Analysis of rainfall threshold curves for flood forecasting in ungauged basins and its application on a real case : Bogot river basin

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POLITECNICO DI MILANO

School of Civil, Environmental and Land Management Engineering

Master Degree in Environmental and Land Planning Engineering

Analysis of rainfall threshold curves for flood forecasting

in ungauged basins and its application on a real case:

Bogotá River Basin

Tutor: Ing. Giovanni Ravazzani

Master degree thesis submitted by:

Luisa Ximena López Tamayo

(832807)

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III

Acknowledgements

Nothing of this could be possible without my family whom encouraged me even when I was down and wanted to give up. Specially to my parents, thank you for all the support that kept me going; all my achievements are for you.

Also, I want to give special thanks to my tutor Giovanni Ravazzani, which is one of the professors who I admire the most, his patience and willing to help were very valuable to me. Thank you for helping me going through this road.

Finally, thank you to all my friends which made this path in Milan way easier. Special thanks to those friends who kept me going and spent hours helping me to understand things about life and university.

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IV

Abstract

Since it is difficult to count on detailed information from basins so it can be used in a distributed model, a lumped model to forecast flood events have been identified and analyzed to be used in ungauged basins. The Rainfall Threshold Curves method have been chosen for the present project due to its simplicity and low quantity demand of basin parameters. With the latter method, the threshold curves were calculated for 39 Italian sub-basins and later compared with the curves obtained by the FEST-WB distributed model to identify how different the results from both models were and therefore have an idea of the accuracy of the lumped model. Later, a comparison of the curves obtained from both models with 40 real events was made to understand how well the lumped model was at identifying heavy rainfall events in comparison with the FEST-WB model. The results were 21 and 19 correctly identified events for the lumped model and the distributed model respectively. Afterwards, a sensitivity analysis was made to identify which parameters were most important for the calculation of the threshold curves with the lumped model to subsequently use it in a real ungauged basin. The results were that the most important parameters were the CN and length of the stream with the maximum order. Finally, considering the information delivered by the previous analysis, the lumped model was applied for the calculation of the threshold curves on Bogotá River Basin which correctly identified more than half of the analyzed events.

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V

Sintesi

Nel documento AR5 (Fifth Assessment Report) rilasciato dal Gruppo intergovernativo sul cambiamento climatico, viene spiegato che a causa del cambiamento climatico gli eventi forti di pioggia saranno più comuni di prima. Se si considera questo oltre al fatto che la maggior parte dei bacini idrografici nel mondo non sono in possesso di informazione di qualità (chiamati bacini non strumentati), la severità degli impatti generati dagli eventi di esondazione potrebbe incrementare.

Parecchie ricerche sono state fatte nel campo delle previsioni di eventi di esondazione risultando in strumenti come i modelli distribuiti e concentrati che sono in grado di identificare gli eventi importanti di pioggia in anticipo. Alcuni modelli distribuiti che sono stati sviluppati, hanno bisogno di informazione dettagliata del bacino il cui è generalmente suddiviso in una griglia che è in grado di catturare la eterogeneità spaziale delle caratteristiche del bacino (CN, pendenza, precipitazione). Dato questo, la quantità e qualità delle informazioni utilizzate da questi modelli può essere elevata.

Siccome può essere difficile avere informazione con un livello molto dettagliato dai bacini non strumentati, è necessario utilizzare modelli che utilizzino pochi dati ma che ugualmente arrivino a risultati affidabili. In questo ambito esistono modelli concentrati che con pochi dati del bacino sono in grado di calcolare diversi parametri che permettono prevedere situazioni di inondazione. Uno dei modelli concentrati utilizzati in questo campo è quello delle soglie pluviometriche, il cui per la sua semplicità nel calcolo della quantità di pioggia necessaria per innescare un evento di piena in una fissata sezione è stato utilizzato nel presente progetto. Per sapere quanto affidabile può essere il modello concentrato scelto, le soglie pluviometriche di 39 bacini italiani localizzati sul Bacino del Po sono state calcolate e poi confrontate con le soglie pluviometriche risultanti dal modello FEST-WB. Fatto questo, una delle scoperte più interessanti è stata quella di vedere che anche se le curve non fossero uguali, gli ordini di grandezza sono stati simili e per alcuni dei bacini più piccoli le curve calcolate con tutti i due modelli sono molto vicine tra loro soprattutto per lo ietotipo III. Inoltre, altro risultato importante per questo analisi è stato che in generale le curve calcolate col modello concentrato sono in generale più basse da quelle fatte col modello distribuito; questo andamento può essere spiegato dal fatto che diversamente dal modello FEST, il modello concentrato non considera diversi processi e percorsi che l’acqua può avere prima di far parte del deflusso superficiale.

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VI Inoltre, un confronto tra le curve calcolate dai due modelli e 40 eventi reali è stato fatto per capire la performance di questi nel momento di identificare eventi di piena; fatto questo, i resultati sono stati riassunti in tabelle di contingenza. Il confronto fatto ha dato dei risultati molto interessanti; da 34 eventi di piena osservati, tutti i due i modelli hanno identificato correttamente 21 e 19 eventi per il caso del modello concentrato e quello distribuito rispettivamente. In aggiunta, solo c’è stata una falsa allarma dai due modelli sui 6 eventi che hanno avuto una portata vicina alla soglia ma non la hanno superata.

Dopo avere questi resultati, con l’obiettivo di utilizzare il modello concentrato in bacini non strumentati reali, un’analisi di sensibilità è stato fatto per capire quali sono i parametri più importanti del modello. Questo analisi è stato fatto perché dato che si è in mancanza di dati, bisogna sapere quali input devono essere precisi per evitare avere grossi errori nel momento di calcolare le soglie pluviometriche col modello concentrato. Il risultato è stato che i parametri più importanti, che se non precisi nel momento del calcolo genererebbero grosse incertezze nelle soglie risultanti, sono il CN e la lunghezza del ramo di ordine massimo.

Finalmente, considerando i risultati dalle analisi fatte, il modello concentrato è stato utilizzato per calcolare le soglie pluviometriche per il Bacino del Fiume Bogotá localizzato nel centro della Colombia. Utilizzato il modello, i risultati sono stati incoraggianti avendo per le aree studiate (Santa Rosita e Puente Florencia) metà degli eventi analizzati sono stati correttamente identificati.

Parole Chiavi: Soglie Pluviometriche, Previsioni di esondazioni, Modelli concentrati, Bacini

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VII

List of Figures

FIGURE 1.RAINFALL THRESHOLD CURVE EXAMPLE TAKEN FROM ROSSI AND VIGANÒ'S ... 9

FIGURE 2.RAINFALL PROFILES WHICH DESCRIBE THE INTENSITY DEVELOPMENT IN TIME FOR THREE CASES: STEADY (TYPE I), LINEARLY INCREASING (TYPE II), LINEARLY DECREASING (TYPE III) ... 10

FIGURE 3. EXAMPLE OF THE COMPARISON BETWEEN THE RAINFALL THRESHOLD CURVE AND A REAL HEAVY RAINFALL EVENT ... 11

FIGURE 4.DETENTION COEFFICIENT CURVE TAKEN FROM ROSSI AND VIGANÒ’S BACHELOR THESIS (2002) AND TRANSLATED ... 14

FIGURE 5.EXAMPLE OF THE RAINFALL THRESHOLD CURVES OBTAINED WITH THE ITERATION METHODOLOGY PROPOSED BY CROLEY (1977) AND USED BY ROSSI AND VIGANÒ (2002) ... 18

FIGURE 6.GENERAL DIAGRAM FOR THE EXPLANATION FOR THE OBTAINMENT OF RAINFALL THRESHOLD CURVES WHICH WAS DESCRIBED AND DEVELOPED BY ROSSI AND VIGANÒ THESIS (2002) ... 19

FIGURE 7.SPECIFIC DIAGRAM FOR THE EXPLANATION FOR THE OBTAINMENT OF RAINFALL THRESHOLD CURVES WHICH WAS DESCRIBED AND DEVELOPED IN ROSSI AND VIGANÒ THESIS (2002) ... 20

FIGURE 8. DIAGRAM ILLUSTRATING THE RAINFALL-RUNOFF DISTRIBUTED HYDROLOGICAL MODEL FEST-WB TAKEN FROM PHDDISSERTATION OF ALESSANDRO CEPPI (2011) ... 22

FIGURE 9.RESULTING RAINFALL THRESHOLD CURVES FROM FEST-WB MODEL FOR CHIUSELLA A PARELLA FOR HYETOGRAPH TYPE I,II AND III RESPECTIVELY, TAKEN FROM THE RESEARCH AGREEMENT BETWEEN ARPA PIEMONTE AND POLITECNICO DI MILANO (2012) ... 23

FIGURE 10.SEEKING OF THE MINIMUM WITH PARABOLIC INTERPOLATION METHOD PRESENTED BY RAVAZZANI (2004) ... 24

FIGURE 11.ALTIMETRY OF PIEDMONT REGION ... 26

FIGURE 12.PO BASIN STATIONS USED IN THE PRESENT PROJECT ... 27

FIGURE 13.RAINFALL THRESHOLD CURVES CALCULATED WITH LUMPED MODEL FOR BASALUZZO ORBA SUB-BASIN ... 34

FIGURE 14.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE CN PARAMETER ... 37

FIGURE 15.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE RB PARAMETER ... 38

FIGURE 16. RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE RA PARAMETER ... 38

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X

FIGURE 18.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE S PARAMETER ... 39

FIGURE 19.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE N PARAMETER ... 40

FIGURE 20.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE B PARAMETER ... 40

FIGURE 21.RAINFALL THRESHOLD CURVES FOR THE SENSIBILITY ANALYSIS OF THE L PARAMETER ... 40

FIGURE 22.COMPARISON BETWEEN THE RAINFALL THRESHOLD CURVES BETWEEN THE LUMPED AND DISTRIBUTED MODEL FOR THE CARTOSIO ERRO SUB-BASIN ... 43

FIGURE 23.COMPARISON BETWEEN THE RAINFALL THRESHOLD CURVES BETWEEN THE LUMPED AND DISTRIBUTED MODEL FOR THE CASAL CERMELLI ORBA SUB-BASIN ... 44

FIGURE 24.COMPARISON BETWEEN THE RAINFALL THRESHOLD CURVES BETWEEN THE LUMPED AND DISTRIBUTED MODEL FOR THE FOSSANO STURA DI DEMONTE SUB-BASIN ... 45

FIGURE 25.GEOGRAPHIC LOCATION OF BOGOTÁ RIVER BASIN (CUENCA DEL RÍO BOGOTÁ)- PRESENTED BY ECOFOREST LTDA –PLANEACIÓN ECOLÓGICA LTDA IN THE POMCA DOCUMENT ... 54

FIGURE 26.SUBDIVISION OF BOGOTÁ RIVER BASIN FOR THE IDENTIFICATION OF MORPHOLOGIC PARAMETERS - PRESENTED BY ECOFOREST LTDA –PLANEACIÓN ECOLÓGICA LTDA IN THE POMCA DOCUMENT... 55

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XI

List of Tables

TABLE 1.SEASONAL RAINFALL LIMITS FOR AMC CLASSES (SOIL CONSERVATION SERVICE 1972) ... 10

TABLE 2.OBTAINED CUMULATIVE HEIGHTS FOR THE CALCULATION OF RTC OF BASALUZZO ORBA SUB-BASIN .. 34

TABLE 3.PARAMETERS OF THE VALUES FOR THE HYPOTHETICAL STATION TO DO THE SENSITIVITY ANALYSIS ... 35

TABLE 4.STANDARD DEVIATION OF THE INPUT PARAMETERS CALCULATED FROM THE 39ITALIAN SUB-BASINS .. 36

TABLE 5.CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH LUMPED MODEL FOR CARTOSIO ERRO SUB-BASIN ... 43

TABLE 6.CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH FEST-WB MODEL FOR CARTOSIO ERRO SUB-BASIN ... 43

TABLE 7.CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH LUMPED MODEL FOR CASAL CERMELLI ORBA SUB -BASIN ... 44

TABLE 8.CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH FEST-WB MODEL FOR CASAL CERMELLI ORBA SUB-BASIN ... 44

TABLE 9.CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH LUMPED MODEL FOR FOSSANO STURA DI DEMONTE SUB-BASIN ... 45

TABLE 10. CUMULATIVE RAINFALL HEIGHTS CALCULATED WITH FEST-WB MODEL FOR FOSSANO STURA DI DEMONTE SUB-BASIN ... 45

TABLE 11.IDENTIFIED CRITIC FLOW EVENTS IDENTIFIED FOR CARTOSIO ERRO SUB-BASIN ... 49

TABLE 12.CONTINGENCY TABLES OF THE RESULTS FROM THE COMPARISON OF CUMULATIVE RAINFALL WITH THE RESULTING RTC CURVES FROM LUMPED AND DISTRIBUTED MODELS IN THE CASE OF CARTOSIO ERRO BASIN ... 49

TABLE 13.IDENTIFIED CRITIC FLOW EVENTS IDENTIFIED FOR CASAL CERMELLI ORBA SUB-BASIN ... 50

TABLE 14.CONTINGENCY TABLES OF THE RESULTS FROM THE COMPARISON OF CUMULATIVE RAINFALL WITH THE RESULTING RTC CURVES FROM LUMPED AND DISTRIBUTED MODELS IN THE CASE OF CASAL CERMELLI ORBA BASIN ... 50

TABLE 15.IDENTIFIED CRITIC FLOW EVENTS IDENTIFIED FOR FOSSANO STURA SUB-BASIN ... 51

TABLE 16.CONTINGENCY TABLES OF THE RESULTS FROM THE COMPARISON OF CUMULATIVE RAINFALL WITH THE RESULTING RTC CURVES FROM LUMPED AND DISTRIBUTED MODELS IN THE CASE OF FOSSANO STURA BASIN ... 52

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TABLE 17.CONTINGENCY TABLES OF THE RESULTS FROM THE COMPARISON OF CUMULATIVE RAINFALL WITH THE RESULTING RTC CURVES FROM LUMPED AND DISTRIBUTED MODELS IN THE CASE OF THE THREE-ANALYZED BASIN ... 52

TABLE 18. MORPHOLOGIC INFORMATION OF BOGOTÁ RIVER BASIN - PRESENTED BY ECOFOREST LTDA – PLANEACIÓN ECOLÓGICA LTDA IN THE POMCA DOCUMENT ... 55

TABLE 19.BOGOTÁ RIVER BASIN-SANTA ROSITA AND PUERTA FLORENCIA AREAS: NECESSARY PARAMETERS TO CALCULATE THE RAINFALL THRESHOLD CURVES ... 58

TABLE 20.CONTINGENCY TABLE FOR THE RESULTS OBTAINED WITH THE LUMPED MODEL IN BOGOTÁ RIVER BASIN ... 60

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1

1. INTRODUCTION AND STATE OF ART

While working in the field of hydrological risk in basins there are three important stages for the assessment of an extreme event (e.g. a flood) in aims to reduce its level of risk. The first step consists in the identification, which aim is to describe the current problem and therefore gather the available information about the affected area (description of the area, identification of past events, identification of possible modifications of the area, etc.). The second stage consists in the risk assessment itself in which it is necessary to identify the probability of hazard (e.g. the probability that the maximum level of a river is exceeded in a certain event), the exposed elements and its related vulnerability in the interest to calculate the level of risk that the studied event generates. Finally, given the level of risk, a mitigation action that can be structural or non-structural is made in aims to reduce the impacts of the hydrological issue.

When these types of assessments are developed, as it can be noticed, the gathered information is a key aspect. With this information, the hydrologist can identify in general terms what is the problem, what is causing it and maybe give some approaches of how to control it. For example, in the case of understanding the hydrological response of a catchment, the first step is to analyze the characteristics of the basin (geology, land use, slope); in that moment, the hydrologist can identify the size of the basin, the location of the area, the differences in slope within the area, the land cover, etc. These latter characteristics can lead to a rough idea of, in case of a heavy rainfall event, how the subsurface runoff could behave. Later, the hydrologist can collect information about the climate and runoff data, as for instance precipitation, air temperature, solar radiation and registered flow in the existing stations. Firstly, the latter data can be analyzed to understand if the area is dry or rainy (among other types) but also gives the possibility to understand the heterogeneity of the meteorological inputs in space and time; afterwards, taking the data into consideration, the hydrologist can make use of developed models that can somehow predict the behavior of the subsurface runoff within the basin and therefore calculate the flow of its streams [1] [2]. Once the prediction is made, it is in hands of the hydrologist to analyze those results and propose different solutions to possible problems as for instance setting an alarm in case of exceedance of water’s critic level of a stream or start to use the gathered data to propose a structural action.

Considering what has been said in the previous paragraphs, to guarantee the correct understanding of the problem in its complexity it is almost necessary to possess all the possible

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2 information about the phenomena. For example, when it comes to flood forecasting, some rainfall-runoff models are based on the input of gross amount of information regarding the precipitation and air temperature in an hourly basis (sometimes quarter of hour), flow of the streams inside the basin (also in an hourly, quarter of hour basis), high resolution DEM, land use cover of the basin, among others. For instance, some models that work with high quality hydrological records are distributed models. The latter subdivide the entire basin in cells segregating the catchment in a kind of “grill” identifying the hydrological parameters for each cell to model the watershed considering its spatial heterogeneity.

Unfortunately, there are several basins across the world that do not have enough information or worst, not information at all. In fact, most of the basins around the world have this issue and this lack of data often increases when the size of the catchment decreases [1]. These type of basins, also called ungauged basins, are difficult to manage because the availability of data can reduce the span of available models that can be used.

Having in mind what has been described in the previous paragraph and considering the 5th report made by the Intergovernmental Panel on Climate Change, there is a warning of the possible changes in the frequency of extreme events. Specifically, the first working group, which is centered in the analysis of the physical science of climate change, states based in several studies “the tendency of increases in heavy precipitation events in the global means even, but there are significant variations across regions” [3]. Therefore, it is necessary to find alternative tools for flood forecast in the basins which do not possess good quantity and quality data with the aim to reduce the negative impacts that floods can impose to the society.

Even though the absence of data for ungauged basins makes difficult modeling those, there have been a lot of research in that field; in fact, the International Association of Hydrological Sciences (IAHS) have developed a working group for Predictions in Ungauged Basins (PUB) with the aim to reduce the uncertainty in hydrological predictions for basins with few or non-available information [4] [5]. Among with the development of this commission there has been several research on the field of ungauged basins modeling; one example can be the use of the open-source semi-distributed hydrologic model of Variable Infiltration Capacity (VIC) developed by Liang et al. (1994) linked with the use of HEC-RAS that has the aim to integrate satellite-based data in river modeling. Briefly, the latter model represents the catchment in a grid composed by cells where daily meteorological drivers are introduced to model the rainfall-runoff transformation of the different streams within a basin located in the border between

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3 Bangladesh and India [6, 7]. Another example of the development of flood forecasting modeling in ungauged basins is the use of SWAT hydrological model which is a distributed model that can simulate water quantity and quality as well as crop growth; furthermore, SWAT model subdivides the catchment into sub-basins using the DEM which are later sub-divided in Hydrological Response Units that are the basis of the water balance calculation. The model allows the simulation of surface flow, sediment generation and nutrient movement [7] [8]. Along with distributed models there have been some other developments in non-structural systems that work with flood forecast as lumped model. This type of models unlike the ones previously mentioned, avoids the discretization of the entire basin in cells describing it with few parameters that generalize the characteristics of all the watershed. Therefore, unlike a distributed model, which for example can produce a hydrograph for each discretized cell, the lumped model results in the production of one hydrograph that represents the runoff excess of the entire basin that corresponds to its outlet. This means that it is not necessary to have very detailed information of the basin but only just certain parameters useful to the development of the lumped model.

One of the lumped models used for flooding forecast are the rainfall threshold curves. This method identifies the quantity of cumulative rainfall that for a certain duration can deliver to the surpass of the critic level of the stream and therefore provoke a flood event [9]. Specifically, this method relies on the rational formula that after different variations provides the cumulative rainfall depth that triggers a flood event for a certain stream section through a relation that considers the soil moisture at the beginning of the event (through the runoff coefficient), the flow of the main stream, the duration, a detention coefficient, and the area of the basin. This model has been used in some projects developed by the Civil and Environmental and Land Planning Engineering faculty in Politecnico di Milano due to the simplicity on its application [10].

Colombia is a country located in northern South America that is distinguished by its large amount of hydric resources. In Colombia, only from 1987 to 2007, 6,326 flooding disasters have been registered and those have been related to the vulnerability of the hydrologic systems to maintain its hydric capacity. Specifically, in the country there are three regions that present hydrologic regimes of flooding (Interandean, Caribbean and Orinoco) meanwhile there are two regions with dry hydrologic regime (Pacific and Amazonian) [11]. During 2010 and 2011 a winter wave took place in Colombia, presenting one of the highest precipitation levels in history

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4 for some regions leading to flood events all over the country mainly for the months of July, November and December of 2010 and March and May for 2011. Only for this time lapse 1,233 flooding events were reported flooding 3.5 million hectares of land mainly occupying the northern area of the country and part of the central and southern areas as well. The result of this winter wave were 3,219,239 affected people across the country, that is almost 7% of Colombia’s population; in fact, focusing only in the department of Cundinamarca located in the Interandean Region, where the capital of the country (Bogotá D.C.) is, Bogotá River affected circa 80,000 people [12].

Colombia is a country where hydrometric data is scarce and not open to the public, therefore its basins are generally ungauged. Difficulties to find information about meteorological and physical properties of Colombian basins have been one of the causes of not counting with warning measures for flood events and having disasters as the ones previously mentioned. Given this, it could be interesting to look for techniques that can be coupled with the scarce information of the catchment and still forecast a heavy rainfall event in an ungauged basin as Bogotá River basin.

Taking into consideration what have been described until here, the aim of this thesis is to analyze an existing lumped model which could be used for ungauged basins to forecast flooding events. Being precise, the selected model which is going to be used and analyzed is the Rainfall Threshold Curves (RTC).

To assess the lumped model and therefore use it in an ungauged basin, the thesis is going to be divided in two main parts from which the first one is subdivided as well. The first section is going to describe four subsections: the first one has the aim to describe and identify the equations of the rainfall threshold curves method to develop a calculation module ready to be used in real basins; here the methodology described by Rossi and Viganò (2002) on their bachelor thesis to develop Rainfall Threshold Curves is going to be presented and used. Later a sensibility analysis is going to be made to understand which parameters of the lumped model are more important and therefore crucial for the calculation of the curves; specifically, this is going to be made because the final objective is to use the model in ungauged basins and so, it is important to know which are the parameters that need to be precise and which of those are not that relevant.

Afterwards a third section is going to describe a validation procedure of the Rainfall Threshold Curves method comparing the resulting curves with existing rainfall threshold curves for a

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5 gauged basin calculated from a distributed model. Specifically, for this aim, the lumped model is going to be applied for 39 Italian gauged basins in Piedmont Region which rainfall threshold curves have been already identified by the FEST-WB physically-based distributed model. Finally, for the first section of the thesis, a comparison of the resulting curves from the Rainfall Threshold Curves of 3 stations with real precipitation data is going to be made. Specifically, this subsection has the aim to understand how many times the lumped model could forecast a heavy rainfall event and therefore identify how much the model can be reliable.

The second section of the thesis is the application of the lumped model on a real ungauged basin which is going to be the Bogotá River located in Colombia. Since there are no rainfall threshold curves for this Colombian basin or models to compare the results, the curves from the lumped model are going to be compared with flood events described in the news or bulletins made from national governmental corporations. Finally, conclusions and recommendations are going to be presented to describe the positive aspects and drawbacks of the lumped model, as well as possible enhancements some other conclusions of the present research activity.

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6

2. METHODOLOGY

As mentioned in the introduction, there are different available methods to forecast floods. Specifically, it is known that some of these methods are the lumped and distributed models which with some information about the characteristics of the basin (with different degree of detail depending on the model) can estimate when a flood event is going to take place and in a general basis how critic its related impacts are going to be.

These methods can be known as hydrological models, which are based on the study of the water cycle and its interaction with the environment. These models have the objective to recreate in a simplified way the hydrologic cycle in different scales, depending on the necessity, to understand how different phenomena develops in time and space. Some uses of hydrological models are the identification of the impacts related to a change of land use on a basin’s runoff response or also for understanding which is going to be the response of a basin giving different amount of rain giving specific meteorological conditions [7] [13]. To do these latter tasks, as stated in Gayathri et al. (2015), hydrological models can have different type of inputs as for instance rainfall, air temperature, soil characteristics, watershed topography, vegetation, hydrogeology, among other parameters which define the characteristics of the model and the possible results it can deliver.

Once the parameters have been identified, each model has embedded mathematical equations that represent the physical processes that the model wants to reproduce. Once the inputs are introduced to the model and the equations have been identified and used, the model must be calibrated and validated with real data to verify that the selected model represents, with a good accuracy, the phenomenon which the model has been created for [13].

As stated before, lumped and distributed models are just some examples of hydrological models; in fact, these types of models can be used to describe the rainfall-runoff phenomenon of a basin. In one hand, the lumped model represents the entire basin as one “big cell” without considering its spatial variability; given this, all the inputs, the calculated parameters within the model and the results are generalized for the whole basin. On the other hand, the distributed model discretizes the entire basin in single cells of a big grid and therefore, each cell is going to have its own parameters considering the spatial variability within the basin.

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7 Taking into consideration the brief description of two hydrological models, the fact that different types of models needs different data resolution can be highlighted. Unlike the lumped model where the parameters represent just one unit (the entire basin) and therefore the number of data needed can be less, the distributed model needs information of each cell in which the basin has been discretized. Therefore, in the case of the distributed model the number of parameters is going to be large and consequently it means a heavy computation load for its application.

Depending the case, some researchers tend to use one model rather than the other due to different facts. For example, when the availability of data in space and time can be gathered, researchers can use that information to develop distributed models which can deliver to a more detailed understanding of a system. Unfortunately, there are worst scenarios where the availability of data is reduced and so researchers should work with lumped models which can work with few data to represent a phenomenon. This, of course not considering different techniques for data interpolation and extraction.

Since basins with few or non-available data can be present very often, given by different economic, technological, social between other conditions of developed and developing countries, the main objective of this project is to describe and use a lumped model that with few data can identify in which conditions a flood event can happen in a basin.

As stated in the introduction, the lumped model which is going to be presented is Rainfall Threshold Curves. To do so, first the theory of the model is going to be described and later the methodology used in Rossi and Viganò’s thesis to obtain these curves is going to introduced to later be used in the studied basins. Particularly, this method is going to be firstly applied in 39 sub-basins of the Po basin located in Piedmont Region of northern Italy.

Subsequently, to understand the accuracy of the results from the lumped model, the threshold curves obtained from a distributed model are going to be used as an accurate reference. Taking this into account, the FEST-WB distributed model is going to be briefly described to understand how the rainfall threshold curves are obtained for that case and therefore show the comparison between the curves from both models.

After the description of both models are presented, a sensitivity analysis is going to be made to identify the most important parameters of the lumped model and therefore make some conclusions about the results. For this aim, to understand better how the analysis was made, a

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8 brief description of the results from the lumped model for some Po sub-basin is going to be presented, analyzing the results of the lumped model

Afterwards, a validation step is going to take place to compare the resulting curves from the lumped and distributed models. For this aim, firstly the resulting curves of the FEST-WB model are going to be plotted along with the lumped model curves so it can be noticed if there is an important difference between the two models. More in detail, this step is made to identify in which cases the curves are similar but also to understand which are the causes of existing differences. Once this is done the verification of the accuracy of both models is going to be made using real events and identifying in which cases the curves forecast a flooding event or in which cases these are not able to do it. To organize this information, contingency tables with the results from 3 sub-basins are going to be presented.

Finally, since the final objective of this project is to analyze if a lumped model as the Rainfall Threshold curves can be used for ungauged basins, in the first place the Bogotá River basin is going to be described as well as a summary of which data from the basin was available. Later, once all the information to use the lumped model is gathered, the curves are going to be calculated for the Colombian basin and then compared with real events to identify if the lumped model can forecast a heavy rainfall event in ungauged basins.

Finally, some conclusions about the comparison from both analyzed models but also its validation and later the used of the lumped model in a real ungauged basin is going to be presented.

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9

3. DESCRIPTION OF USED MODELS

a) Lumped model: rainfall threshold curves

The rainfall threshold curves make possible a graphical identification of the cumulative rain height that for a certain duration in a studied section, can increase the water level until the surpass of the maximum level (Figure 1). In other words, this method can define the quantity of gross rainfall that if fallen above a basin for a certain duration can deliver to the surpass of a water’s critic level that subsequently could release a flood event.

Figure 1. Rainfall Threshold Curve example taken from Rossi and Viganò's thesis (2001) and translated by the author.

To obtain these curves, the model relies on an important hypothesis. As it is known, generally the precipitation that falls over basins is not homogeneous neither in space nor in time; nevertheless, given that a lumped model count on few data that is then generalized for the entire basin, it could be difficult to represent that heterogeneity within the model. Given this, the lumped model hypothesizes that the rainfall which falls above the basin is uniform in space and time.

Along with the latter hypothesis, there are two other conditions that are generalized in the model which are the rainfall profile and the water content of the soil. On one hand, since the model do not considers the variability of the rainfall during time, there is a way in which the model generalizes the rainfall profiles in three categories. The first category represents an event with constant intensity during all the duration which is named steady profile or type I; the second category is the one that represents an event that starts with a weak intensity and later increases in time until a maximum point which is called type II or linearly increasing profile; and finally the third kind considers a rainfall event that starts with its highest intensity to subsequently lose

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10 strength during the event which is called linearly decreasing profile or type III (Figure 2). The latter profiles can be also called hyetographs.

On the other hand, it is necessary to describe the water content of the soil at the beginning of a specific event depending on the total rainfall dropped in the previous days. This condition is important since it has a strong influence on the response of the basin; for instance, if the soil is already humid because the previous days were rainy to the point where there is no space in the soil to retain more water, when there is a rainfall event the precipitation will not be retained but it will turn into runoff immediately. This phenomenon can be described by the AMC (Antecedent Moisture Condition) which results in three classes labeling the soil moisture with the total rainfall that fell in the previous 5 days of the event (Table 1) [14].

Table 1. Seasonal rainfall limits for AMC classes (Soil Conservation Service 1972)

AMC CLASS Resting Season [mm] Development Season [mm]

I 0 ≤ 𝑃 ≤ 12.7 0 ≤ 𝑃 ≤ 35.5

II 12.7 ≤ 𝑃 ≤ 27.9 35.5 ≤ 𝑃 ≤ 53.3

III 𝑃 ≤ 27.9 𝑃 ≤ 53.3

Taking the previous conditions into consideration, the lumped model consists in assessing for all the different combinations between rainfall profiles and water content of the soil (9 combinations in total) the height of cumulative rainfall that if fallen for certain duration causes a flood event in the section of a stream. So, once the curves are calculated, there is a comparison between the results of the model and a real event to identify possible flood events

To do the comparison between the curves and a real event, the process starts identifying the AMC and rainfall profile of the event that is being assessed to pick which of the 9 curves

Figure 2. Rainfall profiles which describe the intensity development in time for three cases: steady (type I), linearly increasing (type II), linearly decreasing (type III)

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11 identifies the event. Then, the cumulative rainfall of the event is graphed along with the threshold curve with the aim to identify if the model can accurately set an alarm; this alarm is triggered when the cumulative rainfall crosses the threshold curve as showed in Figure 3.

Figure 3. Example of the comparison between the rainfall threshold curve and a real heavy rainfall event

Now, to go in more detail and explain how rainfall threshold curves are calculated, it is necessary to start from its principal equation which is the rational formula (Eq. 1).

𝑄 = 𝜑 𝜀 𝑖 𝐴 (𝐸𝑞. 1)

The flow formed in a precipitation event (Q), is determined by 𝜑 which is the runoff coefficient, 𝜀 that represents the detention coefficient, 𝑖 which is the rainfall intensity, and A that is the area of the basin.

Organizing the rational formula in a way where is explicit the relationship between height of cumulative precipitation (h) and the flow, duration of the rainfall event (d), area of the basin and detention coefficient, is necessary to obtain the values that result in the rainfall threshold curves. Hereafter is the transformation of the formula.

ℎ = 𝑖 𝑑 (𝐸𝑞. 2) 𝑄 = 𝜑 𝜀 𝐴 ℎ 𝑑 (𝐸𝑞. 3) ℎ = 𝑄 𝑑 𝜑 𝜀 𝐴 (𝐸𝑞. 4) 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 h [m m ] Duration [h]

Rainfall Threshold Curves

Hyetograph Type II

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12 To calculate the cumulative rainfall height (h) there are some values which are already known as the critic flow of the stream and the area of the basin. Also, the duration can be chosen depending on the event which values can be for example 3, 6, 12, 24 and 48 hours. The only remaining parameters that are not known, which are 𝜑 and 𝜀 must be calculated; therefore, in the next paragraphs the method used for its calculation is going to be described.

Runoff Coefficient (𝝋)

The runoff coefficient is a parameter with values between 0 to 1 that describes how much the soil of a studied area is permeable; numbers near to 0 means that the soil is completely permeable and none of the rainfall turns to runoff, meanwhile values near to 1 means that the land is not permeable and therefore the total amount of water that fell as rainfall turned completely into runoff.

The runoff coefficient can be obtained by different methods that have as their main goal to determine with a number how much a certain type of land use can “retain water” or contribute to runoff. Some of these methods rely on tables that relates a runoff coefficient for each type of land use; as for instance an industrial area can have a runoff coefficient that varies from 0.5 to 0.9 and a forest can have values from 0.05-0.25 [15]. Although it can be easy to use those tables, to pick a value from a table and then use it in the rational formula, as for instance a value from 0.5 to 0.9 in the case of industrial area, is very subjective and can introduce uncertainty in the results [16]. To avoid this, it is advisable to use another approach as the one of the Soil Conservation Service CN (SCS-CN) method which through the Curve Number [dimensionless] identifies the part of rainfall that contributes to runoff; the latter approach is going to be briefly explained in the next paragraph.

The SCS-CN method consist in identifying the gross rainfall that potentially could turn into runoff and then “subtract” from it the fraction of rainfall that is retained in the soil or intercepted by the plants to finally obtain the runoff coefficient. The method, as stated before, uses the Curve Number which is a value between 0 to 100 that indicates the runoff potential of the land being 100 the highest runoff potential; this curve number is stated to be an index that encompass the complex combination of hydrologic soil group (soil) and land use and treatment class (cover) of a specific studied soil [17] . Once the soil-cover complex has been identified for the type of land under study, the CN value can be selected from different tables in the bibliography as for instance the ones available in the “National Engineering Handbook -Hydrology” made by the Department of Agriculture of the United States. The difference between the CN index

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13 and the runoff coefficient C (which could also be taken from tables) is that the CN has been calibrated based on several studies whereas the C is not based on experimental results [16]. Considering the description of the CN number, the gathered values taken from the National Engineering Handbook, Section 4 refers to an average antecedent moisture (AMC II) so to use the index value for wet and dry conditions, it can be adjusted by the following equations [18] [19]. 𝐶𝑁𝐼 = 𝐶𝑁_𝐼𝐼 2.281 − 0.01281𝐶𝑁𝐼𝐼 (𝐸𝑞. 5) 𝐶𝑁_𝐼𝐼𝐼 = 𝐶𝑁𝐼𝐼 0.427 − 0.00573 𝐶𝑁_𝐼𝐼 (𝐸𝑞. 6)

Going more into detail, the SCS-CN method relies on a ratio which relates the gross rainfall (Pg) with net rainfall (Pn) delivering to the runoff coefficient as it can be seen in the following equation.

𝜑 =𝑃𝑛

𝑃_𝑔 (𝐸𝑞. 7)

The ratio considers the total amount of rainfall which is represented by gross rainfall, and abstracts the net rainfall that means the total amount of rainfall minus the water that is “retained” due to the permeability of the soil and the presence of pits and vegetation. The net rainfall (Pn) is calculated using the CN index for the calculation of the potential maximum water retention (S) which considers the permeability of the soil and the initial abstraction (Ia). The latter considers the presence of pits, vegetation and the initial state of water content that can be assumed to be 20% of S [16]. Taking the previous parameters into consideration, the main equation of the SCS-CN method and the ones for S and Ia are showed hereafter.

𝑃_𝑛 = (𝑃𝑔 − 𝐼𝑎) 2 (𝑃_𝑔− 𝐼_𝑎 + 𝑆) (𝐸𝑞. 8) 𝑆 = 𝑆₀(100 𝐶𝑁 − 1) (𝐸𝑞. 9) 𝐼_𝑎 = 0.2 𝑆 (𝐸𝑞. 10)

As it can be noticed, the equation related to the potential maximum water retention has a parameter S0 which aim is to be a scale factor depending on the units; in case where the values of the net rainfall and S are in millimeters, its value turns into 254.

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14

Detention Coefficient (𝜺)

The detention coefficient takes into consideration which fraction of the basin is contributing in the formation of the runoff. To be more clear, this coefficient describes for different durations which fraction of the total runoff generated from the basin is contributing to the formation of the critic flow at the basin outlet. The behavior of 𝜀 can be evidenced in the Figure 4.

Figure 4. Detention Coefficient Curve taken from Rossi and Viganò’s bachelor thesis (2002) and translated

As presented in Rossi and Viganò (2002), the behavior of the detention coefficient is alike to a cumulative distribution of a Gamma function which can be derived from the instantaneous unit hydrograph of Nash model [10]. This model describes the response of a basin by the imposition of 𝛼 reservoirs in series which are characterized by the same time constant 𝑘 [20]. Therefore, using the equation of Nash model (Eq. 11), it is possible to obtain an approximate behavior of the detention coefficient curve where t is the duration.

𝐼𝑈𝐻(𝑡) = 1 𝑘 Г(𝛼)( 𝑡 𝑘) 𝛼−1 𝑒−𝑡/𝑘_{(𝐸𝑞. 11)}

In that order of ideas, to calculate the parameters of the Nash model (α and k), a comparison between a gamma distribution and the Geomorphological Instant Unit Hydrograph (GIUH) proposed by Rodriguez-Iturbe and Valdés in 1979 was made. The latter is a process, as stated by Franchini and O’Connell (1996), “based on Shreve’s theory (1966) of topologically random networks of a given magnitude and on the state-transition approach (Howard, 1971), coupled with Markov process” where the probability distribution of the total time to the outlet, in this case the GIUH, can be obtained [21] .

0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 25 30 ɛ Coef fici en t [-] Duration [h]

Time - Area Curve

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15 The GIUH approach was proposed by Rodríguez-Iturbe and Valdés to understand the role of geomorphologic properties in the hydrology of the basin, but also to enable the development of a flood analysis with insufficient data [22]. The authors presented in their article “The Geomorphologic Structure of Hydrologic Response” all the complexity that their method had; however, the latter was analytically complicated so Gupta et al. (1980) derived the cumulative density function of time of travel to the basin outlet as follows (Eq. 12) [23] [21].

𝑃(𝑇_𝐵 ≤ 𝑡) = ∑ 𝑃(𝑇_𝑠 ≤ 𝑡) 𝑃(𝑠) 𝑠∈𝑆

(𝐸𝑞. 12)

For the latter equation TB is the travel time to the basin outlet, TS is the travel time for the path s, the P(s) the probability of a drop taking the path s and finally S represents all the possible paths that a drop can take within the basin [21].

Nonetheless, the process of obtaining GIUH was still complex and so Rodríguez-Iturbe and Valdés, based on the concept of Henderson (1963), concluded that only the most important characteristics of an Instantaneous Unit Hydrograph are the peak flow (𝑞_𝑝) and the time to peak (𝑡_𝑝) for being able to recreate the hydrograph [22] [24] ; therefore, they were able to reproduce a triangular hydrograph that even with a different form could be satisfactory for their results. Given the previous assumption, after several trials the authors got two equations derived from the GIUH theory to obtain the values of peak flow and time to peak. To do this, the authors involved the use of the Horton Ratios (RA, RB, RL), the length of the principal stream (LΩ) and velocity (v) to finally make a regression with the resulting values. Making different combinations of velocity, lengths and basin’s order, the authors obtained the following equations. 𝑞𝑝 = 1.31 𝐿−1Ω 𝑅𝐿0.43𝑣 (𝐸𝑞. 13) 𝑡_𝑝= 0.44𝐿_Ω𝑣−1(𝑅𝐵 𝑅_𝐿) 0.55 𝑅_𝐿−0.38 (𝐸𝑞. 14)

The expression 𝐿_Ω represents the length of the stream of maximum order and 𝑣 the peak velocity which calculation was proposed by Rodríguez-Iturbe in 1982 as it can be seen in the next equation.

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16 Replacing the equation for the velocity in the equations for peak flow and time to peak previously showed, the respective equations result in the ones hereafter.

𝑞_𝑝 = 0.871

𝜋0.4 (𝐸𝑞. 16) 𝑡𝑝 = 0.585𝜋0.4 (𝐸𝑞. 17)

The parameter 𝜋 is a value that relates the parameters of the triangular hydrograph with some characteristics of the basin which are the mean width of the stream with the maximum order (𝑏Ω), mean slope of the stream with the maximum order 𝑆Ω, net rainfall intensity (𝑖𝑛), area of the basin (𝐴Ω), Manning’s coefficient (𝑛) and its calculated with the following equation.

𝜋 = 𝐿Ω 2.5 𝑖_𝑛𝐴_Ω𝑅_𝐿𝛼_Ω1.5 (𝐸𝑞. 18) And, 𝛼Ω = 1 (𝑛 𝑏_Ω2⁄3_{) (𝑆} Ω 1 2 ⁄ ) (𝐸𝑞. 19)

Once these equations were identified it was necessary to link the peak time and peak flow with the duration of the event (tr) of a uniform net rainfall (in), the base time (tb) and the formation time (tf). To do so, Henderson (1963) found a relationship between those, the peak flow and equilibrium peak (𝑄𝑒 = 𝑖𝑛𝐴Ω) as in it is showed hereafter [25] [10].

𝑄_𝑝 𝑄_𝑒 = 2 𝑡_𝑟 𝑡_𝑏 (1 − 𝑡_𝑟 2𝑡_𝑏) 𝑖𝑓 𝑡𝑟 ≤ 𝑡𝑏 𝑞_𝑝 = 𝑞_𝑒 𝑖𝑓 𝑡_𝑟 > 𝑡_𝑏 (𝐸𝑞. 20)

Since a triangular IUH has 𝑞_𝑝 = 2 𝑡⁄ and using the equation 16, after different unit’s _𝑏 arrangements, Rodríguez-Iturbe et al. (1979) end up with the equation to calculate the peak flow of the hydrograph and the peak time in a case of triangular IUH base on the basin properties for a uniform intensity rainfall.

𝑄_𝑝 = 2.42 𝑖𝑛 𝐴Ω 𝑡𝑟 𝜋0.4 (1 − 0.218 𝑡𝑟 𝜋0.4 ) (𝐸𝑞. 21) 𝑇_𝑝= {𝑡𝑝+ 0.75 𝑡𝑟 𝑖𝑓 𝑡𝑟 ≤ 𝑡𝑏 𝑡_𝑝 𝑖𝑓 𝑡_𝑟 > 𝑡_𝑏 (𝐸𝑞. 22)

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17 Continuing with the research of a relationship that could lead to the calculation of the gamma parameters Rosso (1984) using the GIUH parameters (Qp and Tp) and confronting them with the ones of the IUH of Nash model, obtained an equation that delivers to the conclusion of similarity between both models [26].

Г−1(𝛼)(𝛼 − 1)𝛼_𝑒1−𝛼 _{= 0.58(𝑅}

𝐵⁄𝑅𝐴)0.55𝑅𝐿−0.38(𝐸𝑞. 23)

The equation which resulted from the latter comparison was very complex to calculate, so after using an iterative scheme proposed by Croley in 1977 that resulted in a regression it was possible to obtain the two gamma parameters that are showed hereafter [27]. For further details on the presented equations and the theory behind those, please refer to Rossi and Viganò thesis (2002) [10]. 𝛼 = 3.29 (𝑅𝐵 𝑅_𝐴) 0.78 𝑅_𝐿0.07 (𝐸𝑞. 24) 𝑘 = 0.70 ( 𝑅𝐴 𝑅𝐵𝑅𝐿 ) 0.48 𝑣−1𝐿Ω (𝐸𝑞. 25)

Methodology to develop the threshold rainfall curves

To obtain the rainfall threshold curves, using the previous equations, Viganò and Rossi (2002) created a simple iterative methodology (based on Croley, 1977) that could be easily applied in an excel sheet where only the main characteristics of the basin must be introduced by the user and, with the help of a macro, the curves were calculated and graphed.

The methodology consists in introducing a preliminary value for the rainfall height (h’) [m] to introduce it into the model to calculate, for the first iteration, the net rainfall (Pn) for the different combinations of rainfall profiles and AMC; given this, the result is going to be the attainment of 45 values of rainfall height for each iteration if there are 5 analyzed durations. In the first step, it is necessary to assume that the gross rainfall Pg[mm] has the same value as the preliminary height h’[m] having h’= Pg. Once the value of gross rainfall has been identified, using the SCS-CN method, the value of the net rainfall is going to be obtained and therefore the calculation of the runoff coefficient 𝜑[-] can be made straight forward.

In the next step, the calculation of the detention coefficient 𝜀 [-] is made. As stated in the previous paragraphs, using the theory presented by Rodríguez-Iturbe and Valdés, Nash,

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18 Henderson and Rosso, the equations to calculate the Gamma parameters based on the geomorphologic characteristics of the basin are used to obtain the detention coefficient.

Finally, the last step consists in determine the rainfall height h[m] with Equation 4 for each duration. During the procedure, it is noticed that the values of the preliminary height h’ and the resulting height h are different and therefore is in this step where a goal seek is developed for each of the 45 resulting heights. The macro which is developed consists in recording a goal seek procedure where the difference of each resulting height hi with respect of its corresponding preliminary height value hi’ has to be approximately 0. If the difference is not near to zero, the goal seek tool looks for the hi’ value that after all the calculations results in a null difference. At the end, the value that corresponds to a difference 0 between h and h’ is the result that gives for different durations the critic height that could determine a flood event in a certain stream section of the studied basin. An example of the graphic result can be seen in Figure 5.

Figure 5. Example of the Rainfall Threshold Curves obtained with the iteration methodology proposed by Croley (1977) and used by Rossi and Viganò (2002)

For a better comprehension of the methodology, hereafter are going to be presented to diagrams that explains, the first one roughly the steps to follow (Figure 6) and in the second one in more detail for each step which equation is used and in which step the iteration is developed (Figure 7). 0 5 10 15 20 25 0 10 20 30 40 50 60 h [m m] Duration [h]

Rainfall Threshold Curves

Hyetograph Type II

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19

Figure 6. General diagram for the explanation for the obtainment of rainfall threshold curves which was described and developed by Rossi and Viganò thesis (2002)

START h' φ calculation ɛ calculation h calculation Is h=h'? NO h'=h'' STOP YES

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20

Figure 7. Specific diagram for the explanation for the obtainment of rainfall threshold curves which was described and developed in Rossi and Viganò thesis (2002)

START POTENTIAL MAXIMUM

WATER RETENTION (S) INITIAL ABSTRACTION (Ia) CN I, CN II, CN III, So c=0.2 h'=Pg NET RAINFALL CALCULATION (Pn) RUNOFF COEFFICIENT (φ) RAINFALL INTESITY (i)

NET RAINFALL INTESITY (in) AREA PARAMETER n, bΩ SΩ VELOCITY (v) AΩ GAMMA PARAMETER (k) LΩ, RA Rb Rl GAMMA PARAMETER (α) RA Rb Rl PI PARAMETER (π) AΩ, LΩ, RL

IUH -PEAK TIME (tp) IUH PEAK FLOW (qp)

BASE TIME (tb)

FORMATION TIME (tf)

NET RAINFALL

DURATION (tr) PEAK TIME (Tp)

DETENTION COEFFCIEINT (ɛ)

d RAINFALL HEIGHT (h) Is h=h'? STOP

YES h'=h''

NO

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21

b) Distributed model: FEST-WB model

As stated before, a distributed model is the one that subdivides the basin in a grid to analyze which is the behavior of the flow in each cell. Once the subdivision of the catchment is made, information about the basin (e.g. type of soil, land use, soil cover) but also of meteorological forcings (e.g. precipitation, temperature, solar radiation) must be identified for each cell with the purpose to use those in the equations embedded in the model. For instance, some distributed models use differential equations to recreate mass balances that are developed in the basin and therefore the result can be getting the value of a certain hydrological parameter distributed in space and time [13]. Given this level of detail (since the cell can be as for instance of 1km x 1km) there is an improvement of the predictive skills of the model that facilitates the estimation of the different parameters within the basin but as a drawback there is a greater computational load if it is compared with a lumped model [28].

One physically-based distributed model that has been developed in Italy, is the FEST-WB model, first developed by Politecnico di Milano in the nineties and later improved by the same university until recent years. This model, which acronym stands for “Flash Flood Event-based Spatially-distributed rainfall-runoff Transformations-Water balance” basically recreates the water and energy balance in a basin using differential equations to calculate the water flow in each cell of the grid in which the catchment has been subdivided [29].

Essentially, the FEST-WB model is subdivided in five main components which are: a) The flow paths and channel definition, b) the spatial distribution of site measured meteorological forcings, c) the snow dynamics, d) the runoff calculation and e) the overland and subsurface flow routing [13]. Briefly, each component is going to be explained hereafter.

The first component, that is the flow path calculation is automatically derived from the DEM using the least-coast algorithm; besides, for the hillslope and channel network definition the model uses the constant minimum support area concept which relies in selecting a constant critical support area that defines the minimum drainage area required to initiate a channel. In the second component, there is a spatial interpolation of the ground measured meteorological forcings using the information of precipitation, air temperature, air relative humidity and net solar radiation; specifically, for this aim the data is interpolated using the inverse distance weighting technique. For the snow dynamics component, the model considers that some of the precipitation can be solid or liquid (which is function of air temperature) as well as the dynamics of snow melting and snow accumulation. Next, for the runoff calculation of each cell, the

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22 modified SCS-CN method extended for continuous simulation is used. Here the potential maximum retention is updated in each cell at the beginning of rainfall as a function of the degree of saturation which at the same time depends on the soil moisture [30] [13] [29].

Here it is important to highlight the importance of the estimation of the antecedent soil moisture condition which is a key parameter for flood prediction [28]. A research was carried out by Ravazzani et al. (2008) comparing the previous FEST model, which adopted the classical SCS-CN method to assign the AMC depending in the total rainfall in the five days preceding a rainfall event, with the FEST-WB model that computes AMC with a modified SCS-CN method taking into consideration a mass balance for the soil saturation which later modifies the potential maximum retention. The main conclusion of this research was that the FEST model underestimated significantly both peak and volume flow mainly in cases where flood events were characterized by dry soil at the beginning of the rainfall event [29]. Therefore, it is vital to make a correct identification of the antecedent soil moisture before the event to have a more accurate flood forecast.

Finally, for the fifth component, which is the runoff routing, the model uses the Muskingum-Cunge method in its non-linear form with the time variable celerity. For a more concise description of the FEST-WB model a diagram showing how the model works is going to be showed in Figure 8.

Figure 8. Diagram illustrating the rainfall-runoff distributed hydrological model FEST-WB taken from PhD Dissertation of Alessandro Ceppi (2011)

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23 Considering the complexity of the FEST-WB model, now a description for the calculation of the rainfall threshold curves in the case of the distributed model is going to be done hereafter.

Calculation of the RTC with FEST model

As stated before in the description of the lumped model, the variability of the rainfall development in space and time is complex to model. Given this, also the distributed model hypothesizes two things. The first one is that, even if real rainfall events do not have a steady distribution in space, for the calculation of the threshold curves is necessary to assume a uniform rain above all the catchment; nevertheless, it is important to notice that this assumption can lead to the overestimation of the rainfall that generates the surpass of the critic level [31]. The second assumption is the generalization of rainfall profiles in three types (the same of the lumped model: type I, II and III) even if the evolution of rainfall in time is far from being linear to take rigorously the latter profiles [31].

A third important aspect that is also considered in the distributed model is the initial conditions of soil moisture which are generalized for uniform rainfall in space for a certain rainfall evolution in time. Basically, the model also describes soil moisture in different groups but, unlike the lumped model, uses 11 different types of soil moisture starting from 0 which represents completely dry basin to 1 which stands for the case of completely saturated soil [31]. Therefore, after the calculation of the threshold curves the result are 3 graphs that are going to show the curves for each soil moisture in the three possible rainfall profiles. An example of the resulting curves can be seen in Figure 9 [31].

Figure 9. Resulting Rainfall Threshold Curves from FEST-WB model for Chiusella a Parella for hyetograph type I, II and III respectively, taken from the research agreement between ARPA Piemonte and Politecnico di Milano (2012)

Even if it seems until here that the lumped and the distributed models have the same procedure to calculate the Rainfall Threshold Curves, its major difference resides in the calculation of the cumulative rainfall height (h). Once the hyetograph, the duration and the soil moisture has been defined for the studied event, the h is modified until a peak flow (Qp) with the same value of the critic flow of the basin outlet (Qmax) is obtained. The peak flow, which is the result of a

0 50 100 150 200 250 300 0 6 12 18 24 30 36 42 48 Pioggia (mm ) Durata (ore) Ietotipo 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 0 6 12 18 24 30 36 42 48 Pioggia (mm ) Durata (ore) Ietotipo 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 0 6 12 18 24 30 36 42 48 Pioggia (mm ) Durata (ore) Ietotipo 3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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24 transformation from gross rainfall to runoff, is calculated through FEST-WB model that relies in a unidimensional optimization algorithm to seek the cumulative rainfall that provokes it [32] [33] [9].

As described by Ravazzani (2004), the algorithm starts to change the h value until the deviation between Qp and Qmax is very close to 0 having as the goal function the Equation 26 [9].

𝐹 = (𝑄_𝑝− 𝑄_𝑚𝑎𝑥)2 (𝐸𝑞. 26)

Since the goal function has a parabolic evolution with respect to h, initially a parabolic interpolation method is used to get the threshold curves. This method starts with the calculation of the goal function in three points (P1, P2, P3) which are used to generate a parabolic equation that passes through all the points. Later the x-coordinate value of the lowest point of the resulting parabola (v), which turns into P2’, and the other two P-points with the lowest F values are going to be used again to find the parabola equation. This is repeated until the changes of the resulting parabola are not noticeable [9] [31]. A graphic example of the algorithm is showed by Ravazzani (2004) which is reported in Figure 10 [9].

Figure 10. Seeking of the minimum with parabolic interpolation method presented by Ravazzani (2004)

Finally, a linear interpolation is done with the resulting points from the parabolic interpolation method having thus the rainfall threshold curve for the predefined conditions of rainfall profile and soil moisture [9].

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4. CHARACTERIZATION OF THE ITALIAN BASIN

To use the rainfall threshold curves (RTC) in ungauged basins, as stated in the introduction, it is necessary to first utilize the lumped model in a basin which has already been analyzed and has the threshold curves from a distributed model with the aim to compare the results and understand the accuracy of the lumped one. In the present project, the Po basin which is in Piedmont Region (Italy) has been selected since the threshold curves have been already calculated with FEST-WB model. Given that a distributed model has been carried out in the basin, all the general information from the catchment as for instance area of the basin, critic flow, mean width and length of the maximum order stream, mean slope of the maximum order stream, CN, manning coefficient and Horton ratios are available.

a) General Characteristics

Piedmont Region, located in northern Italy, is surrounded in its south-east area by the Apennines, in the south-west, west and north by the Alps and in the East by Maggiore Lake and Ticino and Sesia River. Furthermore, the region is characterized for having three types of areas; the first one is the mountainous area, which takes 48.6% of the region having 12,380km2 that are mainly the Alpine area; the second one is the hilly area which represents 25.9% covering 6,450km2 of the total area located in the south-east confines including Torino, Monferrato and Langhe hills; finally there is the flat area which represents 25.4% having 6,570km2 which are located between the Alpes and Monferrato and Langhe hills (Figure 11) [34] [13] .

In terms of the climate, the Piedmont region is characterized by a temperate temperature that gets lower approaching the Alpes. During the winter season, there are few precipitation events at low altitudes meanwhile in summer there is more possibility of severe storms. Nevertheless, the seasons when the most incidence of heavy precipitation events take place in an important part of the territory is on autumn and spring. Especially yearly rainfall is significant across the mountainous and northern hilly areas meanwhile lower values are concentrated in the southern plains of Po River [13].

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Figure 11. Altimetry of Piedmont Region

b) Available data of Po Basin

Thirty-nine stations from small to medium sub-basin sizes in the Po basin, varying from 43km2 to 1530km2, have been analyzed. The necessary information for the calculation of the threshold rainfall curves in the studied area were provided by Politecnico di Milano. To be more precise values of critic flow were available in a document made from a research agreement between Politecnico di Milano and ARPA Piemonte in 2012 [31] meanwhile information regarding the area of the sub-basins, maximum order stream length and width, mean slope of maximum order stream, Curve Number, Manning coefficient, and Horton ratios were calculated using different ArcGIS tools. The DEM of the Po basin was available at 100m x 100m resolution as well as the CN map in raster format which enabled the identification of the area of each sub-basin along with the mean CN related to each sub-catchment. For the other values, the surface channels within the Po basin where available in vector format and where used to obtain the values regarding the slope, width, the Horton ratios, length and Manning coefficient of the maximum order stream of each sub-basin.

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27 Is important to underline that not all the values for mean width of the maximum order stream were available from all the sub-basins so in this case, a random value between 10m to 30m was given to those.

Analysis of rainfall threshold curves for flood forecasting in ungauged basins and its application on a real case : Bogot river basin

POLITECNICO DI MILANO

School of Civil, Environmental and Land Management Engineering

Master Degree in Environmental and Land Planning Engineering

Analysis of rainfall threshold curves for flood forecasting

in ungauged basins and its application on a real case:

Bogotá River Basin

Tutor: Ing. Giovanni Ravazzani

Master degree thesis submitted by:

Luisa Ximena López Tamayo

(832807)

Acknowledgements

Abstract

Sintesi

Table of Contents

List of Figures

List of Tables

1. INTRODUCTION AND STATE OF ART

2. METHODOLOGY

3. DESCRIPTION OF USED MODELS

a) Lumped model: rainfall threshold curves

Rainfall Threshold Curves

Hyetograph Type II

Time - Area Curve

Rainfall Threshold Curves

Hyetograph Type II

b) Distributed model: FEST-WB model

4. CHARACTERIZATION OF THE ITALIAN BASIN

a) General Characteristics

b) Available data of Po Basin