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International Conference on the Status and Future of the

World‘s Large Rivers

11-14 April, 2011 Vienna

Villazón, M. F. 1, 2, Willems, P.1, Berlamont, J.1

1. Introduction

Synthetic rainfall events, derived from statistical analysis of rainfall series and used as input to rainfall-runoff models,

will not lead to accurate probability estimation of impacts in the case of non-linear systems. The reason is that the

frequency of the impact is not equal to the frequency of the input. Therefore, continuous simulations are necessary for

the hydrological part of a river model. The produced series of rainfall-runoff discharges can then be statistically

processed to determine representative hydrographs for the river routing. Composite Hydrographs (CH) can be

determined based on QDF-relationships.

The Piraí River Basin, is located in Santa Cruz - Bolivia (Figure 1a). This basin is part of the Amazon basin, and has

tremendous discharge variability from 10 to 2000 m3/s. The river width goes from 50 to 500 meters and the elevation

ranges between 300 to 2700 m.a.s.l.. The study area has been divided in seven sub-basins (Figure 1b) and the

outflow of each sub-basin is calculated after calibration of a conceptual rainfall-runoff model (NAM-DHI). The time

series has been analyzed and filled (Villazón and Willems, 2010a, 2010b). A hydrodynamic model (MIKE11-DHI) for a

total river length of 100 km was implemented and calibrated using flow gauging series at two stations (Angostura and

La Belgica). A long-term simulation has been performed for a period of 14 years (1986-1999); hourly results are

considered.

1 Katholieke Universiteit Leuven, Hydraulics Section, Kasteelpark Arenberg 40, BE-3001 Leuven, Belgium. Phone: +32/16/321656; e-mail: [email protected] [email protected]

2 Universidad Mayor de San Simón, Hydraulics Laboratory, Petrolera Avenue Km 4.2 Cochabamba, Bolivia. phone: +591/4/4217370; fax: +591/4/4330010.

Composition of Representative Hydrographs and Scenario Analysis for Extreme Events

b)a)

Figure 1: a) Location of Pirai Basin in Bolivia, b) Overview of sub-catchments and stations

300 0 300 Kilometers

2. Methods

Extreme value analysis

Based on the selection of the nearly independent peak flow extremes from the long-term river flow series or the long-

term simulation results of the rainfall-runoff model, extreme value analysis can be carried out to obtain the recurrence

rates of important events.

Consider the following set of ordered and independent observations in a sample of the variable X, and xi the ranked

values of a sample having probability distribution FX. If only values of X above a sufficiently high threshold xt are taken

into consideration, the conditional distribution converges to the Generalized Pareto Distribution (GPD) G(x) as xt

becomes higher:

The parameter γ is called the extreme value index and shapes the tail of the distribution and the parameter β is the

slope (scale parameter). Based on different types of quantile plots the shape of the tail of the GPD was analyzed and

the parameters of the extreme value distribution estimated. For the present study case, extreme value index was found

zero (γ=0), such that an exponential distribution was applied.

Composite hydrographs construction

When the extreme value analysis is done for a whole range of aggregation levels (e.g. from the time step of 1, 2, 3, 6,

12, 24, 48, 120, 240, 360, and 720 hours), amplitude/duration/frequency relationships can be set up. For discharge

time series, such amplitude/duration/frequency relationships are called Quantity/Duration/Frequency (QDF)

relationships. Relationships are calibrated between

the parameters θ of the extreme value distribution

and the aggregation level D, using the formula presented:

Katholieke Universiteit Leuven

Universidad Mayor de San Simón

0 if exp1)( 0 if 11)(

1

=

−−−=≠

−+−=

−

γβ

γβ

γγ

tt xxxG

xxxG

qD

AwcDD a

z

aH

+

+=

−−

β

θ )(1)(1

where: A is the area of the catchment upstream of the discharge measuring station, H is called ‘Hurst-exponent’, z

represents the dynamic scaling exponent, a the scaling exponent, q is the mean long-term discharge and c is the linear

regression parameter. The formula is based on scaling properties for the rainfall intensities and consequently of the

river discharges. Figure 2 shows the calibrated relationships between the parameters of the exponential distribution

and the aggregation levels for the Bermejo station.

a) b) c)

Figure 2. Calibrated relationships for the exponential distribution parameters for rainfall-runoff peak flows versus the

aggregation level for the Bermejo station: a) parameter β, b) optimal threshold rank t, c) optimal threshold level xt

0.1

1

10

100

1000

0.1 1 10 100 1000 10000

Agregation level [h]

Slo

pe β

[m

3/s

]

Slope

Slope, calibrated

1

10

100

0.1 1 10 100 1000 10000


Thre

shold

rank t

Threshold rank

Threshold rank, calibrated1

10

100

1000

0.1 1 10 100 1000 10000


Thre

shold

level x t [m

3/s

]

Threshold levelThreshold level, calibrated

In Figure 3a the final result of the QDF curves for different return periods (1, 2, 5, 10, 25, 50, 100, 500 and 1000 years)

is shown. For return periods up to 10 years also the empirical QDF relationships are shown. In Figure 3b, comparison

is shown between the most optimal distribution calibrated for 3 hours of aggregation levels and the distribution

calculated on the basis of the calibrated QDF-relationships.

a) b)

Figure 3: a) empirical and calibrated QDF curves for rainfall-runoff peak flows for Bermejo station, b) Check of the

extreme value distribution for the rainfall-runoff discharges after calibration of the QDF-relationships at Bermejo station.

0

200

400

600

800

1000

1200

1400

1 10 100 1000

Aggregation level [h]

Dis

charg

e [m

3/s

]

empirical, T= 1 years




Calibrated QDF, T= 1 years









25

125

225

325

425

525

625

0.1 1 10 100

Return period [years]

Dis

charg

e [m

3/s

]

POT values, Aggregation level = 3Calibrated QDF-relationshipextreme value distribution

3 Scenarios and results

Due to the size and the heterogeneity of the basin under study, the use of CHs in all the sub-basins at the same time

of occurrence might lead to slightly underestimated return periods (overestimated flows). This can occur at

confluences of river branches with significantly different response times. In the present research different scenarios

have been proposed with different time moments for the CH peaks (with the same return period) in each sub-basin. By

comparing the CHs derived from the long term simulation and those from the observed data against the results from

the scenario analysis, the optimal time shift is calculated between the CHs in each sub-basin (3 scenarios).

a) b)

Figure 4. CH results of the observed river peak flows, the long-term MIKE11 results and the scenarios at the river

gauging stations: a) Angostura TR 10, scenario 3,b) La Belgica TR 5 scenario1.

The shape and the peak of the hydrographs are well represented; the CHs from the scenario analysis have the same

flow variation as the ones derived from the observations and the long-term simulation. This approach provides a

convenient, fast way to study extreme events taking into account the duration of the event for different return periods.

0

250

500

750

1000

1250

200 220 240 260 280 300

Time [hours]

Dis

charg

e [m

3/s

]

CH Angostura

observed

CH Angostura

MIKE11 result

Proposed

scenario result

CH Angostura

basin

CH Bermejo plus

Colorado basins0

700

1400

2100

2800

200 220 240 260 280 300

Time [hours]

Dis

charg

e [m

3/s

]

Original

CH Belgica

observed

CH Belgica

MIKE11 result

Proposed

scenario result

CH Guardia basin

CH Belgica basin

CH Espejos basin

CH Torno basin

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