UNCERTAINTY IN THE RAINFALL INPUT DATA IN A CONCEPTUAL WATER BALANCE MODEL: . EFFECTS ON OUTPUTS AND IMPLICANCES IMPLICATIONS ON THE PREDICTABILITY

Enrique Muñoz, Pedro Tume, Gabriel Ortíz[1]

ABSTRACT

As the use of hHydrological models have been widely used in the water resroucesresources planning and management of water resourceshas become more prevalent, ,an increasing level of detail and precision has been demanded of them. and therefore the model capabilities and limitations have become highly which are being more exploited. in terms of their capabilities and limitations. Currently, it is not only proves indispensable to have reliable models that simulate the hydrological behavior of a basin, but it is also necessary to know the limits of predictability and confiability reliability of the model outputs. The present study evaluates the influence on output uncertainty in a hydrological model produced by uncertainty in the main input variable of the model, rainfall. Using concepts ofTo do this, using concepts of identifiability and sensitivity, the uncertainty of model structure and parameters uncertainty associated with the model structure and calibration parameters was werewas estimated. Then, the output uncertainty , and then, using this foundation, the influence on the outputs produced by uncertainty in caused by uncertainties in i) the rainfall amounts, and ii) the periods of occurrence of these amounts, was determined. Among tThe main conclusions isconclusion is that the model presented greater sensitivity to rainy periods, and therefore, greater uncertainty in rainfall the estimation of rainfall during rainy periods producesa greater output uncertainty in the outputs. On the other hand, in non-rainy periods, the output uncertainty bands are not very sensitive to uncertainty in rainfall. Finally, , is was determined that uncertainties in rainfallduringthe basin filling and emptyingperiods (Apr. – . to Jun., and Sep. to – Nov., respectively) produce an alteration in the uncertainty bands in theofsubsequent periods. Therefore, uncertainties in these periods could result in limited , increasing the average range of uncertainty of the model outputs, and limiting the ranges of model predictabilityility.of the model.

Key words: Uncertainty, hydrological predictability, surface hydrology, conceptual water balance model, water resources.

INTRODUCTION

Considering that i) water is a limited resource, that ii) demand for it increases with development and population growth, that iii) availability changes year to year due to local, global, natural and anthropogenic phenomena, that iv) the impact of climate change on the water supply is of fundamental importance, and that v) a large part of the world population is already experiencing water stress (Vörösmarty et al., 2000), it is necessary to develop tools that allow for an efficient water resources management and to ofwater resources and that support the prediction of future conditions under different scenarios caused by local (orographic), regional (El Niño Southern Oscillation) or global effects (climate change), in order to avoid or reduce water stress in basins around the world..

One alternative for the efficient water resources management and planning of water resources is the use of hydrological models. A model attempts to reproduce a physical phenomenon that occurs in an object or area. Therefore, in hydrology, a model seeks to represent an area defined by a watershed, the phenomenaon ofrainfall-runoffprocesses and the water movement within itthem. The object objective of reproducing these processes is to simulate and predict future conditions with the aim of acting from a perspective of management, administration and optimization of water uses.

Consequently, it not only proves essential not only to have reliable models that adequately represent the hydrological behavior of a basin, but also, in the . For example, in case of using predictive models, or the use of using alternative sources of information (for example using climatological databases interpolated at a global or regional scale as hydrological modele.g global datasets inputs) (Mahe et al., 2008, Muñoz, 2010), proves it is necessary to know the predictability limits of the model outputs, and the uncertainty associated with them.

Normally, a hydrological conceptual hydrological model requires at least two input variables (potential evapotranspiration and rainfall) in order to quantify the inputs and water losses in the water balance. It is known that rainfall is the main input variable in a hydrological model (Olsson and Lindström, 2008). T, and therefore, the potential predictability and the range of predictability of the basin flows will depend on the uncertainty associated with the input variables and their quality, and on the uncertainty produced by the model..

Uncertainty is defined as the degree of the lack of knowledge of or confidence in a certain process or result (Caddy and Mahorn, 1995). Therefore, in a hydrological model the sources of uncertainty are associated with the input variables (lack of knowledge of the quality of the measurements and predictions) and with the model structure and calibration parameters (lack of knowledge and simplification of the simulated hydrological processes in a basin).

The present study aims to quantify how uncertainty in the outputs of a conceptual hydrological model is affected by uncertainty in the main input, the rainfall, while keeping fixed the uncertainty associated with the model structure and calibration parameters fixed, in order to discuss and evaluate the implications on the confiability and hydrological predictability of the model outputs..

MATERIALS AND METHODS

Study Case Study

For the study case, a conceptual water balance model of the Polcura River Basin was constructed (Figure 1).

The Polcura River Basin (Figure 1) is located in the temperate zone of South-Central Chile, between 37°20'S - 71º31'W and 36°54'S - 71°06'W. It is bounded by the Andes on the East, the Nevados de Chillánvolcanic complex on the southnorth, and Lake Laja on the northsouth. It comprises an area of 914 km2 between 700 and 3,090 masl, and it is characterized by steep slopes (≈ 26° on average) and is mainly composed of partially eroded volcano-sedimentary sequences partially eroded (OM2c, PPl3 and M3i) (SERNAGEOMIN, 2006).

. In addition, it is located in one of the extratropical zones most affected by the El Niño Southern Oscillation phenomenon (ENSO) (Grimm et al., 2000), presenting interannual seasonality and interannual variability in the local hydro-meteorological patterns (Grimm et al., 2000;, Montecinos and Aceituno, 2003).

The average annual precipitation in the basin is 2300 mm with a pluvial period during winter and aanice-melt and snow-melt period in spring and the start of summer. The average monthly temperature is9° C, and ranges from 2.5° C in winter to 16.5 ° C in summer.

Due to the location of the basin, its mountainous nature, and its geomorphology (Figure 1), itit presents exhibits high temporal variability with respect to hydro-meteorological characteristics, where the orographic effect on the eastern slope of the Andes produces an increase in rainfall amounts (Garreaud, 2009;, Vicuña et al., 2011). Additionally, it is a zone affected by the ENSO phenomenon. ENSO is a coupled ocean-atmosphere phenomenon that is characterized by irregular periodicity(2 to 7 years), where the alternation between El Niño and La Niña is the main source of interannual variability, where El Niño/La Niña episodes are is associated with above/below average rainfall and warmer/colder than normal air temperature warmer/colder than normal (Garreaud, 2009).

Therefore, due to the characteristics of the basin characteristics such as high rainfall variability and ENSO influence result in an ideal case of study where in which uncertainty in precipitation could be the main source of uncertainty in predictions.

ConceptualDescription of the water balance model description

The model used in this paper is the snow-rain and semi-distributed conceptual water balance model presented in Muñoz (2010) and Muñoz et al. (2011). This model simulates the pluvial and snow-melting processes separately and includes external alterations such as irrigation or transfer canals by adding or subtracting flows. This model has been successfully implemented in Andean basins in south-central Chile (e.g. Zúñiga et al., 2012, Arumí et al., 2012). Therefore, it is an adequate option tofor analyzing e the study area.

The pluvial component is modeled through a lumped monthly rainfall-runoff model that considers the watershed as a double storage system: the subsurface-superficial (SS) and the underground storage (US). The SS represents the water stored into the unsaturated soil layer as soil moisture. The US is the water that covers the saturated soil layer. The model needs requires two inputs, rainfall (PM) and potential evapotranspiration (PET). The model output is the total runoff (ETOT) at the watershed outlet, and includes both the subterraneaneous (ES) and direct runoff (EI). These , whose amounts are calculated through six calibration parameters of calibration, plus two for the input modification (useful in case of non-representative PM and PET data).

The snow-melt model calculates the snowfall (Psnow) based on the rainfall precipitation above the 0 (°C) isotherm. Psnow is stored in the snow storage system (SN), on which the melting calculations are performedbased on the concept ofusing the degree-day method (Rango and Martinec, 1995). Thus, the potential melting (PSP) is estimated, and then based depending on the snow stored,;the real melting (PS) is calculated. LaterThen, PS is distributed into the pluvial model through the calibration the parameter of calibration F.

Every calibration parameter of calibration has a conceptual physical meaning, integrating spatial and temporal variability. Table 1 presents a brief description of the model parameters and their influence on the model.

The external alterations module permitsthe incorporation of alterations such as irrigation or transfer canals, and simulates inflows and/or outflows to/from the basin by adding or subtracting flows as follows:

(1)

Where the basin outflow (Qout) at the time t, is the basin runoff (ETOT) plus the contributions (Qcontributions) and less the extractions (Qextractions) during the same period.

For a further explanation of the model, refer to Muñoz (2010) and Muñoz et al. (2013)..

Model inputs

In order to run the described model, it is necessary to have precipitation, temperature and potential evapotranspiration series, as well as the geomorphological characterization of the basin to allow compute the monthly 0°C isotherm elevation.

For the geomorphological characterization of the basin, a digital elevation model was constructed using data from the Shuttle Radar Topography Mission (SRTM) of 3 arc-seconds (90 m). Fort the inputs,rainfall series from the pluviometric rain gauge stations Las Trancas, San Lorenzo andTrupán were obtained (administrated by the Dirección General de Aguas, DGA), and synthetic temperature seriesfrom theCenter for Climatic Research of the University of Delaware (UD) (, Willmot and Matsuura, 2008) were collected. Then the cClimatological model inputs were constructed using both sources of information.Additionally, the potential evapotranspiration series was were calculated using the Thornthwaite method and UD temperature data series. The spatial distribution of said these variables in the basin was carried out through Thiessen polygons. Based on prior experiences, both, the Thornthwaite and Thiessen polygons methods have been demonstrated to be adequate for estimating potential evapotranspiration and for meteorological spatial distribution of monthly data in basins of south-central Chile (e.g. Muñoz, 2010, Zúñiga et al., 2012). Therefore, both methods were used in this study.

Due to the availability and quality of the input data, the analysis was performed on a monthly time-step for a period of 13 years (1990 – 2002). The fluviometric stream gauge station for controlling the basin outflows was is Polcura Antes de Descarga central El Toro station (Figure 1).

Uncertainty analysis

In order to quantify the uncertainty, the Monte Carlo Analysis Toolbox (MCAT) (, Wagener and Kollat, 2007)was used. MCAT is a tool that operates under the Generalized Likelihood Uncertainty Estimation methodology (Beven and Binley, 1992) and contains a group of analysis tools that allowto explore the identifiability of a model and its parameters to be investigated, where a . well-identified model is considered a model with realistic behavior (Wagener et al., 2001).MCAT was usedchosen because it is an easy-ofto-use tool and is an adequate and simple option tofor evaluateing model behavior and uncertainty.

In order to evaluate identifiability and quantify the uncertainty of a model, MCAT operates by performing repetitive simulations using a set of randomly selected parameters within a range defined by the user. . The program stores the outputs and values of the objective function(s) for posterior subsequent analyses. In this case, due to the lack of knowledge about the parameter distribution, a prior uniform distribution of the model parameters was used.

Parameters of hydrological models cannot be identified as unique sets of values. This is mainly due to the fact thatbecause changes of one parameter can be compensated for by changes of one or more others due to their interdependence (Bárdossy, 2007), and due to thebecause fact that the processes simulated in a hydrological model are commonly interrelated. In the case of the Muñoz (2010) model it is held, for example, that the direct runoff depends on Cmax, and therefore the processes that occur in the subsurface storage layer depend on the amount of rainfall that is not transformed into runoff, that is, on 1-Cmax, and consequently the processes that occur in saidthis storage layer will depend on the identifiability of Cmax. This generates an interconnection betweenamong the identifiability of the calibration parameters of a model. Therefore, it proves necessary to perform various iterations as a means of restricting the range of validity of the parameters that first show identifiability, in order to then observe identifiability in the remaining (dependent) parameters as a means of also reducing the range of identifiability of these parameters.

Because the Monte Carlo method is based on random trials, it normally requires a large number of simulations to cover a wide spectrum of possible simulations. In this case, the number of Monte Carlo simulations was estimated via trial and error, where the stop criterion was met when the correlation (according to the Pearson correlation coefficient) between uncertainty bands (calculated as a linear correlation between the time-series of the upper and lower limits of the bands of uncertainty bands) of two different trials, but with same number of simulations, was equal to or greater than 0.999. Under this criterion, it was determined that the adequate number of simulations for this study is 25,000.

The processes simulated in a hydrological model are normally interrelated. In the case of the Muñoz (2010) model it is seen, for example, that the direct runoff depends on Cmax, and therefore the processes that occur in the subsurface storage layer will depend on the amount of rainwater that does not become runoff, which is to say, on 1-Cmax, and consequently, the processes that occur in said storage layer will depend on the identifiability of Cmax. This generates an interconnection among the identifiability of the calibration parameters of a model. Therefore, it proves necessary to perform various iterations as a means of restricting the range of validity of the parameters that first show identifiability, in order to then observe identifiability in the remaining (dependent) parameters, thereby also reducing the range of identifiability of these parameters.

In the present study, in order to quantify the uncertainty associated with rainfall, 25000 simulations were performed, from which, first, the uncertainty associated with the calibration parameters and model structure, was determined, and then the effect on this uncertainty due to a given uncertainty in rainfall, were determined. The procedure to estimate the rainfall uncertainty and to differentiate it from model structure used and calibration parameters was the as followingfollows:

i) i) throughThrough an identifiability analysis of the factors for the modification of the inputs (a comparison of these factors with the value of the objective function), these factors were limited to precise values. It provesisIt is necessary to set these factors to a unique value, as they fulfill the function of assuring the closure of the long-term water balance, assuring that the precipitation input volumes are equal to the evapotranspiration and flow output volumes during the entire period of the simulation.

ii), and ii) withWith the values of said thoese fixed factors, new simulations were performed, in which the ranges of variation assigned to each parameter were reduced in accord with the observed identifiability. The established stop criterion was when identifiability was not observed in the new defined range (the reduced range) of each parameter.

After the identifiability analysis,the resulting uncertainty in the model outputs was quantified. This uncertainty was defined as the uncertainty associated with the model structure and calibration parameters.

The confidence level used was 90% (a range between 5 and 95% of the outputs for each time-step of the series).The established rejection criterion was set according to the Kling-Gupta efficiency objective function (KGE) (, Gupta et al., 2009). The KGE function (Eq. 2) is an improvement of the Nash-Suctliffe efficiency index (NSE) (, Nash and Sutcliffe, 1970) in which the correlation, deviation and variability are equally weighted, resolving systematic problems of underestimation of the maximum values and low variability identified in the NSE function (Gupta et al., 2009). The KGE varies between to 1 where the betterst model is closerst to 1. Due toBecause the GLUE methodology is based on likelihood, the KGE was transformed asinto 1-KGE. With this change, the 1-KGE represents a measure of likelihood where the best model is closerst to 0 and the worst model tends to .

(2)

Where r is the linear correlation between simulated and observed flows, α is the ration between the standard deviation of simulated and observed flows, representing the variability in the values, and β is the ratio between mean simulated and mean observed flows (i.e. represents the bias).