\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[ngerman,polish,english]{babel}



\usepackage{float}








\begin{document}

\title{To what extent are changes in flood magnitude related to changes in
precipitation extremes?}



\author[1,1]{Hong Xuan Do}%
\author[1,1]{Yiwen Mei}%
\author[1,1]{Andrew Gronewold}%
\affil[1]{University of Michigan}%


\vspace{-1em}



  
  \date{May 21, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Despite increasing evidence of intensification of extreme precipitation
events associated with a warming climate, the magnitude of extreme river
flows is decreasing in many parts of the world. To better understand the
range of relationships between precipitation extremes and floods, we
analyzed annual precipitation extremes and flood events over the
Contiguous United States from 1980 to 2014. A low spatial correlation
(less than 0.2) between changes in precipitation extremes and changes in
floods was found, attributable to a weak causal relationship. The
co-variation between precipitation extremes and floods is also
substantially low, with a majority of catchments having a coefficient of
determination of less than 0.5, even among the catchments with a
relatively strong causal relationship. The findings indicate a need for
more investigations into causal mechanisms driving a non-linear response
of floods to intensified precipitation extremes in a warming climate.%
\end{abstract}%



\sloppy


Hong Xuan Do\textsuperscript{1,2}, Yiwen Mei\textsuperscript{1} and
Andrew Gronewold\textsuperscript{1}

\textsuperscript{1}School for Environment and Sustainability, University
of Michigan, Ann Arbor, MI, USA

\textsuperscript{2}Faculty of Environment and Natural Resources, Nong
Lam University, Ho Chi Minh City, Vietnam

Corresponding author: Hong Xuan Do (hongdo@umich.edu)

Key points

\begin{itemize}
\tightlist
\item
  In contrast to conventional beliefs, we find weak causal linkages
  between precipitation extremes and floods
\item
  The spatial pattern of changes in precipitation extremes explains less
  than 20\% of the spatial pattern of changes in floods
\item
  Most catchments have a co-variation of less than 0.5 between annual
  precipitation extremes and annual floods
\end{itemize}

Abstract

Despite increasing evidence of intensification of extreme precipitation
events associated with a warming climate, the magnitude of extreme river
flows is decreasing in many parts of the world. To better understand the
range of relationships between precipitation extremes and floods, we
analyzed annual precipitation extremes and flood events over the
Contiguous United States from 1980 to 2014. A low spatial correlation
(less than 0.2) between changes in precipitation extremes and changes in
floods was found, attributable to a weak causal relationship. The
co-variation between precipitation extremes and floods is also
substantially low, with a majority of catchments having a coefficient of
determination of less than 0.5, even among the catchments with a
relatively strong causal relationship. The findings indicate a need for
more investigations into causal mechanisms driving a non-linear response
of floods to intensified precipitation extremes in a warming climate.

1 Introduction

Among the most important implications of global climate change is the
intensification of the hydrologic cycle {[}\emph{Huntington} , 2006{]},
including the intensification of rainfall extremes {[}\emph{Westra et
al.} , 2014{]}. As air temperature rises, the water vapor held in the
atmosphere also increases following the Clausius-Clapeyron relation
{[}\emph{Clausius} , 1850{]}. This relationship has been documented
extensively in the climate literature {[}\emph{Donat et al.} ,
2013;\emph{Guerreiro et al.} , 2018; \emph{Papalexiou and Montanari} ,
2019;\emph{Westra et al.} , 2013{]}, and has led to concerns of a future
characterized broadly by an increase in the magnitude of global flood
events.

Large scale investigations into changes in floods, however, indicate a
broader range of global flood response, with many studies documenting
sites with a decrease in flood magnitude {[}\emph{Do et al.} ,
2017;\emph{Do et al.} , 2020b; \emph{Gudmundsson et al.} , 2019;
\emph{Hodgkins et al.} , 2017; \emph{Kundzewicz et al.} , 2004;
\emph{Lins and Slack} , 1999{]}. These somewhat unexpected relationships
between trends in extreme precipitation and trends in extreme discharge
can be attributed to the influence of other flood generation mechanisms
such as soil moisture {[}\emph{Ivancic and Shaw} , 2015; \emph{Wasko et
al.} , 2020;\emph{Ye et al.} , 2017{]} and snow dynamics
{[}\emph{Berghuijs et al.} , 2016; \selectlanguage{ngerman}\emph{Blöschl et al.} , 2017;
\emph{Do et al.} , 2020a;\emph{Ledingham et al.} , 2019; \emph{Stein et
al.} , 2020{]}. Even when floods are triggered by precipitation
extremes, the relationship between precipitation magnitude and flood
magnitude is likely non-linear {[}\emph{Sharma et al.} , 2018{]}, owing
to the complex interactions of many variables which have undergone
substantial changes such as land cover {[}\emph{Keenan et al.} , 2015;
\emph{Lambin et al.} , 2003{]}, river channels {[}\emph{Slater et al.} ,
2015; \emph{Yamazaki et al.} , 2014{]} and evapotranspiration
{[}\emph{Bosilovich et al.} , 2005;\emph{Gronewold and Stow} , 2014{]}.

However, there is still a lack of quantitative understanding of the
relationship between precipitation extremes and floods, partially due to
unavailable discharge observations in many parts of the world
{[}\emph{Do et al.} , 2018; \emph{Do et al.} , 2020b{]}. Even for
regions with relatively good streamflow records, empirical
investigations have primarily focused on the consistency between the
timing of precipitation extremes and that of floods {[}\emph{Berghuijs
et al.} , 2019;\emph{Blöschl et al.} , 2017; \emph{Do et al.} , 2020a;
\emph{Ivancic and Shaw} , 2015; \emph{Stein et al.} , 2020; \emph{Wasko
et al.} , 2020{]} rather than co-variation between precipitation extreme
intensity and flood magnitude. As a result, it is difficult to promote
generalized statements about global relationships between changes in
precipitation extremes and changes in floods, which is essential to the
design of robust flood prevention and mitigation strategies in a warming
climate {[}\emph{Milly et al.} , 2008{]}.

We aim to fill in this gap through an empirical assessment of the
co-variation of precipitation extremes and flood magnitude using a large
sample (671) of catchments across the Contiguous United States (CONUS)
(Section 2.1). We used annual maxima streamflow from 1980 to 2014 from
these catchments as the flood population, and we used three metrics of
annual maxima precipitation to represent precipitation extremes (Section
2.2). Temporal changes in floods and precipitation extremes were then
estimated at each catchment and the correlation between the spatial
patterns of these trends was assessed (Section 2.3). The ordinal date of
precipitation extreme events was then compared to that of annual flood
events (Section 2.4) to assess potential causal relationships between
these hydro-climatic extremes. Finally, the co-variation between the
intensity of precipitation extremes and flood magnitude across
catchments was assessed (Section 2.5) to evaluate the appropriateness of
using changes in extreme precipitation as a proxy for changes in floods.

2 Data and Methods

2.1 Data

Data for our analysis was derived from the Catchment Attributes and
Meteorology for Large-sample Studies (CAMELS) dataset {[}\emph{Addor et
al.} , 2017a; \emph{Newman et al.} , 2015{]}. The CAMELS database
aggregates a variety of hydrometeorological variables (primarily derived
from other studies) for 671 catchments across the CONUS (the outlets of
CAMELS catchments are shown in Figure 1). The catchments in the CAMELS
database are intended to reflect relatively natural hydrologic
conditions (the impervious surface area of each catchment is less than
5\% of the total catchment area; see \emph{Newman et al.} {[}2015{]} for
more information). These catchments have a relatively small size (the
median catchment area is 340.7 km\textsuperscript{2}) and cover a range
of climatic conditions (e.g., dry, temperate, and continental climates)
as well as geographic features (e.g., mountains and deserts). Other
variables in the CAMELS database include daily streamflow (originally
obtained from the United States Geological Survey), catchment-average
daily precipitation and temperature (derived from the Daymet dataset
{[}\emph{Thornton et al.} , 1997{]}), and daily evapotranspiration,
simulated by the conceptual SAC-SMA model {[}\emph{Burnash et al.} ,
1973{]}.

In addition to the hydro-meteorological data available through CAMELS,
we also obtained soil moisture data from the NOAA Climate Prediction
Center {[}\emph{Van den Dool et al.} , 2003{]}. This dataset provides
monthly soil moisture water height equivalent, simulated by a leaky
bucket model, with a spatial resolution of 0.5-degree latitude x
0.5-degree longitude (i.e., a cell-size of about 2,500
km\textsuperscript{2}). We used monthly soil moisture from the cell
containing each catchment outlets as a proxy for catchment-wide to
obtain soil moisture conditions from 1980 to 2014 for each of the CAMELS
catchments. We judged this approach to be appropriate for the CAMELS
catchments, which have a relatively small size.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 1.} Location of CAMELS catchment outlets across the
CONUS. Each catchment was classified into one of the six groups based on
the annual average fraction of annual precipitation falling as snow
(\emph{f\textsubscript{snow}} ; categories were defined at 0.1
intervals, with all catchments having \emph{f\textsubscript{snow}} of at
least 0.5 grouped into one category). The 18 major water resources
regions over the CONUS are also shown (grey polygons bounded by whites
lines) for reference (see also Figure S1 in Supporting Information).

2.2 Identifying streamflow and precipitation extremes

Our approach to quantifying rainfall and streamflow extremes is based on
the annual maxima (AMAX) index, one of the most common indices for
assessing temporal changes in hydro-climatic extremes {[}\emph{Do et
al.} , 2017; \emph{Kundzewicz et al.} , 2004; \emph{Ledingham et al.} ,
2019; \emph{Villarini and Smith} , 2010; \emph{Westra et al.} , 2013{]}.
We first processed streamflow data to obtain the magnitude
(Q\textsubscript{MAX} index) and the timing
(Q\textsubscript{DOYMAX}index; defined as the ordinal date of annual
floods) for each AMAX streamflow event. To reduce the chance of
misattributing flood events, we omitted from our data set any years
missing more than 15\% of daily values (to have a compromise between
data coverage and data quality).

We then processed daily precipitation to derive three sets of variables,
each representing a different approach to quantifying precipitation
extremes (a threshold of 15\% missing data was also used to remove years
with insufficient data.). The first variable representing precipitation
extremes is AMAX precipitation (P), which is defined using the same
approach to that of AMAX streamflow. The second precipitation variable
allows us to assess whether constraining the timing of precipitation
extremes to seasons where the catchments are wet, and when floods are
more likely to occur {[}\emph{Ivancic and Shaw} , 2015{]}, could improve
the relationship between precipitation extremes and floods. This second
variable is calculated as the annual maximum precipitation based only on
months in which soil moisture was above-average. The third precipitation
variable is intended to take into account catchment saturation and snow
dynamics, and how they affect the propagation of precipitation
into\emph{effective} precipitation {[}\emph{Berghuijs et al.} ,
2016;\emph{Berghuijs et al.} , 2019{]}. We calculated this variable
using a coupled soil-snow routine {[}\emph{Berghuijs et al.} , 2016;
\emph{Hock} , 2003; \emph{Stein et al.} , 2020; \emph{Woods} , 2009{]}
based on daily precipitation, temperature, and evapotranspiration (all
readily available in the CAMELS dataset). Details of this routine is
provided in the Supporting Information; for further reading, see
\emph{Stein et al.}{[}2020{]}.

Finally, we calculate the intensity and timing of each of the three
precipitation AMAX variables, leading to a total of six new
precipitation indices; P\textsubscript{MAX} and
P\textsubscript{DOYMAX}(for the first precipitation variable),
P\textsubscript{sm.MAX} and P\textsubscript{sm.DOYMAX} (for the second),
and P\textsubscript{eff.MAX} and P\textsubscript{eff.DOYMAX} (for the
third). Note that evapotranspiration is only available from October 1980
onward, thus P\textsubscript{eff.MAX} and P\textsubscript{eff.DOYMAX}are
not available for 1980.

2.3 Assessing the correlation between the spatial pattern of changes in
precipitation extremes and the spatial pattern of changes in floods

We calculated temporal changes in the magnitude of AMAX streamflow
(Q\textsubscript{MAX}) and changes in precipitation extreme intensity
(P\textsubscript{MAX}, P\textsubscript{sm.MAX}, and
P\textsubscript{eff.MAX}) using normalized Theil--Sen slope
{[}\emph{Gudmundsson et al.} , 2019; \emph{Stahl et al.} , 2012{]} as
follows:

\(\tau_{c}=\text{median}\left(\frac{x_{j}-x_{i}}{j-i}\right)\)(1)

\(T_{c}=\frac{\tau_{c}\times 10\text{years}}{{\overset{\overline{}}{x}}_{c}}\times 100\)(2)

where \(\tau_{c}\) is the Theil-Sen slope estimator for
catchment\emph{c} , which is defined as the median of the average annual
difference in AMAX values (\emph{x} ) between all possible pairs of
years. The indices \emph{i} and \emph{j} represent year numbers such
that \emph{i} \(\in\) {[}1, \emph{n\textsubscript{c}} -1{]},
j \(\in\){[}2, \emph{n\textsubscript{c}} {]}, \emph{i}
\textless{} \emph{j} , and\emph{n\textsubscript{c}} is the number of
years in the data record (after the screening process described above)
for each catchment.\(T_{c}\) is the normalized trend, expressed
as a percentage of change per decade relative to the mean of all AMAX
values in a catchment (\({\overset{\overline{}}{x}}_{c}\)). This approach leads to
four\emph{T\textsubscript{c}} values for each catchment, one for
Q\textsubscript{MAX} and three for precipitation intensity
(P\textsubscript{MAX}, P\textsubscript{sm.MAX}, and
P\textsubscript{eff.MAX}).

To evaluate whether the spatial pattern of changes in floods can be
explained by the spatial pattern of changes in precipitation extremes,
we calculated the coefficient of determination
R\textsuperscript{2}{[}\emph{Rao} , 1973{]}, which is the square of the
correlation between the T\textsubscript{c} values of
Q\textsubscript{MAX} and the T\textsubscript{c} values of a
precipitation extreme metric (e.g., T\textsubscript{c} of
P\textsubscript{MAX}). The value of R\textsuperscript{2} ranges from 0
to 1, and a high R\textsuperscript{2}indicates a strong correlation.

2.4 Assessing the causal relationship between precipitation extremes and
floods

To quantify the strength of a potential relationship between
precipitation extremes and floods, we calculated the probability that an
annual precipitation extremes would be followed closely (in time) by an
annual flood extreme in each catchment (referred to as the co-occurrence
probability hereafter). We matched the timing of AMAX precipitation
events (represented by P\textsubscript{DOYMAX},
P\textsubscript{sm.DOYMAX}, and P\textsubscript{eff.DOYMAX} indices) and
the timing of annual floods (represented by
Q\textsubscript{DOYMAX}index) and derive the fraction of annual flood
events that can be directly attributed to a preceding precipitation
extremes. To account for travel time required for precipitation to reach
a catchment outlet, we adopted a previous approach {[}\emph{Ivancic and
Shaw} , 2015{]} and allowed a lag of up to 5 days. Specifically, we
presume that there is a causal relationship between AMAX precipitation
and AMAX streamflow if 0 [?] Q\textsubscript{DOYMAX} -
P\textsubscript{DOYMAX} [?] 5. If precipitation extremes and floods were
independent, the random chance of a match, on average, would be less
than 2\%, which is the random chance of Q\textsubscript{DOYMAX} having a
value between P\textsubscript{DOYMAX} and P\textsubscript{DOYMAX} + 5
(six days) of all possible days in a non-leap year (365 days).

Given the important relationship between the precipitation type (i.e.
snow or rain) over much of the CONUS {[}\emph{Berghuijs et al.} ,
2016{]}, as well as other parts of the world {[}\selectlanguage{ngerman}\emph{Blöschl et al.} ,
2017; \emph{Do et al.} , 2020a{]}, we also assessed whether
relationships between precipitation extremes and floods vary by
precipitation type. To achieve this objective, we used the annual
average proportion of precipitation that falls as snow, readily
available in \emph{Addor et al.} {[}2017a{]}, and is denoted as
\emph{f\textsubscript{snow}} . Each catchments was classified into one
of the six categories; the first five are defined by
\emph{f\textsubscript{snow}} values between 0.0 and 0.5 at intervals of
0.1; the sixth category includes all catchments with
an\emph{f\textsubscript{snow}} value between 0.5 and 1.0 (see Figure 1).
We then assessed whether there are significant differences in the
co-occurrence probability of precipitation extremes and floods
across\emph{f\textsubscript{snow}} classification categories.

2.5 Assessing temporal correlation between the intensity of
precipitation extremes and flood magnitude

We identified catchments with similar causal relationship between
precipitation extremes and flood by grouping catchments into seven
groups according to the co-occurrence probability at 0.1 intervals (note
that all catchments with at least 0.6 co-occurrence probability grouped
into one category). We then measured the co-variation between the
intensity of precipitation extremes (e.g., P\textsubscript{MAX} index)
and the magnitude of floods (Q\textsubscript{MAX} index) at each
catchment using the coefficient of determination R\textsuperscript{2}.
The R\textsuperscript{2} values were then analyzed alongside the
co-occurrence probability to quantify the extent of which changes in
precipitation extremes are useful to infer changes in flood magnitude.

3 Results and Discussions

3.1 A low correlation between the spatial pattern of changes in floods
and the spatial pattern of changes in precipitation extremes

Figure 2 shows temporal changes in the magnitude of annual floods and
precipitation extremes across the CAMELS catchments. We note that the
effective precipitation (P\textsubscript{eff}) could be equal to zero
throughout the year wherever precipitation could not make the catchment
fully saturated. Specifically, there are 110 catchments having zero
P\textsubscript{eff.MAX} over more than 20 years, leading to a zero
Thiel-Sen slope estimated as shown in Figure 2h (see also Supplementary
Figure S2). We also removed 33 catchments that have
P\textsubscript{eff.MAX} equal to zero across all years from our
analyses, leading to a sample size of 638 catchments for
P\textsubscript{eff.MAX} assessment.

Over the reference period, more CAMELS catchments (53\%) experienced a
decrease in Q\textsubscript{MAX} index (Figure 2a), consistent with
recent investigations {[}\emph{Do et al.} , 2017; \emph{Do et al.} ,
2020b; \emph{Gudmundsson et al.} , 2019; \emph{Hodgkins et al.} ,
2019;\emph{Hodgkins et al.} , 2017{]}. On the contrary,
P\textsubscript{MAX}index shows an increasing trend (Figure 2b) over the
majority of catchments (67\%), although some interior water resources
regions exhibited a more prominent decreasing trend (e.g. Missouri
Region; see Figure S2 in the Supporting Information for trends in annual
floods and precipitation extremes over individual regions). There is a
low correlation between the spatial pattern of changes in
P\textsubscript{MAX} and the spatial pattern of changes in
Q\textsubscript{MAX} (Figure 2f) with an R\textsuperscript{2} of 0.11,
indicating that only 11\% of the spatial variation of trends of
Q\textsubscript{MAX} can be explained using trends of
P\textsubscript{MAX}.

The spatial pattern of P\textsubscript{sm.MAX} trends (Figure 2c) is
generally consistent with that of P\textsubscript{MAX} trends, while the
spatial pattern of P\textsubscript{eff.MAX} trends (Figure 2d) shows a
substantial difference relative to that of P\textsubscript{MAX} trends,
and appears to be more consistent with the spatial pattern of
Q\textsubscript{MAX} trends. The coefficient of determination between
precipitation extreme trends and Q\textsubscript{MAX} trends support
this finding, with an R\textsuperscript{2} of 0.06 and 0.17 for
P\textsubscript{sm.MAX} trends and P\textsubscript{eff.MAX} trends
respectively (Figure 2g and Figure 2h). These results are generally
expected, as the snow-soil routine underlying
P\textsubscript{eff.MAX}can be seen as a simple conceptual model that
takes into account several catchment processes.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

More importantly, the R\textsuperscript{2} values between
Q\textsubscript{MAX} trends and precipitation extreme trends are less
than 0.2 across all precipitation extreme metrics. This result means
that the spatial variation of precipitation extreme trends can explain
less than 20\% of the spatial variation of Q\textsubscript{MAX} trends
across the CONUS from 1980 to 2014. The R\textsuperscript{2} calculated
over individual water resources regions (Figure 2e; also see Figure S2
of the Supporting Information) also indicates a low correlation.
Specifically, more than 60\% of the regions having an
R\textsuperscript{2} value of less than 0.2, indicating the limitation
of using trends of precipitation extremes to infer trends of floods.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 2.} Trends in (a) Q\textsubscript{MAX}, (b)
P\textsubscript{MAX}, (c) P\textsubscript{sm.MAX} and (d)
P\textsubscript{eff.MAX} across each of the 671 CAMELS catchments based
on the normalized Thiel-Sen slope (\emph{T\textsubscript{c}} ). Scatter
plots between \emph{T\textsubscript{c}} values of
Q\textsubscript{MAX}and \emph{T\textsubscript{c}} values of (f)
P\textsubscript{MAX}, (g) P\textsubscript{sm.MAX}, (h)
P\textsubscript{eff.MAX} are also shown. (d) Boxplot of the
R\textsuperscript{2} between \emph{T\textsubscript{c}} values of annual
floods and \emph{T\textsubscript{c}} \textsubscript{}values of annual
precipitation extremes (see Figure S2 in the Supporting Information for
disaggregation of results across water resources regions).

3.2 Co-occurrence probability of precipitation extremes and floods
across the CONUS

The low correlation between the spatial pattern of changes in
precipitation extremes and that of floods (discussed in Section 3.1) is
potentially attributable to a weak causal relationship, as there are
other mechanisms that could trigger floods {[}\selectlanguage{ngerman}\emph{Blöschl et al.} ,
2019; \emph{Merz and Blöschl} , 2003{]}. To assess this hypothesis, we
assessed the co-occurrence probability between precipitation extremes
and floods over individual catchments and the results are shown in
Figure 3 (See Figure S3-S5 in the Supporting Information for
Q\textsubscript{DOYMAX}, P\textsubscript{DOYMAX},
P\textsubscript{sm.DOYMAX}, and P\textsubscript{eff.DOYMAX} across all
catchments). The averaged co-occurrence probability across all
catchments is 32\%, 30\%, and 37\% for P\textsubscript{DOYMAX},
P\textsubscript{sm,DOYMAX}, and P\textsubscript{eff.DOYMAX} extremes
respectively. This number is consistent with a previous investigation
{[}\emph{Ivancic and Shaw} , 2015{]}, indicating that annual
precipitation extremes are only responsible for about one-third of the
annual flood population.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 3.} Co-occurrence probability between AMAX streamflow and
(a) AMAX precipitation, (b) AMAX wet-month precipitation, (c) AMAX
effective precipitation across all CAMELS catchments; and (d) when
grouped into six categories using the fraction of precipitation falling
as snow (\emph{f\textsubscript{snow}} ). Note that
P\textsubscript{eff}were available for only 638 catchments.

The vast majority of catchments (more than 95\%) have a co-occurrence
probability that is much higher than random chance (i.e., 2\%), which is
generally expected. Catchments with a relatively high co-occurrence
probability are mostly located in coastal regions (e.g., South
Atlantic-Gulf Region, Texas Gulf Region, California Region, and Pacific
Northwest Region) while co-occurrence probability tends to be low
whenever the fraction of precipitation falling as snow
(\emph{f\textsubscript{snow}} ) is high (e.g., Upper Colorado Region and
Great Basin Region). More importantly, only a small fraction (14\%) of
all catchments having a co-occurrence probability between
Q\textsubscript{DOYMAX} and P\textsubscript{DOYMAX} (Figure 3a; see
Figure S6 for regional results) of at least 0.5, indicating a weak
causal linkage.

The co-occurrence probability between Q\textsubscript{DOYMAX} and
P\textsubscript{sm.DOYMAX} (Figure 3b; see Figure S7 for regional
results) is higher than or equal to 0.5 over 7\% of all catchments,
indicating a weaker causal linkage relative to that between
Q\textsubscript{DOYMAX} and P\textsubscript{DOYMAX}. A possible reason
is that soil moisture has a relatively strong seasonal cycle
{[}\emph{Eltahir} , 1998; \emph{Findell and Eltahir} , 1997{]},
contrasting to a weak seasonality of short-duration precipitation
extremes {[}\emph{Do et al.} , 2020a{]}. Using P\textsubscript{sm} has
potentially masked out many short-duration flood-induced rainfall events
that spread throughout the years, leading to a lower co-occurrence
probability. This finding suggests that precipitation extremes
constrained over wet months has a potentially lower predictability for
changes in annual floods.

Among the three precipitation extreme metrics, effective precipitation
(Figure 3c; see Figure S8 for regional results) is the variable with the
strongest causal relationship to floods. Specifically, 26\% of all
catchments have a co-occurrence probability between
P\textsubscript{eff.DOYMAX} and Q\textsubscript{DOYMAX} of at least 0.5.
A simple approach to take into account snow-soil interaction has led to
a substantial increase in co-occurrence probability, suggesting that
catchment processes potentially play a more important role in modulating
floods relative to precipitation intensity.

When catchments are divided into different categories
using\emph{f\textsubscript{snow}} (Figure 3d), there is a notable
decrease of co-occurrence probability when \emph{f\textsubscript{snow}}
increases. We note that the co-occurrence probability between
precipitation extremes and floods is not consistently low across all
catchments with a high \emph{f\textsubscript{snow}} . For instance, of
all 73 catchments with an \emph{f\textsubscript{snow}} of higher than or
equal to 0.5, six catchments (located in the Pacific Northwest Region)
have a co-occurrence probability between P\textsubscript{DOYMAX} and
Q\textsubscript{MAX} of at least 0.5. As a result, catchments with a
high snow-to-rain ratio are likely to have floods not driven by
precipitation extremes, but there are exceptions such as catchments
strongly influenced by atmospheric rivers which are responsible for
flood-induced rainfall events.

3.3 To what extent are changes in precipitation extremes useful to
explain changes in floods?

The co-variation between precipitation extremes and
Q\textsubscript{MAX}is relatively low, with 81\%, 85\%, and 66\% of all
catchments having an R\textsuperscript{2} of less than 0.5 for
P\textsubscript{MAX}, P\textsubscript{sm.MAX}, and
P\textsubscript{eff.MAX} respectively. When catchments are grouped into
different categories according to co-occurrence probability, a strong
positive correlation between co-variation and co-occurrence probability
is observed (Figure 4). Of all catchments with co-occurrence probability
of less than 0.5, the averaged R\textsuperscript{2} is 0.28 (for
P\textsubscript{MAX}), 0.24 (for P\textsubscript{sm.MAX}) and 0.33 (for
P\textsubscript{eff.MAX}) respectively, indicating that only about 30\%
of the temporal variability of floods can be explained by precipitation
extremes.

Focusing on the catchments with the strongest causal relationship (i.e.,
a co-occurrence probability of at least 0.6), a low-to-moderate
correlation is observed, with the median of R\textsuperscript{2} between
Q\textsubscript{MAX} and precipitation extremes is 0.41, 0.43 and 0.52
for P\textsubscript{MAX}, P\textsubscript{sm.MAX} and
P\textsubscript{eff.MAX} respectively. The causal relationship between
P\textsubscript{eff.MAX} and Q\textsubscript{MAX} is the strongest, with
34 out of 63 catchments (54\%) have an R\textsuperscript{2} of above
0.5. The relationship between P\textsubscript{MAX} and
Q\textsubscript{MAX} is the lowest, with 10 out of 28 catchments (36\%)
have an R\textsuperscript{2} value of above 0.5, further confirming the
need for considering catchment processes (e.g., snow-soil interaction)
to explain changes in annual floods.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 4.} Coefficeint of determination (R\textsuperscript{2})
between AMAX discharges and AMAX precipitation across all CAMELS
catchments, grouped by the co-occurrence probability. Results are showed
for precipitation (P), wet-month precipation (P\textsubscript{sm}) and
effective precipitation (P\textsubscript{eff}). Note that
P\textsubscript{eff} were available for only 638 catchments.

\textbf{4 Summary and Conclusions}

Using annual maxima precipitation and streamflow across a large sample
of catchments, this study has empirically assessed the relationship
between temporal changes in precipitation extremes and changes in annual
flood magnitude. The spatial pattern of trends detected from
precipitation extremes is weakly correlated to the spatial pattern of
trends detected from AMAX streamflow over 671 CONUS catchments, with a
coefficient of determination of less than 0.2.

A weak linkage between annual precipitation extremes and annual floods
is apparent across the CAMELS catchments, with the vast majority of
catchments have less than 50\% of annual flood events directly linked to
precipitation extremes (85\%, 90\%, and 73\% of all catchments for AMAX
precipitation, AMAX wet-month precipitation and AMAX effective
precipitation respectively). Catchments with a high snow-to-rain ratio
(indicated by \emph{f\textsubscript{snow}} value) generally have a low
causal relationship between precipitation extremes and floods, but the
impact of snow presence is not uniform. The co-variation between extreme
precipitation intensity and flood magnitude is also low, with more than
60\% of catchments having an R\textsuperscript{2} of less than 0.5,
regardless of which precipitation extreme metrics being used. Using a
snow-soil routine to correct the actual amount of precipitation
modulating floods has led to a substantially improved predictability for
changes in floods, suggesting that future trend detection studies should
focus more on the catchment attributes such as soil profile and
impervious area.

Notwithstanding the complex processes driving floods, this study has
quantitatively assessed the limitation of using changes in precipitation
as a proxy for potential changes in floods. The findings indicate that
the intensity of daily precipitation extremes is a weak predictor for
temporal changes in annual maxima of daily streamflow, even for
catchments with a relatively high causal relationship. This study
highlights the need for additional efforts to investigate the non-linear
responses of floods to climate changes using a larger sample of
catchments, which would hopefully achieve a universal understanding of
how floods might evolve. For instance, the approach presented in this
study can be applied for other large sample datasets {[}\emph{Addor et
al.} , 2019; \emph{Alvarez-Garreton et al.} , 2018; \emph{Coxon et al.}
, 2020; \emph{Gudmundsson et al.} , 2018{]} to quantify the contribution
of extreme precipitation to historical changes in floods for other parts
of the world.

Acknowledgments and Data

Hong Xuan Do is currently funded by the School for Environment and
Sustainability, University of Michigan through Grant U064474. The
authors appreciate the developers of the CAMELS dataset for making this
asset publicly available. Hydrometerological data is freely available at
https://dx.doi.org/10.5065/D6MW2F4D {[}\emph{Newman et al.} , 2014{]}
while the catchment attributes, including the fraction of precipitation
falling as snow is freely available at https://doi.org/10.5065/D6G73C3Q
{[}\emph{Addor et al.} , 2017b{]}.

Reference

Addor, N., A. J. Newman, N. Mizukami, and M. P. Clark (2017a), The
CAMELS data set: catchment attributes and meteorology for large-sample
studies, \emph{Hydrol. Earth Syst. Sci. Discuss.} , \emph{2017} , 1-31.

Addor, N., A. J. Newman, N. Mizukami, and M. P. Clark (2017b), Catchment
attributes for large-sample studies, Boulder, CO: UCAR/NCAR, edited.

Addor, N., H. X. Do, C. Alvarez-Garreto, G. Coxon, K. Fowler, and P.
Mendoza (2019), Large-sample hydrology: recent progress, guidelines for
new datasets and grand challenges, \emph{Hydrological Sciences Journal}
.

Alvarez-Garreton, C., et al. (2018), The CAMELS-CL dataset: catchment
attributes and meteorology for large sample studies -- Chile
dataset,\emph{Hydrol. Earth Syst. Sci.} , \emph{22} (11), 5817-5846.

Berghuijs, W. R., R. A. Woods, C. J. Hutton, and M. Sivapalan (2016),
Dominant flood generating mechanisms across the United
States,\emph{Geophysical Research Letters} , \emph{43} (9), 4382-4390.

Berghuijs, W. R., S. Harrigan, P. Molnar, L. J. Slater, and J. W.
Kirchner (2019), The Relative Importance of Different Flood-Generating
Mechanisms Across Europe, \emph{Water Resources Research} , \emph{55}
(6), 4582-4593.

Bl\selectlanguage{ngerman}öschl, G., J. Hall, A. Viglione, R. A. Perdigão, J. Parajka, B. Merz,
D. Lun, B. Arheimer, G. T. Aronica, and A. Bilibashi (2019), Changing
climate both increases and decreases European river floods,\emph{Nature}
, \emph{573} (7772), 108-111.

Blöschl, G., et al. (2017), Changing climate shifts timing of European
floods, \emph{Science} , \emph{357} (6351), 588.

Bosilovich, M. G., S. D. Schubert, and G. K. Walker (2005), Global
Changes of the Water Cycle Intensity, \emph{Journal of Climate}
,\emph{18} (10), 1591-1608.

Burnash, R. J. C., R. L. Ferral, and R. A. McGuire (1973), \emph{A
generalized streamflow simulation system: Conceptual modeling for
digital computers} , US Department of Commerce, National Weather
Service, and State of California~\ldots{}.

Clausius, R. (1850), Über die bewegende Kraft der Wärme und die Gesetze,
welche sich daraus für die Wärmelehre selbst ableiten
lassen,\emph{Annalen der Physik} , \emph{155} (3), 368-397.

Coxon, G., et al. (2020), CAMELS-GB: Hydrometeorological time series and
landscape attributes for 671 catchments in Great Britain, \emph{Earth
Syst. Sci. Data Discuss.} , \emph{2020} , 1-34.

Do, H. X., S. Westra, and L. Michael (2017), A global-scale
investigation of trends in annual maximum streamflow, \emph{Journal of
Hydrology} .

Do, H. X., L. Gudmundsson, M. Leonard, and S. Westra (2018), The Global
Streamflow Indices and Metadata Archive (GSIM) -- Part 1: The production
of a daily streamflow archive and metadata, \emph{Earth Syst. Sci. Data}
, \emph{10} (2), 765-785.

Do, H. X., S. Westra, M. Leonard, and L. Gudmundsson (2020a),
Global-Scale Prediction of Flood Timing Using Atmospheric
Reanalysis,\emph{Water Resources Research} , \emph{56} (1),
e2019WR024945.

Do, H. X., et al. (2020b), Historical and future changes in global flood
magnitude -- evidence from a model--observation
investigation,\emph{Hydrol. Earth Syst. Sci.} , \emph{24} (3),
1543-1564.

Donat, M. G., et al. (2013), Updated analyses of temperature and
precipitation extreme indices since the beginning of the twentieth
century: The HadEX2 dataset, \emph{Journal of Geophysical Research:
Atmospheres} , \emph{118} (5), 2098-2118.

Eltahir, E. A. B. (1998), A soil moisture--rainfall feedback mechanism:
1. Theory and observations, \emph{Water resources research} ,\emph{34}
(4), 765-776.

Findell, K. L., and E. A. B. Eltahir (1997), An analysis of the soil
moisture-rainfall feedback, based on direct observations from
Illinois,\emph{Water Resources Research} , \emph{33} (4), 725-735.

Gronewold, A. D., and C. A. Stow (2014), Water loss from the Great
Lakes, \emph{Science} , \emph{343} (6175), 1084-1085.

Gudmundsson, L., H. X. Do, M. Leonard, and S. Westra (2018), The Global
Streamflow Indices and Metadata Archive (GSIM) - Part 2: Time Series
Indices and Homogeneity Assessment, edited, PANGAEA.

Gudmundsson, L., M. Leonard, H. X. Do, S. Westra, and S. I. Seneviratne
(2019), Observed trends in global indicators of mean and extreme
streamflow, \emph{Geophysical Research Letters} , \emph{46} (2),
756-766.

Guerreiro, S. B., H. J. Fowler, R. Barbero, S. Westra, G. Lenderink, S.
Blenkinsop, E. Lewis, and X.-F. Li (2018), Detection of
continental-scale intensification of hourly rainfall
extremes,\emph{Nature Climate Change} , \emph{8} (9), 803-807.

Hock, R. (2003), Temperature index melt modelling in mountain
areas,\emph{Journal of Hydrology} , \emph{282} (1), 104-115.

Hodgkins, G. A., R. W. Dudley, S. A. Archfield, and B. Renard (2019),
Effects of climate, regulation, and urbanization on historical flood
trends in the United States, \emph{Journal of Hydrology} , \emph{573} ,
697-709.

Hodgkins, G. A., et al. (2017), Climate-driven variability in the
occurrence of major floods across North America and Europe,\emph{Journal
of Hydrology} , \emph{552} , 704-717.

Huntington, T. G. (2006), Evidence for intensification of the global
water cycle: Review and synthesis, \emph{Journal of Hydrology}
,\emph{319} (1), 83-95.

Ivancic, T., and S. Shaw (2015), Examining why trends in very heavy
precipitation should not be mistaken for trends in very high river
discharge, \emph{Climatic Change} , 1-13.

Keenan, R. J., G. A. Reams, F. Achard, J. V. de Freitas, A. Grainger,
and E. Lindquist (2015), Dynamics of global forest area: Results from
the FAO Global Forest Resources Assessment 2015, \emph{Forest Ecology
and Management} , \emph{352} , 9-20.

Kundzewicz, Z. W., D. Graczyk, T. Maurer, I. Przymusi\selectlanguage{polish}ń\selectlanguage{english}ska, M.
Radziejewski, C. Svensson, and M. Szwed (2004), Detection of change in
world-wide hydrological time series of maximum annual flow, Global
Runoff Date Centre, Koblenz, Germany.

Lambin, E. F., H. J. Geist, and E. Lepers (2003), Dynamics of land-use
and land-cover change in tropical regions, \emph{Annual review of
environment and resources} , \emph{28} (1), 205-241.

Ledingham, J., D. Archer, E. Lewis, H. Fowler, and C. Kilsby (2019),
Contrasting seasonality of storm rainfall and flood runoff in the UK and
some implications for rainfall-runoff methods of flood
estimation,\emph{Hydrology Research} , \emph{50} (5), 1309-1323.

Lins, H. F., and J. R. Slack (1999), Streamflow trends in the United
States, \emph{Geophysical research letters} , \emph{26} (2), 227-230.

Merz, R., and G. Bl\selectlanguage{ngerman}öschl (2003), A process typology of regional
floods,\emph{Water Resources Research} , \emph{39} (12).

Milly, P. C. D., J. Betancourt, M. Falkenmark, R. M. Hirsch, Z. W.
Kundzewicz, D. P. Lettenmaier, and R. J. Stouffer (2008), Stationarity
Is Dead: Whither Water Management?, \emph{Science} , \emph{319} (5863),
573-574.

Newman, A. J., K. Sampson, M. P. Clark, A. Bock, R. J. Viger, and D.
Blodgett (2014), A large-sample watershed-scale hydrometeorological
dataset for the contiguous USA, \emph{UCAR/NCAR, doi} , \emph{10} ,
D6MW2F4D.

Newman, A. J., et al. (2015), Development of a large-sample
watershed-scale hydrometeorological data set for the contiguous USA:
data set characteristics and assessment of regional variability in
hydrologic model performance, \emph{Hydrol. Earth Syst. Sci.} ,\emph{19}
(1), 209-223.

Papalexiou, S. M., and A. Montanari (2019), Global and Regional Increase
of Precipitation Extremes Under Global Warming, \emph{Water Resources
Research} , \emph{55} (6), 4901-4914.

Rao, C. R. (1973), \emph{Linear statistical inference and its
applications} , 625 pp., Wiley New York.

Sharma, A., C. Wasko, and D. P. Lettenmaier (2018), If Precipitation
Extremes Are Increasing, Why Aren't Floods?, \emph{Water Resources
Research} , \emph{0} (0).

Slater, L. J., M. B. Singer, and J. W. Kirchner (2015), Hydrologic
versus geomorphic drivers of trends in flood hazard, \emph{Geophysical
Research Letters} , \emph{42} (2), 370-376.

Stahl, K., L. M. Tallaksen, J. Hannaford, and H. A. J. van Lanen (2012),
Filling the white space on maps of European runoff trends: estimates
from a multi-model ensemble, \emph{Hydrol. Earth Syst. Sci.} ,\emph{16}
(7), 2035-2047.

Stein, L., F. Pianosi, and R. Woods (2020), Event-based classification
for global study of river flood generating processes, \emph{Hydrological
Processes} , \emph{34} (7), 1514-1529.

Thornton, P. E., S. W. Running, and M. A. White (1997), Generating
surfaces of daily meteorological variables over large regions of complex
terrain, \emph{Journal of Hydrology} , \emph{190} (3), 214-251.

Van den Dool, H., J. Huang, and Y. Fan (2003), Performance and analysis
of the constructed analogue method applied to US soil moisture over
1981--2001, \emph{Journal of Geophysical Research: Atmospheres}
,\emph{108} (D16).

Villarini, G., and J. A. Smith (2010), Flood peak distributions for the
eastern United States, \emph{46} (6).

Wasko, C., R. Nathan, and M. C. Peel (2020), Changes in Antecedent Soil
Moisture Modulate Flood Seasonality in a Changing Climate, \emph{Water
Resources Research} , \emph{56} (3), e2019WR026300.

Westra, S., L. A. Alexander, and F. W. Zwiers (2013), Global Increasing
Trends in Annual Maximum Daily Precipitation, \emph{Journal of Climate}
,\emph{26} (11), 15.

Westra, S., H. J. Fowler, J. P. Evans, L. V. Alexander, P. Berg, F.
Johnson, E. J. Kendon, G. Lenderink, and N. M. Roberts (2014), Future
changes to the intensity and frequency of short-duration extreme
rainfall, \emph{Reviews of Geophysics} , \emph{52} (3), 522-555.

Woods, R. A. (2009), Analytical model of seasonal climate impacts on
snow hydrology: Continuous snowpacks, \emph{Advances in Water Resources}
, \emph{32} (10), 1465-1481.

Yamazaki, D., F. O'Loughlin, M. A. Trigg, Z. F. Miller, T. M. Pavelsky,
and P. D. Bates (2014), Development of the Global Width Database for
Large Rivers, \emph{Water Resources Research} , \emph{50} (4),
3467-3480.

Ye, S., H.-Y. Li, L. R. Leung, J. Guo, Q. Ran, Y. Demissie, and M.
Sivapalan (2017), Understanding Flood Seasonality and Its Temporal
Shifts within the Contiguous United States, \emph{Journal of
Hydrometeorology} , \emph{18} (7), 1997-2009.

\selectlanguage{english}
\FloatBarrier
\end{document}

