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View of Ponte Vecchio in Florence before the 1966 flood.

The Arno River Flood Study (1971-1976) Lorenzo Panattoni IBM Scientific Center of Pisa, Pisa, Italy

James R. Wallis IBM T. J. Watson Research Center, Yorktown Heights, New York

T h e Arno River in Italy intermittently inundates large a r e a s of the Arno River basin . Th is paper invest igates flood p h e n o m e n a and eva lua tes s o m e of the proposals that have been m a d e to a l lev iate the flood hazard using procedures deve loped at the IBM Pisa Scient i f ic Center .

C o v e r The design on the cover is the new AGU logo. The logo is the culminat ion of the work of the staff art department , Dae Sung Kim, Pamela Thompson, and Ed Pitts, supervisor. In the spir i t of de gustibus non dispu-tandum, the logo was presented to the Counci l last month in San Francisco, where it was not d isapproved.

0 0 9 6 - 3 9 4 1 / 7 9 / 0 0 0 1 / 0 0 0 1 $01.00 Copyright 1 9 7 9 by the American Geophysical Union.

The Arno River Flood Phenomena

On the morning of November 4 , 1 9 6 6 , the residents of Florence, Italy, awoke to f ind themselves the v ict ims of a major f lood. Thir ty-s ix people were drowned in Florence, and replaceable real property damage has been est imated at $640 ,000 ,000 (C. Pandolf i , personal communicat ion , 1977) . In addi t ion, art t reasures of inest imable value were lost, and the c leanup operat ion at t racted wor ldwide at tent ion, sympathy, and help [Judge, 1967] . As an i l lustrat ion of the amount of damage caused by this catast rophic f lood, we have reproduced a photograph of Ponte Vecchio (Figure 1) taken on the morning of November 5 , 1 9 6 6 .

Engineer ing st ructures and land use zoning measures can usual ly be des igned so as to prevent most downstream f lood damages, but the protect ion is rarely absolute and may become prohibi t ively expen­sive as one designs to protect against ever larger and rarer catast rophic events. For a rational evaluat ion of al ternat ive f lood control measures it is helpful to have an accurate f lood f requency analysis that is an est imate of the probabi l i t ies of f lood events of var ious mag­nitudes. Customari ly, these f lood probabi l i t ies are expressed as recurrence interval Tfor f loods of a given magnitude. To est imate Tfor a major f lood, it is cus tom­ary to consider only the highest annual f low recorded for each year at a convenient s t ream-gaging stat ion. A little upstream from Florence, at Nave di Rosano, there is a s t ream-gaging stat ion that was establ ished in 1920. The records are d iscont inuous dur ing the war

years, but the instantaneous peak f low for each of 35 years is known. These data were ranked and plotted on Gumbel ext reme value probabi l i ty paper (Figure 2). The data were f i t ted to a Gumbel ext reme value d is t r ibut ion using both the method of moments (line A, Figure 2) and the method of maximum l ikel ihood (line B, Figure 2). By the method of moments the 1966 f lood is est imated at T = 633 , whi le the method of maximum l ikel ihood yie lds T = 3 6 7 5 . Whi le Gumbel extreme value procedures are commonly used in Europe, they are not favored by the U.S. Water Resources Counci l , wh ich has recom­mended using a Pearson type III d is t r ibut ion wi th the method of moments and the ca lcu la t ions conduc ted in logar i thmic space [U.S. Water Resources Council, 1 9 7 6 , 1 9 7 8 ] .

If this procedure is fo l lowed for the Nave di Rosano data, the th i rd moment in logar i thmic space is + 0 . 1 , wh ich leads to very high probabi l i t ies for extremely large f loods. As can be seen from line C in Figure 2, the recurrence interval for the 1966 f lood is est imated at T = 200 years, and several even more extreme f loods cou ld be expected in a 1000-year interval, but as we shal l see, the histor ic information does not accord wel l wi th such a supposi t ion.

It is clear f rom the above analyses that the 1966 f lood was an extreme event, but there have been many other f loods in Florence before the Nave di Rosano s t ream-

gaging program was ini t iated. Spurred by the 1966 f lood, the histor ical ev idence for past major f loods has been the subject of extensive invest igat ions [Losacco, 1 9 6 7 ; Cawna, 1969] . For the 1333 and 1557 f lood events the water levels reached were marked on many of the bui ld ings of Florence. Some of these markers sti l l exist (Figures 3 and 4). Vi l lani, a prominent wr i ter of the per iod, stated that the f lood of 1333 was less than that of 1269 , a l though the damage in the c i ty was reported to have been greater. No markers exist for the 1269 or 1171 inundat ion levels, but they are both bel ieved to have been at least comparab le to those of 1333 and 1557 .

This exogenous information on the f requency of past large Arno River f loods can be incorporated into the f lood f requency analysis in the manner suggested by the U.S. Water Resources Counci l (WRC). Their report, Bul let in 17A, conta ins addi t ional procedura l ad just ­ments that are recommended when it is known that a large f lood in a short record has not been exceeded in a longer histor ical period. For the Arno River it appears cer ta in beyond reasonable doubt that no f lood s ince 1 270 has exceeded the 1966 event. Using the WRC procedures and ass ign ing T = 7 0 0 for the 1966 event y ie ld a negat ive third moment in log space (—0.1), wh ich assures that a maximum certa in f lood can be ca l ­cu la ted from the data and that ext reme events wil l be

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Fig. 1 . View of Ponte Vecchio in Florence after the 1966 flood.This photograph was taken on the morning of Novem­ber 5 , 1 9 6 6 . The boutiques that line both sides of the bridge were totally destroyed. Three bridges have occupied this site. The first, a Roman bridge, disappeared sometime before 1080 . The second, built in 1080 , withstood major floods in 1171 and 1269 and was destroyed by the flood of 1333 . The present bridge was designed by Tad-deo Gaddi and completed in 1345 ; it has withstood major floods in 1557 as well as in 1966.

Fig. 2. Plot of the available 24 years of highest instantaneous peak flows for the Nave di Rosano Gaging Station. The graph paper is Gumbel extreme value type 1, and a Weibull plotting position was used. A, Gumbel method of moments; B, Gumbel method of maximum likelihood; C, log Pear­son III, and D, log Pearson III 'adjusted' for WRC historic procedures.

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assigned extremely low probabi l i ty of occur rence (line D, Figure 2). In fact, by fo l lowing the WRC Bul let in 1 7 A procedures for histor ic f loods, one is led to an erroneous and inconsistent est imate of T = 200 ,000 for the 1966 event.

It is evident that the WRC procedures lead to either r id iculously high or incredib ly low est imates of the Arno River f lood hazard, and they should therefore be d is­regarded. However, the Gumbel extreme value pro­cedures and the histor ical ev idence conf i rm that the f lood hazard in the Arno River basin is severe and that whi le f loods of the 1 966 magni tude may be rare, others of only sl ight ly less severi ty have occur red on four occas ions wi th in the last 8 0 0 years. It was the f lood of 1966 , however, wh ich ini t iated and gave impetus to the Arno River f lood study wh ich is to be descr ibed in the fo l lowing sect ions.

Overview of the Arno River Flood Study

To predict, contro l , and evaluate the Arno River f lood hazard, it would be useful to have an accurate, easy to use model of the f lood phenomena. With this goal in mind a group of sc ient is ts at the IBM Scient i f ic Center of Pisa and the Ist i tuto di Idraul ica of the Universi ta di Pavia set up a joint research study that lasted 6 years and took approximately 15 man-years of work. Whi le the init ial and the f inal goal a lways remained the con ­st ruct ion of a computer -based model of the Arno River f lood phenomena, the study also inc luded a cr i t ical examinat ion of theory and exper imental evaluat ion of many numer ical al ternat ives. Computer programs for most of these al ternat ive a lgor i thms did not exist and had to be prepared, debugged, and evaluated as the work Droceeded [Ciriani et a/., 1977] .

Partial and intermediate results and f ind ings of the study have been publ ished in numerous journals and books wr i t ten in both Engl ish and Ital ian, and a massive three volume documentat ion of the whole model has been publ ished in Italian [IBM, 1976] . This paper is an attempt at an Engl ish language summary of the model ; much has been omit ted, but it is hoped that suf f ic ient remains for the interested reader to understand the model, its implementat ion, and its usefulness.

The Arno River Basin

The Arno River basin has an area of about 8229 km 2 , with a main channel of approximately 245 -km length from its source on the slopes of Mount Falterona to the Tyrrhenian Sea below Pisa. Proceeding downstream, the Arno River receives several major t r ibutar ies as well as many minor ones (Figure 5).

For the purpose of f lood forecast ing at Florence, the major contr ibutors are the Sieve and Casent ino regions, both of wh ich are steep mounta inous regions that can exper ience heavy fall and winter rains and have shal low impermeable soi ls developed from calcar ious marine sediments. In part icular, the d isastrous f lood at Florence on November 4 , 1 9 6 6 , was precipi tated by the nearly contemporaneous arrival of major f lood waves from both the Sieve and Casent ino regions. The other subregions of the Arno River basin have a more rol l ing topography with verdant hil ls that are largely devoted to agr icul ture and gently s loping r ivers that usual ly meander quiet ly on wide al luvial plains.

The Data Base

A hydrologic model is only as good as the data that are avai lable to cal ibrate and operate it, and for the Arno River the data were far from opt imal . In fact, model choice was often restr icted by the def ic iencies in the exist ing data base and by the prohibi t ive costs in t ime and money associated with obta in ing more and better machine readable data.

The main channel from Levane (Figure 5) to the sea was surveyed at intervals of —200 m in the period 1 9 5 2 - 1 9 6 3 by the Uff ic io Idrograf ico. The main chan­nel from Levane (Figure 5) to the sea was surveyed at intervals of —200 m in the period 1 9 5 2 - 1 9 6 3 by the Uff icio Idrograf ico deH'Arno. The avai labi l i ty of these 9 0 0 preexist ing cross sect ions al lowed the main chan ­nel rout ine of the Arno model to use gradual ly varied nonuni form partial di f ferent ial equat ions; the so-cal led equat ions of Saint Venant. (Note: the mathemat ical descr ipt ion of the model has been relegated to an appendix, avai lable on microf iche, where the interested reader can obtain detai ls about the Saint Venant equa­t ions and other matters.)

There are eight cont inuous record ing stream gages in the Arno River basin, three cont inuous stage recorders, and a few other staff gages that are only observed in t imes of high flow. Information about these gages is conta ined in Table 1, and their locat ions are shown in Figure 5. The records for all of these stat ions are d is ­cont inuous dur ing the war years and also somet imes for other periods. More data are avai lable for the postwar per iod than for the prewar per iod, and the postwar per iod was selected for the model cal ibrat ion. The avai labi l i ty of data for the pr incipal stream gages in the postwar per iod is shown in Figure 6. None of these hydrologic data were avai lable in machine readable form at the start of the project, and preparing the data for computer process ing was a major chore for the pro­jec t staff.

Fifty recording rain gages were operat ional wi thin the Arno River basin dur ing the postwar per iod (see Figure 5 for locat ions). Not all gages were operat ional at all

t imes, the data were co l lec ted by more than one agency, and none of the measurements were avai lable in machine readable form at the commencement of the project. Obta in ing usable weighted average storm ra in­fal ls for all the subbas ins l ikewise proved to be a tedious and t ime consuming process.

The Arno Model

As previously ment ioned, the main channel f lood rout­ing model is based upon the solut ion of a set of one-d imensional part ial di f ferent ial equat ions that were f irst formulated by Barre de Saint Venant in 1871 and which have become known as the Saint Venant equat ions. The Saint Venant equat ions are der ived from conservat ion pr inc ip les, the f irst from the conservat ion of mass and

Fig. 3. Plaques marking the depth of inundation for the 1 3 3 3 and 1 9 6 6 floods at the intersec­tion of Via S. Remigio and Via dei Neri.

the second from the conservat ion of momentum (detai ls are to be found in the appendix (on m ic ro f i che ) . 1

Various numer ical solut ions of the Saint Venant equa-t idns are possible, and the these along with an exp lana­t ion of the cal ibrat ion procedures for them are also c o n ­ta ined in the appendix.

The rainfal l runoff procedures used for each con ­t r ibut ing subcatchment area were based upon a l inear sys tems approach. For the cr i t ical f lood producing ca tchments , the Casent ino and Sieve, and also the less cr i t ica l Chiana, suf f ic ient data were avai lable to al low for the coup l ing of two paral lel l inear systems, wh ich

gave a very good approximat ion to the overal l nonl inear ca tchment responses [Todiniand Wallis, 1977] .The basis for choos ing between which of the two linear systems to use was by reference to a threshold value and the cumulat ive precip i tat ion in a preceding t ime period (see appendix for fur ther detai ls). The other sub-basins of the Arno River do not have stream gages near their conf luences with the Arno River, and the extra accuracy of using coupled l inear systems could not be just i f ied. However, the other subbasins have histor ical ly been of comparat ively minor importance to the overall f lood phenomena of the Arno River. Figure 7 is a schemat ic diagram of the overal l model.

1 Appendix is available with entire article on microfiche. Order from American Geophysical Union, 2 0 0 0 Florida Avenue, N.W., Washington, D.C. 2 0 0 0 9 . Document E 7 9 - 0 0 1 ; $1 .00. Payment must accompany order.

Computer Implementation of the Model

All of the computer programs for the Arno model were wri t ten in Fortran IV and were designed to be used in an

Fig. 4. Plaques marking the depth of inundation for the 1557 and 1 9 6 6 floods in Piazza di Santa Croce.

interact ive manner from a remote terminal . The model has been extensively tested under the CP/CMS monitor system on an IBM 3 7 0 - 1 6 8 owned and operated by Centro Nazionale Universi tar io Calcolo Elet t ronico at Pisa. Some of the rout ines are also avai lable in a batch mode formulat ion. All the operat ional detai ls needed to run the model are conta ined in volume 3 of the pre­viously ment ioned report [Cirianiera/., 1977 ] , and the necessary data to reproduce the cal ibrat ion and tests of the model are given in volume 2. A s impl i f ied f lowchar t for the main program of the Arno model is d isp layed in Figure 8.

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Tests of the Model

The completed model has been tested on three of the larger histor ic f loods that were not used for cal ibrat ion of the model, the f loods of 1 9 5 1 , 1 9 6 0 , and 1968. The f lows predicted by the model for three cross sect ions ol the Arno River, Nave di Rosano, Bruc ianesi , and San Giovanni , are shown with dot ted l ines in Figure 9a for 1 9 5 1 , Figure 9bfor 1960, and Figure 9 c for 1968. These predicted f lood waves were made on the basis of the precip i tat ion data for the events and the previously ca l ibrated model, that is, no on- l ine updat ing of model

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predict ions was involved. For compar ison purposes the measured f lood heights have been shown in the f igures (when avai lable), and it can be seen that the predicted values are comparat ively good for the 1951 and 1960

f lood events and much less good for the 1968 event. The channel c ross-sect ion surveys for the Arno River

were accumula ted in the period 1 9 5 2 - 1 963 , and it is bel ieved that t ime and the 1966 f lood have caused

Table 1. Stream Gages in the Arno River Basin

Name of Station Area of Basin, km 2 Beginning Year of Record Elevation Above Sea Level, m Type of R

Arno River at 738.0 1929 249.95 Mr Subbiano

Chiana River at 1271.0 1926 229 .80 Mr Ponte Ferrovia

Sieve River at 830.6 1921 92.47 Mr Fornacina

Arno River at 4082 .8 1920 72.33 Mr Nave di Rosano

Greve River at 124.1 1953 90 .00 Mr Strette di Bifonica

Bisenzio River at 150.0 1937 92.98 Mr Gamberame

Arno River at 5469 .0 1927 25.93 Ir Brucianesi

Elsa River at 805 .8 1950 46 .00 Mr Castelfiorentino

Arno River at 7007 .00 1916 12.73 I Callone

Arno River at 8181 .2 1916 9.22 I Leoncini

Arno River at 8186 .8 1922 6.78 Mr S. Giovanni

Arno River at 8212 .0 1916 3.08 I Pettori

Arno River at 8220 .0 1916 0.02 I Politeama

Arno River at 8224 .7 1916 - 0 . 0 6 Ir Sostegno

Arno River at 8226 .0 1916 0.31 Ir Cascine Nuove

Mr, continuous recording stream gage of volumes of flow, Ir, continuous recording stream gage of stage height. I, intermittent (high flow) stream gage of stage height.

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Fig. 8. Simplified flowchart of the Arno River model.

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many changes in channel geometry. New measure­ments of the channel geometry are current ly being made, and recal ibrat ion with the newer measurements should great ly increase the accuracy of subsequent model predict ions.

It should be noted that whi le wai t ing for these new cross-sect iona l surveys to be completed, model

forecasts for later per iods could be made more accur­ate by a recal ibrat ion of the roughness coef f ic ients using pos t -1966 data for the cal ibrat ion per iod. In essence, the roughness coeff ic ient is a f i t t ing para­meter that incorporates model errors as well as energy losses, and this approach might obviate much of the need for expensive resurveys of the channel .

Uses of the Model

Catastrophic f loods generate numerous commit tees and panels of experts who make plans and recommen­dat ions to avert simi lar damages in the future. The 1966 Arno f lood has not proved an except ion, and there is now a plethora of proposals and counterproposals to al leviate the f lood hazards of the Arno River valley, It is not yet clear which of the many al ternat ive proposals wi l l emerge as the f inal cho ice. However, a model such as the one that has been presented in this paper can be used to help in the evaluat ion process. An example of this type of use for the Arno model has been reported by Todiniand Buffoni [1976] .

Todini and Buffoni used a s impl i f ied version of the Arno model to predict the indiv idual and combined ef fects of three proposed f lood control measures. The proposals wh ich they s tudied were (1) channel improvements in the v ic in i ty of Florence, (2) a dam on the Sieve River of 20,000 * 1 0 6 m 3 capaci ty in the v ic in i ty of Dicomano, and (3) a dam of 70 ,000 * 1 0 6 m 3

capac i ty at Laterina ( jus t above Levane and below the

Chiana-Arno conf luence) . For the 1966 f lood event their conc lus ions were that the combinat ion of both dams with channel improvements to al low 3 5 0 0 m 3 / s of f low with in the banks would have total ly prevented inundat ion in Florence. Lesser channel improvements, or the const ruct ion of only one of the dams would give lesser protect ion, the results being summarized in Figure 10. By Monte Carlo s imulat ion of numerous other f lood events it was determined that the f requencies of possible future inundat ions of any given size would be reduced by these measures, as wel l as the ratios of inundat ion amounts to total f lood f low. A further interest ing d iscovery of this use of the Arno model was the f inding that if the f lood gages instal led at the base of t t ie proposed reservoirs were made suf f ic ient ly large, then the protect ion against inundat ion at Florence was not a funct ion of whether the reservoirs were full or empty at the commencement of the f lood-produc ing storms. This f ind ing is important, as it appears to al low for the reservoirs to be managed for water supply, recreat ion, or pol lut ion abatement, wi thout compromis­ing their f lood reduct ion propert ies.

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ADMISS IBLE IN BANK CHANNEL FLOW ( m 3 / s )

Fig. 10. Evaluation of proposed flood control measures for the Arno River at Florence with reference to the Novem­ber 4 , 1 9 6 6 , event for different admissi­ble in-bank flows: (1) without reser­voirs, (2) Sieve reservoir, (3) Arno reservoir at Laterina, and (4) both reservoirs.

Simulat ion models such as the Arno model are used in many parts of the wor ld for making real t ime forecasts of f lood events, a l though other techn iques are l ikely to be super ior for this purpose. An early warn ing f lood forecast ing system was not a pol i t ical ly or economica l ly feasib le appl icat ion for the original Arno River model ing group to undertake. However, the relevant publ ic author i t ies have expressed some interest in such an appl icat ion, and the complete model has been made avai lable to them. The f inal on- l ine f lood forecast model when it emerges wil l hopeful ly be a highly modi f ied ver­sion of the Arno model with a Kalman f i l ter updat ing capac i ty and telemeter to upstream sensors [Todini and IYa// /s.1978].

References

Cavina, G., Le Grandi Inondazioni dell'Arno Attraverso i Secoli, 225 pp., Bonechi Editore, Firenze, Italy, 1969 .

Cir iani , T. A., U. Maione, and J. R. Wal l is (Eds.), Mathematical Models for Surface Water Hydrology, John Wiley, New York, 1977.

IBM, Modello Matematico delle Piena dell'Arno, vol. 1 , Studi e Ricerche per la Slmulazione dei Fenomeni di Formazione e Propagazione delle Onde di Piena nel Bacino dell'Arno, 210 pp., vol. 2, Dati Utilizzati per la Costruzione del Modello, 496 pp., vol. 3, Organizzazione dei Dati e Programmi di Calcolo, 79 pp., Pisa, Italy, 1976 .

Judge, J., F lorence r ises from the f lood, Nat Geogr., 732(1), 1 - 4 3 , 1 9 6 7 .

Losacco, U., Notizie e considerazioni sul le inondazioni d ' Arno in Firenze, 140 pp., Coi t ipi del l ' ls t i tu to Geograf ico Mil i tare, Florence, Italy, 1967.

Todin i , E., and E. Buffoni ; Studio su una possib le d i fesa di Firenze dal le al luvioni mediante serbatoi , No. 6, Bol let t ino degl i Ingegneri , Florence, Italy, 1976 .

Todin i , E., and J. R. Wal l is ; Using CLS for dai ly or longer period rainfal l model l ing, in Mathematical Models for Surface Water Hydrology, edi ted by T. A. Cir iani , U. Maione, and J. R. Wall is, pp. 1 4 9 - 1 6 8 , John Wiley, New York, 1977.

Todini , E., and J. R. Wall is, A real t ime rainfal l runoff model for an onl ine f lood warn ing system, paper pres­ented at the Chapman Conference on Appl icat ions of Kalman Filters to Hydrology, Hydraul ics, and Water Resources, AGU, Pi t tsburgh, Pa., May 1978.

U. S. Water Resources Counc i l , Guidel ines for f lood f re­quency analysis, Bull. 17, Washington, D. C , 1976 .

U.S. Water Resources Counc i l , Guidel ines for de te rmin­ing f lood f low f requency, Bull. 17A, Washington, D. C , 1978.

Lorenzo Panattoni, a native of Pisa, Italy, received his degree in phys ics from the Universi ty of Pisa in June 1968 . He jo ined the IBM Scient i f ic Center of Pisa in 1969 , where he is a researcher in the f ield of river model ing, wi th specia l emphasis on f luvial hydraul ics.

James R. Wal l is received his B.S. from the Universi ty of New Brunsw ick , his M.S. from Oregon State Universi ty and his Ph.D. from the Universi ty of Cal i fornia at Berkeley. He served as advisor for the Arno River Flood Study Group at the IBM Scient i f ic Center, Pisa. At pre­sent he is a research staff member wi th the IBM T. J. Watson Research Center, York town Heights, New York. He is also president-e lect of AGU's Hydrology Sect ion.