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Frontiers of Earth Science

Front. Earth Sci.    2017, Vol. 11 Issue (2) : 203-213     https://doi.org/10.1007/s11707-017-0622-7
RESEARCH ARTICLE |
Simple statistical models for relating river discharge with precipitation and air temperature—Case study of River Vouga (Portugal)
T. STOICHEV1(), J. ESPINHA MARQUES2, C.M. ALMEIDA1, A. DE DIEGO3, M.C.P. BASTO4, R. MOURA2, V.M. VASCONCELOS1
1. Interdiscplinary Center of Marine and Environmental Research (CIIMAR/CIMAR), University of Porto, 4450-208 Matosinhos, Portugal
2. Institute of Earth Sciences (ICT) and Department of Geosciences, Environment and Land Planning, Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal
3. Department of Analytical Chemistry, Faculty of Science and Technology, University of the Basque Country UPV/EHU, 48080 Bilbao, Basque Country, Spain
4. CIIMAR/CIMAR and Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal
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Abstract

Simple statistical models were developed to relate available meteorological data with daily river discharge (RD) for rivers not influenced by melting of ice and snow. In a case study of the Vouga River (Portugal), the RD could be determined by a linear combination of the recent (PR) and non-recent (PNR) atmospheric precipitation history. It was found that a simple linear model including only PR and PNR cannot account for low RD. The model was improved by including non-linear terms of precipitation that accounted for the water loss. Additional improvement of the models was possible by including average monthly air temperature (T). The best model was robust when up to 60% of the original data were randomly removed. The advantage is the simplicity of the models, which take into account only PR, PNR and T. These models can provide a useful tool for RD estimation from current meteorological data.

Keywords multiple regression      atmospheric precipitation      river discharge      runoff      Aveiro Lagoon     
Corresponding Authors: T. STOICHEV   
Just Accepted Date: 09 January 2017   Online First Date: 28 February 2017    Issue Date: 19 May 2017
 Cite this article:   
T. STOICHEV,J. ESPINHA MARQUES,C.M. ALMEIDA, et al. Simple statistical models for relating river discharge with precipitation and air temperature—Case study of River Vouga (Portugal)[J]. Front. Earth Sci., 2017, 11(2): 203-213.
 URL:  
http://journal.hep.com.cn/fesci/EN/10.1007/s11707-017-0622-7
http://journal.hep.com.cn/fesci/EN/Y2017/V11/I2/203
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T. STOICHEV
J. ESPINHA MARQUES
C.M. ALMEIDA
A. DE DIEGO
M.C.P. BASTO
R. MOURA
V.M. VASCONCELOS
Fig.1  Regional framework of the Aveiro Lagoon. DEM obtained from the Shuttle Radar Topography Mission (SRTM), in 90 meter resolution, produced by NASA and distributed by the United States Geological Survey [USGS], 2009.
FitP1; P2_11a)P1; P2_31b)P3; P4_13c)P3; P4_33d)P5; P6_15e)P5; P6_35f)
a02.4701.4752.2351.3442.0251.159
a10.1160.1050.2540.2160.3580.298
a20.4190.3630.3470.3700.3030.355
a1,2–2.38×103–3.59×103–4.41×103–6.35×103–6.83×103–1.05×102
λ0.26260.14140.26260.18180.26260.1818
R20.6810.6730.7330.7380.7520.775
Tab.1  Significant (p<0.05) regression coefficients (Eq. (1)), Box-Cox parameter l and adjusted R2 for model 1
FitP1; P2_11a)P1; P2_31b)P3; P4_13c)P3; P4_33d)P5; P6_15e)P5; P6_35f)
a02.0220.53501.8770.44901.58300.3427
a17.23×1036.37×1020.2220.1400.3100.226
a20.5500.6810.5660.6980.4630.722
a1,2–1.35×103–2.44×103–3.67×103
a1,1–3.54×104–7.92×104–7.13×104–1.81×103–1.05×103
a2,2–7.99×103–2.17×102–9.68×103–2.23×102–8.49×103–2.28×102
λ0.22220.06060.26260.10100.22220.1414
R20.6900.7890.7630.8260.7680.852
Tab.2  Significant (p<0.05) regression coefficients (Eq. (2)), Box-Cox parameter λ and adjusted R2 for model 2
Fig.2  Success (a) and failure (b) frequencies for RD estimation (models without temperature). Ratio (c) between RDMODEL and actual RD for failure of the best fit (P5, P6_35) as a function of RD and DRDi=RDi+1−RDi. PR=Pi (i = 1, 3, 5) are recent daily precipitations Pi, averaged for periods of one, three or five days before the day i; PNR are non-recent daily precipitations averaged for ten (P2_11, P4_13, P6_15) or thirty day (P2_31, P4_33, or P6_35) periods prior to each PR period. The frequency of success (%) shows how often the calculated RDMODEL differed from the real value of RD no more than twice. The frequency of failure (%) shows how often the calculated RDMODEL differed from the real value of RD more than four times.
FitP1; P2_11; Ta)P1; P2_31; Tb)P3; P4_13; Tc)P3; P4_33; Td)P5; P6_15; Te)P5; P6_35; Tf)
a09.3956.3938.9246.7828.8616.487
a13.28×1023.45×1020.1500.152
a2–0.284–4.81×102–0.352–0.125–0.358
a3–0.426–0.301–0.403–0.327–0.405–0.316
a1,29.46×1043.63×1031.61×1026.23×1031.93×102
a1,3–3.93×1036.67×1031.02×102
a2,31.56×1023.74×1021.68×1024.31×1022.06×1024.34×102
a1,2,3–3.69×104–1.50×103–6.09×104–1.96×103
λ0.18180.02020.18180.06060.18180.0606
R20.8520.8370.8700.8620.8740.879
Tab.3  Significant (p<0.05) regression coefficients (Eq. (3)), Box-Cox parameter λ and adjusted R2 for model 3
FitP1; P2_11; Ta)P1; P2_31; Tb)P3; P4_13; Tc)P3; P4_33; Td)P5; P6_15; Te)P5; P6_35; Tf)
a08.8205.1798.4885.3148.2565.009
a12.70×1024.06×1027.91×1028.29×102
a20.1580.1347.94×102
a3–0.395–0.241–0.382–0.254–0.376–0.243
a1,21.40×1031.89×1031.61×1028.70×1032.07×102
a1,36.49×1034.89×1031.02×102
a2,37.72×1032.44×1028.72×1032.76×1021.17×1022.76×102
a1,2,3–1.48×103–7.56×104–2.09×103
a1,1–1.76×104
a2,2–2.56×103–6.65×103–3.87×103–8.53×103–3.87×103–8.69×103
λ0.18180.02020.18180.06060.18180.0606
R20.8540.8460.8760.8720.8810.888
Tab.4  Significant (p<0.05) regression coefficients (Eq. (4)), Box-Cox parameter λ and adjusted R2 for model 4
Fig.3  Success (a) and failure (b) frequencies for RD estimation (models including temperature). Ratio (c) between RDMODEL and actual RD for failure of the best fit (P5, P6_35, T) as a function of RD and DRDi=RDi+1−RDi. PR=Pi (i = 1, 3, 5) are recent daily precipitations Pi, averaged for periods of one, three or five days before the day i; PNR are non-recent daily precipitations averaged for ten (P2_11, P4_13, P6_15) or thirty day (P2_31, P4_33, or P6_35) periods prior to each PR period. The frequency of success (%) shows how often the calculated RDMODEL differed from the real value of RD no more than twice. The frequency of failure (%) shows how often the calculated RDMODEL differed from the real value of RD more than four times.
Fig.4  Success (a) and failure (b) frequencies for RD estimated by model 4 (PR=P5, PNR=P6_35, T) for randomly selected fractions f of the original dataset; comparison (c) of the time dependence of RD estimated by model 4 (PR=P5, PNR=P6_35, T) using all data (f=1) and three replicates of randomly selected data (f=0.06). The frequency of success (%) shows how often the calculated RDMODEL differed from the real value of RD no more than twice. The frequency of failure (%) shows how often the calculated RDMODEL differed from the real value of RD more than four times. PR=P5 are recent daily precipitations Pi, averaged for periods of five days before the day i; PNR=P6_35 are non-recent daily precipitations averaged for thirty day periods prior to the PR period.
Fig.5  Variation over time of the real RD (continuous line) and of the RD estimated by model 4 fit (red dots, all data, PR=P5, PNR=P6_35, T). The same model was also developed using the training set 10/03/1978?20/06/1979 and applied for the remaining data as the validation set (black hyphens). PR=P5 are recent daily precipitations Pi, averaged for periods of five days before the day i; PNR=P6_35 are non-recent daily precipitations averaged for thirty day periods prior to the PR period.
A user-built function, called “select1” was developed under R for random selection without replacement of fraction f<1 (from “fbegin” to “fend” increasing with step “fstep”) of the original data from the dataframe x:
select1<-function(x, fbegin, fend, fstep){
rounded<-function(n) floor(n+0.5)
f<-fbegin
sel<-x[sample(1:length(x[,1]),rounded(f*length(x[,1]))),]
repeat{
t<-max(f)
if (t>=fend) break
sel1<-x[sample(1:length(x[,1]),rounded((t+fstep)*length(x[,1]))),]
f<-c(f, (t+fstep))
sel<-list(sel, sel1)}
return(list(f, sel))}
  
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