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\begin{document}

\title{Allometric scaling of hyporheic respiration across basins in the Pacific
Northwest USA}



\author[1]{Peter Regier}%
\author[1]{Kyongho Son}%
\author[1]{Xingyuan Chen}%
\author[1]{Yilin Fang}%
\author[1]{Peishi Jiang}%
\author[1]{Micah Taylor}%
\author[1]{Wilfred M Wollheim}%
\author[1]{James Stegen}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\textbf{Authors:} Peter Regier\textsuperscript{1}, Kyongho
Son\textsuperscript{2}, Xingyuan Chen\textsuperscript{2}, Yilin
Fang\textsuperscript{2}, Peishi Jiang\textsuperscript{2}, Micah
Taylor\textsuperscript{2}, Wilfred M Wollheim\textsuperscript{3}, James
Stegen\textsuperscript{2}

\textbf{Affiliations} \textsuperscript{1}Marine and Coastal Research
Laboratory, Pacific Northwest National Laboratory, Sequim, WA, United
States\textsuperscript{2}Pacific Northwest National Laboratory,
Richland, WA, United States\textsuperscript{3}University of New
Hampshire, Durham, NH, United States

\textbf{Abstract:} Hyporheic zones regulate biogeochemical processes in
streams and rivers, but high spatiotemporal heterogeneity makes it
difficult to predict how these processes scale from individual reaches
to river basins. Recent work applying allometric scaling (i.e.,
power-law relationships between size and function) to river networks
provides a new paradigm to develop a scalable understanding of hyporheic
biogeochemical processes. We used reach-scale hyporheic aerobic
respiration estimates to explore allometric scaling patterns across two
basins, and related these patterns to watershed characteristics. We
found consistent scaling behaviors at lowest and highest exchange flux
(HEF) quantiles, and consistent but HEF-dependent relationships to
watershed elevation, precipitation, and land-cover. Our results also
suggest variability of hyporheic respiration allometry for middle
exchange flux quantiles, and in relation to land-cover. Our findings
provide initial evidence that allometric scaling may be useful for
predicting hyporheic biogeochemical dynamics across watersheds from
reach to basin scales.

\textbf{Scientific Significance Statement:} The hyporheic zone is a
biogeochemical control point in streams and rivers, and processes like
hyporheic respiration are important determinants of how watersheds move
and process carbon and nutrients. However, the hyporheic zone is also
characterized by high spatial heterogeneity, which makes it difficult to
predict how hyporheic functions like respiration change across
watersheds from reach to basin scales. This study applies allometric
scaling theory, which suggests that function scales in a predictable way
with size, to determine if hyporheic respiration scales with watershed
area in two basins with contrasting watershed characteristics. We found
some consistent patterns between basins that suggest allometric scaling
of hyporheic respiration may be a tool for transferable knowledge of
hyporheic function between basins, but also note some site-specific
relationships may constrain the generalizability of this method to other
regions and watersheds.

\textbf{Introduction:}

The hyporheic zone (HZ), which forms the interface between surface water
and groundwater in streams and rivers, is a spatially heterogeneous and
temporally dynamic biogeochemical zone (McClain et al. 2003; Gomez-Velez
et al. 2014; Bernhardt et al. 2017; Lee-Cullin et al. 2018). Because the
HZ regulates nutrient exchange/processing and provides habitat to
diverse biological communities, it serves as a biogeochemical reactor
for aquatic metabolism across river corridors (Gomez-Velez and Harvey
2014; Sackett et al. 2019). In fact, the hyporheic zone accounts for the
majority of ecosystem metabolism in some aquatic systems (e.g., Naegeli
and Uehlinger 1997; Fulton et al. 2024). Although characterizing
hyporheic metabolism is key for understanding river corridor
biogeochemistry, high spatiotemporal heterogeneity and interacting
environmental drivers in the HZ makes it difficult to develop predictive
relationships for hyporheic metabolism at reach-to-basin scales
(Buser-Young et al. 2023; Stegen et al. 2023; Ture\selectlanguage{polish}ţ\selectlanguage{english}caia et al. 2023).

Allometry, or a power-law relationship between function and size, is a
central theory of metabolism in ecology (e.g., Brown et al. 2004), and
has been applied to understand how biogeochemical properties scale
across freshwater and estuarine environments (Bertuzzo et al. 2017;
Nidzieko 2018). A recently introduced theoretical framework for an
idealized watershed suggests ecosystem metabolism scales allometrically,
meaning that cumulative metabolism of entire river networks relates
predictably to cumulative watershed area via a power law (Wollheim et
al. 2022). However, it remains unknown how, or even if this relationship
is transferable to non-idealized cases (real watersheds), or if
cumulative hyporheic metabolism exhibits equivalent allometric scaling
behavior. Additionally, although we know that watershed characteristics
relate to hyporheic metabolic processes, (Son et al. 2022a; Buser-Young
et al. 2023), it is unknown how differences in watershed characteristics
within or between watersheds might impact such scaling relationships.

We addressed this knowledge gap by asking two questions: 1) Does
hyporheic respiration follow an allometric relationship to watershed
area across reach-to-basin scales?, and 2) What ecosystem properties
relate to any observed differences in allometry? We used previously
modeled reach-scale hyporheic respiration for more than 16,000
individual reaches in two river basins to calculate allometric
relationships between cumulative hyporheic aerobic respiration and
cumulative watershed area. We then explored allometric relationships in
the context of watershed characteristics to understand common patterns
between our study basins. Finally, we identified next steps for
developing a more generalizable understanding of the controls on
hyporheic respiration allometry across basins.

\textbf{2. Methods}

2.1 Basin descriptions

We focused on the Willamette river basin (WRB) and the Yakima river
basin (YRB), which are both part of the Columbia River watershed,
located in the Pacific Northwest United States (Figure 1). The WRB in
central Oregon covers an area of 29,636 km\textsuperscript{2} with 9756
individual stream reaches, while the YRB in southeastern Washington
state is approximately half the size (15,887 km\textsuperscript{2}) and
contains 6538 individual stream reaches (Figure 1). We selected these
two basins because they differ in terms of biophysical characteristics
(Fulton et al. 2024), including precipitation, topography, and land
cover (Figure S1).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 1} : A) Locations of the Yakima and Willamette river
basins within the Pacific Northwest United States, and long-term
cumulative network scale respiration at each reach across B) the YRB and
C) the WRB.

2.2 Modeling hyporheic respiration

We used previously published modeled estimates for aerobic respiration
in the hyporheic zone at the scale of NHDPLUS reaches within the two
study basins using the River Corridor Model (RCM) presented in (Son et
al. 2022b). The model computes at hourly timesteps, using reactions,
rate formulations and kinetic rate parameters described in (Fang et al.
2020). Key model inputs for each reach include hyporheic exchange fluxes
(HEF) and hyporheic residence times estimated by the Networks with
Exchange and Subsurface Storage (NEXSS) model (Gomez-Velez and Harvey
2014), and long-term (\textasciitilde{}10 year) average stream solute
concentrations (dissolved organic carbon, dissolved oxygen, and nitrate)
estimated from available observations. We gap-filled missing HEF and
residence time values for \textasciitilde{}13\% of reaches, using a
eXtreme Gradient Boosting (XGBoost, v1.5.0.2) machine learning model
(Chen and He 2023) ingesting topography variables (watershed/catchment
area, maximum/mean/minimum elevation), hydrology (mean annual
streamflow/runoff/velocity and mean annual precipitation), and stream
geometry (slope, width and length, sinuosity, bankfull depth, median
particle size, roughness, total stream length) with high accuracy
(R2\textgreater{}0.94). Detailed descriptions of the stream solute input
data and our XGBoost approach are included in Supplemental Information.

2.3 Allometric scaling relationships

Prior to calculating allometric scaling relationships, we calculated
cumulative values for respiration and HEFs as the sum of all reaches
upstream of a given reach (and inclusive of the reach), as visualized
for a watershed in Figure S2. We calculated allometric scaling
relationships as cumulative hyporheic aerobic respiration (g
CO\textsubscript{2} d\textsuperscript{-1}) against watershed area
(km\textsuperscript{2}) following Wollheim et al. (2022). First, we
grouped reaches in each basin into 10 equally sized quantiles based on
cumulative HEF, which is an important control of hyporheic respiration
(Son et al. 2022). Within each quantile, a random subset of 50 reaches
representing the range of stream orders was selected via bootstrapping
and used to calculate slope, intercept, and R\textsuperscript{2} values
for the relationship between cumulative respiration and watershed area.
This process was then iterated 1000 times for each combination of bin
and basin to calculate average values and confidence intervals for
slope, intercept, and R\textsuperscript{2} values.

2.4 Statistics

We categorized the scaling behavior of relationships between cumulative
hyporheic respiration and watershed area for each combination of HEF bin
and basin (n = 20) into four allometric classes: Uncertain, Sublinear,
Linear, and Super-linear. We visually identified an
R\textsuperscript{2}threshold of 0.8 where values exhibit a ``knee''
(Figure S3A), which we interpret as weak or no allometric scaling.
Relationships with mean R\textsuperscript{2} values \textless{} 0.8 were
classified as ''Uncertain''. To classify allometric behavior for
relationships with R\textsuperscript{2} values \textgreater{}= 0.8, we
used 2.5\% and 97.5\% confidence intervals (CI\textsubscript{2.5} and
CI\textsubscript{97.5}) of slope values as follows: ``Linear'' for
CI\textsubscript{2.5} \textless{} 1 and
CI\textsubscript{97.5}\textgreater{} 1, ``Sublinear'' for
CI\textsubscript{2.5} \textless{} 1 and CI\textsubscript{97.5}
\textless{} 1, and ``Super-linear'' for CI\textsubscript{2.5}
\textgreater{} 1 and CI\textsubscript{97.5}\textgreater{} 1 (Figure
S3C). We capitalize these terms throughout the rest of this study to
clarify when we are referring to these definitions.

To reveal watershed features explaining variation in the spatial
distributions of hyporheic respiration across each basin, we used mutual
information analysis, which calculates the dependency between two
variables based on Shannon's entropy. Mutual information analysis was
conducted with the \emph{infotheo} R package (Meyer 2022), where each
variable was first discretized, then calculated mutual information was
normalized to the maximum information within each analysis. For
statistical tests with significance values, we used a p-threshold of
0.05. All analyses were performed in R (R Core Team 2023) or Python 3.

\textbf{3. Results}

3.1 Allometric scaling of aerobic respiration by hyporheic exchange flux
quantiles

We observed allometric scaling in both basins, particularly for higher
HEF quantiles (Figure 2, Figure S3). However, we also observed a range
of behaviors in the strength (R\textsuperscript{2} values) and slopes of
cumulative allometric scaling patterns across the study basins. For
lowest HEF quantiles (Q10-Q30) we observed lower mean
R\textsuperscript{2} values (R\textsuperscript{2} \textless{} 0.80)
consistently across basins, indicating weaker (``Uncertain'') allometric
scaling behavior (Figure 2, Figure S3). Conversely, mean
R\textsuperscript{2} values for Q40-100 were consistently higher than
the 0.80 threshold across both basins, indicating stronger allometric
scaling relationships (Figure 2, Figure S3). For Q40-100, we observed
increasing trends in slopes, with consistent Super-linear scaling at
highest quantiles (Q80-100) in both basins. For the WRB, scaling shifted
from Linear (Q40-Q50) to Super-linear (Q60-Q100), while in the YRB,
scaling shifted from Sublinear (Q40-Q60) to Linear (Q70) to Super-linear
(Q80-100) (Figure S3). We also fitted single regression lines to each
basin as a whole, independent of HEF (Figure S4). Based on our
R\textsuperscript{2} threshold of 0.8, both basins exhibited Uncertain
scaling behavior, though we note that both slopes are well above 1 (3.2
and 1.9 for WRB and YRB, respectively).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 2 : Allometric scaling of log-transformed cumulative aerobic
respiration with log-transformed cumulative watershed area for all
reaches in A) the Willamette river basin (WRB) and B) the Yakima river
basin (YRB). Colored lines show least-squares regression lines for each
HEF quantile. Dashed lines represent a 1:1 relationship corresponding
with linear scaling. Inset plots show mean (points) and ranges
(whiskers) for best-fit line slopes and R\textsuperscript{2}values by
HEF quantile.

3.2 Spatial patterns of scaling across basins

In contrast to relatively similar scaling relationships by HEF quantile
observed in Figure 2, spatial arrangement of scaling behaviors differed
notably between the two study basins (Figure 3). Uncertain relationships
predominantly occurred for smaller stream reaches (with less cumulative
HEF) in the lower-elevation portions of each catchment (Figure 3, Figure
S1), and was more widespread across the majority of the YRB, but largely
present near the main stem in the WRB (Figure 3). No Sublinear scaling
was observed in WRB, while reaches with Sublinear scaling in the YRB
were primarily located in the higher-elevation headwaters (Figure S1).
Linear scaling in the YRB was limited to higher-elevation reaches, but
distributed fairly widely across the WRB (Figure 3). Super-linear
scaling occurred primarily along the main stem in YRB, but largely in
the forested headwaters in WRB (Figure 3). Across each basin, the WRB
transitioned from Super-linear in headwaters through Linear into
Uncertain for tributaries near the main-stem, while the YRB transitioned
from a mix of Super-linear, Linear, and Sublinear to Uncertain, and both
main-stems were consistently Super-linear.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 3} Spatial distributions of relationships in Figure 2
across the two study basins. The main stems of the Yakima and Willamette
rivers are shown as blue lines, and the outlet of each basin is marked
with a black point.

3.3 Relating allometric behavior to watershed characteristics

We calculated mutual information to understand which watershed
characteristics in Figure S1 (elevation, precipitation, and land-cover)
share the most information with cumulative respiration (Figure 4). We
selected HEF quantile groupings with consistent scaling behavior between
basins (Q10-30: Uncertain, and Q80-100: Super-linear) to assess whether
basin or scaling had more consistent relationships to respiration.
Maximum elevation consistently shared the most information with
cumulative respiration (i.e., shared the strongest spatial relationship
with respiration as quantified by mutual information) for Uncertain
(Q10-Q30) quantiles in both basins, followed by forested land-cover
(Figure 4). Precipitation shared the most information with cumulative
respiration for Super-linear Q80-100) quantiles in both basins, followed
by maximum elevation in the WRB, and human-influenced land-cover in the
YRB. The consistency in highest mutual information between HEF quantiles
(maximum elevation for Q10-30 and precipitation for Q80-100) indicates
these factors may share similar relationships to allometric scaling
patterns in the two study basins, which suggests the potential to
generalize these relationships to other basins.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 4:} Mutual information between cumulative hyporheic
respiration and watershed characteristics. We include Q10-30 as HEF
quantiles that were consistently Uncertain in both basins and Q80-100 as
quantiles that were consistently Super-linear in both basins. Mutual
information values were normalized to the maximum value for each
analysis.

As a first step towards generalizing scaling behavior, we plotted the
relationship between maximum elevation, precipitation, and cumulative
respiration within the context of scaling (Figure 5). Both basins showed
significant (p \textless{} 0.0001) positive relationships between
maximum elevation and cumulative respiration (R\textsuperscript{2} =
0.48 for WRB and 0.68 for YRB, with a relatively consistent transition
from Uncertain scaling behavior at lower respiration and maximum
elevation values to Linear then Super-linear with increasing respiration
and maximum elevation (Figure 5). This is expected, as maximum elevation
positively correlates to watershed area (R\textsuperscript{2} values of
0.19 and 0.20 for WRB and YRB respectively, Figure S5). We did observe
some non-linear behavior in WRB, where reaches with the lowest maximum
elevation had a much steeper relationship (higher slope) compared to
reaches with higher maximum elevations (Figure 5). In contrast, the YRB
showed remarkably consistent linear behavior across the full elevation
range with no apparent breakpoint in the relationship. For both basins,
we observed a divide along the elevation axis for Uncertain reaches
between lower-precipitation and higher-precipitation values. In the YRB
in particular, drier reaches generally fall close to the best-fit line,
while wetter reaches consistently diverge from the dominant linear
pattern.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 5} : Relationships between cumulative respiration and
maximum watershed elevation for the A) WRB and B) YRB. Each reach is
color-coded based on scaling as described above, and the size of each
point corresponds to mean annual precipitation.

\textbf{4. Discussion}

To the authors' knowledge, this study presents the first basin-scale
analysis of allometric scaling of hyporheic respiration. Our analysis
demonstrates considerable variability in scaling relationships, both
within and across the study basins, but also suggests some common
responses, which provides initial evidence that some aspects of
hyporheic respiration allometry may generalize across basins. Below, we
discuss where we found consistent relationships, where we did not,
potential connections to watershed characteristics, limitations of our
approach, and future research directions.

4.1 Hydrologic exchange fluxes organize hyporheic respiration allometry

We found two consistent patterns in allometric relationships across the
study basins where 1) scaling was consistently weakest (Uncertain) for
lowest HEF quantiles, and 2) scaling was consistently strong (high
R\textsuperscript{2}) and Super-linear at highest HEF quantiles (Figure
2). Because HEFs regulate carbon and nutrient delivery to the hyporheic
zone and residence times, which in turn influence respiration (Krause et
al. 2022), consistency in the relationship between cumulative
respiration and cumulative HEF is expected. Reaches with Uncertain
scaling relationships were largely clustered in the lower elevation
headwater portions of each watershed (Figure 3, Figure S1), which is
likely linked to weaker vertical hydrologic connectivity between surface
water and the hyporheic zone. Lower water velocities may result in
higher heterogeneity in subsurface biogeochemistry, including redox
(Briggs et al. 2013), resulting in more variable (and therefore weaker)
allometric behavior. Conversely, strong, Super-linear scaling at highest
cumulative HEFs is consistent with previous observations that HEFs
control biogeochemical processes in the hyporheic zone, including
denitrification (Son et al. 2022a).

We observed differences in scaling patterns between the YRB and WRB for
middle HEF quantiles, which may indicate that basin-specific
characteristics limit our ability to effectively predict respiration
scaling behavior for these portions of the basin. Specifically, we
suggest that, at lower HEF quantiles, hyporheic respiration is
transport-limited where movement of carbon, nutrients, and oxygen
between surface and subsurface is the limiting factor, but also highly
variable. In contrast, at highest HEF quantiles, hyporheic respiration
is reaction-limited because of high exchange fluxes between the surface
and subsurface. This is consistent with observations from surface water
biogeochemistry that watersheds are generally transport-limited for
smaller watershed areas and transition to reaction-limited as watershed
area grows (Liu et al. 2022), and likely also varies in time, where
baseflow conditions are more transport-limited while quickflow
conditions are more reaction-limited (Raymond et al. 2016; Wollheim et
al. 2022).

4.2 Generalizability of relationships between allometry and watershed
characteristics

Consistent with findings in Figure 2, the spatial distribution of
allometric scaling of hyporheic respiration (Figure 3) and its
relationships to various watershed characteristics (Figure 4, Figure 5)
suggest some common patterns between the study basins, which may
generalize to other basins, but also identify several basin-specific
behaviors. Specifically, maximum elevation shared the most mutual
information with cumulative respiration for low-HEF reaches (Figure 4),
with relatively strong linear relationships in both basins (Figure 5).
This finding matches previous reports that elevation organizes
biogeochemical processes related to metabolism in other watersheds,
including greenhouse gas concentrations and ratios, and oxygen content
(Chiriboga and Borges 2023). We also observed consistent relationships
to precipitation for high HEF quantiles, consistent with other findings
that precipitation influences hyporheic processes, including respiration
(Leggieri et al. 2013; Son et al. 2022b; Coulson et al. 2022).

Although it is beyond the scope of this study, we suggest that common
responses to general watershed characteristics like elevation and
precipitation may also organize hyporheic respiration scaling in other
river basins. However, we also note that there are distinct differences
in respiration allometry between the study basins, notably in spatial
patterns (Figure 3). Thus, while the relationships we observed in our
two basins may be transferable to others, basin-specific patterns may
limit our ability to generalize a predictive understanding of hyporheic
respiration allometry to other basins.

4.3 Study limitations

Our estimates of hyporheic respiration allometry are based on a largely
mechanistic model which incorporates numerous assumptions. We also note
that our approach simplifies some important, yet complex, watershed
characteristics that influence hyporheic respiration. First, our
cumulative respiration rates are time-averaged, and likely smooth over
hot moments of respiration within different portions of the basin, or
potentially across the entire basin. Additionally, we did not explicitly
incorporate stream intermittency, which can dramatically alter how
nutrients and associated biogeochemical processes like respiration
change not only across space but through time (Newcomer et al. 2018;
Coulson et al. 2022). Together, integration of spatial and temporal
information, and explicitly investigating the role of variable
inundation on disconnecting and reconnecting portions of the watershed
will lead to a more realistic representation of watershed
biogeochemistry, and more accurate predictions of allometric scaling of
hyporheic respiration across spatial and temporal scales under shifting
climate regimes.

4.4 Future research directions

Ground-truthing model outputs by collecting samples or deploying sensors
in every reach is logistically impossible even for a small
sub-catchment. Our RCM approach already incorporates a model-experiment
(Modex) loop (Serbin et al. 2021), where laboratory measurements of
sediments collected within the Columbia River Basin were used to
parameterize biogeochemical rates in the model. We suggest implementing
a subsequent ModEx loop, where our allometry estimates can identify
sub-catchments that may function as biogeochemical control points
(\emph{sensu} Bernhardt et al. 2017). For instance, by examining
residuals in Figure 2 for each regression line fit, we could iteratively
identify which reaches adhered most poorly to scaling relationships
(where the largest positive residuals represent respiration hot spots)
and target field sampling campaigns to confirm whether outliers are due
to high respiration or heterogeneity poorly captured by the model.

Predictions of how hyporheic respiration allometrically scales across
watersheds that can generalize between basins will dramatically improve
our ability to model and therefore forecast how biogeochemical processes
influencing, and influenced by, aquatic metabolism will respond to
natural or anthropogenic changes in watershed dynamics. Our findings
present an initial attempt to characterize how hyporheic respiration
scales allometrically with watershed area across two environmentally
distinct basins. We found that, while some commonalities exist in
allometric scaling patterns and relationships to watershed
characteristics, particularly precipitation and elevation,
basin-specific patterns suggest that the factors driving hyporheic
respiration scaling require additional study. We suggest future studies
incorporate a larger number of study basins to more effectively assess
generalizability of patterns and relationships to watershed
characteristics. Further, incorporation of key disturbances, including
non-perenniality, wildfires, and urbanization, whose downstream impacts
increasingly influence, and are influenced by, surface and hyporheic
biogeochemical processes (Lawrence et al. 2013; Zhao et al. 2021; Ball
et al. 2021; DelVecchia et al. 2022), will improve our ability to model
the hyporheic zone and more accurately represent river corridor function
in earth system models.

\textbf{Acknowledgements:} The authors thank Francisco Guerrero for
contributions in conceptualization and analysis to an earlier draft of
this manuscript. This research was supported by the U.S. Department of
Energy (DOE), Office of Biological and Environmental Research (BER),
Environmental System Science (ESS) Program as part of the River Corridor
Science Focus Area (SFA) at the Pacific Northwest National Laboratory
(PNNL). PNNL is operated by Battelle Memorial Institute for the DOE
under Contract No. DE-AC05-76RL01830. The authors declare no conflict of
interest.

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