Supplementary information to “Technical Note: Statistical generation of climate-perturbed flow duration curves” Veysel Yildiz 1 , Charles Rougé 1 , Robert Milton 2 , and Solomon Brown 2 1 Department of Civil and Structural Engineering, The University of Sheffield 2 Department of Chemical and Biological Engineering, The University of Sheffield Correspondence: Charles Rougé (c.rouge@sheffield.ac.uk) Abstract. This supplementary information demonstrates that for triplets (M,V,L) of streamflow statistics representing average behavior, variability, and low flows, there is unique parameterisation of the flow duration curve (FDC) according to the Kosugi model. We consider the “mean” case where (M,V,L)=(μ,σ,q low ) where μ is the mean, σ is the standard deviation and q low is the 1 st or 5 th percentile of flow, and the “median case” (M,V,L)=(m,CV,q low ) where m is the median and CV = μ/σ is the coefficient of variation. It also provides conditions on (M,V,L) for the existence of a parameterisation. 5 1 Kosugi function reminders We model the flow duration curve (FDC) with the Kosugi equation, as proposed by Sadegh et al. (2016). The equation models streamflow q as a function of the flow quantile u ∈ [0, 1]: q(u)= c +(a − c) z(u) b (1) where (a,b,c) are parameters, with a and c in the same units as q, and b unitless. We need a − c> 0 and b> 0 for z(u) is 10 defined as follows, and represented in Figure S1: z(u) = exp  √ 2 erfc −1 (2u)  (2) This supplementary information will relate parameters (a,b,c) to triplets of streamflow statistics (M,V,L) representing average behavior, variability, and low flows. It will do so first in the “mean” case of (M,V,L)=(μ,σ,q low ) in Section 2, where μ is the mean, σ is the standard deviation and q low is the 1 st or 5 th percentile of flow. Then in Section 3 we will examine 15 the “median case” of (M,V,L)=(m,CV,q low ), where m is the median and CV = μ/σ is the coefficient of variation. 2 “Mean” case (M,V,L)=(μ, σ, q low ) In this section we assume we know the streamflow mean μ, standard deviation σ and low flow percentile q low . We assume we have μ>q low . We will prove the (a,b,c) triplet of Kosugi parameter is unique, and give a sufficient condition on (μ,σ,q low ) for its existence. 20 1