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Water Resources Research
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Supporting Information for
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Reliability, Return Periods, and Risk under Nonstationarity
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Laura K. Read1 and Richard M. Vogel1
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1Department
of Civil and Environmental Engineering, Tufts University, Medford, MA, USA
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Contents of this file
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Introduction
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This supporting information provides the derivation and details of the coefficient of variation of
the nonstationary flood series resulting from nonstationary 2-parameter lognormal model
presented in the paper.
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Derivation of Coefficient of Variation of X in nonstationary trend model
Figure S1
Derivation of Coefficient of Variation of X in nonstationary lognormal model.
The nonstationary LN2 flood hazard model employed in this paper assumes a linear trend in the
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mean of logarithms of the annual maximum flood series Y=ln(X). . Interestingly, this single trend
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model of Y implies a trend in both the mean and the variance of the annual maximum floods X.
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Importantly, although the coefficient of variation of the flood series X, denoted Cx, is assumed
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fixed, the nonstationary LN2 model implies a different coefficient of variation of X which we
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term the conditional coefficient of variation and denote as Cx|t and for which we derive an
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expression below. Recall from equation (13b) the well-known relationship between the stationary
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standard deviation of Y=ln(X) and Cx:
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 y  ln(1  C x2 )
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The nonstationary LN2 model introduces a trend in the mean of Y so that the variance of Y,
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conditioned on the time t, is obtained by simply taking the variance of Y, given in (15), assuming
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,  and t are constants which leads to:
(S1)
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 2y |t    2   y 2   2 t2
(S2)
This is equivalent to
 y|t   y (1   2 )
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
 t
y
(S3)
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because ρ is the Pearson correlation coefficient defined as
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strength of the linear relationship between the flood series Y and the covariate time, t. Consider
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two extreme cases. If the regression model has perfect explanatory power, then =1 and the
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conditional variance of Y in (S3) is equal to zero. If the regression model has no explanatory
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power, then =0 and the conditional variance in (S3) is exactly equal to the unconditional
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variance of Y. Of course the unconditional variance of Y is always the same, regardless of the
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explanatory power of the regression. Since the residuals are assumed to be normally distributed,
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the values of exp(ε) are lognormal, thus analogous to (S1) :
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which measures the
 y|t 2  ln( 1  C x|t 2 )
(S4)
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where C x|t is the coefficient of variation of the nonstationary variable X.
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Combining equations (S1), (S3) and (S4) leads to a relationship between the coefficient of
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variation of the stationary series Cx and the coefficient of variation of the nonstationary series,
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C x |t
given by
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2
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C x |t  (C x 2  1) (1  )  1
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(S5)
For a reasonable range of ρ, we explore the relationship between Cx and C x|t in Figure
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S1. As expected, there is a reduction in the coefficient of variation of the nonstationary series Cx|t
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as ρ increases, i.e. as the trend increases. Since the true value of ρ is unknown, is generally quite
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small in practice, and will depend on the level of sophistication of the trend model employed, we
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assume that
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(S5) be used in any particular application of the nonstationary lognormal model.
C x  C x |t
for all trends analyzed in this paper, but we recommend that in practice
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Figure S1. Relationship between coefficient of variation of the stationary series Cx and the
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coefficient of variation of the nonstationary series x|t for a feasible range of correlation
coefficients, ρ (grey lines). Note that the black solid line represents the 1:1 line for stationary
conditions where ρ=β=0.
C
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