What drives <em>F<sub>yw</sub></em> variations with elevation in Alpine catchments?

Gentile, Alessio; Canone, Davide; Ceperley, Natalie; Gisolo, Davide; Previati, Maurizio; Zuecco, Giulia; Schaefli, Bettina; Ferraris, Stefano

doi:https://doi.org/10.5194/egusphere-2022-921

Preprints

https://doi.org/10.5194/egusphere-2022-921

Preprints

20 Sep 2022

| 20 Sep 2022

What drives F_yw variations with elevation in Alpine catchments?

Alessio Gentile, Davide Canone, Natalie Ceperley, Davide Gisolo, Maurizio Previati, Giulia Zuecco, Bettina Schaefli, and Stefano Ferraris

Abstract. The young water fraction (F_yw), defined as the fraction of catchment outflow with transit times of less than about 2–3 months, is increasingly used in hydrological studies, replacing the widely used Mean Transit Time (MTT), which is subject to aggregation error. The use of this new metric in catchment intercomparison studies is helpful to understand and conceptualize the relevant processes controlling catchment’s hydrological function. Past work has shown the remarkable and counterintuitive evidence that steep (and generally high elevation) catchments worldwide reveal small F_yw values. However, the topographic slope only partially explains the observed F_yw variance, and the mechanisms hidden behind this lowering with slope remain basically unclear.

The main aim of this paper is to investigate what drives F_yw variations with elevation in Alpine catchments clarifying why F_yw is low at high altitudes. In this regard, we use a dataset composed of 27 study catchments, located both in Switzerland and in Italy, that we categorize as rainfall-dominated, hybrid or snow-dominated according to a proposed formal classification scheme that considers both a common-used monthly streamflow ratio and the snow cover regime. We analyze three not previously investigated variables that could potentially explain the F_yw elevation gradients: the fractional snow cover area (F_SCA), the fraction of quaternary deposits (F_qd), and the fraction of baseflow (F_bf). We also consider a fourth variable, namely the Winter Flow Index (WFI), for comparing our results about the groundwater contribution to streamflow with those of previous scientific publications.

Our results suggest that unconsolidated sediments could play a role in modulating F_yw elevation gradients via their capacity to store groundwater, but further geological information, such as the portion of fractured bedrocks, would be desirable for a complete picture of the role of geology. Based on our analysis concerning the F_SCA, we develop a perceptual model that explains how the increasing duration of the snowpack promotes a progressive emptying of the groundwater storage during winter, thereby increasing the streamwater age, while ephemeral snowpack generally favors rapid flow paths that increase F_yw. Finally, our work highlights that F_bf, considered as a proxy for groundwater flow, is roughly the one’s complement of F_yw. In harmony with the model, we find high F_bf during all low-flow periods, which underlines that streamflow is mainly sustained by groundwater in such flow conditions. For catchments where the winter low-flow period is long compared to the summer high-flow period, this results in low F_yw.

In conclusion, our data set suggests that F_bf is the best explanatory variable of F_yw elevation gradients in Alpine catchments, implying the key-role of major groundwater storages that, with the increasing snowpack duration, are actively involved in streamflow generation processes.

Received: 13 Sep 2022 – Discussion started: 20 Sep 2022

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Journal article(s) based on this preprint

22 Jun 2023

Towards a conceptualization of the hydrological processes behind changes of young water fraction with elevation: a focus on mountainous alpine catchments

Alessio Gentile, Davide Canone, Natalie Ceperley, Davide Gisolo, Maurizio Previati, Giulia Zuecco, Bettina Schaefli, and Stefano Ferraris

Hydrol. Earth Syst. Sci., 27, 2301–2323, https://doi.org/10.5194/hess-27-2301-2023,https://doi.org/10.5194/hess-27-2301-2023, 2023

Short summary

Alessio Gentile et al.

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2022-921', Anonymous Referee #1, 11 Oct 2022

Please find my comments in the attached file

Citation: https://doi.org/10.5194/egusphere-2022-921-RC1
- AC1: 'Reply on RC1', Alessio Gentile, 17 Nov 2022
  
  The work of Gentile et al. investigated the causes for young water fraction (F_yw) variations with elevation (F_yw is low at high altitudes) in Alpine catchments. The study areas are 27 catchments in Switzerland and Italy. The authors proposed new criteria for catchment classification into different hydro-climatic regimes. To gain insight into the reason for F_yw variations with elevation, this author used a new set of hydrological variables, namely the fractional snow cover area (F_SCA), the fraction of quaternary deposits (F_qd), and the fraction of baseflow (F_bf). In general, the idea of this paper about what drives F_yw variations with elevations is novel and of interest for understanding the functioning of catchments in Alpine regions as well as for understanding flow and transport in this region and potentially in other areas. However, the methodology and results do not fully support this idea. The text was not well written. Please find my main comments and line-by-line comments below.
  
  Dear referee #1,
  
  We would like to thank you for the overall positive assessment and the numerous detailed comments, which will contribute to our manuscript’s improvement considerably.
  Please find below a point-by-point response to your main comments. The responses to both your main and minor comments are reported in the Supplement.
  We will incorporate all your constructive feedback once we receive the editor’s response.
  
  Sincerely,
  
  The Authors
  Main comments
  
  • Why did the authors need to propose a new criterion for catchment classification? The authors used two variables: (1) streamflow ratio between different months and (2) snow cover fraction for the proposed catchment classification, but later they adjusted the threshold of these two variables to have consistent results with Staudinger et al. (2017). Why didn’t they just use the method of Staudinger et al. (2017)?
  • We propose a new criterion for the regime classification because our dataset includes catchments outside the Swiss borders (i.e., the four Italian catchments) for which the Weingartner and Aschwanden (1992) and Staudinger et al. (2017) classification scheme cannot be strictly applied since they were designed for the Swiss hydro-climatic regimes. We “manually calibrate” the thresholds of F_SCA and Q_June/Q_DJF for classifying catchments in “rainfall-dominated”, “hybrid” and “snow-dominated” as in the work of Staudinger et al. (2017). In this way, the classification scheme is “calibrated” on the Staudinger et al. (2017) catchments and we can apply it also outside the Swiss borders. According to the referees’ comments, we will consider the possibility of modifying the classification scheme to make it more straightforward to link to previous classification (e.g., using streamflow and topographical data), but it will remain transferable to other regions.
  
  • The objective is to investigate what drives F_ywvariation with elevation. The authors proposed using a new set of hydrological variables, but what are the relations between these variables with elevation? For example, what are the relations between F_SCA, F_qd, F_bf with elevation? With F_SCA, I can infer from the text, but it was not explained in the text until the last sections (Section 5.2) of the manuscript. F_SCA cannot be directly related to elevation, instead, it needs to be related to the catchment classification then from catchment classification to mean elevation. However, in other areas, can we still relate F_SCA to elevation? With the other variables (F_qd and F_bf), it is unclear to me what are their relations to elevations. In addition, F_qddoes not seems to be a good variable because there is no significant relation between F_ywand F_qd.
  • Thank you for this comment: this is a good point. We will add, for each variable (F_SCA, F_qd and F_bf), a figure that shows the relation with mean catchments elevation. The three figures are reported in the Supplement, and we will include them in the revised manuscript.
  
  a) The F_SCA increases with the mean catchment elevation in our data set, revealing a positive, statistically significant correlation. This suggests the increasing snow cover persistence at high altitudes.
  
  b) F_qd decreases with the mean catchment elevation in our data set, revealing a negative, statistically significant correlation. This negative correlation reflects the fact that F_qd decreases when the mean slope increases (Arnoux et al., 2021) (mean slope increases with mean elevation for the catchments analyzed in this study, as shown in Fig. 4a of the manuscript). We have decided to use F_qd because Arnoux et al. (2021) demonstrated a strong positive correlation between F_qd and Winter Flow Index (WFI) highlighting the role of unconsolidated deposits in storing groundwater (in terms of age, old water). The missing information about the portion of fractured bedrocks, the thickness of quaternary deposits and the bedrock topography will demand future attention for a complete picture of the role of geology (potentially resulting in a statistically significant correlation with F_yw).
  
  c) F_bfreveals an opposite behavior with respect to F_yw: it decreases until 1500 m and it increases at higher elevations.
  • The manuscript needs to be restructured and revised. There is a lack of clarification in the text. More description of the study area characteristics is needed. Much of the information provided in Study Sites, and Material and Methods is not relevant (e.g., shape file, detailed source of data, etc.). Instead, citing the sources of the various data (both from individuals and organizations) can be moved to either the Authors' Contributions or Acknowledgements, or in the supporting information Sections or to a table rather than describe them within the text of the article, making it very difficult to read such detailed information. If possible, I would also suggest the authors publish their data in an open repository.
  • Thank you for these suggestions. We will revise the “Study sites” and “Material and Methods” sections accordingly. We will move all the data sources in the “Data availability” section and remove irrelevant information. We will describe the study sites in a more concise manner using a Table and some descriptive figures: e.g., mean slope against mean elevation, mean annual precipitation and mean annual discharge against mean elevation, variations of mean monthly flow with elevation. These changes should make the text more fluent.
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC1
RC2:
'Comment on egusphere-2022-921', Jana von Freyberg, 25 Oct 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-921/egusphere-2022-921-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-921-RC2
- AC3: 'Reply on RC2', Alessio Gentile, 01 Dec 2022
  
  Dear Jana von Freyberg,
  thank you for your care and attention during your reading of the manuscript, your positive remarks and your comments that will help to improve the work.
  Please find in the Supplement the responses to all your comments.
  We will take into account all your constructive feedback in the revised version of the manuscript once we receive the editor’s response.
  With kind regards,
  The Authors
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC3
CC1:
'Comment on egusphere-2022-921', Peter Jansson, 04 Nov 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-921/egusphere-2022-921-CC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-921-CC1
- AC2: 'Reply on CC1', Alessio Gentile, 01 Dec 2022
  
  Dear Peter Jansson and Ryan Teuling,
  we would like to thank you for having selected our paper for this class.
  The input is highly appreciated and will help to improve our paper. We would like to point out here that we find the review extremely helpful but overly harsh at instances. The wording implies that we made mistakes. While this wording style is omnipresent (unfortunately!), we take the opportunity to call for a change in wording style in scientific reviews. A simple example is the following: “the authors should further stress the relevance of this study” could be reformulated to “the study would gain from a more concise presentation of its relevance”.
  Please find in the Supplement a point-by-point response to both your general and specific comments.
  With kind regards,
  The Authors
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2022-921', Anonymous Referee #1, 11 Oct 2022

Please find my comments in the attached file

Citation: https://doi.org/10.5194/egusphere-2022-921-RC1
- AC1: 'Reply on RC1', Alessio Gentile, 17 Nov 2022
  
  The work of Gentile et al. investigated the causes for young water fraction (F_yw) variations with elevation (F_yw is low at high altitudes) in Alpine catchments. The study areas are 27 catchments in Switzerland and Italy. The authors proposed new criteria for catchment classification into different hydro-climatic regimes. To gain insight into the reason for F_yw variations with elevation, this author used a new set of hydrological variables, namely the fractional snow cover area (F_SCA), the fraction of quaternary deposits (F_qd), and the fraction of baseflow (F_bf). In general, the idea of this paper about what drives F_yw variations with elevations is novel and of interest for understanding the functioning of catchments in Alpine regions as well as for understanding flow and transport in this region and potentially in other areas. However, the methodology and results do not fully support this idea. The text was not well written. Please find my main comments and line-by-line comments below.
  
  Dear referee #1,
  
  We would like to thank you for the overall positive assessment and the numerous detailed comments, which will contribute to our manuscript’s improvement considerably.
  Please find below a point-by-point response to your main comments. The responses to both your main and minor comments are reported in the Supplement.
  We will incorporate all your constructive feedback once we receive the editor’s response.
  
  Sincerely,
  
  The Authors
  Main comments
  
  • Why did the authors need to propose a new criterion for catchment classification? The authors used two variables: (1) streamflow ratio between different months and (2) snow cover fraction for the proposed catchment classification, but later they adjusted the threshold of these two variables to have consistent results with Staudinger et al. (2017). Why didn’t they just use the method of Staudinger et al. (2017)?
  • We propose a new criterion for the regime classification because our dataset includes catchments outside the Swiss borders (i.e., the four Italian catchments) for which the Weingartner and Aschwanden (1992) and Staudinger et al. (2017) classification scheme cannot be strictly applied since they were designed for the Swiss hydro-climatic regimes. We “manually calibrate” the thresholds of F_SCA and Q_June/Q_DJF for classifying catchments in “rainfall-dominated”, “hybrid” and “snow-dominated” as in the work of Staudinger et al. (2017). In this way, the classification scheme is “calibrated” on the Staudinger et al. (2017) catchments and we can apply it also outside the Swiss borders. According to the referees’ comments, we will consider the possibility of modifying the classification scheme to make it more straightforward to link to previous classification (e.g., using streamflow and topographical data), but it will remain transferable to other regions.
  
  • The objective is to investigate what drives F_ywvariation with elevation. The authors proposed using a new set of hydrological variables, but what are the relations between these variables with elevation? For example, what are the relations between F_SCA, F_qd, F_bf with elevation? With F_SCA, I can infer from the text, but it was not explained in the text until the last sections (Section 5.2) of the manuscript. F_SCA cannot be directly related to elevation, instead, it needs to be related to the catchment classification then from catchment classification to mean elevation. However, in other areas, can we still relate F_SCA to elevation? With the other variables (F_qd and F_bf), it is unclear to me what are their relations to elevations. In addition, F_qddoes not seems to be a good variable because there is no significant relation between F_ywand F_qd.
  • Thank you for this comment: this is a good point. We will add, for each variable (F_SCA, F_qd and F_bf), a figure that shows the relation with mean catchments elevation. The three figures are reported in the Supplement, and we will include them in the revised manuscript.
  
  a) The F_SCA increases with the mean catchment elevation in our data set, revealing a positive, statistically significant correlation. This suggests the increasing snow cover persistence at high altitudes.
  
  b) F_qd decreases with the mean catchment elevation in our data set, revealing a negative, statistically significant correlation. This negative correlation reflects the fact that F_qd decreases when the mean slope increases (Arnoux et al., 2021) (mean slope increases with mean elevation for the catchments analyzed in this study, as shown in Fig. 4a of the manuscript). We have decided to use F_qd because Arnoux et al. (2021) demonstrated a strong positive correlation between F_qd and Winter Flow Index (WFI) highlighting the role of unconsolidated deposits in storing groundwater (in terms of age, old water). The missing information about the portion of fractured bedrocks, the thickness of quaternary deposits and the bedrock topography will demand future attention for a complete picture of the role of geology (potentially resulting in a statistically significant correlation with F_yw).
  
  c) F_bfreveals an opposite behavior with respect to F_yw: it decreases until 1500 m and it increases at higher elevations.
  • The manuscript needs to be restructured and revised. There is a lack of clarification in the text. More description of the study area characteristics is needed. Much of the information provided in Study Sites, and Material and Methods is not relevant (e.g., shape file, detailed source of data, etc.). Instead, citing the sources of the various data (both from individuals and organizations) can be moved to either the Authors' Contributions or Acknowledgements, or in the supporting information Sections or to a table rather than describe them within the text of the article, making it very difficult to read such detailed information. If possible, I would also suggest the authors publish their data in an open repository.
  • Thank you for these suggestions. We will revise the “Study sites” and “Material and Methods” sections accordingly. We will move all the data sources in the “Data availability” section and remove irrelevant information. We will describe the study sites in a more concise manner using a Table and some descriptive figures: e.g., mean slope against mean elevation, mean annual precipitation and mean annual discharge against mean elevation, variations of mean monthly flow with elevation. These changes should make the text more fluent.
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC1
RC2:
'Comment on egusphere-2022-921', Jana von Freyberg, 25 Oct 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-921/egusphere-2022-921-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-921-RC2
- AC3: 'Reply on RC2', Alessio Gentile, 01 Dec 2022
  
  Dear Jana von Freyberg,
  thank you for your care and attention during your reading of the manuscript, your positive remarks and your comments that will help to improve the work.
  Please find in the Supplement the responses to all your comments.
  We will take into account all your constructive feedback in the revised version of the manuscript once we receive the editor’s response.
  With kind regards,
  The Authors
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC3
CC1:
'Comment on egusphere-2022-921', Peter Jansson, 04 Nov 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-921/egusphere-2022-921-CC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-921-CC1
- AC2: 'Reply on CC1', Alessio Gentile, 01 Dec 2022
  
  Dear Peter Jansson and Ryan Teuling,
  we would like to thank you for having selected our paper for this class.
  The input is highly appreciated and will help to improve our paper. We would like to point out here that we find the review extremely helpful but overly harsh at instances. The wording implies that we made mistakes. While this wording style is omnipresent (unfortunately!), we take the opportunity to call for a change in wording style in scientific reviews. A simple example is the following: “the authors should further stress the relevance of this study” could be reformulated to “the study would gain from a more concise presentation of its relevance”.
  Please find in the Supplement a point-by-point response to both your general and specific comments.
  With kind regards,
  The Authors
  
  Citation: https://doi.org/10.5194/egusphere-2022-921-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (29 Dec 2022) by Rohini Kumar

AR by Alessio Gentile on behalf of the Authors (08 Feb 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (08 Feb 2023) by Rohini Kumar

RR by Anonymous Referee #1 (24 Feb 2023)

Suggestions for revision or reasons for rejection

The authors have made significant changes to the manuscript. I am satisfied with the authors‘ responses as well as the changes. I think the quality of the manuscript was significantly improved. In general, I only have one major comment regarding the F*yw and the age threshold of F*yw (please see below).

Major comments

As I understood from the manuscript. The authors use F*yw from 22 Swiss catchments from Freyberg et al. (2018). I am not sure if the authors and Freyberg et al. (2018) used the same code (method) for the calculation F*yw or not. In the attached data along with this review, I demonstrated that using different source codes (methods) could result in different values of Fyw (or F*yw) and the age thresholds of F*yw. If so, the differences in F*yw among catchments could be driven by using different methods (source code) rather than the physical factors (e.g., precipitation, elevation, LFD, Fbf,...). Therefore, I would suggest the authors to use the same source code (method) for calculating F*yw for all catchments and provide the source code for review/or make it publicly accessible.

When applying the sine wave fitting, the authors will get F*yw corresponding to a specific age threshold (τyw), depending on the shape factor (alpha) of the gamma distribution (Equations 13-14 from Kirchner 2016). The age threshold could be 1 to 3 months (Figure 10, Kirchner 2016) and even beyond this range. Therefore, the variation in F*yw might be because of different age thresholds rather than physical factors. In this sense, the F*yw or Fyw might not be a useful matrix for catchment intercomparison studies. Therefore, I suggest showing the alpha and age thresholds for each catchment in the results (these values were often not shown in previous studies). Discuss the consequences of the differences found with the age threshold in the limitation section of the paper (e.g., will the relations between F*yw and other factors (e.g., precipitation, elevation, LFD, Fbf,...) change if somehow we can calculate F*yw that corresponds to the same age threshold).

Minor comments

Figure 2c: Might be using a triangle (for P) and a circle (for Q) in combination with different colors (for rain, hybrid, and snow) for easier to differentiate between them

L237-238: Should be noted that this demonstration uses thought experiments (not from real catchments)

Figure 6: Why did the authors only show quaternary deposits for 7 catchments, I think either showing for all catchments or not showing the figure or not showing (Table 2 is sufficient)

L207-211: “Our data set includes NBPV, whose area is 42% glaciercovered and consequently shows a characteristic glacier-dominated streamflow regime with a monthly peak in late summer. Thus, NBPV may belong to a fourth category of glacier-dominated catchments, for which the effect of glacier-melt on F*yw cannot be neglected, and this was partially discussed by Ceperley et al. (2020)”: Why did the author still classify NBPV as “snow-dominated” catchment? Does glacier-melt account for when calculating F*yw?

Section 4.1. Why did the authors discuss the Winter Flow Index (WFI) - F*yw relation in this section “The role of Quaternary deposits”?
Section 4.2. To be consistent, I would suggest using another title, e.g., “the role of groundwater flow (baseflow) in F*yw”

L414: I think the author mentions the fraction of based flow, not the amount of stored water

#-------------------------------------
#Rscript for Fyw and alpha
#1. How to run this script
# Copy this code to a text file and save as "youngfraction.R"
# Then open this file in R
# All of the required functions to run this Rsript was attached at the end of this file (please run these function before running this code)

#2. Download isotope data from the below link and extract in the same folder with the file "youngfraction.R"
# https://zenodo.org/record/4057967#.Y00oMHZBxPY
# then you will have two folders "Precipitation" and "isotopes_streamflow"

#3. The IRLS_hess-2017-720.R file was downloaded from
#https://hess.copernicus.org/articles/22/3841/2018/hess-22-3841-2018-supplement.zip
#I made 3 modifications in this function (please search with keywords "original code", where modifications were made)

#-------------------------------------
# Load library
library(lubridate)
library(lhs)

# set working directory - PLEASE CHANGE TO YOUR WORKING DIRECTORY --
setwd("./isotope_data_sine_fitting")

#-------------------------------------------------------------------------------
# 1. Fit sine wave to isotope in precipitation and streamflow (not using IRLS method)
# 20k parameter sets were generated within a given range
# best parameter set was selected based on weighted root mean square error
# this procedure is repeated n times to have the uncertainty in Fyw and alpha
#-------------------------------------------------------------------------------

# Read isotope in precipitation and convert date to decimal
isotope_P <- read.csv("./Precipitation/RIA.isoPrcp", header = TRUE, sep = "", skip = 12)
isotope_P$date <- toDecimal(as.Date(isotope_P$date, format = "%Y-%m-%d"))

# Read isotope in streamflow and convert date to decimal
isotope_Q <- read.csv("./isotopes_streamflow/RIA.isoStrm", header = TRUE, sep = "", skip = 22)
isotope_Q$Sampling_date <- toDecimal(as.Date(isotope_Q$Sampling_date, format = "%Y-%m-%d"))

# I cannot find Q data, now calculate with weight is = 1, in other words, flow non-weighted Fyw
weight_P <- rep(1, nrow(isotope_P))
weight_Q <- rep(1, nrow(isotope_Q))

# Initialize result
Fyw <- c()
alpha <- c()

# Make n iterations to get the uncertainty range of Fyw and alpha
n = 5
for (iter in 1:n){
# Generate parameter set
nsim <- 20000
parameter <- randomLHSrange(data.frame(
A = c(0,15),
phi = c(0,2*pi),
k = c(-14,-5)),
nsim)

# Fit isotope in P to sinewave - Plot result - parameter(A, phi, k)
bestParameterP <- sinefitbest(parameter, isotope_P$d18O_prcp, isotope_P$date, weight_P)
# plot(isotope_P$d18O_prcp) lines(isotope_Q$d18O_mean, add = TRUE)
# lines(sinefit(bestParameterP[1], bestParameterP[2], bestParameterP[3], isotope_P$date), col = "blue", lwd = 3)

# Fit istope data in Q
bestParameterQ <- sinefitbest(parameter, isotope_Q$d18O_mean, isotope_Q$Sampling_date, weight_Q)
# plot(isotope_Q$d18O_mean)
# lines(sinefit(bestParameterQ[1], bestParameterQ[2], bestParameterQ[3], isotope_Q$Sampling_date), col = "blue", lwd = 3)

# Find young water fraction and alpha value of the gamma distribution
Fyw <- c(Fyw, bestParameterQ[1]/bestParameterP[1])
alpha <- c(alpha, findAlpha(bestParameterQ,bestParameterP))
}

# Get the range of the estimated Fyw and alpha
summary(Fyw)
summary(alpha)

# Write result to result.csv file
write.csv(data.frame(Simulation = c(1:length(Fyw)), Fyw = Fyw, Alpha = alpha),
file = "Fyw_alpha.csv", row.names = FALSE)

#-------------------------------------------------------------------------------
# 2. Fit sine wave to isotope in precipitation and streamflow (using IRLS method)
# https://hess.copernicus.org/articles/22/3841/2018/hess-22-3841-2018-supplement.zip
#-------------------------------------------------------------------------------

# Fit sine wave to isotope concentrations in Q
fitQ <- IRLS(isotope_Q$d18O_mean,
data.frame(cos = cos(2*pi*isotope_Q$Sampling_date),sin = sin(2*pi*isotope_Q$Sampling_date)),
weight_Q,
type="Cauchy")
AQ <- sqrt(sum(fitQ$fit$coefficients[2:3]^2))
phiQ <- atan(fitQ$fit$coefficients[3]/fitQ$fit$coefficients[2])

# # Fit sine wave to isotope concentrations in P
fitP <- IRLS(isotope_P$d18O_prcp,
data.frame(cos = cos(2*pi*isotope_P$date),sin = sin(2*pi*isotope_P$date)),
weight_P,
type="Cauchy")
AP <- sqrt(sum(fitP$fit$coefficients[2:3]^2))
phiP <- atan(fitP$fit$coefficients[3]/fitP$fit$coefficients[2])

# Find Fyw
Fyw = AQ/AP
Fyw

# The phase shift must be positive and iwthin the range [0, 2*pi]
phiP <- phiP%%(2*pi)
phiQ <- phiQ%%(2*pi)

alpha <- findAlpha(c(AQ, phiQ%%(2*pi)), c(AP, phiP%%(2*pi)))
alpha

# Please run all functions after this line before running the main code

##############################################
#iteratively reweighted least squares (IRLS) with optional point weights
# (in addition to residual weights that are adjusted to downweight data with unusually large residuals,
# as in conventional IRLS)
#
#Author: James Kirchner, ETH Zurich
#Public use allowed under GNU Public License v. 3 (see http://www.gnu.org/licenses/gpl-3.0.en.html)
#
##############################################
IRLS <- function(Y, X, ww=rep(1,length(Y)), wt=rep(1,length(Y)), type="Cauchy") {

# Y is a numeric vector representing a response variable
# X is a numeric vector or array representing one or more explanatory variables
# ww is an optional numeric vector of point weights (such as masses for mass-weighted regressions)
# the default weight vector gives all ww's the value of 1
# wt is an optional numeric vector of initial values for the residual weights for the IRLS
# the default weight vector gives all initial wt's the value of 1
# type is an optional string constant indicating the weight function to use.
# If type is specified, it must be "bisquare", "Welsch", or "Cauchy". Anything else generates an error.
# the default weight function is Cauchy

# Y, ww and X (or each vector comprising X, if X is two-dimensional) must be of the same length, otherwise an error results.
# Y and X must not be exactly collinear (that is, there must be some nonzero residuals). This is not checked.

# IRLS returns an object of class "lm", generated by a call lm(Y ~ X, weights=wt, na.action="na.omit") with the iteratively determined weights.
# This object can then be handled just like any other return from lm.
# However, the variable names in this "lm" object will be Y and X (as defined internally here) regardless of what names were passed to this function!

##############################################
#define bisquare weight function
##############################################
bisquare <- function(x, MAR) {
if (MAR==0) w <- ifelse(x==0, 1, 0) #if MAR is zero, only 0 or 1 weights are possible
else {
w <- (1-(x/(6*MAR))^2)^2 #bisquare function
w[x>6*MAR] <- 0 #replace with zero whenever X is more than 6 times the median absolute residual
}
return(w)
}

##############################################
#define Welsch weight function
##############################################
Welsch <- function(x, MAR) {
if (MAR==0) w <- ifelse(x==0, 1, 0) #if MAR is zero, only 0 or 1 weights are possible
else w <- exp(-(x/(4.4255*MAR))^2) #Welsch weight function
return(w)
}

##############################################
#define Cauchy weight function
##############################################
Cauchy <- function(x,MAR) {
if (MAR==0) w <- ifelse(x==0, 1, 0) #if MAR is zero, only 0 or 1 weights are possible
else w <- 1/(1+(x/(3.536*MAR))^2) #Cauchy weight function
return(w)
}

if (type!="bisquare") {
if (type!="Welsch") {
if (type!="Cauchy") stop("IRLS stopped: no valid weight type specified. Valid functions are 'bisquare', 'Welsch', and 'Cauchy'")
}
}

if (length(Y) != length(ww)) stop("IRLS stopped: supplied weight vector must be same length as Y")

wwwt <- ww*wt
# original code fit <- lm(Y ~ X, weights=wwwt, na.action="na.omit")
fit <- lm(Y ~ X$cos + X$sin, weights=wwwt, na.action="na.omit") #initial least-squares fit

wt_chg <- 999.0 #initialize weight change
iter <- 0 #initialize the iteration counter

##############################################
#Here's the iteration loop, which runs until the largest weight change for any point is less than 0.01, or iteration limit is exceeded
##############################################
while ( (max(wt_chg,na.rm=TRUE) > 0.01) & !all(iter>10, summary(fit)$r.squared>0.999) ){
if (iter>1000) stop("IRLS stopped: more than 1000 interations, sorry!")
# This error can arise when Y is perfectly collinear with X for more than half the points,
# and thus the MAR fluctuates near zero, with the weights never stabilizing.
# That should normally be handled by the r-squared criterion for loop exiting as defined above.

iter <- iter+1 #increment the iteration counter
old_wt <- wt #save the old vector of weights for comparison with the next one

const <- fit$coefficients[1] #this is the intercept
slope <- fit$coefficients[2:length(fit$coefficients)] #this is the vector of regression coefficients

#explicit calculation of residuals
# original code if (length(slope)==1) resid <- Y - const - X*slope
if (length(slope)==1) resid <- Y - const - as.matrix(X)*slope
else resid <- as.vector(Y - const - as.matrix(X) %*% slope)
#can't use residuals(fit) because missing values will mess up the assignment of weights in the steps that follow
#note %*% is matrix multiplication in R

abs_resid <- abs(resid)
abs_resid_nonzero <- ifelse(Y==0, NA, abs_resid) #exclude residuals corresponding to exact zeroes from median, to avoid blowup when there are many repeated zeroes in any flow decile
MAR <- median(abs_resid_nonzero, na.rm=TRUE)

if (MAR==0.0) stop("IRLS stopped. Solution has collapsed: median absolute residual is zero!")

if (type=="bisquare") wt <- bisquare(abs_resid,MAR) #use bisquare weights
else if (type=="Welsch") wt <- Welsch(abs_resid,MAR) #use Welsch weights
else if (type=="Cauchy") wt <- Cauchy(abs_resid,MAR) #use Cauchy weights

wwwt <- ww*wt

# original code fit <- lm(Y ~ X, weights=wwwt, na.action="na.omit")
fit <- lm(Y ~ X$cos + X$sin, weights=wwwt, na.action="na.omit") #run multiple regression

wt_chg <- abs(wt-old_wt) #calculate change in weights from previous iteration

} #end while

qq <- list("fit"=fit, "wt"=wt)

return(qq)

}
##############################################
#END OF iteratively reweighted least squares
##############################################

##############################################
# The below Rscript was written by the reviewer
# This script contains functions for fitting the sinewave isotope data in streamflow and precipitation
# There is also functions for finding youngwater fraction and alpha value
##############################################

# Create a sine function describing isotope in precipitation or streamflow
sinefit <- function(A, phi,k,t){
sinefit <- A * sin(2*pi *t - phi) + k
return(sinefit)
}

# Find best fit parameter to sinewave fitting given parameter sets and objective function is RMSE
sinefitbest <- function(parameter, isotopeData, inputDate, weight){
modelPerformance <- c()
for (i in 1:nrow(parameter)){
modelPerformance <- c(modelPerformance,
weighted_least_square(isotopeData, sinefit(parameter[i,1], parameter[i,2], parameter[i,3], inputDate), weight))
}

#return best par
return(parameter[which(modelPerformance == min(modelPerformance)),])
}

# Model performance
weighted_least_square <- function(obs, sim, weight){
missing <- which(is.na(obs))
if(length(missing)>0){
weighted_least_square <- sqrt(mean(((obs[-missing] - sim[-missing])*weight)^2))
} else {
weighted_least_square <- sqrt(mean(((obs - sim)*weight)^2))
}
return(weighted_least_square)
}

# Convert date to decimal
toDecimal <- function(inputDate){
toDecimal <- decimal_date(as.Date(inputDate, format = "%Y-%m-%d"))
toDecimal <- toDecimal - trunc(toDecimal)
return(toDecimal)
}

# Latin hypercube with input range
randomLHSrange <- function(inputRange, n){
randomLHSrange <- randomLHS(n,ncol(inputRange))
for(i in 1:ncol(inputRange)){
randomLHSrange[,i] <- inputRange[1,i] + (inputRange[2,i] - inputRange[1,i])*randomLHSrange[,i]
}
return(randomLHSrange)
}

# find alpha value
findAlpha <- function(parameterQ, parameterP){
error <- c()
alpha <- seq(0.01,20,0.01)
for (i in 1:length(alpha)){
error <- c(error, (parameterQ[2] - parameterP[2]) - alpha[i] * atan(sqrt((parameterQ[1]/parameterP[1])^(-2/alpha[i]) -1)))
}
error <- abs(error)
return(alpha[which(error == min(error))])
}

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RR by Jana von Freyberg (07 Mar 2023)

ED: Publish subject to revisions (further review by editor and referees) (14 Mar 2023) by Rohini Kumar

AR by Alessio Gentile on behalf of the Authors (22 Apr 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (24 Apr 2023) by Rohini Kumar

RR by Anonymous Referee #1 (11 May 2023)

ED: Publish as is (11 May 2023) by Rohini Kumar

AR by Alessio Gentile on behalf of the Authors (17 May 2023) Manuscript

Post-review adjustments

AA: Author's adjustment | EA: Editor approval

AA by Alessio Gentile on behalf of the Authors (21 Jun 2023) Author's adjustment Manuscript

EA: Adjustments approved (21 Jun 2023) by Rohini Kumar

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22 Jun 2023

Towards a conceptualization of the hydrological processes behind changes of young water fraction with elevation: a focus on mountainous alpine catchments

Alessio Gentile, Davide Canone, Natalie Ceperley, Davide Gisolo, Maurizio Previati, Giulia Zuecco, Bettina Schaefli, and Stefano Ferraris

Hydrol. Earth Syst. Sci., 27, 2301–2323, https://doi.org/10.5194/hess-27-2301-2023,https://doi.org/10.5194/hess-27-2301-2023, 2023

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What drives the young water fraction (i.e. the fraction of water in streamflow younger than roughly 2–3 months) variations with elevation? Why is F_yw counterintuitively low in high-elevation catchments, in spite of steeper topography compared with lower elevations? In this paper it is shown how the long duration of the snowpack favors the groundwater contribution to streamflow, thus increasing the streamwater age. A dataset of 27 Swiss and Italian catchments between 472 and 3049 m asl is used.


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What drives Fyw variations with elevation in Alpine catchments?

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What drives F_yw variations with elevation in Alpine catchments?