POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA
Linking Long-term Water Balances and Statistical Scaling to Estimate River Flows along the Drainage Network of Colombia Germ´an Poveda, Jaime I. V´elez, Oscar J. Mesa, Adriana Cuartas, Janet Barco, Ricardo Mantilla, John Freddy Mej´ıa, Carlos D. Hoyos, Jorge M. Ram´ırez, Lina I. Ceballos, Manuel D. Zuluaga, Paola A. Arias, Blanca A. Botero, Mar´ıa I. Montoya, Juan D. Giraldo, and Diana I. Quevedo Posgrado en Recursos Hidr´ aulicos, Escuela de Geociencias y Medio Ambiente, Universidad Nacional de Colombia, Medell´ın, Colombia Revised version submitted to Journal of Hydrologic Engineering May 24, 2005
Abstract. Colombia has climatic influences ranging from the Caribbean Sea and the Pacific and Atlantic Oceans, through the tropical forest of the Amazon basin, the savannas of the Orinoco River, amidst the orographic and hydro-climatic effects introduced by the southwesterly crossing of the Andes. Such environmental complexity presents a challenging problem to predicting hydrological processes in commonly ungauged river basins. Here we estimate river flows along the entire river network of Colombia through the joint use of the long-term water balance in river basins and statistical scaling of hydrological processes. The method is used to estimate long-term average river discharges, as well as peak and low flows of different return periods. The calculation of the long-term water balance considers the spatial variability of hydrologic fields, in which drainage basins are considered the basic hydrological control volumes for estimation. This work presents a systematic effort to estimate long term averages of main hydrologic variables using available observations, physical principles and statistical estimation techniques. Such estimation process requires consideration of physical controls. For instance, topography is a major control for precipitation, temperature, wind velocity, and air humidity. The precipitation field was estimated combining raingauge measurements with existing hand-made expert maps as an input trend for a universal Kriging interpolation technique. The estimation of actual evaporation was a major challenge due to the variety of methods available and because of conceptual problems related to the long-term and regional scale estimation required. Most of the methods are short term point estimation equations of potential evaporation, and may not be suitable for larger scales. A systematic evaluation of climatic evapotranspiration estimation was carried out for diverse methods. The results were tested using the long term water balance equation against 200 streamflow gauging stations. No method for actual evapotranspiration showed significant superiority over the rest. Overall, we conclude that the magnitude of errors arises fundamentally from deficiencies in the data and the paucity of the observations. In addition to long term average fields we estimated floods and low flow fields for different return periods, for the entire river networks of Colombia using regionalization and scaling concepts with the long-term flow field as the scaling variable. All data sets, methods, and results are included in HidroSIG, an available interactive hydrologic atlas of Colombia.
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1. Introduction
tribution of this work. Additionally, the newly created distributed data sets of fundamental hydrological variables constitute another contribution of this work. The PUB problem becomes relevant to estimating long-term mean and extreme river discharges along the drainage network of river basins, guiding the scientific and practical advancements of hydrology for decades. Estimation of floods and their associated return periods have been approached from diverse points of views including the probabilistic flood frequency analysis [Chow, 1951, 1964], the so-called regionalization method [Kirby and Moss, 1987; Bobee et al., 1993; Hoskings, 1990], which involves extrapolation from one catchment to another within a “homogeneous” region, and also the so-called “derived flood frequency” approach [Eagleson, 1972], that combines statistical models of rainfall with dynamical modeling of the physical processes underlying runoff generation and routing. A comprehensive account of the suite of computer models of watershed hydrology is reviewed in Singh [1995]. The lack of sufficiently long records of river flows and rainfall measurements, and the lack of detailed small scale measurements of parameters and processes precludes these thechniques for reasons stated previously. The framework of statistical scaling is based upon scale-invariant behavior of hydrological processes in time and/or space [Feder, 1988; Rodriguez-Iturbe and Rinaldo, 1997; Sposito, 1998]. There is a growing body of literature on the scaling properties of floods [Gupta and Waymire, 1990; Smith, 1992; Turcotte and Greene, 1993; Gupta et al., 1994; Gupta and Dawdy, 1995; Tessier et al., 1996; Goodrich et al., 1997; Bloschl and Sivapalan, 1997; Gupta and Waymire, 1998; Pandey et al., 1998; Vogel and Sankarasubramanian, 2000; Menabde et al., 2001; Troutman and Over, 2001; Schertzer et al., 2002; Labat et al., 2002; Odgen and Dawdy, 2003; Gupta, 2004; Mantilla et al., 2005]. Statistical scaling provides an reasonable compromise between the physical description of detailed hydrological processes and a more simplified, larger-scale approach with a much smaller number of degrees of freedom [Menabde and Sivapalan, 2001; Gupta, 2004]. This approach has been used to study “downstream hydrological processes” [Leopold and Maddock, 1953; Jarvis and Woldenberg, 1984], where basin area is used as the scale parameter. The scaling framework is an attractive methodology to tackle the PUB problem for average [Vogel and Sankarasubramanian, 2000], and minimum annual river flows [Furey and Gupta, 2000]. In particular, important hydroecological processes controlling the space-time variability of river flows in our region of interest, exhibit scaling
Colombia, located in northwestern tropical South America boasts a wide variety of climates, landscapes and ecological environments, including rainforests, mountain forests, deserts, savannas, and tropical glaciers. The country exhibits complex hydrological, geomorphological and climatological features not only due to its tropical setting, but also because of: (i ) the strong topographic gradients of the three branches of the Andean Mountains crossing from southwest to northeast, (ii ) the hydro-climatic and ecological dynamics of the Amazon and Orinoco River basins, (iii ) the atmospheric circulation patterns over the neighboring tropical Pacific and Atlantic oceans, and (iv ) the strong landatmosphere feedbacks [Poveda and Mesa, 1997]. The concomitant high spatial hydro-climatological variability includes one of the wettest regions on Earth [Snow, 1976; Poveda and Mesa, 2000], and time variability spans from diurnal [Poveda et al., 2005a], to interannual timescales [Poveda et al., 2001, 2005b]. On top of this daunting hydro-climatological complexity, additional hurdles impede the advancement of hydrologic engineering research and practice: (i ) the pervasive lack of adequate information of relevant hydrologic variables in space and time, (ii ) the existence of poor-quality, limited (space and time) and costly data sets sold by local hydro-meteorological services, (iii ) the lack of appropriate methodologies to predict hydrologic variables over a wide range of space-time scales in tropical environments, and (iv ) the prohibitive licensing costs of commercial geographical information systems to model the spatial dynamics of hydrological processes. This situation is the rule throughout the developing world in general. Thus, prediction of hydrologic processes in ungauged basins (“PUB problem”; http://www.cig.ensmp.fr/∼iahs/), an important focus of the International Association for Hydrological Sciences (IAHS), becomes an even more challenging task. This paper represents an effort to overcome most of the aforementioned problems of hydrologic engineering, for the case of Colombia. In particular, we aim to estimate mean and extreme (floods and minimum) river flows for the entire drainage river network of the country, using a simple yet powerful methodological approach that links the long-term water balance equation within the framework of regionalization and statistical scaling, for estimation purposes. Both methodologies have been long known in the hydrological literature, but their joint usage to tackle the PUB problem at the space-time scales of interest, constitutes an original con-
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POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA properties in space. These include, mesoscale convective systems [Poveda and Mej´ıa, 2004], vegetation activity (NDVI) [Poveda and Salazar, 2004], and minimum streamflows [Poveda et al., 2002]. Our methodological approach to estimate peak flows for different return periods is based on the classical theory of quantile analysis [Chow, 1951]. This method is used in combination with scaling ideas to estimate statistics of annual floods and low-flows of return periods of 2.33, 5, 10, 25, 50 and 100 years throughout the entire drainage river network of Colombia. There exist clear power law relations between mean and maximum flows, that arise from the well-known relations that link annual flood estimates with basin area [Gupta and Waymire, 1990; Smith, 1992; Gupta et al., 1994; Gupta and Dawdy, 1995], and through the power law relation between long-term average river flows and basin area [Vogel and Sankarasubramanian, 2000]. From this, the long-term water balance equation for river basins can be used to estimate long-term average flows throughout the entire river network [Eagleson, 1994] and connect them to annual floods. Towards this end, we developed maps of long-term mean annual precipitation and evapotranspiration (actual and potential), using diverse input data sets and Kriging for interpolation purposes. The results of this work have been incorporated into an Interactive Digital Hydrological Atlas, supported on HidroSIG. HidroSIG, developed in-house, is a GIS capable of calculating and visualizing fields of long-term average precipitation, actual and potential evapotranspiration, river flows (mean and extreme), and other hydro-climatological variables. HidroSIG includes tools for time series analysis of monthly and annual hydroclimatic records. Algorithms to extract river networks from Digital Elevation Maps (DEM) and integrating random fields over river basins have been incorporated as well. The graphical user interface allows the estimation of mean annual discharges at any desired location on Colombia’s drainage network. Currently, the Ministry of Mining and Energy of Colombia is using HidroSIG to re-asses the hydropower generation capacity of the country. This paper proceeds as follows. Section 2 describes the data sets. Section 3 presents the methods used for estimation purposes. Section 4 contains the results, including the maps constructed for precipitation, and evapotranspiration (actual and potential), and a discussion of estimation errors and validation of results. A brief description of HidroSIG is presented at the end of section 4, and the conclusions are drawn in the final section.
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2. Data 2.1. Digital Elevation Model and River Network Extraction Topography is an essential component of this work. We have used GTOPO30, a Digital Elevation Map (DEM) developed by the U.S. Geological Survey, to extract the portions of Colombia and neighboring countries. GTOPO30 provides regularly spaced elevations at 30-arc seconds (≈ 1 km). Extraction of river networks and drainage basins from DEMs is a state-of-the-art problem in hydrology and geophysics, which demands high algorithmic efficiency [Band , 1986; Garbrecht and Martz, 1994]. We used the steepest descent method to extract river networks from the topographic DEM, starting with the construction of a direction matrix that defines the path of stream channels over terrain. We performed rigorous quality-control procedures to: check for problems within the DEM; for consistency with observed river networks at finer spatial scales; to account for geologic controls; to eliminate errors; and to resolve the appearance of spurious sinks in the DEM, especially on low-slope terrains and flood plains. Details of the procedures developed to extracting the river network used in this work and its quality-control may be found in Ram´ırez and V´elez [2002]. 2.2. Precipitation Precipitation data were obtained from Colombian meteorologic Institute (IDEAM), Empresas P´ ublicas de Medell´ın (EPM), Corporaci´on del Valle del Cauca (CVC), and Centro Nacional de Investigaciones del Caf´e de Colombia (Cenicaf´e). A total of 688 raingauges were used to estimate the distributed average rainfall field, using monthly data for the period 1965-1987. Raingauges are generally concentrated along the Andes, with sparser coverage over central, western and northern Colombia. Long-term data over the Colombian portions of the Orinoco and Amazon River basins are very scarce. To overcome this limitation we digitized and used previous studies and maps developed by the National Water Assessment Study (Estudio Nacional de Aguas, DNP [1984]), and by Oster [1979], and Snow [1976], with the aim of incorporating the expert knowledge contained in those maps, as a baseline for our estimation. In the ungauged regions of the Colombian Amazon basin, we also used the long-term average precipitation map developed by the EOS-Amazon Project (INPE-University of Washington; http://boto.ocean.washington.edu/eos/). Information from neighboring countries was used to estimate the interpolated rainfall field, including some
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA data from Brazil (LBA-HydroNET; http://www.lbahydronet.sr.unh.edu/) and Ecuador [Pourrut, 1994]. 2.3. Temperature Given the tropical setting of Colombia, average air temperature (T in Celsius) is strongly liked to altitude above sea level (h in meters), with no major changes throughout the year. Chaves and Jaramillo [1998] regionalized such relationship for Colombia, using information of 1002 gauging stations in Colombia, as follows Andean: Caribbean: Pacific: Eastern Plains and Amazon:
T T T T
= 29.42 − 0.0061h = 27.72 − 0.0055h = 27.05 − 0.0057h = 27.37 − 0.0057h
The number of stations used to fit these equations was 626, 239, 46 and 91 respectively with good correlations, R2 higher than 0.9 in all the cases. The empirical linear dependence between average air temperature and height is in close agreement with the theoretical adiabatic saturated lapse rate which is typical in tropical regions [Wallace and Hobbs, 1977], although with minor regional differences associated with air humidity. The combination of these relationships with the DEM, allowed the estimation of the average temperature map for Colombia. For those methods that estimate evapotranspiration based on temperature data at 2 meters above ground, we combine the above with information from the Global Data Sets for Land-Atmosphere Models [Sellers et al., 1995; Meeson et al., 1995]. Similar relations (not shown) were used for minimum temperatures, which play a role in the estimation of dew point condensation, an important portion of precipitation, not accounted for by raingauges, but of significance in high altitude humid tropical ecosystems known as p´ aramos. 2.4. Average Atmospheric Pressure Data pertaining to surface atmospheric pressure and height at 153 gauging stations were obtained from Eslava [1995], covering the period 1951-1980. The resulting estimated relationship between surface atmospheric pressure, (P in hP a), and altitude (h in meters above sea level), is P = 1009.28 exp{−h/8631}; R2 = 0.978 .
(1)
It is worth noting that this equation is in close agreement with the theoretical hydrostatic approximation, with a scale height of 8, 600m [Wallace and Hobbs, 1977].
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2.5. Radiation We used monthly data obtained from the Surface Radiation Budget as derived by NASA/Langley Research Center [Darnell et al., 1996], during the period July 1983-June 1991. The data set includes the following parameters: all-sky surface downward short-wave (SW) and long-wave (LW) radiation, as well as SW and LW net fluxes, clear-sky downward SW and LW fluxes, and cloud fraction, gridded at 1◦ x 1◦ . 2.6. Atmospheric Moisture Dew point temperature and relative humidity are observed in ordinary climatic stations. But in Colombia the number of those stations and the quality of their records are of very limited use, probably because air humidity is a variable with less direct application and fewer users than precipitation, temperature or river flows. For this work, data belonging to 45 ground stations from IDEAM were complemented by the large scale maps (1◦ x 1◦ ) produced by the Global Energy and Water Experiment (GEWEX) [Sellers et al., 1995; Meeson et al., 1995]. In general, long term average atmospheric moisture in Colombia shows dependence upon altitude and distance from the neighboring oceans. 2.7. Winds Long term records of wind speed and direction are very scarce in Colombia, limited to a few stations located in major cities and airports. We complemented those records with the GEWEX Global Data Sets for Land-Atmosphere Models [Sellers et al., 1995; Meeson et al., 1995]. From the original six-hour resolution data, the corresponding daily, monthly and yearly averages were computed. 2.8. Vegetation Cover According to Holdridge’s classification [Holdridge, 1978], Colombia harbors 26 life zones. According to this pioneering theory, the ecology of each zone is the result of climatic factors such as bio-temperature, precipitation and the relation between potential and actual evapotranspiration. Field studies of vegetation and biota, in general, allow the identification of these zones, following well established standards. Colombia’s Geographic Agency produced a life zone’s map [IGAC, 1988]. In view of the lack of evapotranspiration observations, we utilize this ecological classification as a proxy to estimate evaporation in combination with the rest of the climatic variables. The life zones map was digitized to use in the computations and as an independent variable
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA
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of interest in the constructed atlas. See details in V´elez et al. [2000]. 2.9. Soil Water Holding Capacity One of the methods to estimate actual from potential evapotranspiration requires quantification of the plant extractable water holding capacity [V´elez et al., 2000]. The Colombian portion was extracted from the data set produced by Dunne and Willmott [1996].
3. Estimation 3.1. Long-term Annual Precipitation Interpolation From long-term point averages, the interpolated longterm average precipitation field was estimated using Kriging with drift [Bras and Rodr´ıguez-Iturbe, 1984; Webster and Oliver, 2001]. Kriging is an optimal linear interpolation method that incorporates the spatial correlation structure of the random field to define required interpolation weighting factors. Such correlation function is represented by the semi-variance between pairs of data, or semi-variogram defined as, γ(h) =
1 V ar [z(xi ) − z(xi + h)] , 2
(2)
where z(xi ) is the value of the function (annual rainfall depth) to be interpolated at location i, and h represents separation distance between gauging stations. The range of the semi-variogram identifies the distance at which the interpolation algorithm searches for neighboring stations, consistent with the spatial threshold defined by the decay of the correlation function of average rainfall in space. There are diverse types of theoretical semi-variograms, e.g., circular, spherical, exponential, Gaussian, power functions, etc. To estimate regional semi-variograms exponential and Gaussian models were used since both produced the best fit to observations. In addition to the information provided by the semi-variogram, the Kriging method has been mofied to include so called drift. A variable’s drift is a single trend function defined at each location based on external information or a secondary variable. Topographic information was chosen as the drift function for interpolating precipitation fields. This decision was made based on the strong orographic forcing that the Andean Mountains impose on tropical rainfall over northern South America at different timescales [Poveda and Mesa, 2000; Poveda et al., 2005a]. Literature also reports improvement on interpolation of hydro-climatological variables through the
Figure 1. Estimated semi-variograms for annual precipitation for (a) the whole country), and three separate regions: (b) Andes region, (c) Caribbean, and (d) Pacific. use of topographic parameters as a source of auxiliary information [G´ omez-Hern´ andez and Cassiraga, 2000; Goovaerts, 2000; Diodato, 2005], especially for longterm averages, as is the case in the present work. The empirical and fitted semi-variograms for Colombia and three of its regions are shown in Figure 1. The principal direction of spatial correlation for the rainfall field, which corresponds to the direction of maximum isotropy or continuity in the precipitation field, was determined to be N30E, the main orientation of the Andean Mountains across Colombia, strengthening the argument of using topographic information as drift on the interpolation algorithm. The estimated regional semivariograms were used for interpolation purposes, and the country-scale precipitation map was constructed with careful analysis of continuity and consistency of estimates. A detailed analysis of different options examined to develop the long-term average annual precipitation map for Colombia may be found in Mej´ıa et al. [1999]. 3.2. Long-term Annual Evapotranspiration Long-term actual and potential evapotranspiration were estimated using the methods introduced by Turc [1955], Turc [1962], Coutagne [1954], Thornwaite [1948], Holdridge [1978], Meyer [1942], Penman [1948], Budyko [1974], Morton [1983], and Cenicaf´e [Chaves and Jaramillo, 1998]. Except for Cenicaf´e’s, all methods are well detailed in the literature and they will not be reviewed here. Cenicaf´e method estimates potential evaporation via an exponential relationship with elevation. The
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA equation parameters were estimated from regression analysis between evaporation estimates1 obtained at Cenicaf´e climatological stations throughout the Colombian Andes and elevation. The equation gives Ep = 1700 exp(−0.0002h), where Ep represents annual potential evaporation [mm/yr], and h represents the height above sea level [m]. For those methods which solely provide estimates of potential evapotranspiration, estimates of actual evaporation, E, were obtained using the pioneering formulation proposed by Budyko [1974], ·µ E=P
1 − cosh
Ep Ep + sinh P P
¶
P Ep tanh P Ep
¸1/2 .
(3) Table 1 contains details of references, parameters needed, type of method and units for the different methods of evaporation used in this work. 3.3. Long-term mean river flows The differential equation for water balance of a drainage basin is given by [Manabe, 1969; Schaake, 1990], dS(t) = P (t) − E(t) − R(t) , (4) dt where S(t) represents soil and ground water storage as a function of time, P (t) and E(t) represent basinintegrated precipitation and actual evapotranspiration rates, and R(t) represents the total runoff leaving the basin. Total runoff R(t) includes the streamflow at the basin outlet and the net integrated lateral subsurface runoff. Integrating equation (4) over long time scales, say of time length T gives 1 [S(T ) − S(0)] T
=
1 T
Z
T
[P (t) − E(t) − R(t)] dt 0
= P −E−R
(5)
where the bars denote time average according to the mean value theorem. Since soil and groundwater storage, S, is finite as T is increased, the quantity [S(t) − S(0)]/T goes to zero. Thus, the long-term approximation for the water balance gives, R = P − E. Therefore, estimation of mean annual runoff requires basinintegrated estimates of precipitation and actual evapotranspiration. To illustrate the validity of R = P −E, as an adequate long-term approximation, it can be shown 1 Obtained via the Penman equation using measured variables at the station
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that the magnitude of the maximum error is bounded above by, Smax ≤ 15mm/year; , (6) n where n is the number of years (22 in this study, period 1966-1987) used to estimate the long-term averages for P , E, and R; and max [S(t) − S(0)] = Smax is less than the maximum soil water holding capacity, which we take as 300 mm, a high value according to the Soil Map and Soil Climate Map (USDA-NRCS, Soil Survey Division, World Soil Resources, Washington D.C, http://www.nrcs.usda.gov/technical/worldsoils/). This large value of water holding capacity is used to illustrate the largest error inherent to equation (5). Using this value we obtain that the maximum error for the equation is 15 mm/yr corresponding to less than 8% of runoff in arid zones in Colombia (200 mm/yr), and less than 2% in wet regions, where runoff is in the order of 103 mm/yr. The validity of our estimation method is implied, since the approximation produces smaller errors than those associated with each of the water balance components, P , E, and R. An additional simplification is introduced by letting R be represented by surface runoff, Q, such that R = Q/A, where A is the basin area. This simplification avoids the difficulty of estimating the net subsurface water budget, and its almost exact for large basins where groundwater flow enters the basin channels as baseflow. Although, in some regions this assumption is not valid, the validity of this approximation for the Colombian case will be discussed below in the results section. To simplify notation and because of ergodicity, one can replace time averages for expected values. Therefore, over bars will be dropped hereafter. The equation Q = A[P − E] is taken as the methodological basis of this study, through integration of P and E over the spatial domain. Long-term river discharge is therefore estimated by integration of the long-term averages of precipitation and actual evapotranspiration over the corresponding drainage river basin area, A, extracted from HidroSIG, as Z Z Q=A [P (x, y) − E(x, y)]dxdy, (7) | P − E − R |≤
whose numerical approximation is given as, X Q∼ (Pi,j − Ei,j ). =A
(8)
i,j∈A
For validation purposes, discharge records from more than 200 gauged sites throughout Colombia were compared with estimated long-term river discharges through
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA
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Table 1. Methods used to estimate Evaporation Method
Type
Units
Turc
Actual
mm/year
Turc Modified
Potential
mm/month
Coutagne
Actual
m/year
Cenicaf´e Thornwaite
Potential Potential
mm/yr cm/month
Holdridge
mm/year
Meyer
Actual & Potential Potential
Penman
Potential
mm/day
Morton
Potential
W/m2
Budyko
Actual Actual
E = 2EW − Ep mm/yr
cm/month
Input Parameters
Sources
Annual Precipitation Mean Annual Temperature Mean Annual Temperature Monthly Net Solar Radiation Monthly Mean Rel. Humidity Annual Precipitation Mean Annual Temperature Height above sea level Monthly Mean temperature Annual Mean Temperature Mean Annual Temperature Latitude Mean saturation vapor pressure Mean vapor pressure Mean wind speed Monthly Net Solar Radiation Atmospheric pressure Albedo Atmospheric surface pressure Mean temperature Mean Dew Point Mean saturation vapor pressure
This work 1, 2, 3, 4, 5 1, 2, 3, 4, 5 6, 7 Estimated This work 1, 2, 3, 4, 5 Penman method 1, 2, 3, 4, 5 1, 2, 3, 4, 5 1, 2, 3, 4, 5
Ep
Each Method
f (Tdew ) f (Tair ) GEWEX (8, 9) 6, 7 3 6, 7 10 1, 2, 3, 4, 5 IDEAM (10) f (Tdew )
Sources: 1, Chaves and Jaramillo [1998]; 2, Cort´es [1989]; 3, Eslava [1995]; 4, Eslava et al. [1986]; 5, Stanescu and Diaz [1971]; 6, Rodr´ıguez and Gonz´ alez [1992]; 7, Morton [1983]; 8, Meeson et al. [1995]; 9, Sellers et al. [1995]; 10, Instituto de Hidrolog´ıa, Meteorolog´ıa y Estudios Ambientales, Colombia.
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA
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the water balance equation using different methods to estimate actual evapotranspiration. 3.4. Regionalization of floods Our methodological approach to estimate peak flows for different return periods is based on the classical quantile analysis [Chow, 1951]. This method is used in combination with scaling theories to estimate statistics of annual floods in terms of mean annual flows. Observations show the existence of power law relations between mean and maximum flows. In addition, both long-term average river flows as well as annual floods exhibit scaling properties with basin areas [Gupta and Waymire, 1990; Smith, 1992; Gupta et al., 1994; Goodrich et al., 1997; Vogel and Sankarasubramanian, 2000]. Those references contain the basis for a theory to explain observations, both in the framework of simple scaling or multiscaling. Thus, we estimated annual floods, Qmax (Tr ), for a given return period (Tr = 1/p, the inverse of the exceedence probability), as [Chow, 1951], Qmax (Tr ) = µQmax + k(Tr , γ)σQmax ,
Figure 2. Location of river gauging stations used to estimate annual floods and low flows. The boundaries correspond to the identified homogeneous hydroclimatological regions.
(9)
where µQmax , and σQmax are the mean and standard deviation of annual floods, and k(Tr , γ) is the frequency factor, which is a function of the return period and possibly of other parameters that are generically represented in γ. For different probability distribution functions (PDF) assigned to annual floods, the functional form of k(Tr , γ) is different, as is long known in hydrologic literature [Chow, 1951, 1964]. After some exploration with various distributions and the standard statistical tests, we chose lognormal distribution, which is a particular case of the family of Log-Stable distributions whose structure is consistent with the theory of multi-scaling [Zolotarev, 1986]. The frequency factor for the lognormal distribution may be found in Chow [1964, p. 8-25]. We assume a power law relating the statistical parameters of annual floods and mean annual flows, Q, which in turn can be expressed in terms of the long-term water balance equation, µQmax = αµ Qθ1 = αµ [A(P − E)]θ1 ,
(10)
σQmax = ασ Qθ2 = ασ [A(P − E)]θ2 .
(11)
where the pre-factors, αµ and ασ , and the scaling exponents, θ1 and θ2 [Mandelbrot, 1998], are fitted from observed data. Figure 2 shows the location of the 225 river gauging stations used to estimate pre-factors and scaling exponents in (10) and (11). Towards that end, different hydro-climatological regions in Colombia were
defined according to similarities among topographical, climatic and vegetation conditions, within a context of nested river basins, and taking into account location of gaging stations and climatic conditions. The identified sub-regions are shown in Figure 2. From the data set, 65% of all stations have at least 24 years of record, with less than 5% of missing records, and half of the data set having observation periods longer than 30 years. The procedure implies estimating the parameters in (10) and (11) that minimize the sum of squared errors between observed and estimated mean and standard deviation of the annual floods in each case, and within each hydroclimatic region. Then, for estimation purposes inside a single hydro-climatic region, we use the estimated prefactors and scaling exponents into the right hand side terms of (10) and (11). For a specific site comprised of sub-basins from different hydro-climatic regions, we use the area to weight the pre-factors for the corresponding regions. With those estimates, and the integrated maps of precipitation and evaporation, we developed maps corresponding to the mean and standard deviation of annual floods using equations (10) and (11). Then, we can estimate floods associated with different return periods, for any site along the channel network, using the corresponding frequency factors in equation (9). The range of basin areas utilized in the regressions go from 0.2 km2 to 4500 km2 .
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA 3.5. Regionalization of low flows A similar approach was adopted towards estimating annual (daily) low flows, using the mean and standard deviation of the annual minima fitted by the lognormal distribution. The low flow statistics are related to mean annual discharge as [Poveda et al., 2002], µQmin = βµ Qθ3 = βµ [A(P − E)]θ3 ,
(12)
σQmin = βσ Qθ4 = βσ [A(P − E)]θ4 ,
(13)
Similarly, the pre-factors βµ and βσ , and the scaling exponents, θ3 and θ4 , in (12) and (13), were also regionalized according to the same procedure followed for floods, and for the same hydrologic regions (see Figure 2). A total of 240 gauging stations having more than 20 years (less restricted than flood data), were used in the analysis. Figure 2 shows the distribution of gauging stations used in the procedure. Once the scaling exponents and pre-factors of (12) and (13) were estimated, we produced maps for the mean and the standard deviation of minimum flows. Such maps and the corresponding equation analogous to (9) for low flows, allowed estimation for different return periods, over the whole river network.
4. Results 4.1. Long-term Annual Precipitation Hydrological fields were estimated with data sets for the period 1966-1987. Figure 3 shows the estimated long-term average precipitation field for Colombia at 5 arcmin resolution. Visual inspection reveals features consistent with characterization of precipitation in Colombia reported in literature: (i ) Extremely high precipitation occurs along the Pacific coast including one of the rainiest regions in the world [Poveda and Mesa, 2000; Mapes et al., 2003], with 8, 000 − 12, 000 mm/year; but also the middle Magdalena river valley and the eastern piedmont of the Andes (5, 000 − 6, 000 mm/year); (ii ) the so-called “pluviometric optimum” inside the intra-Andean valleys [Hastenrath, 1991; Oster, 1979], and (iii ) the dry Guajira peninsula (300 − 400 mm/year) by the Caribbean Sea. As was expected, measures of the variance of the estimation provided by Kriging’s method indicate that the largest interpolation errors correspond to regions with scarcer data (not shown). The main shortcomings of the estimated precipitation map are the paucity of long-term basic data. Almost half of the country lacks information for the type of estimations required in this study.
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Most data are derived from the populated Andean region, while large portions of the Amazon and Orinoco basins in Colombia lack adequate basic data. 4.2. Long-term Annual Actual Evaporation Turc Method. Actual evapotranspiration is based on mean temperature and precipitation, both strongly associated with topography in the tropics. Figure 4a shows the derived long-term average actual evapotranspiration map for Colombia. Cenicaf´e Method. Estimates of evapotranspiration are based on a simple linear regression between estimates of evaporation (through Penman equation) and elevation above sea level. It represents the influence of topography, terrain and vegetation complex along the Andes. As Penman estimates potential evaporation, Budyko’s equation (3) is used to estimate actual evaporation. See map in Figure 4b. Penman Method. This method tended to underestimate evapotranspiration, when compared to estimates of E = P − Q, with observed values of P and Q. This could be due to the fact that most input variables are obtained from very coarse satellite information. The approximation given by Priestley and Taylor [1972], produced much better closure of the water balance equation. From the obtained potential evapotranspiration map, the actual evapotranspiration map was estimated through Budyko’s equation (3). Figure 4c shows the map for E using the Priestley-Taylor approximation. Holdridge Method. This method produced acceptable evapotranspiration estimates under the water balance closure; see map in Figure 4d. Morton Method. The estimates of both actual and potential evaporation arising from this method tended to overestimate evaporation, perhaps due to very coarse input data for solar radiation and wind velocity. Recent studies have revisited the rationale of this method [Szilagy, 2001], which contributed to explaining the socalled evaporation paradox of climate change [Brutsaert and Parlange, 1998], and which performs adequately for humid environments [Hobbins et al., 2001], like Colombia’s. The performance of the different methods used to estimate actual evaporation were compared with estimates from the long-term water balance equation in gauged basins. The results are presented in the following section.
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Precipitation 14 Caribbean Sea
12
mm/year 6000 5000 4500
10 Venezuela
4200 3900
8 3600 Pacific Ocean
6
3300 3000
4 2700 2
2400 2100
0 Ecuador -2
Brazil Peru
-78
-76
-74
1500 1200
-4 -80
1800
-72
-70
-68
-66
800 300
Figure 3. Long-term average annual precipitation field for Colombia, obtained through interpolation using Kriging with drift. See text for details.
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Figure 4. Long-term average annual actual evapotranspiration estimated using the methods by (a) Turc-Budyko, (b) Cenicaf´e, (c) Priestley-Taylor, and (d) Holdridge.
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4.3. Long-term Average Annual Stream Flows Estimation of average flows starts by extracting the drainage basin from the DEM, and then HidroSIG integrates precipitation and evapotranspiration values over the basin. For instance, we estimate 7, 439 m3 /s as the long-term river flow of the Magdalena River near its mouth (255, 586 km2 ; see Figure 5), for a long-term average annual precipitation of 2, 049 mm/yr, and actual evaporation of 1, 131 mm/yr. The user can choose among the available methods to estimate actual evapotranspiration. Water balance estimates were compared to observed long-term river flows at more than 200 gauging stations in Colombia. Our results confirm that the water balance equation provides very good estimations for long-term average river flows. Figure 6 shows observed (abscissa) and estimated (ordinates) long-term river flows, using different evapotranspiration methods. Our results indicate that the methods of Turc (22.8% r.m.s.e.), Cenicaf´e (23.3%), Morton (23.4%), Holdridge (24.1%), and Priestley-Taylor (24.6%) exhibit the lowest errors in closing the water balance equation. An investigation of the geographical distribution of errors indicated no systematic errors throughout the country. Figure 7 shows the distribution of errors for estimated discharges at all gauged sites, using the Morton’s method to estimate actual evapotranspiration. Average errors are on the order of 20-30%, distributed all over Colombia. There is an obvious need for improved spatial coverage of the existing networks of all hydroclimatological parameters, in order to improve the quality of estimations. One possibility for a spurious agreement between Q and A(P − E) is that errors in P and in E could compensate each other. Also, because of the space integration within each basin, there may be cancellation of errors in one or both of the variables, P or E. For the first case, one needs to consider that such errors would be reflected in the energy balance, too. In fact, the available net solar radiation is partitioned into warming (sensible heat) and evaporation (latent heat). Therefore, assuming that good-quality solar radiation data are available, an error in evaporation estimates would reflect in opposite sign errors in air temperature; moreover in the Colombian humid climate, evaporation is mostly energy-limited not water-limited. With the kind of data sets available for both solar radiation and surface air temperatures, we do not consider this first possibility to be of first order in our analysis. The second possibility is that spatial compensation of errors will be reflected in the water balance of the sub-basins. One
Figure 5. River basin and channel network of the Magdalena river at a site close to its mouth by the Caribbean Sea, extracted from the Digital Elevation Map. The river network has been checked for consistency and precision inside the Colombian territory.
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA
Figure 6. Comparison of observed and estimated longterm average river flows at more than 200 river gauging stations in Colombia, using diverse evapotranspiration methods. r.m.s. refers to error estimation.
Figure 7. Distribution of errors between observed and estimated long-term river discharges at gauging stations, using Morton’s method to estimate actual evapotranspiration.
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cannot discard this possibility altogether but there is no evidence of an overall tendency in the errors either with basin area, altitude or geographical location. Regarding this second possible source of false agreement, here we address our neglect of groundwater runoff, as a source of real discordance between A(P −E) and Q. One should notice first that, in the long run, sources should be compensated with sinks. Such sources correspond with the so-called recharge areas from groundwater, which are more important in arid or semiarid climates, but not so in humid climates. Evidence in regards to this issue is the linear dependence of low flows with basin areas, as will be discussed below. This fact indicates that groundwater flows into the river network are mostly uniformly in space. From these analyses we conclude that the main source of errors in the closure of the water balance reside more in the data bases (quality and space coverage), than in the methods themselves. Few antecedents on the systematic validation of methods to estimate actual evapotranspiration at this scale appear in the works of V¨ or¨ osmarty et al. [1998], and Choudhury [1999], is contemporaneous with most of the work reported in this paper. The former study compares a set of potential evaporation functions as a precursor to estimating actual evapotranspiration in the US. To that end, they use a simple water budget model to derive actual from potential evapotranspiration, whereas we computed actual evapotranspiration directly using Budyko’s climatic relation [Budyko, 1974], or the advection aridity method of Morton [1983]. Their test was on a 0.5◦ x0.5◦ grid, coarser than the spatial resolution used in the present work. Their comparison was made between the estimated evapotranspiration and P − R for a representative site of each grid point, whereas we used the extracted basin corresponding to the streamflow gage and the integrated water balance. The study by Choudhury [1999] uses the longterm water balance equation to fit the optimal parameter of an equation for long-term annual actual evapotranspiration in major river basins of the world, using the results of a complex global model that couples water, energy and carbon budgets. Our results are similar to both aforementioned studies, taking into consideration obvious differences in the amount and quality of the observational database. 4.4. Annual floods Estimates of the parameters of equation (10) showed little variation among different hydro-climatic regions, with average pre-factors αµ = 6.71, and scaling ex-
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Figure 9. Maps of αµ (left, equation 10) and ασ (right, equation 11), estimated for the various hydrological regions of Colombia.
Figure 8. Regressions between average river flows and statistical parameters of annual floods at 225 stations in Colombia. Results are shown for the mean (top), and standard deviation (bottom). ponents θ1 = 0.82. See Figure 8. A similar analysis was performed to estimate the standard deviation of annual maximum flows. Regression analysis yielded ασ = 3.29 (varying regionally), and highly stable scaling exponent, θ2 =0.648. Maps of αµ and ασ are shown in Figure 9. From these maps, we estimated annual floods with different return periods through eqn. (9). For instance, the 100-yr flood at the chosen site on the Magdalena river was estimated as 14, 197 m3 /s, with mean and standard deviation of annual maximum flows estimated as 10, 527 m3 /s, and 1, 169 m3 /s, respectively. HidroSIG provides these results at the click of the mouse. 4.5. Low annual flows Estimation of statistical parameters for (12) and (13) produced very similar scaling exponents θ3 ≈ 1, and θ4 ≈ 1, throughout the country. Figure 10 shows the results for the estimation of both scaling exponents in gauged basins. These results suggest the existence of simple scaling in the statistics of low flows with basin area, which was confirmed through analysis of linearity
in the structure functions of moments, up to the fourthorder moment. A more in-depth analysis of the simple scaling nature of low flows in Colombia may be found in Poveda et al. [2002]. Simple scaling of regional low flows has been demonstrated in Furey and Gupta [2000]. The pre-factors βµ , and βσ exhibited remarkable stability among regions, as shown in Figure 11. For instance, using the lognormal distribution, the estimated 50-yr return period low flow of the Magalena River at the chosen site (Fig 5) is 1, 294 m3 /s, while mean and standard deviation of annual low flows are 2, 539 m3 /s, and 794 m3 /s, respectively.
4.6. HidroSIG Java All data sets, methods and results of the present work have been incorporated into a geographic information system for hydrological analysis, HidroSIG, developed in JavaTM . HidroSIG contains a complete hydroclimatological database for Colombia, and a series of tools for visualization, interpolation and analysis of the spatially distributed hydro-climatic variables, to perform time series analysis of data at a station. Additionally, HidroSIG contains modules to estimate, store and display geomorphological information and parameters from DEM’s, including extraction of the stream channel network, identification of river basin divides and areas, stream channel ordering using the Strahler scheme, estimation of Horton ratios, width function, hypsometric curve, aspects, etc. It was developed for our research in Colombia, but any user is able to use their own DEMs, databases and maps at any spatial resolution, and get results at the “click” of the mouse. HidroSIG has been developed during the last
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10 years, and constitutes the hydrological and computational basis for a recently developed tool to study hydrological processes in real and virtual basins [Mantilla and Gupta, 2005]. It is freely available to the scientific community. Detailed information can be found in http://cancerbero.unalmed.edu.co/∼hidrosig/index.php.
5. Conclusions
Figure 10. Regressions between average river flows and the mean annual low flow in Colombia.
Figure 11. Maps of βµ (left, equation 12) and βσ (right, equation 13), estimated for the various hydrological regions of Colombia.
Estimates of river flows (mean and extremes) along the entire river network of Colombia were developed through the conjoint use of the long-term water balance equation in river basins, and the framework of statistical scaling of hydrological processes. Both methods are long known in hydrological practice and research, but their combined usage constitutes a novel contribution of this work. To that end, interpolated maps of longterm mean annual rainfall and evapotranspiration, both actual and potential evapotranspiration maps were constructed for the country, using Kriging with drift for interpolation of at-a-station information, and maps from previous studies. We tested the validity of the long-term water balance equation with independent measurements of river discharges at more than 200 river gaging station throughout the country. Estimation errors on the order of 15-30% rmse confirm the quality of our estimates at the chosen spatial scale of resolution. No method for estimating evapotranspiration showed clear superiority between the different methods. Results of the water balance to estimate long-term river flows were used to introduce a methodology that estimates both peak and low flows for different return periods. The method combines the traditional quantile analysis within a scaling framework that relates average and extreme river flows through hydro-climatological parameters. We have developed a hydro-climatic atlas for Colombia, HidroSig, as a custom-made GIS that stores, estimates, and deploys all calculations and results from the present study. It can be easily used and adapted elsewhere. The software and the hydro-climatological database of Colombia constitute additional contributions of this work.
Acknowledgments. This work has been supported by Unidad de Planeaci´ on Minero-Energ´etica (UPME) from the Ministry of Mining and Energy of Colombia, by COLCIENCIAS, by DIME from Universidad Nacional de Colombia, and by the Interamerican Institute for Global Change Research (IAI). G. Poveda is grateful to CIRES (U. Colorado, Boulder) for their support as a Visiting Fellow. We thank to H. Diaz, V. K. Gupta, P. R. Waylen, B. Kindel, E. Machado and A. Jaramillo for their insights. To the three anonymous
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA reviewers for helpful comments that improved the manuscript. To the Distributed Active Archive Center (DAAC) at the Goddard Space Flight NASA Center, for providing access to radiation data sets, to the USGS for access to the GTOPO30 data, to LBA project for precipitation data from the Amazon, to K. Dunne for providing the water holding capacity data, and to Empresas P´ ublicas de Medellin (EPM), Corporaci´ on del Valle del Cauca (CVC), and Cenicaf´e for providing free access to hydrologic data sets.
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G. Poveda, J. I. V´elez, O. J. Mesa, L. A. Cuartas, O. J. Barco, R. Mantilla, J. F. Mej´ıa, C. D. Hoyos, J. M. Ram´ırez, L. I. Ceballos, M. D. Zuluaga, P. A. Arias, B. A. Botero, M. I. Montoya, J. D. Giraldo, and D. I. Quevedo, Posgrado en Recursos Hidr´aulicos, Escuela de Geociencias y Medio Ambiente, Universidad Nacional de Colombia, Carrera 80 x Calle 65, Bloque M2-315, Medell´ın, Colombia. e-mail:
[email protected]
POVEDA ET AL.: WATER BALANCES, SCALING AND RIVER FLOWS IN COLOMBIA This preprint was prepared with AGU’s LATEX macros v5.01, with the extension package ‘AGU++ ’ by P. W. Daly, version 1.6b from 1999/08/19.
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