1996). Scaling, Self-sim ilar it y and int er m ediat e asym pt ot ics. L ondon: Cam br idge U niver sit y Pr ess.

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Fur ey, P., Tr out m an, B., Gupt a, V. K., & Kr ajew sk i, W. F. (2016). Co n n ec t i n g even t -b ased sc al i n g o f f l o o d p eak s t o r eg i o n al f l o o d f r eq u en c y r el at i o n sh i p s. Jour nal of H ydr ologic Engineer ing, 21(10), 1–11. Fur ey, P. R., & Tr out m an, B. M . (2008). A consist ent fr am ew or k for H or t on r egr ession st at ist ics t hat leads t o a m odified H ack ’s law. Geom or phology, 102(3–4), 603–614. Govindar aju, R. S., Cor r adini, C., & M or bidelli, R. (2012). L o c al - an d f i el d -sc al e i n f i l t r at i o n i n t o ver t i c al l y n o n -u n i f o r m so i l s w i t h sp at i al l y var i ab l e su r f ac e h yd r au l i c c o n d u c t i vi t i es. H ydr ologic Pr ocesses, 26, 3293–3301. Gupt a, V. K. (2004a). Em er gence of st at ist ical scaling in floods fr om com plex r unoff dynam ics on channel net w or k s. Fr act als, Chaos and Solit ons, 19, 357–365. Gupt a, V. K. (2004b). Pr edict ion of st at ist ical scaling in peak flow s for r ainfall–r unoff event s: A new fr am ew or k for t est ing physical hypot heses. I n Scales in hydr ology and w at er m anagem ent , Kovacs Colloquium I nt er nat ional Associat ion of H ydr ologic Sciences. Pub. 287, 97–110. Gupt a, V. K., Ayalew, T. B., M ant illa, R., & Kr ajew sk i, W. F. (2015). Cl assi c al an d g en er al i zed H o r t o n l aw s f o r p eak f l o w s i n r ai n f al l -r u n o f f even t s. Chaos, 25, 075408. Gupt a, V. K., Cast r o, S., & Over, T. M . (1996). On scaling exponent s of spat ial peak flow s fr om r ainfall and r iver net w or k geom et r y. I n P. Bur lando, G. M enduni, & R. Rosso (Eds.), Fr act als, Scaling and N onlinear Var iabilit y in H ydr ology (Special issue). Jour nal of H ydr ology, 187(1–2), 81–104. Gupt a, V. K., & Daw dy, D. (1995). Physical int er pr et at ion of r egional var iat ions in t he scaling exponent s in flood quant iles. H ydr ologic Pr ocesses, 9, 347–361. Gupt a, V. K., M ant illa, R., Tr out m an, B., Daw dy, D., & Kr ajew sk i, W. F. (2010). Gen er al i zi n g a n o n l i n ear g eo p h ysi c al f l o o d t h eo r y t o m ed i u m -si zed r i ver n et w o r k s. Geophysical. Resear ch L et t er s, 37(11), L 11402. Gupt a, V. K., & M esa, O. J. (1988). Runoff gener at ion and hydr ologic r esponse via channel net w or k geom or phology: Recent pr ogr ess and open pr oblem s. Jour nal of H ydr ology, 102, 3–28. Gupt a, V. K., & M esa, O. J. (2014). H o r t o n l aw s f o r H yd r au l i c -Geo m et r i c var i ab l es an d t h ei r sc al i n g exp o n en t s i n sel f -si m i l ar To k u n ag a r i ver n et w o r k s. N onlinear Pr ocesses in Geophysics, 21, 1007–1025. Gupt a, V. K., M esa, O. J., & Daw dy, D. R. (1994). M ult iscaling t heor y of flood peak s: Regional quant ile analysis. Wat er Resour ces Resear ch, 30, 3405–3421.

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Gupt a, V. K., Tr out m an, B., & Daw dy, D. (2007). Tow ar ds a nonlinear geophysical t heor y of floods in r iver net w or k s. An over view of 20 year s of pr ogr ess. I n A. A. Tsonis & J. B. Elsner (Eds.), Tw ent y year s of nonlinear dynam ics in geosciences (pp. 121–152). N ew Yor k : Elsevier. Gupt a, V. K., & Waym ir e, E. (1998). Spat ial var iabilit y and scale invar iance in hydr ologic r egionalizat ion. I n G. Sposit o (Ed.), Scale dependence and scale invar iance in hydr ology (pp. 88–135). Cam br idge, U.K.: Cam br idge U niver sit y Pr ess. Gupt a, V. K., Waym ir e, E., & Wang, C. T. (1980). A r epr esent at ion of an inst ant aneous unit hydr ogr aph fr om geom or phology. Wat er Resour ces Resear ch, 16(5), 855–862. I bbit t , R. P., M cKer char, A. I ., & Duncan, M . J. (1998). Taier i r iver dat a t o t est channel net w or k and r iver basin het er ogeneit y concept s. Wat er Resour ces Resear ch, 34, 2085– 2088. Jar vis, R. S., & Woldenber g, M . J. (Eds.). (1984). Benchm ar k paper s in geology (Vol. 80). St r oudsbur g, PA: H ut chinson Ross. Jor dan, A. C., & Kean, J. W. (2010). Est ab l i sh i n g a m u l t i -sc al e st r eam g ag i n g n et w o r k i n t h e W h i t ew at er r i ver b asi n , K an sas. Wat er Resour ces M anagem ent , 24, 3641–3664. Kean, J. W., & Sm it h, J. D. (2005). Gen er at i o n an d ver i f i c at i o n o f t h eo r et i c al r at i n g c u r ves i n t h e W h i t ew at er r i ver b asi n , K an sas. Jour nal of Geophysical. Resear ch, 110, F04012, 1–17. Kir k by, M . J. (1976). Test s of t he r andom net w or k m odel, and it s applicat ion t o basin hydr ology. Ear t h Sur face Pr ocesses, 1, 197–212. Kr ajew sk i, W. et al. (2017). R eal t i m e f l o o d f o r ec ast i n g an d i n f o r m at i o n syst em f o r t h e st at e o f I o w a. Bullet in Am er ican M et eor ological Societ y, 98(3), 1–16. L ee, M . T., & Delleur, J. W. (1976). A var iable sour ce ar ea m odel of t he r ainfall-r unoff pr ocess based on t he w at er shed st r eam net w or k . Wat er Resour ces Resear ch, 12(5), 1029– 1035. L eopold, L . B., & M iller, J. P. (1956). Epher m al st r eam s—H ydr aulic fact or s and t heir r elat ion t o t he dr ainage net . U S Geological Sur vey Pr of. Paper 282-A. Washingt on, DC: U.S. Govt . Pr int ing Office. L eopold, L . B., Wolm an, M ., & M iller, G. (1964). Fluvial pr ocesses in geom or phology. San Fr ancisco: Fr eem an. L im a, C. H . R., & L all, U. (2010). Spat ial scaling in a changing clim at e: A hier ar chical bayesian m odel for nonst at ionar y m ult i-sit e annual m axim um and m ont hly st r eam fl ow. Jour nal of H ydr ology, 383(3–4), 307–318.

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s L insley, R. K., Jr., Kohler, M . A., & Paulhus, J. H . L . (1982). H ydr ology for engineer s (3d ed.). N ew Yor k : M cGr aw -H ill. M andapak a, P. V., Kr ajew sk i, W. F., M ant illa, R., & Gupt a, V. K. (2009). Dissect ing t he effect of r ainfall var iabilit y on t he st at ist ical st r uct ur e of peak flow s. Advances in Wat er Resour ces, 32, 1508–1525. M andelbr ot , B. (1982). The fr act al geom et r y of nat ur e. N ew Yor k : Fr eem an. M ant illa, R., & Gupt a, V. K. (2005). A GI S num er ical fr am ew or k t o st udy t he pr ocess basis of scaling st at ist ics on r iver net w or k s. I EEE Geoscience and Rem ot e Sensing L et t er s, 2(4), 404–408. M ant illa, R., Gupt a, V. K., & M esa, O. J. (2006). Role of coupled flow dynam ics and r eal net w or k st r uct ur es on H or t onian scaling of peak flow s. Jour nal of H ydr ology, 322, 155– 167. M ant illa, R., Gupt a, V. K., & Tr out m an, B. M . (2012). Ext ending gener alized H or t on law s t o t est em bedding algor it hm s for t opological r iver net w or k s. Geom or phology, 151–152, 13–26. M ant illa, R., Tr out m an, B. M ., & Gupt a, V. K. (2011). Scaling of peak flow s w it h const ant flow velocit y in r andom self-sim ilar r iver net w or k s. N onlinear Pr ocesses in Geophysics, 18, 489–502. M ant illa, R., Tr out m an, B. M ., & Gupt a, V. K. (2010). Test i n g st at i st i c al sel f -si m i l ar i t y i n t h e t o p o l o g y o f r i ver n et w o r k s. Jour nal of Geophysical Resear ch, 115, 1–12. M cConnell, M ., & Gupt a, V. K. (2008). A pr oof of t he H or t on law of st r eam num ber s for t he Tok unaga m odel of r iver net w or k s. Fr act als, 16(3), 227–233. M cKer char, A. I ., I bbit t , R. P., Br ow n, S. L . R., & Duncan, M . J. (1998). Dat a for Ashley r iver t o t est channel net w or k and r iver basin het er ogeneit y concept s. Wat er Resour ces Resear ch, 34, 139–142. M enabde, M ., & Sivapalan, M . (2001). L ink ing space-t im e var iabilit y of r ainfall and r unoff fields: A dynam ic appr oach. Advances in Wat er Resour ces, 24, 1001–1014. M ilne, B., & Gupt a, V. K. (2017a). H ydr o-ecological t heor y of int er m it t ent r ipar ian diver sit y on st r eam net w or k s. M ilne, B., & Gupt a, V. K. (2017b). H o r t o n R at i o s L i n k Sel f -Si m i l ar i t y w i t h M axi m u m En t r o p y o f Ec o -Geo m o r p h o l o g i c al Pr o p er t i es i n St r eam N et w o r k s. Ent r opy, 19, 1– 15. O’Conner, J. E., Gr ant , G. E., & Cost a, J. E. (2002). The geology and geogr aphy of floods. I n K. P. H ouse, R. H . Webb, V. P. Bak er, & D. R. L evish (Eds.), Ancient floods, m oder n

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s hazar ds: Pr inciples and applicat ions of paleoflood hydr ology (pp. 359–385). Washingt on, DC: Am er ican Geophysical U nion. Ogden, F. L ., & Daw dy, D. R. (2003). Peak dischar ge scaling in sm all H or t onian w at er shed. Jour nal of H ydr ologic Engineer ing, 8(2), 64–73. Peck ham , S. (1995). N ew r esult s for self-sim ilar t r ees w it h applicat ions t o r iver net w or k s. Wat er Resour ces Resear ch, 31(4), 1023–1029. Peck ham , S., & Gupt a, V. (1999). A r efor m ulat ion of H or t on’s law s for lar ge r iver net w or k s in t er m s of st at ist ical self-sim ilar it y. Wat er Resour ces Resear ch, 35(9), 2763– 2777. Pielk e, R., Sr., et al. (2009). Clim at e change: t he need t o consider hum an for cings besides gr eenhouse gases. Eos, 90(45), 413. Poveda, G., et al. (2007). L ink ing long-t er m w at er balances and st at ist ical scaling t o est im at e r iver fl ow s along t he dr ainage net w or k of Colom bia. Jour nal of H ydr ologic Engineer ing, 12(1), 4–13. Reif, F. (1965). Fundam ent als of st at ist ical and t her m al physics. N ew Yor k : M cGr aw -H ill. Rodr iguez-I t ur be, I ., & Rinaldo, A. (1997). Fr act al r iver basins. L ondon: Cam br idge U niver sit y Pr ess. Rodr iguez-I t ur be, I ., & Valdez, J. B. (1979). The geom or phologic st r uct ur e of hydr ologic r esponse. Wat er Resour ces Resear ch, 15(6), 1409–1420. Shr eve, R. L . (1966). St at ist ical law of st r eam num ber s. Jour nal of Geology, 74, 17–37. Shr eve, R. L . (1967). I nfinit e t opologically r andom channel net w or k s. Jour nal of Geology, 75, 178–186. Sm it h, J. A., Baeck , M . L ., Villar ini, G., Wr ight , D. B., & Kr ajew sk i, W. (2013). Ext r em e flood r esponse: The June 2008 flooding in I ow a. Jour nal of H ydr om et eor ology, 14(6), 1810–1825. Tok unaga, E. (1966). The com posit ion of dr ainage net w or k s in Toyohir a r iver basin and valuat ion of H or t on’s fir st law [ in Japanese w it h English sum m ar y] . Geophysics Bullet in H ok k aido U niver sit y, 15, 1–19. Tok unaga, E. (1978). Consider at ion on t he com posit ion of dr ainage net w or k s and t heir evolut ion. Geogr aphy Repor t 13, Tok yo M et r opolit an U niver sit y, 1–27. Tr out m an, B. M . (2016). h t t p s://w at er .u sg s.g o v/n r p /p u b l i c at i o n s.p h p ? p I D = 5 0 4 2 1 6 b 7 e4 b 0 4 b 5 0 8 b f d 3 3 4 f & sc i N am e= B r en t %2 0 M .%2 0 Tr o u t m an .

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Sc al i n g T h eo r y o f Fl o o d s f o r D evel o p i n g a Ph ysi c al B asi s o f St at i st i c al Fl o o d Fr eq u en c y R el at i o n s Tr out m an, B., & Kar linger, M . (1984). On t he expect ed w idt h funct ion of t opologically r andom channel net w or k s. Jour nal of Applied Pr obabilit y, 22, 836–849. Tr out m an, B., & Kar linger, M . (1988). Asym pt ot ic Rayleigh inst ant aneous unit hydr ogr aph. St ochast ic H ydr ology and H ydr aulaulics, 2, 73–78. Tr out m an, B., & Kar linger, M . (1989). Pr edict or s of peak w idt hs for net w or k s w it h exponent ial link s. St ochast ic H ydr ology and H ydr aulics, 3, 1–16. Tr out m an, B., & Kar linger, M . (1998). Spat ial channel net w or k m odels in hydr ology. I n O. E. Bar ndor ff-N ielsen, V. K. Gupt a, V. Per ez-Abr eu, & E. C. Waym ir e (Eds.), St ochast ic m et hods in hydr ology: Rain, landfor m s and floods. Singapor e: Wor ld Scient ific. Tr out m an, B., & Over, T. (2001). River flow m ass exponent s w it h fr act al channel net w or k s and r ainfall. Advances in Wat er Resour ces, 24, 967–989. Tr out m an, B. M . (2005). Scaling of flow dist ance in r andom self-sim ilar channel net w or k s. Fr act als, 13(4), 265–282. Tur cot t , D. (1997). Fr act als and chaos in geology and geophysics (2d ed.). N ew Yor k : Cam br idge U niver sit y Pr ess. Veit zer, S. A., & Gupt a, V. K. (2000). Random self-sim ilar r iver net w or k s, sim ple scaling and gener alized H or t on law s. Wat er Resour ces Resear ch, 36(4), 1033–1048. Veit zer, S. A., & Gupt a, V. K. (2001). St at ist ical self-sim ilar it y of w idt h funct ion m axim a w it h im plicat ions t o floods. Advances in Wat er Resour ces, 24, 955–965. Viglione, A., M er z, B., Viet Dung, N., Par ajk a, J., N est er T., & Bloschl, G. (2016). At t r i b u t i o n o f r eg i o n al f l o o d c h an g es b ased o n sc al i n g f r am ew o r k . Wat er Resour ces Resear ch, 52, 5322–5340. Waym ir e, E. (2016). h t t p ://m at h .o r eg o n st at e.ed u /p eo p l e/vi ew /w aym i r ee. Zhang, J., & Wu, Y. (2007). K-Sam ple t est s based on t he lik elihood r at io. Com put at ional St at ist ics & Dat a Analysis, 51(9), 4682–4691.

Vi j ay Gu p t a

Civil, Envir onm ent al and Ar chit ect ur al Engineer ing, U niver sit y of Color ado

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Scaling Theory of Floods for Developing a Physical

Scaling Theory of Floods for Developing a Physical

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