Hydrologic moment of a watershed area - …Geomorphological Unit Hydrograph [9-10] conceptualizes...

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Proceedings of the IMProVe 2011

International conference on Innovative Methods in Product Design

June 15th – 17th, 2011, Venice, Italy

Hydrologic moment of a watershed area

Mikel Goñi (a), José Javier López (a), Faustino N. Gimena (a) (a) Department of Projects Engineering. Public University of Navarre.

Article Information

Keywords: Hydrologic moment, Gis, sub-watersheds and iso-distances, Unit Hydrograph

Corresponding author: Faustino N. Gimena Tel.: +34 948 169 225 e-mail: [email protected] Address: Campus de Arrosadía, s/n. 31006 Pamplona. Spain

Abstract

Hydrologic moment of the watershed area is defined as the static moment with respect to the outlet of the watershed. In this paper we show two different ways to get the hydrologic moment through GIS tools: by dividing the watershed into sub-watersheds from the drainage network, or by dividing the watershed into areas related to the distance to the watershed’s

outlet. The Reservoir Geomorphological Instantaneous Unit Hydrograph is determined for each value of hydrologic moment previously obtained, and these hydrographs are applied to two watershed events. The results are analysed in order to show an appropriate and accurate way to determine the hydrologic moment.

1 Introduction In recent years, there has been a major effort to

characterize the average response of a watershed based on geomorphological properties using the unit hydrograph technique [1]. The first to introduce the concept of Geomorphological Instantaneous Unit Hydrograph were Rodriguez-Iturbe and Valdes [2], who determined it from the probability distribution function of surface water’s travel time along the watershed. Another way of looking at the geomorphology of the watershed is to express the parameters of the Instantaneous Unit Hydrograph of Nash [3] on the basis of morphological indices of Horton’s watershed [4], as did Rosso [5]. Another is making the Geomorphological Instantaneous Unit Hydrograph depend on the function of width of the watershed [6]. Karlinger and Troutman [7] studied different approaches to the unit hydrograph under the consideration of a random topological model of the drainage channel network.

On the other hand, applying the linear reservoir model has been and still is very frequent in determining the Unit Hydrograph of a watershed. Singh [8] gives a detailed collection of models using this theory. The Reservoir Geomorphological Unit Hydrograph [9-10] conceptualizes the watershed as a cascade of linear storage reservoirs with two variants: one in which each reservoir represents the area of land that sheds a channel of the drainage network, also called sub-watershed, and one in which each reservoir represents the area of land between iso-distances to the drainage point or watershed outlet. In both cases, the geomorphology of the watershed itself is introduced into the formulation of the model, coming to an expression that only depends on an uncertain parameter. With the development of Geographical Information System (GIS) tools, it is now possible to determine hydrological and watershed geomorphological parameters using Digital Elevation Models (DEMs) [11-13]. In this paper we analyze the different ways to characterize the geomorphology through the hydrologic moment and

subsequently define the sensitivity of the single parameter associated with the Reservoir Geomorphological Unit Hydrograph. This requires the use of geographic information systems to obtain both sub-watersheds and for the layout of the iso-distances.

2 Hydrologic moment and hydrograph unit of a watershed

In this communication hydrologic moment is defined as the static moment of the area of a watershed in respect to its outlet. Its mathematical expression may be noted as follows

[ ] [ ]1 1

n n

i i

i i

iA Aa= =

=å å (1)

Where iA represents the area of an element that the

watershed has been divided into. To analyse the geomorphology of a study watershed

and determine the hydrologic moment, this paper presents different possibilities: — Partition the watershed into sub-watersheds from the permanent drainage system represented in the mapping.

— Structuring the watershed from the drainage network drawn based on values of the Critical Source Area (CSA).

— Partition the watershed in orders through areas between two consecutive iso-distances depending on the distance to the outlet.

In addition to determining the area iA with the three

previous procedures, you may present other divisions of the watershed to obtain the hydrologic moment.

The hydrologic model called Reservoir Geomorphological Instantaneous Unit Hydrograph, RGIUH, has the following expression [9-10]:

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M. Goñi et al. Hydrologic moment of a watershed area

June 15th – 17th, 2011, Venice, Italy Proceedings of the IMProVe 2011

( ) ( )

1

1 1 !

t

i

n

i

i

e A th t

i

ata

at t

- -

=

=-

é ùé ùê úê úê úë ûë û

å (2)

The geomorphologic value a represents the hydrologic moment of the watershed area. The only uncertain parameter t represents the centre of gravity of the hydrograph. To obtain the direct runoff hydrograph simulated from the Instantaneous Unit Hydrograph we implement the convolution equation [14] to each of the effective rainfall hyetographs.

3 Description of the watersheds and events

The application and analysis of the Instantaneous Unit Hydrograph, IUH, have been carried out in two watersheds, Oiartzun and Aixola [15], located in northern Spain in the province of Gipuzkoa (fig. 1). Both watersheds have a gauging station as part of the hydro-meteorological network of Gipuzkoa Provincial Council which records the flow of the river every ten minutes. The precipitation data are also recorded every ten minutes by a rain-gauge located at the gauging station in Aixola’s watershed and by three distributed throughout the watershed in Oiartzun. Aixola’s watershed, located on the western boundary of the province of Gipuzkoa and of mostly forest use (> 85% area) has an area of 4.70 km ², with extreme levels of 315 and 740 metres, an average slope of 44.25 % and an annual average rainfall of 1600 mm. Oiartzun’s watershed located in the north-east of the province, which is larger than Aixola, has an area of 56.07 km ², extreme levels of 11 and 831 meters, an average slope of 43.11% and an average annual rainfall between 1700 mm in the lower part of the watershed to 2100 mm in the highest part of the watershed. In fig. 1 the location of both study watersheds is shown in Gipuzkoa province and in Spain.

From the data recorded in each of the stations we have selected a series of twenty events in the watershed of Aixola and twelve in the watershed of Oiartzun which can be adapted to the implementation hypothesis of the Unit Hydrograph method. The base flow has been extracted from each of these events and the effective rainfall hyetograph has been obtained. The base flow extraction was performed using a recursive filter, namely the one proposed by Eckhardt [16] of two parameters. After removing the base flow the effective rainfall hyetograph

has been obtained using the curve number method developed by the Soil Conservation Service [17] adjusting the observed direct runoff volume and the beginning of the observed direct runoff hydrograph. The main features of each of these selected events are in the doctoral thesis entitled "Development of a rainfall-runoff simulation model in humid areas. Implementation and evaluation in headwater catchments located in Gipuzkoa"[15].

4 Hydrologic moment and study watershed’s unit hydrograph

We present different ways to get the hydrologic moment through GIS tools for the two study watersheds, Oiartzun and Aixola, described in paragraph 2. First divide the watershed into sub-watersheds from the permanent drainage network represented on maps of 1:5000 and CSA values of 10%, 5%, 2% and 1% of total watershed area. Then in orders between consecutive iso-distances. The order numbers used for this second analysis of the geomorphology of the watershed have been 3, 6, 9, 12 y 15. From the hydrologic moment you can determine the Reservoir Geomorphological Instantaneous Unit Hydrograph. The dimensionless hydrograph is one that expresses the flow and time depending on response time and is achieved when defining the instantaneous unit hydrograph for 1t = .

Fig. 1 Location of Aixola and Oiartzun watershed

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In fig. 2 the generated divisions are represented as their related areas in Aixola’s watershed, we show the value of the hydrologic moment a and the dimensionless Reservoir Geomorphological Instantaneous Unit Hydrograph. These representations are made when using in sub-watershed partitions as in the case of using iso-distances.

Fig. 2 Hydrologic moment and unit hydrograph in Aixola’s watershed

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In fig. 3 the generated divisions are represented as their related areas in the Oiartzun’s watershed, we show the value of the hydrologic moment a and the dimensionless Reservoir Geomorphological Instantaneous Unit Hydrograph.

Fig. 3 Hydrologic moment and unit hydrograph of Oiartzun’s watershed

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It is observed in the different dimensionless calculated RGIUH that the largest peak flow is reached with the geomorphology represented by sub-watersheds. The differences in the magnitude of peak flow are significant, being the minimum up to 18% and 19% lower than the peak in the watersheds of Aixola and Oiartzun respectively. With regard to the form, in the case of the iso-distances, the increase in division degree results in a delay in the peak time and an increase in peak flow. However in the case of the sub-watersheds there is no clear trend.

5 Influence of the divisions in the determination of the hydrologic moment

Once generated the instantaneous unit hydrographs have been applied to twenty and twelve events of the watersheds of Aixola and Oiartzun respectively. The kindness of the adjustment in the application of the simulated hydrographs, derived from the hydrologic moments obtained in the previous section, has been performed in respect to the efficiency defined by Nash and Sutcliffe [18] whose expression is:

[ ]22

, , ,

1 1

1m m

ob j sij ob j ob

j j

E Q Q Q Q

= =

= - - -é ùë ûå å (3)

where: ,ob j

Q is the flow is observed at the time j, ,si j

Q

is the flow simulated at the time j and ob

Q is the

average flow observed. In each of the events parameter t is optimised. In fig.

4 we represent the distribution bars with the percentiles 10, 25, 50, 75 and 90 of the highest efficiency for each of the geomorphologies.

Fig. 4 Efficiency of the geomorphologies of Aixola and

Oiartzun’s watersheds

In general the performance of the simulated hydrographs is satisfactory and achieves high efficiencies. In Aixola’s watershed the best performances are those of the geomorphology obtained by sub-watersheds with 2% CSA followed by topographic channels and 1% CSA. In Oiartzun’s watershed, however, the geomorphology

defined by iso-distances of order 3n= presents the best efficiencies and lower dispersions, followed by the iso-distance order 6n= and sub-watersheds with 2% CSA.

In fig. 5 distribution bars of parameter are represented for each of the studied geomorphologies.

Fig. 5 Optimal parameter t values for the geomorphologies

of Aixola and Oiartzun’s watersheds In the case of parameter t the two watersheds show

clearly a different average value, being higher in Oiartzun’s watershed in accordance with its higher response time. Yet the dispersion in the value within one watershed is large, with the variability of time response within one watershed becoming apparent. As for the differences due to the geomorphology, the obtained parameter t average values are very similar, although it is observed that the average value of the parameter increases as the value of peak time decreases. The differences in dispersion do not seem significant, showing higher dispersions those geomorphologies with higher t average value.

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Fig. 6 shows the distribution bars of the peak times for

optimum unit hydrographs for each geomorphology.

Fig. 6 Optimum values of

pt for the geomorphologies of

Aixola and Oiartzun’s watersheds

It is observed that unlike parameter t peak time P

t is

not constant in different geomorphologies. From this observation we can conclude that using the first order time to characterize the response time of a watershed is more appropriate than using the peak time.

6 Conclusions We performed a study of the influence of the watershed

geomorphology in determining the hydrologic moment and the Reservoir Geomorphological Instantaneous Unit Hydrograph. For this we have obtained the areas that define the watershed structure by sub-watersheds and iso-distances with different division degrees. We have implemented and optimized the various RGIUH obtained by analysing the geomorphology of twenty direct runoff events in Aixola’s watershed and twelve in Oiartzun’s

watershed. We have presented the efficiency values, E ,

parameter t and peak time, p

t , optimum in each case.

This study concludes that: The Reservoir Geomorphological Instantaneous Unit Hydrograph associated to the hydrologic moment of a single parameter works properly for modelling the direct runoff.

Although there are no major differences between the

ways of looking at the geomorphology, in the case of Aixola the geomorphology representation with sub-watersheds with CSA (critical source area) of 2% has been the most appropriate. In the case of Oiartzun the watershed’s iso-distances with an order of three divisions has been the most suitable. A sub-watershed structure generated with a 2% CSA is considered a satisfactory way to obtain the hydrologic moment values that determine the Reservoir Geomorphological Instantaneous Unit Hydrograph. Analysing both parameter t and peak

time p

t , it is concluded that the response time of a unit

hydrograph is better characterised with the first, as it is maintained regardless the shape.

Acknowledgement The authors of this paper wish to thank the Sustainable

Development Department of the Provincial Council of Gipuzkoa, specially Patxi Tames and Andoni Da Silva with whom we have worked on the project “Análisis y cuantificación de la escorrentía superficial en pequeñas cuencas del territorio histórico de Gipuzkoa” for all the

information and assistance provided.

References [1] L.K. Sherman. Streamflow from rainfall by the unit-graph method. Engineering News Record 108 (1932) pp 501-505. [2] I. Rodriguez-Iturbe, J.V. Valdes. The geomorphologic structure of hydrologic response. Water Resources Research 15, 6 (1979) pp 1409-1420. [3] J.E. Nash. The form of the instantaneous unit hydrograph. International Association of Scientific Hydrology 45, 3 (1957) pp 114–121 [4] R.E. Horton. Erosional development of streams and their drainage basins: Hydrophysical approach to quantitative morphology. Geological Society of America Bulletin 56 (1945) pp 807-813. [5] R. Rosso. Nash model relation to Horton order ratios. Water Resources Research 20, 7 (1984) pp 914-920. [6] V.K. Gupta, E. Waymire, C.T. Wang. A representation of an instantaneous unit hydrograph from geomorphology. Water Resources Research 16, 5 (1980) pp 855-862. [7] M.R. Karlinger, B.M. Troutman. Assessment of the unit hydrograph derived from the theory of topologically random networks. Water Resources Research 21, 11 (1985) pp 1693-1702. [8] V.P. Singh. Hydrologic Systems, Vol. 1: Rainfall–Runoff Modeling. Prentice Hall: Englewood Cliffs, New Jersey 1988. [9] U. Agirre, M. Goñi, J.J. Lopez, F.N. Gimena. Application of a unit hydrograph based on subwatershed division and comparison with Nash's instantaneous unit hydrograph. Catena 64, 2-3 (2005) pp 321-332. [10] J.J. López, F.N. Gimena, M. Goñi, U., Agirre, Analysis of a unit hydrograph model based on watershed geomorphology represented as a cascade of reservoirs. Agricultural Water Management, 77, 1-3 (2005) pp 128-143.

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[11] S.K. Jenson, J.O Domingue. Extracting topographic structure from digital elevation data. Photogrammetric engineering and remote sensing 54, 11 (1988) pp 1593-1600. [12] D.R. Maidment, J.F. Olivera, A. Calver, A. Eatherall, W. Fraczek. A unit hydrograph derived from a spatially distributed velocity field. Hydrological Processes. 10, 6 (1996) pp 831-844. [13] F. Olivera, D.R. Maidment. Geographic information systems based spatially distributed model for runoff routing. Water Resources Research 35, 4 (1999) pp 1155-1164. [14] V.T. Chow, D.R. Maidment, L.W. Mays. Applied Hydrology. McGraw Hill, New York 1988. [15] M. Goñi Garatea. Desarrollo de un modelo de simulación lluvia-escorrentía en zonas húmedas. Aplicación y evaluación en cuencas de cabecera ubicadas en Gipuzkoa. Doctoral Thesis. E.T.S.I.A. Universidad Pública de Navarra, Pamplona 2009. [16] K. Eckhardt. How to construct recursive digital filters for baseflow separation. Hydrological Processes 19, 2 (2005) pp 507-515. [17] V. Mockus. 1972. Estimation of direct runoff from storm rainfall. SCS National Engineering Handbook. NEH Notice. 1972. [18] J.E. Nash, J.V. Sutcliffe. River flow forecasting through conceptual models part I - A discussion of principles. Journal of Hydrology 10, 3 (1970) pp 282-290.

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