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Other model :

• Elvira model : model 4 with river >2nd order
  

Log10(Wet width + 1) = 0.22734+ 0.20045 (log10 catchment area) + 0.25939 (log 10 Shreve index)

• Thornton model : W = 4.98 CA^0.47
  

4. Models tested:
Linear regression

• Model 1 : width ~ a*catchment_area + b
• Model 2 : log(width) ~ a*log(up_area)+ b
• Model 3 : log(width) ~ a*log(up_area) + uga + b
• Model 4 : log(width) ~ a*log(up_area) + b*log(shreve index) + c
• Model 5 : log(width) ~ a*log(up_area) + b*log(shreve index) + uga + c

Non linear regression with library(nlme)

• Model 1: width~alpha*up_area^beta
• Model 2 : width~gamma*shreve^{lambda (à tester : width~gamma*strahler}lambda

• Data extraction :
  • year > 1987 (RHP)
  • Period : april-october
  • complete sampling method
  • largeur lame eau >0
  • CV largeur lame eau <= 5%
  • shreve >1 ? (pris en compte)

NB : f=a(x^{b )
On pose A=log(a), F=log(f), X=log(x)
F=log(f)=log(a)+b log(x)=A+bX
La représentation graphique donne une droite, ce qui permet de déterminer A et b, puis de calculer a grâce à la relation A=log(a). La méthode dite "de régression linéaire par les moindres carrés" est utilisée.}

Model Validation for linear regression

• Residuals versus fitted values to verify homogeneity
• QQ-plot or histogram of the residuals for normality
• Residuals versus each explanatory variable to check independence

River width graphics

Model results 1618 stations

df	AIC
lm1	3	9706.075
lm2	3	2595.026
lm3	12	2472.912
lm4	4	2375.609
lm5	13	2354.614

--> model lm2 width=1.88*area^0.24

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.63593 0.03924 16.20 <2e-16 *
log(ers0_station2$st_up_area) 0.24624 0.00892 27.61 <2e-16 *

--> model lm4 retenu avec 41% de la variation expliquée

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.90824 0.04071 22.312 < 2e-16 *
log(ers0_station2$st_up_area) 0.09331 0.01297 7.195 9.53e-13 *
log(ers0_station2$shreve) 0.20489 0.01331 15.390 < 2e-16 *

• nls1<-nls(st_cs_largeurlameeau~gamma*st_up_area^{lambda, data=ers0_station2,start=list(gamma=5,lambda=0.5))}

summary(nls1)

Estimate Std. Error t value Pr(>|t|)

gamma 2.25662 0.12533 18.00 <2e-16 *
lambda 0.23981 0.01031 23.26 <2e-16 *

• nls2<-nls(st_cs_largeurlameeau~gamma*shreve^{lambda, data=ers0_station2,start=list(gamma=3.33,lambda=0.28))}

summary(nls2)

Estimate Std. Error t value Pr(>|t|)

gamma 4.108061 0.121048 33.94 <2e-16 *
lambda 0.227287 0.008639 26.31 <2e-16 *

• nls3<-nls(st_cs_largeurlameeau~alpha*st_up_area^{beta +gamma*shreve}lambda, data=subset(ers0_station2,ers0_station2$st_up_area<=8000),ers0_station2,start=list(alpha=1.37,beta=0.32,gamma=3.33,lambda=0.28))
  

summary(nls3)
Parameters:

Estimate Std. Error t value Pr(>|t|)

alpha 0.44572 0.32403 1.376 0.169
beta 0.36923 0.09191 4.017 6.16e-05 *
gamma 2.60166 0.62737 4.147 3.55e-05 *[[BR] lambda 0.23017 0.03480 6.614 5.07e-11 *
--- Signif. codes: 0 '*' 0.001 '' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.077 on 1611 degrees of freedom

Number of iterations to convergence: 22
Achieved convergence tolerance: 8.651e-06

--> st_cs_largeurlameeau ~ 0.44 * st_up_area^{0.36+ 2.6 * shreve}0.23

• loess

loess.smooth(ers0_station2$st_up_area[ers0_station2$st_up_area<800],ers0_station2$st_cs_largeurlameeau[ers0_station2$st_up_area<800])

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